a temporal neuro-fuzzy approach for time series …

A TEMPORAL NEURO-FUZZY APPROACH FOR TIME SERIES ANALYSIS

A THESIS SUBMITTED TO

THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES

OF

THE MIDDLE EAST TECHNICAL UNIVERSITY

BY

NURAN ARZU SISMAN YILMAZ

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

IN

THE DEPARTMENT OF COMPUTER ENGINEERING

JULY 2003

Approval of the Graduate School of Natural and Applied Sciences.

Prof. Dr. Canan OzgenDirector

I certify that this thesis satisfies all the requirements as a thesis for the degree ofDoctor of Philosophy.

Prof. Dr. Ayse KiperHead of Department

This is to certify that we have read this thesis and that in our opinion it is fullyadequate, in scope and quality, as a thesis for the degree of Doctor of Philosophy.

Assoc. Prof. Dr. Ferda NurAlpaslan

Supervisor

Examining Committee Members

Prof. Dr. Fatos Yarman Vural

Prof. Dr. Adnan Yazıcı

Prof. Dr. Varol Akman

Assoc. Prof. Dr. Aysen Akkaya

Assoc. Prof. Dr. Ferda Nur Alpaslan

ABSTRACT

A TEMPORAL NEURO-FUZZY APPROACH FOR TIME SERIES

ANALYSIS

Sısman Yılmaz, Nuran Arzu

Ph.D., Department of Computer Engineering

Supervisor: Assoc. Prof. Dr. Ferda Nur Alpaslan

July 2003, 198 pages

The subject of this thesis is to develop a temporal neuro-fuzzy system for fore-

casting the future behavior of a multivariate time series data.

The system has two components combined by means of a system interface.

First, a rule extraction method is designed which is named Fuzzy MAR (Multivari-

ate Auto-regression). The method produces the temporal relationships between

each of the variables and past values of all variables in the multivariate time series

system in the form of fuzzy rules. These rules may constitute the rule-base in a

fuzzy expert system.

Second, a temporal neuro-fuzzy system which is named ANFIS unfolded in -

time is designed in order to make the use of fuzzy rules, to provide an environment

that keeps temporal relationships between the variables and to forecast the future

behavior of data. The rule base of ANFIS unfolded in time contains temporal

TSK(Takagi-Sugeno-Kang) fuzzy rules. In the training phase, Back-propagation

learning algorithm is used. The system takes the multivariate data and the num-

ber of lags needed which are the output of Fuzzy MAR in order to describe a

variable and predicts the future behavior.

iii

Computer simulations are performed by using synthetic and real multivariate

data and a benchmark problem (Gas Furnace Data) used in comparing neuro-

fuzzy systems. The tests are performed in order to show how the system efficiently

model and forecast the multivariate temporal data. Experimental results show

that the proposed model achieves online learning and prediction on temporal data.

The results are compared by other neuro-fuzzy systems, specifically ANFIS.

Keywords: neuro-fuzzy system, unfolding-in-time, fuzzy linear regression, back-

propagation algorithm

iv

OZ

ZAMAN SERISI COZUMLEMEDE ZAMANA DAYALI BULANIK

SINIR AGI YAKLASIMI

Sısman Yılmaz, Nuran Arzu

Doktora, Bilgisayar Muhendisligi Bolumu

Tez Yoneticisi: Doc. Dr. Ferda Nur Alpaslan

Temmuz 2003, 198 sayfa

Tezin konusu zaman serileri icin tasarlanmıs bulanık bir sinir agıdır.

Bir sistem arayuzu kullanılarak birlestirilen iki farklı bolumden olusur.

Oncelikle Fuzzy MAR (Cok degiskenli kendiliginden geri baglanım) adında bir ku-

ral cıkarım yontemi tasarlanmıstır. Yontem herbir degiskenle tum degiskenlerin

gecmisteki degerleri arasında bulanık kurallar seklinde zamana baglı iliskiler

uretir. Bu kurallar bir bulanık uzman sistemin kural tabanını olusturabilir.

Ayrıca bulanık kuralları kullanmak, degiskenlerin arasındaki iliskileri tut-

mak ve zaman serisinin gelecekteki davranısını tahmin etmek icin AN-

FIS Unfolded in time diye adlandırılmıs olan zamana baglı bir bulanık sinir agı

sistemi tasarlanmıstır. Kural tabanı TSK bulanık kuralları icerir. Ogretme

asamasında geri yayılım ogrenme yordamı kullanılmıstır. Sistem, Fuzzy MAR’ın

cıktısı olan cok degiskenli veriyi ve bir degiskeni tanımlamak icin gereken gerideki

an sayısını alır ve gelecekteki davranısı tahmin eder.

Yapay veri, gercek cok degiskenli veri ve bulanık sinir agı sistemlerini

karsılastırmada kullanılan bir denektası sorunu (Gaz odası verisi) kullanılarak

bilgisayar benzetimleri gerceklestirilmistir. Testlerdeki amac sistemin cok

v

degiskenli zamana baglı verileri ne duzeyde etkin modelledigini ve tahmin ettigini

gostermektir. Deneysel sonuclar onerilen modelin zaman icinde surekli gelen ve-

rileri cevrimici ogrenmeyi ve tahmin etmeyi basardıgını gosterir. Sonuclar ANFIS

basta olmak uzere diger bulanık sinir agları ile karsılastırılmıstır.

Anahtar Kelimeler: bulanık sinir agı sistemi, zamana yayma, bulanık dogrusal

baglanım, geri yayılım yordamı

vi

ACKNOWLEDGMENTS

First of all, I would like to thank Assoc. Prof. Dr. Ferda Nur Alpaslan for her

continuous guidance and motivation throughout my graduate studies that made

this thesis possible. She has been a model supervisor.

I would like to thank Prof. Dr. Fatos Yarman Vural and Assoc. Prof. Dr.

Aysen Akkaya for their valuable comments and guidance.

I want to thank all my friends, especially Burcin Bostan for her warm friend-

ship. I also want to thank my colleagues in Central Bank of Turkey, Serdar

Murat Oztaner, Ali Yılmaz and Murat Yucel for their patience and continuous

motivation.

I want to thank my family for their precious love and patience that gave me

the strength to finish my study. Especially my mother, thanks a lot for standing

always by me and taking care of me with all your love. Finally, I would like to

dedicate this thesis to my husband Ugur who provided me everything I needed.

Without his patience, continuous assistance and long-time friendship, I would not

have the strength to complete this work.

vii

TABLE OF CONTENTS

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

OZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

TABLE OF CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv

LIST OF ABBREVIATIONS . . . . . . . . . . . . . . . . . . . . . . . . . xvi

CHAPTER

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Problem Definition . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Contribution of Thesis . . . . . . . . . . . . . . . . . . . . 3

1.3 Scope of the Thesis . . . . . . . . . . . . . . . . . . . . . 4

2 BACKGROUND ON FUZZY LINEAR REGRESSION . . . . . . 5

2.1 Fuzzy Data Analysis . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Fuzzy Regression . . . . . . . . . . . . . . . . . . 5

2.1.2 Fuzzy Time Series Analysis . . . . . . . . . . . . 8

2.2 Fuzzy Linear Regression (FLR) . . . . . . . . . . . . . . . 10

2.2.1 Basic Definitions . . . . . . . . . . . . . . . . . . 10

2.2.2 Linear Programming Problem . . . . . . . . . . 11

2.2.2.1 Non-fuzzy Data . . . . . . . . . . . . 11

2.2.2.2 Fuzzy Data . . . . . . . . . . . . . . 12

viii

3 BACKGROUND ON NEURO-FUZZY SYSTEMS . . . . . . . . . 14

3.1 Neuro-Fuzzy Systems . . . . . . . . . . . . . . . . . . . . 14

3.1.1 Examples of Neuro-Fuzzy Systems . . . . . . . . 15

3.2 ANFIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 A TEMPORAL NEURO-FUZZY SYSTEM . . . . . . . . . . . . 23

4.1 Fuzzy Multivariate Auto-regression(MAR) . . . . . . . . . 23

4.1.1 Multivariate Auto-regression (MAR) Algorithm . 24

4.1.2 Model Selection . . . . . . . . . . . . . . . . . . 26

4.1.3 Motivation for FLR in Fuzzy MAR . . . . . . . 27

4.1.4 Fuzzification of MAR . . . . . . . . . . . . . . . 29

4.1.5 BIC in Fuzzy MAR . . . . . . . . . . . . . . . . 30

4.1.6 Obtaining a Linear Function for a Variable . . . 31

4.1.7 Processing of Multivariate Data . . . . . . . . . 32

4.2 ANFIS unfolded in time . . . . . . . . . . . . . . . . . . . 34

4.2.1 Motivation for Unfolding in Time . . . . . . . . 34

4.2.2 Temporal Back-Propagation Algorithm . . . . . 36

4.3 The Integrated System . . . . . . . . . . . . . . . . . . . . 38

4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5 EXPERIMENTAL RESULTS . . . . . . . . . . . . . . . . . . . . 43

5.1 Fuzzy MAR Algorithm . . . . . . . . . . . . . . . . . . . 43

5.1.1 Synthetic Data . . . . . . . . . . . . . . . . . . . 44

5.1.2 Real Data . . . . . . . . . . . . . . . . . . . . . 49

5.1.2.1 AAA Corporate Bonds and Commer-cial Paper Interest Rates Data . . . . 49

5.1.2.2 Monthly Agriculture Data . . . . . . 54

5.1.2.3 Monthly Flour Price Indices Data . . 67

5.1.2.4 Monthly Forestry Data . . . . . . . . 72

5.1.2.5 Gas Furnace Data . . . . . . . . . . . 87

5.1.2.6 Grain Price Data . . . . . . . . . . . 95

5.1.2.7 Housing Starts and Sold Data . . . . 102

5.1.2.8 Monthly Interest Rates Data . . . . . 106

5.1.2.9 Fixed Investment and Changes in Busi-ness Inventories Data . . . . . . . . . 114

ix

5.1.2.10 Annual Sales of Mink Furs and MuskratFurs Data . . . . . . . . . . . . . . . 118

5.1.2.11 Power Station Data . . . . . . . . . . 122

5.1.2.12 Production Schedule and Billing Fig-ures Data . . . . . . . . . . . . . . . 128

5.1.2.13 Unemployment and GDP Data . . . 132

5.1.3 Discussion of Experimental Results . . . . . . . 137

5.2 ANFIS unfolded in time . . . . . . . . . . . . . . . . . . . 138

5.2.1 Gas-Furnace Data Experiment . . . . . . . . . . 138

5.2.1.1 Gas Furnace Data . . . . . . . . . . . 138

5.2.1.2 Benchmark Problem . . . . . . . . . 138

5.2.1.3 Discussion of Training Results . . . . 138

5.2.1.4 Experiment using Fuzzy MAR resultfor Gas Furnace Data . . . . . . . . . 140

5.2.2 Comparison of MAR and Fuzzy MAR Algorithm 143

5.2.3 Real Data Experiments . . . . . . . . . . . . . . 145

6 CONCLUSION AND FUTURE WORK . . . . . . . . . . . . . . 168

6.1 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . 169

6.2 Future Directions . . . . . . . . . . . . . . . . . . . . . . . 170

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

A TIME SERIES DATA . . . . . . . . . . . . . . . . . . . . . . . . 175

VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

x

LIST OF TABLES

4.1 Multivariate Temporal Data for n variables and m time instances 244.2 Fuzzy MAR Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 334.3 Forward and Backward Phases in ANFIS Unfolded in time . . . . 37

5.1 Fuzzy equations for Synthetic Data . . . . . . . . . . . . . . . . . 445.2 Accuracy Rates for Synthetic Noisy Data . . . . . . . . . . . . . . 455.3 Expected and Obtained Output for Variable x (Synthetical Data

with Random Noise) . . . . . . . . . . . . . . . . . . . . . . . . . 465.4 Fuzzy equations for AAA CP Data . . . . . . . . . . . . . . . . . 495.5 Expected and Obtained Output for Variable x0 (AAA-Corporate

Bonds) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.6 Expected and Obtained Output for Variable x1(AAA-Corporate

Bonds) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.7 Fuzzy equations for Agriculture Data . . . . . . . . . . . . . . . . 545.8 Expected and Obtained Output for Variable x0 (Agriculture Data) 565.9 Expected and Obtained Output for Variable x1(Agriculture Data) 585.10 Expected and Obtained Output for Variable x2(Agriculture Data) 605.11 Expected and Obtained Output for Variable x3(Agriculture Data) 625.12 Fuzzy equations for Flour Price Indices Data . . . . . . . . . . . . 675.13 Expected and Obtained Output for Variable x0 (Flour Price Data) 685.14 Expected and Obtained Output for Variable x1(Flour Price Data) 695.15 Expected and Obtained Output for Variable x2(Flour Price Data) 705.16 Fuzzy equations for Forestry Data . . . . . . . . . . . . . . . . . . 725.17 Expected and Obtained Output for Variable x0 (Forest Data) . . 735.18 Expected and Obtained Output for Variable x1(Forest Data) . . 765.19 Expected and Obtained Output for Variable x2(Forest Data) . . 795.20 Expected and Obtained Output for Variable x3(Forest Data) . . 825.21 Fuzzy equations for Gas Furnace Data . . . . . . . . . . . . . . . 875.22 Expected and Obtained Output for Variable x0(Gas Furnace Data) 885.23 Expected and Obtained Output for Variable x1(Gas Furnace Data) 915.24 Fuzzy equations for Grain Price Data . . . . . . . . . . . . . . . . 955.25 Expected and Obtained Output for Variable x0 (Grain Prices Data) 965.26 Expected and Obtained Output for Variable x1 (Grain Price Data) 975.27 Expected and Obtained Output for Variable x2 (Grain Price Data) 985.28 Expected and Obtained Output for Variable x3 (Grain Prices Data) 995.29 Fuzzy equations for Housing Data . . . . . . . . . . . . . . . . . . 1025.30 Expected and Obtained Output for Variable x0 (Housing Data) . 103

xi

5.31 Expected and Obtained Output for Variable x1 (Housing Data) . 1045.32 Fuzzy equations for Interest Rates Data . . . . . . . . . . . . . . 1065.33 Expected and Obtained Output for Variable x0 (Interest Rate

Data) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075.34 Expected and Obtained Output for Variable x1(Interest Rate Data)1095.35 Expected and Obtained Output for Variable x2(Interest Rate Data)1115.36 Fuzzy equations for Investment and Inventories Data . . . . . . . 1145.37 Expected and Obtained Output for Variable x0 (Investment and

Inventories Data) . . . . . . . . . . . . . . . . . . . . . . . . . . . 1155.38 Expected and Obtained Output for Variable x1(Investment and

Inventories Data) . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165.39 Fuzzy equations for Mink and Muskrat Furs Data . . . . . . . . . 1185.40 Expected and Obtained Output for Variable x0 (Mink and Muskrat

Furs Data) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1195.41 Expected and Obtained Output for Variable x1 (Mink and Muskrat

Furs Data) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205.42 Fuzzy equations for Power Station Data . . . . . . . . . . . . . . 1225.43 Expected and Obtained Output for Variable x0 (Power Station

Data) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1235.44 Expected and Obtained Output for Variable x1 (Power Station

Data) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1245.45 Expected and Obtained Output for Variable x2 (Power Station

Data) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1255.46 Fuzzy equations for Production and Billing Data . . . . . . . . . . 1285.47 Expected and Obtained Output for Variable x0 (Production and

Billing Data) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1295.48 Expected and Obtained Output for Variable x1 (Production and

Billing Data) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1305.49 Fuzzy equations for Unemployment and GDP Data . . . . . . . . 1325.50 Expected and Obtained Output for Variable x0 (Unemployment

and GDP Data) . . . . . . . . . . . . . . . . . . . . . . . . . . . 1335.51 Expected and Obtained Output for Variable x1 (Unemployment

and GDP Data) . . . . . . . . . . . . . . . . . . . . . . . . . . . 1345.52 Accuracy Rates for Fuzzy MAR Algorithm Experiments with Real

Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1365.53 Comparison of RMSE values for different Neuro-Fuzzy Systems

and ANFIS unfolded in time . . . . . . . . . . . . . . . . . . . . 1395.54 Training output in tabular format . . . . . . . . . . . . . . . . . . 1415.55 Training and Recognition Results for Gas Furnace Data Experi-

ment by Fuzzy MAR . . . . . . . . . . . . . . . . . . . . . . . . . 1425.56 Training and Recognition Results for Agriculture Data . . . . . . 1445.57 RMSE for Experiments with ANFIS and ANFIS Unfolded in time

using Real Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

A.1 Syntectical Data produced by xt = 0.67xt−1 + 0.47xt−2 . . . . . . 175A.2 Syntectical Data with Uniform Random Noise . . . . . . . . . . . 176A.3 Quarterly AAA Corporate Bonds and Commercial Interest Rates 177

xii

A.4 Monthly Agriculture Data . . . . . . . . . . . . . . . . . . . . . . 178A.5 Monthly US Interest Rate Series for the Federal Funds Rate, 90-

Day Treasury Bill Rate, and the One-Year Treasury Bill Rate . . 180A.6 Monthly Flour Price Indices for Three US Cities . . . . . . . . . 183A.7 Monthly forestry data. Q: Lumber Production, P: Lumber price,

H: Housing Starts, I: Disposable Income . . . . . . . . . . . . . . 185A.8 Gas furnace data (columnwise order). X: Gas Flow Rate, Y: Car-

bon dioxide concentration . . . . . . . . . . . . . . . . . . . . . . 189A.9 Monthly US Grain Price Data . . . . . . . . . . . . . . . . . . . 190A.10 Monthly Housing Starts and Housing Sold Data . . . . . . . . . . 192A.11 Quarterly Fixed Investment and Changes in Business Inventories 193A.12 Natural Logarithms of the Annual Sales of Mink Furs and Muskrat

Furs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194A.13 Power Station Data from a 50 Megawatt Turbo-Alternator of In-

Phase Current Deviations, Out-of-Phase Current Deviations andFrequency Deviations of Voltage Generated . . . . . . . . . . . . . 195

A.14 Weekly Production Schedule and Billing Figures . . . . . . . . . . 196A.15 Quarterly Unemployment and GDP in UK (UN: unemployment,

GDP: gross domestic product) . . . . . . . . . . . . . . . . . . . . 197

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LIST OF FIGURES

2.1 A triangular fuzzy number . . . . . . . . . . . . . . . . . . . . . . 112.2 A fuzzy linear regression model . . . . . . . . . . . . . . . . . . . 13

3.1 Basic ANFIS structure . . . . . . . . . . . . . . . . . . . . . . . . 18

4.1 ANFIS Unfolded in time . . . . . . . . . . . . . . . . . . . . . . . 354.2 Error computation for a time interval . . . . . . . . . . . . . . . . 384.3 Integrated System . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.1 Linearity Figures for Synthetic Data (xt = 0.67xt−1 + 0.47xt−2) . . 485.2 Linearity Figures for Synthetic Data (xt = 0.67xt−1 + 0.47xt−2)

with uniform random noise . . . . . . . . . . . . . . . . . . . . . . 485.3 Linearity Figures for AAA corporate bonds . . . . . . . . . . . . . 535.4 Linearity Figures for Agriculture data . . . . . . . . . . . . . . . . 645.5 Linearity Figures for Monthly Flour Price Indices . . . . . . . . . 715.6 Linearity Figures for Forestry Data . . . . . . . . . . . . . . . . . 855.7 Linearity Figures for Gas Furnace Data . . . . . . . . . . . . . . . 945.8 Linearity Figures for Monthly Grain Price Data . . . . . . . . . . 1005.9 Linearity Figures for Monthly Housing Data . . . . . . . . . . . . 1055.10 Linearity Figures for Interest Rates for the Federal Funds Rate . . 1135.11 Linearity Figures for Quarterly Invest-Invent Data . . . . . . . . . 1175.12 Linearity Figures for Annual Mink-Muskrat Data . . . . . . . . . 1215.13 Linearity Figures for Power Station Data . . . . . . . . . . . . . . 1265.14 Linearity Figures for Weekly Production Data . . . . . . . . . . . 1315.15 Linearity Figures for Quarterly Unemployment and GDP Data . . 1355.16 Train results of ANFIS unfolded in time for Gas-Furnace data. . . 1405.17 ANFIS unfolded in time output for gas furnace data . . . . . . . 1435.18 Output for AAA-CP Bonds data (a) x0 (b) x1 . . . . . . . . . . . 1495.19 Output for Agriculture data (a) x0 (b) x1 . . . . . . . . . . . . . 1505.20 Output for Flour Price data (a) x0 (b) x1 . . . . . . . . . . . . . 1525.21 Output for Forestry data (a) x0 (b) x1 . . . . . . . . . . . . . . . 1545.22 Output for Gas Furnace data variable x0 . . . . . . . . . . . . . . 1565.23 Output for Grain Price data (a) x0 (b) x1 . . . . . . . . . . . . . 1575.24 Output for Housing data (a) x0 (b) x1 . . . . . . . . . . . . . . . 1595.25 Output for Interest Rate data (a) x0 (b) x1 . . . . . . . . . . . . 1605.26 Output for Investment and Inventories data (a) x0 (b) x1 . . . . 1625.27 Output for Mink and Muskrat Furs data (a) x0 (b) x1 . . . . . . 1635.28 Output for Power Station data (a) x0 (b) x1 . . . . . . . . . . . . 164

xiv

5.29 Output for Production and Billing data (a) x0 (b) x1 . . . . . . . 1665.30 Output for Unemployment and GDP data (a) x0 (b) x1 . . . . . 167

xv

LIST OF ABBREVIATIONS

ANFIS Adaptive Network Based FuzzyInference System

AIC Akaike Information Criterion

BIC Bayesian InformationCriterion

FLR Fuzzy Linear Regression

FLSLR Fuzzy Least Squares LinearRegression

FMLS Fuzzy Membership Least SquaresRegression

FSOM Fuzzy Self-organizing Map

LP Linear Programming

LSE Least Squares Estimation

MAR Multivariate Auto-Regression

RMSE Root Mean Square Error

RSS Residual Sum of Squares

xvi

CHAPTER 1

INTRODUCTION

1.1 Problem Definition

The subject of this thesis is to develop a temporal neuro-fuzzy system for process-

ing multivariate time series data. The aim is to use the system in forecasting and

identification problems involving temporal data. The system contains two main

components: fuzzy rule extraction and prediction modules. A hybrid approach is

employed where both neural network and fuzzy logic theory are utilized in time

series analysis.

Time series analysis deals with tabular data in the form of columns varying

in time intervals. A time series is a collection of observations made sequentially

in time. The analysis is performed to understand the behavior of the data over

time. The data is interpreted in making predictions about future behavior of

the observed system. The basic characteristics of time series analysis is the fact

that successive observations are usually not independent and that the analysis

must take into account the time order of the observations. When successive ob-

servations are dependent, future values may be predicted from past observations.

Moreover, when there are two or more time series dependent on each other, one

of the time series can be expressed in terms of previous values of itself and the

other time series.

1

If there is only one time series to predict the future values, this model is called

univariate time series. An analysis of several data sets for the same sequence of

time intervals is called multivariate time series analysis. Multivariate time series

model is studied in this study.

In multivariate time series analysis, it is possible to define each time series

in terms of previous values of itself and previous values of other time series in

the same system. The definitions of each time series can be represented as a

rule which can be used in a rule-based system. These rules can be utilized for

forecasting the future values of the system.

Neural Networks are suitable structures for function approximation having

learning ability. Fuzzy systems are used for enhancing the neural network’s ex-

planation capability. This hybrid approach yields in a neuro-fuzzy system which

is being widely used for rule-based expert systems. Neuro-fuzzy methods are

applicable to forecasting problems.

Recurrent Neural Network is a convenient structure for processing time series

data [33]. In Recurrent Neural Networks, if the input sequence is of a maximum

length T, the recurrent network can be turned into an equivalent feed-forward

neural network defined over T time intervals. The idea is called unfolding in

time [37]. The feed-forward neural network is duplicated for T times so that each

of the neural network is kept for a time interval. In other words, each neural

network represents a state of the input sequence in time. The connection weights

between the same nodes at the duplicated neural network are identical in such

a way that the neural networks at different time intervals will behave identical.

A modified version of back-propagation algorithm is used for training the neural

network. The weight updates are summed up and applied to the same weights

at different time intervals.

In the literature, there are various examples of unfolding in time applications

such as [9] and [37]. These examples are mostly knowledge-based systems.

In this thesis, the proposed neuro-fuzzy model contains both fuzzy rule ex-

traction by using fuzzy multivariate time series analysis and prediction by using

neuro-fuzzy systems.

2

Fuzzy multivariate time series analysis is based on fuzzy linear regression and

multivariate auto-regression algorithm. Fuzzy rules are obtained at the end of

the processing. These rules can be used by any rule-based system.

Unfolding in time approach is utilized to construct the knowledge-base in a

neuro-fuzzy model. A neuro-fuzzy system is used and it is duplicated for the

number of time intervals needed to forecast the system output accurately. A

modified temporal back-propagation algorithm is used as the learning algorithm.

It is aimed to present the temporal data to the neuro-fuzzy system continuously.

In other words, the learning algorithm is for online learning. The neuro-fuzzy

model unfolded in time is treated as a single neural network, rather than the

duplication of the basic neural network. The connections between neural networks

at different time intervals are also taken into account while network parameters

are updated.

1.2 Contribution of Thesis

In this thesis, two models are proposed which are Fuzzy MAR (Multivariate Auto-

regression) and ANFIS unfolded in time. Fuzzy MAR inputs multivariate tempo-

ral data and produces fuzzy temporal rules. ANFIS unfolded in time inputs fuzzy

temporal data and predicts the future behavior of the system. The following are

accomplished in this thesis.

• Multivariate auto-regression is fuzzified by using fuzzy regression principles.

An application of fuzzy linear regression (FLR) to multivariate time series

analysis is accomplished in this way. Linear programming (LP) method is

used as it is used in FLR.

• The information criteria used in comparing the models obtained is modified.

• Experiments are performed by using both the synthetic and real data. The

fuzzy equations are obtained which forms the fuzzy rules. The relationship

between the antecedent and consequent variables are presented.

• A well-known neuro-fuzzy system is modified in order to model time series

data.

3

• A model that is convenient for online learning is obtained when temporal

data is available.

• The idea of unfolding-in-time is modified and applied to neuro-fuzzy system.

A new error propagation technique is developed for that purpose.

• Simulations are performed on various domains in order to test the neuro-

fuzzy model.

1.3 Scope of the Thesis

Fuzzy regression, fuzzy time series and advances in neuro-fuzzy systems are in-

vestigated in the literature in order to initiate the study. Previous work found in

the literature about fuzzy regression and fuzzy time series is presented in Chapter

2. Fuzzy Linear Regression (the model which forms the basics of Fuzzy MAR) is

described in detail in the same chapter.

Background on neuro-fuzzy systems is presented in Chapter 3. ANFIS model

is also described in this chapter.

The temporal neuro-fuzzy approach of the thesis is explained in Chapter 4.

Fuzzy MAR method is described and fuzzy rule extraction algorithm is given.

The modifications applied to ANFIS are described in order to obtain the model

ANFIS Unfolded in time. The temporal back-propagation algorithm applied is

given.

Chapter 5 includes the experimental results of computer simulation of the

model. Tests are performed using both of the models. Fuzzy MAR is tested by

using synthetic and real data. For ANFIS Unfolded in time, tests are performed

using the benchmark problem of forecasting the Box-Jenkins gas-furnace data.

The experimental results are compared with some neuro-fuzzy models. Real data

are also used in the experiments. ANFIS is used in the comparisons.

Chapter 6 presents Conclusions and the Future Work which include the future

directions about the temporal neuro-fuzzy model.

4

CHAPTER 2

BACKGROUND ON FUZZY LINEAR

REGRESSION

Fuzzy MAR is based on fuzzy linear regression (FLR) and MAR (Multivariate

Auto- Regression) algorithm which is a non-fuzzy method.

In this chapter, previous works related to FLR are summarized. Section 1

describes the methods of fuzzy data analysis briefly. Fuzzy Linear Regression is

investigated in detail in Section 2.

2.1 Fuzzy Data Analysis

In order to explain the relationship between variables having ambiguity fuzzy

regression method is applied. In order to show how fuzzy data changes in time

fuzzy time series analysis is used. The studies related to fuzzy regression and

fuzzy time series analysis is presented in the following sections.

2.1.1 Fuzzy Regression

Tanaka, Uejima and Asai [49] introduced the pioneering work called Fuzzy Linear

Regression (FLR). They defined a fuzzy structure represented as a fuzzy linear

function whose parameters are given by fuzzy sets. Unlike the conventional linear

regression function where the differences between the observed and the expected

values are treated as observational errors, the differences are treated as the fuzzi-

5

ness in the system. In the fuzzy model, each fuzzy parameter is represented by

a triangular fuzzy number (a number having a center and a width equal at the

both sides of the center), and, they are coefficients in the fuzzy linear function.

The vagueness in the system is represented by the total width of the parameters

(i.e. coefficients). The problem is to find the parameters in a linear programming

problem where the vagueness is to be minimized. The Fuzzy MAR model devel-

oped in this thesis uses fuzzy linear regression. Fuzzy MAR accepts non-fuzzy

data as input and outputs fuzzy values. For that reason, FLR will be described

in detail in the following chapters.

Tanaka also described how to apply possibilistic linear systems to fuzzy data

analysis in [45]. Possibilistic linear systems are based on extension principle. The

resulting approach is called fuzzy interval analysis. Tanaka et al. continued pos-

sibilistic linear regression study in [48] and [46]. They claim that the probabilistic

models take central tendency into account and ignore some observations whereas

all data are assumed to occur in possibilistic models.

Savic et al. [38] proposed a two-step procedure named Fuzzy Least Squares

Linear Regression (FLSLR), which provides an enhancement of minimal vague-

ness criterion of Tanaka’s fuzzy linear regression model. Since the center values

are not taken into account in the objective function, least-squares method is ap-

plied in the first step to fit a regression line for the center values. The input data

must be non-fuzzy. The method is a variation of fuzzy linear regression model.

In the study by Kim et al. [27], another modification to fuzzy linear regression

analysis is investigated which is namely Fuzzy Membership Least Squares Re-

gression (FMLS). The new method uses minimization of the difference of fuzzy

membership values between observed and estimated fuzzy numbers. It is stated

that the evaluation of a model is based on the difference between the observed

data and estimated data in conventional regression analysis. On the other hand,

the fuzzy linear regression analysis deals with membership functions so the eval-

uation should be based on the difference between the membership functions. The

error of fitting the membership functions is also defined as the ratio of the differ-

ence of membership values to the observed membership values. It is claimed that

6

FLR and FLSLR have a disadvantage of taking widths of fuzzy coefficients into

account without considering the center points. Moreover, the objective function,

which minimizes the total width of the fuzzy numbers, seems to be contradictory

to fuzzy theory. In Fuzzy MAR, the basic objective function is also the total

width of the fuzzy coefficients.

Ishibuchi et al. aims to define the interval numbers by means of two multi-

layer feed-forward neural networks [19]. Back-propagation algorithm (BP) is used

for training. The approach supplies a nonlinear approximation ability; because,

neural networks can be used for modeling more complex systems. Ishibuchi et

al. suggested non-symmetric fuzzy number coefficients for the fuzzy regression

problem[20]. They claim that symmetric triangular fuzzy numbers are not flex-

ible enough to represent all data types. They modified the model to handle

non-symmetric trapezoidal fuzzy number coefficients. They used fuzzified neural

network with fuzzy number coefficient weights.

Kim et al. compared statistical and fuzzy regression models by using the

criteria which contains the number of data points, quality of data (variance is

small or large), aptness, heteroscedasticity, autocorrelation between error terms,

and non-randomness of error terms [28]. The results show that fuzzy regression is

appropriate when the observations are fuzzy numbers or some data are collected

by measurements yet other are estimated subjectively (qualitative human expert

knowledge) i.e. there is not enough experiments for deriving a valid statistical

model. They claim that fuzzy model is less effective in forecasting, when data

quality is bad and some outliers exist in the model. It is suggested that fuzzy

regression may construct a better model for the quality function deployment

in product design. It is important to evaluate the customer preference (totally

qualitative) and gives vague information for a new product.

The study by Wang et al. [53] uses the objective function, which aims to

minimize the total width of the estimated fuzzy number. This model accepts fuzzy

values as expected output which is a variation to the original fuzzy regression

model.

7

Buckley et al. have suggested an evolutionary model for finding a fuzzy re-

gression function [6]. The regression function can be linear or nonlinear. An

evolutionary algorithm searches a library of fuzzy functions, which can be linear,

polynomial, exponential or logarithmic. The function, which best fits, is found

at the end of the procedure. Functions containing multiple independent variables

can also be searched for the linear model. For each of the function types the best

function is found, i.e. the coefficients of the regression equation are determined.

Best function is the one with the minimum error value. The error function min-

imizes the distance between the function value and the observed value. If the

minimum error of the best function is sufficiently small, it can be said that a

fuzzy function from the function library is found, which reasonably explains the

data. In the evolutionary process different types of fuzzy functions are tested. If

evolutionary algorithm decides certain type of function cannot model the data,

this function is discarded from further study. This is another approach to fuzzy

regression. It fits the best model to the data. The set of models is restricted

where a univariate model is found. Fuzzy MAR assumes the data is defined by

a linear model, in which a variable is defined by previous values of itself and as

well as the previous values of other variables.

2.1.2 Fuzzy Time Series Analysis

There are different approaches to fuzzy time series analysis. Some of the models

applies classical time series analysis methods directly to data. Some others ap-

plies fuzzy regression theory to fuzzy time series analysis and another group of

researchers uses fuzzy relations in order to describe temporal relationships.

Fuzzy time series analysis by Bintley [3] deals with the fuzzy representation of

a classical time series. The model relates gas flow with carbon dioxide concentra-

tion for gas furnace data. The model aims to construct an expert system having

fuzzy rules. The model uses classical time series analysis methods to obtain a set

of fuzzy rules.

Watada also presents a fuzzy time series model [54], where fuzzy time series

is formulated by using a fuzzy function of time whose parameters are fuzzy num-

8

bers. The fuzzy numbers recorded at different time instances are possibilistic

observation data and represented by fuzzy sets. They are used for forecasting

the future behavior. Linear trend and seasonal cycles are also described in the

model. Chang [7] introduces fuzzy trend and fuzzy seasonality concepts to fuzzy

time series. Fuzzy trend is found by using fuzzy regression model and later fuzzy

seasonality index set is calculated for each season.

Tseng et al. introduced a fuzzy ARIMA(auto-regression integrated moving

average) model in [51]. The algorithm has two phases: Phase 1 is fitting the

ARIMA model by using available observations considering the input data as non-

fuzzy. The result of Phase 1 is used as one of the input data sets in Phase 2. Phase

2 is the determination of the fuzzy coefficient widths using the non-fuzzy output

of the first phase as the center values of the parameters. Minimal fuzziness is the

objective function as in fuzzy linear regression model. The data around the upper

and lower bounds are deleted in the last phase of the procedure. The model is

used for predicting the exchange rate of NT (Taiwan) dollar to US dollar. They

utilize basic concept of ARIMA for finding a model. The model is using classical

time series analysis methods and applying fuzzy regression theory on it.

Unlike conventional time series models, linguistic values (fuzzy sets) are stud-

ied as observations in the fuzzy time series model introduced by Song et al. in

[39], [40] and [41], Fuzzy relational equations are employed in order to describe

the dynamic process. The fuzzy time series model aims to forecast future values

by using fuzzy relational equations. Each value (observation) is represented by

a fuzzy set. The transition between consecutive values is taken into account in

order to model the time series data. These transition data are stored in a rela-

tion matrix. Song suggested the seasonality forecast for fuzzy time series in [42]

based on [40]. Fuzzy sets in the same seasonal relationship are used in finding

the fuzzy relationships. The model described in these studies make use of a re-

lational model. It is different than using time series analysis methods. At every

time instance, a fuzzy linguistic value is kept as time series data and the rela-

tionship between different time instances is accepted as the time series model. It

is required to find a robust method for obtaining fuzzy linguistic values for each

9

variable concerned. Hwang et al. [18] makes some modifications to the fuzzy

time series model in [40]. The new method is based on time-variant fuzzy time

series model. The experiments are performed on university enrollment data. The

method for the determination of the transition matrix is changed, so that time

complexity of the fuzzy time series model decreases. The model is compared

with both the older version of fuzzy time series model and Markov model. In

[44] Sullivan et al. compare fuzzy time series modeling to Markov modeling. The

analogy between two models is achieved by replacing the membership functions

in fuzzy approach with the probability density functions.

2.2 Fuzzy Linear Regression (FLR)

In this section, fuzzy linear regression (FLR) is described in detail. The algorithm

Fuzzy MAR utilizes FLR in finding fuzzy equations being converted into fuzzy

rules.

The FLR model is first introduced by Tanaka et al. in [49]. In the following

sections, definitions related to FLR are presented, then the problem is defined.

2.2.1 Basic Definitions

The conventional regression methods treats the difference between the observed

data and the obtained output as the error. In FLR method, this difference is

accepted as the ambiguity in the data [50]. This ambiguity is represented as the

fuzzy coefficients in the fuzzy linear equations. These fuzzy coefficients are fuzzy

numbers.

A fuzzy number A is represented by a center α and a width c. The membership

value for a fuzzy number is given by the following equation.

µA(x) = L((x − α)/c) (2.1)

The membership function must be symmetric. In Figure 2.1, a triangular

fuzzy number can be seen where the membership function is

µA(x) = 1 − |x − α|/c (2.2)

10

1.0

µ

α

µ ( ).

x

A

cc

Figure 2.1: A triangular fuzzy number

2.2.2 Linear Programming Problem

Fuzzy linear regression finds an equation for a dependent variable in which fuzzy

numbers are used as coefficients of independent variables. The data handled can

be either standard (with non-fuzzy output) or fuzzy data.

2.2.2.1 Non-fuzzy Data

The input data is given as ~xi=(xi1, xi2, · · · , xin) for non-fuzzy output yi where i

denotes the sample index.

The fuzzy linear regression model produces the equation

Yi = A0 + A1 ∗ xi1 + A2 ∗ xi2 + · · · + An ∗ xin (2.3)

where Yi is the fuzzy output and ~xi is the input vector. * denotes the scalar

multiplication operation. The coefficients Aj, where j = 1, · · · , n are triangular

fuzzy numbers with (αj, cj).

The coefficients in the model are computed by using the following linear pro-

gramming (LP) problem. The objective function of the LP problem is

minimize J(c) =∑

i=1

Tct|xi| (2.4)

11

and the constraints are

yi ≤ xtiα + |L−1(h)|ct|xi|

yi ≥ xtiα − |L−1(h)|ct|xi|

where T is the size of data set, |xi| = (|x1|, · · · , |xn|)t, L(x) = 1 − |x| , c ≥ 0 and

i = 1, · · · , n. h denotes the degree to which the given data (yi, xi) is included in

the output fuzzy number Yi [50].

If a coefficient has a width c closer to 0, then the variable has less ambiguity,

since the coefficient is similar to a non-fuzzy value.

An example is given in Figure 2.2. The model in the figure is Yi = A0 + A1xi

where A0 = (2,1) and A1 = (3,0.5) and h = 0.5. Y1, Y2, Y3 are three fuzzy outputs

for the non-fuzzy input-output pairs (y1, x1), (y2, x2), (y3, x3).

2.2.2.2 Fuzzy Data

Fuzzy output Yi is defined as (yi, ei). The fuzzy coefficients of the regression

model are computed by solving the following two LP problems which are Min

and Max Problems. The solution to Min problem gives the greatest lower bound

and the solution to the Max problem gives the least upper bound of the fuzzy

estimate.

Min Problem:

The LP problem is as follows

minimize J(c) =∑

ct|xi| (2.5)

and

yi + |L−1(h)|ei ≤ xtiα + |L−1(h)|ct|xi|

yi − |L−1(h)|ei ≥ xtiα − |L−1(h)|ct|xi|

Max Problem:

The LP problem is

minimize J(c) =∑

ct|xi| (2.6)

12

and

yi + |L−1(h)|ei ≥ xtiα + |L−1(h)|ct|xi|

yi − |L−1(h)|ei ≤ xtiα − |L−1(h)|ct|xi|

In the scope of this thesis, FLR for non-fuzzy data is investigated. The adap-

tation of FLR to non-fuzzy MAR algorithm is explained in Chapter 4.

Y1

h1.0µ

h1.0µ

Y2

Y3

c0+c1|x1|

c0+c1|x2|

c0+c1|x3|

6

10

14

18

0

1.0 hµ

: Given Data (y i,xi)

(y1,x1)

(y2,x )

(y3,x3)

2

2 4 6 x

y

Figure 2.2: A fuzzy linear regression model

13

CHAPTER 3

BACKGROUND ON NEURO-FUZZY

SYSTEMS

ANFIS unfolded in time is based on a neuro-fuzzy model named ANFIS. Section

1 includes basic summary of neuro-fuzzy literature. In Section 2, ANFIS structure

will be described in detail.

3.1 Neuro-Fuzzy Systems

It is well known that, neural networks have relatively less explanation ability

compared to the expert systems. In other words, when we consider using a neural

network to represent the knowledge-base in a rule-based expert system, we may

have difficulty in explaining the behavior of the model explicitly. In conventional

expert systems, expert rules are employed for this purpose.

Ambiguity of human reasoning is another problem to be solved in rule-based

expert systems. Fuzzy models can be employed in rule-based systems for rep-

resenting the ambiguity. Neural networks and fuzzy systems can be merged in

order to have a hybrid methodology both having explanation and human reason-

ing capabilities.

Expert rules can be used in order to form a knowledge-base for a neuro-fuzzy

system. Here is an example of an expert rule to be represented by means of a

neuro-fuzzy system.

14

IF Advertisement is Low AND Budget is Medium

THEN Sales is Low

In this expert rule, Advertisement and Budget are antecedent part variables

(independent variables), and Sales is the consequent part variable (dependent

variable). Moreover, Low and Medium are linguistic values that the variables

can take. The input data is presented to the neuro-fuzzy system as either fuzzy

linguistic terms or raw numerical data. The output of the system also can be

either a numerical or a linguistic value.

A neuro-fuzzy system represents the knowledge-base of a rule-based expert

system. Most of the neuro-fuzzy systems have mainly three layers:

• Fuzzification Layer: Fuzzification is a process to convert the discrete values

of the input variables to the corresponding fuzzy sets. Each node in Fuzzi-

fication layer is a membership function denoting a fuzzy linguistic variable

such as SMALL, MEDIUM, LARGE. This layer represents the variables in

the antecedent part of an expert rule.

• Inference Layer: This part produces the strength of the fuzzy rule. In

other words, the strength of the antecedent part is computed by using an

inference mechanism. AND operator is used to form the relationship in

the antecedent part in the previous rule example. AND represents the

multiplication of the membership values of the variables to some linguistic

values given in the fuzzification layer.

• Defuzzification Layer: Defuzzification is the inverse process of Fuzzification.

It deduces an output value from the rule antecedents. In this layer, the

result or the value of consequent part is computed.

In the next section, some neuro-fuzzy system examples in the literature are

presented.

3.1.1 Examples of Neuro-Fuzzy Systems

In a survey study by Buckley and Hayashi, the fuzzy neural networks are stated

as having different characteristics [5]. The neural models can be classified as

15

hybrid neural nets, fuzzy neural nets, and hybrid fuzzy neural nets. Fuzzy neural

networks can have fuzzy input and/or fuzzy weight. Different learning algorithms

can be applied depending on the model. Different application areas are there in

the literature such as fuzzy regression, fuzzy controller, fuzzy expert systems,

fuzzy hierarchical analysis and fuzzy matrix equations.

As an example of a neuro-fuzzy model with fuzzy input values, a connection-

ist model is designed by Lin and Lee and it is used as fuzzy logic control and

decision system in [30] . Input layer consists of linguistic variables. Inner layers

contain membership function computation and fuzzy rule base for preconditions

and consequences. A hybrid learning algorithm having two phases is employed.

Phase one is a self-organizing learning scheme for understanding the presence of

a rule and for locating initial membership functions. Phase two is a supervised

learning phase where back-propagation is utilized.

Another model by Keller et al. [26] is a multi-layer neural network architecture

that represent the fuzzy rules explicitly. The weights between connections are

the antecedent and consequent parts. The model is used for fuzzy logic inference.

Fuzzy sets are supplied in the input layer.

Lin and Cunningham introduced a simple neural network structure that rep-

resents fuzzy rules [32]. The neural network has three-layers excluding the input

layer. Sigmoid membership function is employed for fuzzification layer of the

fuzzy-neural model. Multiplicative inference is used in the inference layer. In the

output layer, the output is computed by using the weighted sum. They developed

a method for computing initial weights. The model is close to ours, but there is

no normalization of rule strengths, which can lead to unexpected output results.

There is a different approach to neuro-fuzzy systems by Pal and Mitra. They

say that human reasoning is fuzzy in nature [35]. Fuzzy logic provides a means

for solving uncertainty. Compared to binary logic where only 0 and 1 are possible

values, it needs more computation so that the highly-parallel structure of neural

networks may be helpful. A mechanism for a fuzzy multi-layer perceptron using

back-propagation algorithm is introduced by the authors. They use their model

in pattern recognition problems. Linguistic data can be input as well as numerical

16

values. Class membership values are output by the neural network. Mitra and Pal

further directed the research through the connectionist expert system framework

[34]. According to them, the major components of an expert system are knowl-

edge base, inference engine, and user interface. The model suggested also handles

partial data which introduces ambiguity to the system. Partial input data is pro-

cessed by using the user interface. The knowledge base is constructed by training

the neural network. It is in the form of IF-THEN rules in expert systems whereas

neural network stores the knowledge-base as the connection weights. When we

compare this model to ours, it is seen that ANFIS unfolded in time adapts a set

of parameters instead of connection weights.

A different model is a fuzzy version of Kohonen’s self organizing map which is

developed by Vuorimaa et al. [52]. They told about a version of fuzzy self-

organizing map (FSOM) which contains one layer excluding the input layer.

Fuzzy rules embedded in the neural network are tuned. Learning Vector Quan-

tization is used as the learning algorithm in tuning phase. An application on

chemical agent detection is found in their study.

Another study by Jang [22] is also a five-layer feed-forward neural network.

It obeys general structure of a neuro-fuzzy system. In our study, this model is

used as the basic model and modifications are made on the model. The model is

described in detail in the next section.

3.2 ANFIS

ANFIS (Adaptive Neuro-Fuzzy Inference System) is a neuro-fuzzy system devel-

oped by Roger Jang [21], [22], [23], [24]. It has a feed-forward neural network

structure where each layer is a neuro-fuzzy system component (Figure 3.1). It

simulates TSK (Takagi-Sugeno-Kang) fuzzy rule [43] of type-3 where the conse-

quent part of the rule is a linear combination of input variables and a constant.

The final output of the system is the weighted average of each rule’s output. The

form of the type-3 rule simulated in the system is as follows:

IF x1 is A1 AND x2 is A2 AND . . . AND xp is Ap

THEN y = c0 + c1x1 + c2x2 + · · · + cpxp

17

Σ

A1

A2

B1

B2

x 1

x 0

f

x 0 x 1

wi wi

wi f i.

3 41 2 5

µ

0

Layer

Figure 3.1: Basic ANFIS structure

The neural network structure contains 5 layers excluding input layer.

• Layer 0 is the input layer. It has n nodes where n is the number of inputs

to the system.

• Layer 1 is the fuzzification layer in which each node represents a membership

value to a linguistic term as a Gaussian function with the mean

µAi(x) =

1

1 + [(x−ci

ai)2]bi

(3.1)

where ai, bi, ci are parameters of the function. These are adaptive parame-

ters. Their values are adapted by means of the back-propagation algorithm

during the learning stage. As the values of the parameters change, the

membership function of the linguistic term Ai changes. These parameters

are called premise parameters.

In that layer there exists n ∗ p nodes where n is the number of input vari-

ables and p is the number of membership functions. For example, if size is

an input variable and there exists two linguistic values for size which are

18

SMALL and LARGE then two nodes are kept in the first layer and they

denote the membership values of input variable size to the linguistic values

SMALL and LARGE.

• Each node in Layer 2 provides the strength of the rule by means of multi-

plication operator. It performs AND operation.

wi = µAi(x0) ∗ µBi

(x1) (3.2)

Every node in this layer computes the multiplication of the input values and

gives the product as the output as in the above equation. The membership

values represented by µAi(x0) and µBi

(x1) are multiplied in order to find

the firing strength of a rule where the variable x0 has linguistic value Ai

and x1 has linguistic value Bi in the antecedent part of Rule i.

There are pn nodes denoting the number of rules in Layer 2. Each node

represents the antecedent part of the rule. If there are two variables in the

system namely x1 and x2 that can take two fuzzy linguistic values SMALL

and LARGE, there exist four rules in the system whose antecedent parts

are as follows:

IF x1 is SMALL AND x2 is SMALL

IF x1 is SMALL AND x2 is LARGE

IF x1 is LARGE AND x2 is SMALL

IF x1 is LARGE AND x2 is LARGE

• Layer 3 is the normalization layer which normalizes the strength of all rules

according to the equation

wi =wi

∑Rj=1 wj

(3.3)

where wi is the firing strength of the ith rule which is computed in Layer

2. Node i computes the ratio of the ith rule’s firing strength to the sum of

all rules’ firing strengths. There are pn nodes in this layer.

• Layer 4 is a layer of adaptive nodes. Every node in this layer computes

a linear function where the function coefficients are adapted by using the

19

error function of the multi-layer feed-forward neural network.

wifi = wi(p0x0 + p1x1 + p2) (3.4)

pi’s are the parameters where i = n + 1 and n is the number of inputs to

the system (i.e. number of nodes in Layer 0). In this example, since there

exists two variables (x1 and x2), there are three parameters (p0, p1 and p2)

in Layer 4. wi is the output of Layer 3.

The parameters are updated by a learning step. Least squares approxima-

tion is used in ANFIS. In the temporal model, back-propagation algorithm

is used for training.

• Layer 5 is the output layer whose function is the summation of the net

outputs of the nodes in Layer 4. The output is computed as:

∑

i

wifi =

∑

i wifi∑

i wi

(3.5)

where wifi is the output of node i in Layer 4. It denotes the consequent part of

rule i. The overall output of the neuro-fuzzy system is the summation of the rule

consequents.

ANFIS uses a hybrid learning algorithm in order to train the network. For

the parameters in the layer 1, back-propagation algorithm is used. For training

the parameters in the Layer 4, a variation of least squares approximation is used.

In the next chapter, the modifications to ANFIS model for using temporal data

will be described.

The following example describes the processing of ANFIS over a data set.

EXAMPLE:

GAS FURNACE DATA PROCESSED BY ANFIS

ANFIS accepts the input data in the (GasFlowRate(t-4), CO2Concentration

(t-1)) format.

1. An input data pair is given to the network.

2. The network performs the forward pass, i.e. the output of the function

which is CO2Concentration(t) is computed.

20

3. Another input data pair is presented to the network and the above com-

putation continues until the network is trained with sample size-4 data

points(last four pairs cannot be used since the expected output is not

known) where sample size denotes the total number of data points in the

training data set.

4. Error is computed for this epoch by using an error measure to compare the

expected output to the output of the system.

5. Training is performed by updating the parameters in layer 1 (a, b, c) and

in layer 4 (~pi). This is offline learning, because all data set is presented to

the network at once and the parameters are updated.

6. After a predetermined number of training epochs is reached, the training

process terminates.

The fuzzy rules produced in terms of parameters are as follows.

Rule 1:

IF GasF lowRate(t − 4) is SMALL1 AND

CO2Concentration(t − 1) is SMALL2

THEN

CO2Concentration(t) =

p11 ∗ GasF lowRate(t − 4) + p12 ∗ CO2Concentration(t − 1) + p13

Rule 2:

IF GasF lowRate(t − 4) is SMALL1 AND

CO2Concentration(t − 1) is LARGE2

THEN



Rule 3:

IF GasF lowRate(t − 4) is LARGE1 AND

CO2Concentration(t − 1) is SMALL2

THEN



21

Rule 4:

IF GasF lowRate(t − 4) is LARGE1 AND

CO2Concentration(t − 1) is LARGE2

THEN



In this example there are two fuzzy values SMALLi and LARGEi for both

variables (GasF lowRate(t − 4) and CO2Concentration(t − 1)) where i denotes

the index of the variable. Each fuzzy value such as SMALLi is denoted by

the parameters in the first layer (ai, bi, ci). pjk is the parameter in the fourth

layer where j denotes the rule and k denotes the parameter index. It is used in

computing the output of the system which is CO2Concentration(t).

22

CHAPTER 4

A TEMPORAL NEURO-FUZZY SYSTEM

In this chapter, temporal neuro-fuzzy approach to the time series analysis will be

described. The system includes two main components which are namely Fuzzy

Multivariate Auto-Regression (MAR) and ANFIS unfolded in time. The aim of

this approach is to construct a system that successfully forecasts the multivariate

time series data by using a neuro-fuzzy approach.

Fuzzy MAR algorithm is used for obtaining fuzzy relationships between the

variables of a multivariate time series data. These fuzzy relationships are used

in ANFIS unfolded in time in order to create a knowledge-base for a prospective

expert system.

These two components have different properties and purposes. A system

interface is developed to combine the components.

In the following sections, Fuzzy MAR Algorithm and ANFIS unfolded in time

will be described. The function of the system interface unit between these two

components will be explained.

4.1 Fuzzy Multivariate Auto-regression(MAR)

Fuzzy MAR is based on the non-fuzzy Multivariate Auto-Regression (MAR) algo-

rithm, which is used to model multivariate time-series data and obtain the rules

necessary for predicting the values of the variables in the future time intervals.

23

4.1.1 Multivariate Auto-regression (MAR) Algorithm

Multivariate auto-regression (MAR) method is a statistical analysis method for

understanding the dynamics and temporal structure of several data sets for the

same sequence of time intervals.

MAR which is introduced in [1] is a stepwise procedure producing IF-THEN

rules for a rule-based system. Each of the resulting IF-THEN rule is an expla-

nation of a variable in terms of possibly all variables in the system and having

also temporal relationships. The rules are extracted from time series data which

is in the form of a table of numerical values defined over different time intervals.

There are n variables and T time instances in multivariate temporal data shown

in Table 4.1.

Table 4.1: Multivariate Temporal Data for n variables and m time instances

VariableTime x1 · · · xn

1 x1,1 · · · xn,1

2 x1,2 · · · xn,2

· · · · · · · · · · · ·T x1,T · · · xn,T

In a multivariate time series data, the dependent variables must be defined

in terms of previous values of itself and the other variables in the system. For

that purpose, previous time instances of the other variables must also be con-

sidered. An information criterion must be used in order to find the best model

describing the variables or to choose the best variable set among candidates in

order to describe a dependent variable. The information criterion compares the

models containing different combinations of the variables and different lags (time

intervals). The information criterion used in testing the MAR Algorithm is BIC

(Bayesian Information Criterion) which will be described in the next section. For

each iteration of the algorithm, a linear equation is found for a dependent variable

by using a regression method. BIC value of the equation is computed by using the

24

RSS(residual sum of squares), the number of regressors (predictor) variables and

the number of observations. The model having the minimum BIC is determined

to be the model needed at the end of processing.

The final output of the system is the set of rules having the variables and

their corresponding time intervals in the antecedent part and the variable to be

described in the consequent part of the IF-THEN rule as in the following example.

EXAMPLE:

GAS FURNACE DATA PROCESSED BY MAR ALGORITHM

The Gas Furnace data contains two time series (multivariate time series data),

namely X and Y . X is GasF lowRate and Y is CO2Concentration. There is one

dependent series CO2Concentration and one independent series GasF lowRate.

The MAR algorithm processes the time series CO2Concentration as follows:

1. An equation set containing previous intervals of CO2Concentration is found.

2. The equation set containing previous instances of GasF lowRate is also

found.

3. BIC information criterion is used to compare the equations and to find the

best equation.

The resulting equation found is

CO2Concentration(t) = f(GasF lowRate(t − 4), CO2Concentration(t − 1)).

where t is representing the present time instance. This result shows

that the variable CO2Concentration is most appropriately explained by

using GasF lowRate(t − 4) (GasF lowRate at time instance (t-4)) and

CO2Concentration(t − 1) (CO2Concentration at time instance (t-1)).

The result of the process is described by means of an IF-THEN rule.

IF GasF lowRate(t − 4) is a AND

CO2Concentration(t − 1) is b

THEN


f(GasF lowRate(t − 4), CO2Concentration(t − 1))

25

A dependent variable can be represented by using many different combina-

tions of dependent variables and different lags of these variables. The Bayesian

Information Criterion (BIC) is utilized to choose the best variable set among the

candidates. The variable set providing the minimum BIC value is chosen among

the candidates and set as the antecedent part.

In the example above, CO2Concentration can be described by a linear func-

tion of previous time instances of itself and GasF lowRate. For each variable-lag

combination a regression equation is found. The BIC values of different regression

equations are compared and the one with smallest BIC value is defined as the

linear equation describing CO2Concentration. CO2Concentration is defined in

terms of its value at the time instance (t-1) and the value of the other variable

GasF lowRate at time instance (t-4). a and b are numerical values.

4.1.2 Model Selection

Model selection is the process of choosing the best set of regressors (predictor

variables) among a set of candidate variables. In forecasting the linear regression

models, some criteria are used to compare competing models and choosing the

best among them [15]. In-sample forecasting and out-of-sample forecasting are

concerned with determining a model which forecasts future values of the regres-

sand (dependent variable) given the values of the regressors. In-sample forecasting

shows how the chosen model fits the data in a given sample. Out-of-sample fore-

casting is concerned with determining how a fitted model forecasts future values

of the regressand, given the values of the regressors.

There are various criteria that are used for this purpose. R2, Adjusted R2,

Akaike Information Criterion (AIC) and Schwarz or Bayesian Information Cri-

terion (BIC) are some examples. The aim of the model selection criteria is to

minimize RSS (residual sum of squares). All the criteria except R2 aim min-

imizing the number of regressors by putting penalty on inclusion of increas-

ing number of regressors. The resulting regression model includes the least

number of predictor variables as well as minimum RSS possible. R2 is an in-

sample criterion which does not take into account adding more variables in the

26

model. It depends on RSS of the regression model for a regressand variable. Ad-

justed R2 is the adaptation of R2 to a criterion which puts penalty on choosing

larger number of variables. Akaike Information Criterion (AIC) applies putting

penalty on the increasing number of regressors. The mathematical formula is

AIC = e2k/nRSS/n where k is the number of regressors and n is the number

of observations. For mathematical convenience, ln (natural logarithm) transform

of AIC (ln(AIC) = (2k/n + ln(RSS/n))) is used. AIC is preferable in both

in-sample and also out-of-sample forecasting performance of a regression model.

Schwarz (or Bayesian) Information Criterion (BIC) is similar to AIC. The

mathematical formula is BIC = nk/nRSS/n and its log transform is ln(BIC =

(k/n)ln(n) + ln(RSS/n)) where penalty factor is [(k/n)ln(n)]. The model with

the smallest BIC is chosen as the best model. It is also convenient to use BIC to

compare in-sample or out-of-sample forecasting performance of a model.

In multivariate time series data, too many set of variables and time instances

can be used in defining a regressand variable. For that reason, it becomes impor-

tant to decide on the best model that describes the data set. Moreover, use of

redundant variables must be prevented. In order to chose the best set of variables

and time instances in the data set an information criterion is used which prevents

use of redundant variables by putting heavier penalty on them. The information

criterion used is BIC (Schwarz or Bayesian Information Criterion).

The mathematical formula of BIC is as follows:

BIC = T ∗ ln(RSS) + K ∗ ln(T ) (4.1)

In this formula, K stands for the number of regressors (i.e. the predictor

variables on the right hand side of the model equation), T number of observations

and RSS for residual sum of squares.

As K or T increases, BIC puts heavier penalty on the model. The variable set

producing the minimum information criterion is accepted as the definitive model.

4.1.3 Motivation for FLR in Fuzzy MAR

FLR model is applied to a set of input-output data pairs in order to obtain a

relationship between input and output data. The relationship is an equation in

27

which fuzzy numbers are used as coefficients. In this model, there exists an output

variable and one or more input variables. The output equation of model contains

all the variables defining the regressand (dependent) variable. The coefficients

Ai = (αi, ci) can have value greater than or equal to 0 (zero) for α (center) and

c (width).

In our problem domain, we assume the availability of a multivariate model

where there is more than one output variable. The output can be defined by

using more than one variable changing over time. Each variable in the model

is processed by MAR algorithm to obtain a function of regressors (predictor

variables) that defines the variable concerned. Moreover, more than one variables

in the multivariate system can be described in terms of other variables in the

system.

In order to find the linear function for each variable in the multivariate system,

a variant of least squares estimation (LSE) is used. Least squares approximation

uses the error between the observed and computed output variables. FLR treats

the computational error as the ambiguity in the data observed. The total width

of the fuzzy output is the total ambiguity in the output of the system. So, FLR

is similar to LSE where total observational error is replaced by total width of the

fuzzy coefficients.

The reasons why FLR is applied to MAR algorithm are as follows:

• Since MAR is a non-fuzzy method, its application to fuzzy systems is not

possible. The adaptation of the method is required.

• There exists a parallelism between obtaining a linear function for a depen-

dent variable in LSE and obtaining a fuzzy linear function in FLR.

• The ambiguity in the temporal data can be handled by fuzzy regression.

Consequently, FLR method is used in MAR algorithm to handle ambiguity

in multivariate data.

Compared to FLR, Fuzzy MAR is a new approach for processing the multi-

variate autoregressive time series data. When considering from fuzzy modeling

perspective, MAR algorithm

28

• deals with standard (non-fuzzy input and output) data and yields fuzzy

rules in order to cover the ambiguity in the temporal data

• produces fuzzy rules that can be used in any fuzzy system

• provides fuzzy rules for forecasting applications.

4.1.4 Fuzzification of MAR

MAR algorithm is a non-fuzzy method which produces linear functions that define

the variables in a multivariate system. Fuzzification is used to obtain fuzzy rules

that can be stored in fuzzy rule-bases where FLR is used to obtain fuzzy linear

functions for each variable. These functions can then be used as fuzzy rules in

fuzzy expert systems.

The resulting equations contain fuzzy coefficients as in a conventional fuzzy

linear regression equation such as

Y = A1 ∗ x1 + A2 ∗ x2 + · · · + Ap ∗ xp (4.2)

where Y is the output, ~x is the input vector and each Ai is a fuzzy number as

stated in Chapter 2. The method of finding a fuzzy linear regression is based

on minimization of total width of fuzzy coefficients in the equation. The model

is the solution of an LP (linear programming) problem in which the objective

function is to minimize total width of fuzzy coefficients.

Before considering multivariate temporal data, the modifications needed for

auto-regressive model will be explained. Given the linear function to define the

value of the variable xi at time instance t is

xi,t = f(xi,t−1, · · · , xi,t−p) (4.3)

which indicates that xi,t is dependent on p previous time instances of itself which

is explained as

xi,t = A1 ∗ xi,t−1 + A2 ∗ xi,t−2 + · · · + Ap ∗ xi,t−p (4.4)

In the equation (4.4), the value Ai is a fuzzy number with center α and width

c.

29

The objective function of the LP problem is also modified as follows:

minimize J(c) =∑

ct|xi| (4.5)

where |xi| = (|xi,t−1|, · · · , |xi,t−p|) and the constraints are

xi,t ≤ xtiα + |L−1(h)|ct|xi|

xi,t ≥ xtiα − |L−1(h)|ct|xi|

where L(x) = 1 − |x| , c ≥ 0 and i = 1, · · · , n

Each of the variables in the fuzzy linear regression equation is a previous value

of the variable to be described or all other possible variables.

In the scope of the thesis, FLR for standard (non-fuzzy input-output) data

will be applied since the observations are non-fuzzy values.

For multivariate time series analysis, Multivariate Auto-regression Algorithm

is adapted to the fuzzy linear regression analysis.

4.1.5 BIC in Fuzzy MAR

A modified BIC is used as criteria to evaluate the models obtained by using

different set of variables. Originally RSS is used as the main item in model

selection criteria. Since the difference between the expected and obtained output

is treated as the ambiguity in FLR, the measure of ambiguity is used instead of

RSS in modified BIC. As a result, modified BIC is as follows:

modified BIC = T ∗ ln(J(c)) + K ∗ ln(T ) (4.6)

where T is the number of observations used in FLR process in order to find a

linear function containing fuzzy coefficients, K is the number of regressors in

the model function and J(c) =∑

ci|xi| (total width of fuzzy coefficients in the

function).

modified BIC is used as the criterion in order to compare the models obtained

for a variable. The best model is the one with the minimum modified BIC value.

30

4.1.6 Obtaining a Linear Function for a Variable

In order to obtain a model for a variable xi in a multivariate temporal data, all

the variables (including their p previous time instances) are tested. It should be

noted that p is always less than the total number of time instances in the time

interval. At each step, the obtained model (a temporary model) is compared with

the previous best model (the model with minimum modified BIC).

The auto-regressive model is given in Section 4.1.4. In a multivariate auto-

regressive model, all variables must also be considered. If there is any input-

output relationship between the regressand variables and the regressors, these

relationships must be embedded within the functions.

The model for a variable xi in a vector of multivariate data ~xt of size n for T

time instances (as shown in Table 4.1) is defined as follows.

xi,t = f(x1,t−1, · · · , x1,t−p1 , · · · , xj,t−1, · · · , xj,t−pj, · · · , xn,t−1, · · · , xn,t−pn

) (4.7)

xi,t = A11∗x1,t−1+· · ·+A1p1 ∗x1,t−p1 +· · ·+An1∗xn,t−1+· · ·+Anpn∗xn,t−pn

(4.8)

which indicates that xi,t is dependent on pj previous time instances of xj.

The objective function of the LP problem by using the Equation 4.8 is for-

mulated as follows:

minimize J(c) =∑

ct|xi| (4.9)

where |xi| = |x1,1|, · · · , |x1,p1 |, · · · , |xj,1|, · · · , |xj,pj|, · · · , |xn,1|, · · · , |xn,pn

| and

xj,pj= xj,t−pj

and the constraints are

xi,t ≤ xtiα + |L−1(h)|ct|xi|

xi,t ≥ xtiα − |L−1(h)|ct|xi|

where L(x) = 1 − |x| , c ≥ 0 and i = 1, · · · , n.

It should be noted that the above LP problem is a general multivariate auto-

regressive model which is formulated for a variable xi and for a specific set of

input variables x1,1, · · · , x1,p1 , · · · , xj,1, · · · , xj,pj, · · · , xn,1, · · · , xn,pn

. In the set of

input variables, j = 1, · · · , n and pj ≤ T where n is the number of variables and

T is the number of time instances in the time interval.

31

Since for a variable all possible time intervals of all the variables are tested, the

general LP problem described in this section will be repeated for each variable.

4.1.7 Processing of Multivariate Data

Fuzzy MAR algorithm gets multivariate data and outputs fuzzy linear functions

stored as fuzzy rules. The multivariate data is input to the algorithm in table

format as in Table 4.1.

The algorithm in Table 4.2 runs for each variable xi in a vector of multivariate

data ~xt. The crucial point in the process is that the variables are processed one

by one by the algorithm. At the end, all the variables in the system are defined

by a linear function with fuzzy coefficients. From the viewpoint of complexity, it

can be thought that the algorithm is exponential, where all variables are tested

for all time instances. On the other hand, the model search is terminated when

a predefined p time instances are reached for a variable. Moreover, the algorithm

does not compare all possible models.

Another limitation is applied to the number of variables. An auto-regressive

model is found first. Then the model containing other variables are searched.

All the models are compared as they are obtained with the previous ideal model

(the model with minimum modified BIC). As a model of one variable (input) is

found, this specific variable is not processed any more. FLR is formulated for all

possible set of variables for all possible p time instances. Whenever a model is

found, FLR is called, i.e. LP problem is solved.

By using the algorithm, a regression equation is found for each variable in

the multivariate auto-regressive system. Since the variable depends on its values

at previous time instances, the algorithm first finds an auto-regressive equation

for that variable. The function f(xi,t−1, · · · , xi,t−k) found in Step 2 is an auto-

regressive function such that k = 1, · · · , p and p is a predetermined limit of

previous time instances to be used in the algorithm.

Whenever a function is found, modified BIC is computed and the function

with the minimum criterion is assigned to fi (the multivariate auto-regressive

function for the variable xi) where i is the index of the variable xi.

32

Table 4.2: Fuzzy MAR Algorithm

1. Input ~xt of size n for T time intervals such that eachvariable is xi where i = 1, · · · , n.

For each xi,

2. Find a linear function fii(xi,t−1, · · · , xi,t−k) for xi,t such thatthe function uses p previous time instances of xi. Notethat k changes from 1 to p.

(a) For each k, compute modified BIC.

(b) Find the minimum modified BIC value, choose thefunction fii for previous time instance k as the fii.

fii is an auto-regressive function such that

xi,t = A1 ∗ xi,t−1 + A2 ∗ xi,t−2 + · · · + Ak ∗ xi,t−k (4.10)

The equation above describes xi,t in terms of its p previousvalues.

3. Repeat Step 2 for all variables, j = 1, · · · , n such that

(a) Find a function for xi,t likefij(xj,t−1, · · · , xj,t−k) where k =1, · · · , p.

(b) Compute modified BIC for all the variables j = 1, · · · , n

(c) Choose the function fij with minimum modified BIC.

4. Compare fii with fij

(a) Choose the one with minimum modified BIC and assign itto fi.

(b) Note the index ind for which fi is found.

5. Find a function fij as follows:

(a) Concatenate xj,k to fi where j = 1, · · · , n and j 6= ind andk = 1, · · · , p.

(b) Compute modified BIC for fij

(c) Compare fi with fij.

(d) Choose the one with minimum modified BIC and assign itto fi.

(e) Continue for allj = 1, · · · , n. Find the function containing all possiblevariables for all possible time instances.

6. Find equations for all variables in the multivariate databy using Steps 2-5.

33

Since in the multivariate auto-regressive model, the variable can depend on

the previous values of the other variables, the auto-regressive functions containing

other variables as regressors are found, and their modified BIC values are com-

puted. These values are compared with fi’s and the function with the minimum

modified BIC is assigned as the auto-regressive equation.

In order to supply other variables in addition to the auto-regressive model,

the variables other than the auto-regressive model are added as in Step 5.a and

modified BIC is computed. The ultimate goal is to obtain the equation having

all the variables possible with the minimum modified BIC.

The above steps are repeated for all variables. Number of variables in the

regression equation, total number of observations, and total width of fuzzy coef-

ficients (J(c)) are the three values used in modified BIC in order to compare the

equations.

The output of the Fuzzy MAR Algorithm is the set of equations which describe

all of the dependent variables in the system. The following is an example of a

rule.

IF xt−4 is B1 AND yt−1 is B2

THEN yt = A0 + A1xt−4 + A2yt−1

It can be seen that yt is described by means of a function of previous instances

of itself and other variables which are xt−4 and yt−1.

4.2 ANFIS unfolded in time

4.2.1 Motivation for Unfolding in Time

The neuro-fuzzy systems in the literature are mostly multi-layer feed-forward

neural network structures. When temporal data is concerned it is needed to con-

struct a neural network structure which uses temporal relationships. Recurrent

neural network structures are more convenient for that purpose. Unfolding in

time is a method used for training the recurrent neural network structures. The

neuro-fuzzy approach in the study utilizes this method.

34

Y

X0

Y1

X1

Y2 Y3

Y4

X3X2

NN2NN1 NN3 NN4

0

Figure 4.1: ANFIS Unfolded in time

Within the scope of this thesis, a temporal neuro-fuzzy system is designed for

processing temporal data, which is based on the ANFIS neuro-fuzzy system. Un-

folding in time approach is applied to the neuro-fuzzy system in order to construct

a temporal multi-layer feed-forward neural network. The feed-forward neural net-

work is duplicated for T times where T is the number of time intervals needed

in the specific problem. The resulting system is called ANFIS unfolded in time.

The neural network structure enables us to define a problem where the input can

be a vector such that (~x, y) and the system produces only one output which is y.

Sample system can be seen in Figure 4.1. Each of the boxes represent one

ANFIS structure defined in Figure 3.1. In the problem given, it is assumed that

the output of the system depends on four previous input values. In order to

achieve the network structure, ANFIS is duplicated for 4 time intervals. The

input of the neuro-fuzzy system is composed of two elements which are X and Y .

There is only one output of the system which is Y . Initially (at time t = 0), X0

and Y0 are input to NN1 (network component for time interval 1). The output

of NN1 is obtained as Y1 (at time t = 1). Then, it is input to the NN2. Another

input is needed which is X1 (external input). So it is supplied externally, since

the system does not produce X1. Output for the second time interval is obtained

as Y2. The same process is performed for the rest of the time intervals (two more

time intervals). Finally, Y4 is obtained as the output of NN4. It is treated as

the output of the system ANFIS unfolded in time for t = 0. In other words, the

input is supplied at time 0, and the output for time 4 is obtained.

The same process is repeated for time t = 1, 2, . . . until the end of the sample

data set is reached.

35

4.2.2 Temporal Back-Propagation Algorithm

The algorithm used in the neural network structure is a modified back-propagation

algorithm. Since the basic neuro-fuzzy system is a feed-forward neural network,

back-propagation algorithm is convenient to use. Because the neural network

is duplicated T times the basic back-propagation learning algorithm is modified

accordingly.

The neuro-fuzzy system is treated as a black-box containing T neural net-

works. The connections between neural networks are also taken into account,

representing the temporal relationships. The parameters in the last network are

updated according to the error in the last interval. The error in the previous net-

works are updated by using the error in the specified time interval and the errors

propagated from the following intervals. The error coming from the following

intervals are back-propagated to the error in the specific interval. Unlike the con-

ventional unfolding in time method, the parameters are updated independently.

The algorithm in Table 4.3 describes the steps for processing the data for

one specific time interval. The data is presented to the neuro-fuzzy system at

each time interval. The data is processed and the output is obtained for T

time intervals ahead. The error is computed and back-propagated through the

network, updating the parameters of each node (online learning).

The algorithm contains two phases: Forward phase and Backward phase.

In the Forward phase, the data at specific time k is introduced to the system

and the computations are performed according to the input value. The important

feature of the temporal neuro-fuzzy system is that at the end of the computation

for a data at time interval k, it yields into the output of the system at time

interval k+T.

In the Backward phase, the parameters in all networks are updated according

to the output produced by the neuro-fuzzy system. The error in network T is used

to update only the parameters in network T. But, for updating the parameters

in network T-1, the error of the following network (which is network T) is back-

propagated to network T-1. The same process is applied to all previous time

intervals until all the parameters in all networks are updated.

36

Table 4.3: Forward and Backward Phases in ANFIS Unfolded in time

Part 1 (Forward Phase)

1. The data at time T=k is given as input to system ( ~xk, yk).2. The output yk+1 is computed for network at time interval 1.

3. If T is greater than 1, the last element in the inputvector ykbecomes yk+1 (y = y1). The output of each step will be oneof the elements in the input vector until t = T.

4. The output of the last neural network is the output ofANFIS unfolded in time. In other words, Yk = yT+k

Part 2 (Backward Phase)

1. Compute the error for the output response of the system Yk

and the given input vector ( ~xk, yk)

error = (yk − Yk)2 (4.11)

2. Back-propagate the error to the parameters in network T.3. Back-propagate the error to the error in the network T-1.

Then update the parameters in network T-1 by using thepropagated error.

4. Repeat the above step until t = 0.

The output of the forward phase is accepted as the output of AN-

FIS unfolded in time. At the end of the Backward Phase all parameters are

updated and the data in the next time interval is presented to the system.

The method of back-propagating the error is shown in Figure 4.2. If the error

computed at time interval t is E(t) then the error is back-propagated through the

neural network at time t (NN(t)). Moreover, it is also back-propagated through

the neural network at time t−1 (NN(t−1)). For that purpose, partial derivative

of E(t) is taken over E(t − 1) which is ∂Et

∂Et−1. The error is propagated through

the parameters in the next time interval such that the partial derivative of the

error E(t) over parameters (a, b, c) and (p0, p1, p2, · · ·) in time interval t summed

to the error in time interval t which is E(t − 1). The procedure goes on like this

for the previous time intervals.

37

E(t)∂E(t−1)∂

E(t−1) +

y(t)

E(t)

y(t−1)

NN(t−1) NN(t)

Figure 4.2: Error computation for a time interval

4.3 The Integrated System

The neuro-fuzzy method proposed in this study has two main components which

are Fuzzy MAR algorithm and ANFIS unfolded in time.

The Fuzzy MAR Algorithm produces fuzzy rules which explain each of the

dependent variables in the multivariate time series data. These fuzzy rules contain

explanatory variables in antecedent parts. The following is a sample rule produced

by Fuzzy MAR Algorithm:

IF X(t − 4) is A1 AND Y (t − 1) is B1

THEN Y (t) = f(X(t − 4), Y (t − 1))

In this example, two explanatory variables can be seen which are X(t−4) and

Y (t− 1). This resulting rule shows that the output part is determined according

to the values of two variables. Moreover, this rule gives information about how

many previous time intervals must be taken into account. In the example, the

value of X at four previous time intervals and the value of Y at one previous time

interval are concerned while computing the output.

As a result, the system interface module will produce the variables and the

number of time intervals to be processed by ANFIS unfolded in time. The original

ANFIS model is duplicated as many times as the number of time intervals. The

input variables will be the variables extracted from antecedent part by the System

Interface.

Figure 4.3 shows the interconnection between the modules. The interface

behaves like a pipe between the two modules of the neuro-fuzzy system. The

steps of the overall processing of the integrated system is as follows:

38

Fuz

zy

MA

R

Alg

orith

m

Sys

tem

Inte

rfac

e

AN

FIS

Unf

olde

d

in

Tim

e

Y t−

kX

t−k

()

,Y t

Tim

e S

erie

s D

ata

(Tab

ular

For

m)

Exp

ert

Sys

tem

Ant

eced

ent P

art

Var

iabl

esO

utpu

t Val

ue:

Time Series Data

Fuz

zy R

ules

RU

LE E

XT

RA

CT

ION

PR

ED

ICT

ION

Fig

ure

4.3:

Inte

grat

edSyst

em

39

1. Multivariate Time Series Data (in tabular form) is given as input to the

Fuzzy MAR Algorithm.

2. Fuzzy MAR Algorithm processes the input data and outputs Fuzzy Rules

in the form of fuzzy linear regression equations.

3. System Interface inputs the fuzzy equations and outputs the independent

variables and the previous time intervals.

4. ANFIS unfolded in time inputs the variables and number of time intervals

needed and processes the multivariate time series data. The output of the

process is the output of the dependent variable in number of time intervals

time.

EXAMPLE:

GAS FURNACE DATA PROCESSED BY INTEGRATED SYSTEM

As an example, Gas Furnace data in Table A.8 is processed. X is GasFlowRate

and Y is CO2Concentration. The system processes the data as follows:

1. In the data set there is one dependent variable, Y (CO2Concentration).

For the dependent variable, Y , the fuzzy linear regression equation Yt =

A0 + A1Xt−4 + A2Yt−1 is obtained at the end of Fuzzy MAR Algorithm.

Yt is the CO2Concentration at time t. A0, A1 and A2 are triangular fuzzy

numbers (which is denoted by a center and a width equal at both sides of the

center) which are fuzzy coefficients of variables Xt−4 (GasFlowRate at four

previous intervals) and Yt−1 (CO2Concentration at one previous interval).

These are the variables that determine the value of CO2Concentration at

time t. The resulting fuzzy rule produced by the Fuzzy MAR Algorithm for

Gas Furnace Data is as follows:

IF GasF lowRate(t − 4) is HIGH AND

CO2Concentration(t − 1) is LOW

THEN

CO2Concentration(t) = A0 + A1 ∗ GasF lowRate(t − 4) + A2 ∗CO2Concentration(t − 1)

40

2. The System Interface takes fuzzy rules as input and extracts the antecedent

part variables, which are GasF lowRate(t− 4) and CO2Concentration(t−

1). The value of number of time intervals which is 4, is provided to the

ANFIS unfolded in time.

3. ANFIS unfolded in time duplicates the ANFIS 4 times. It inputs the train-

ing data in tabular form (X,Y ). ANFIS unfolded in time starts processing

with time interval = 0. The data is given pair by pair. For time interval

= 0, CO2Concentration in time interval = 4 is the output. The process-

ing of ANFIS unfolded in time continues until time interval = 292. AN-

FIS unfolded in time is trained by using four-step ahead prediction. The

output of each input pair is compared to the four-time-interval ahead out-

put value. Sample data set size is 296 and 292 data pair is used in obtaining

the output.

4.4 Discussion

The temporal neuro-fuzzy approach designed for processing multivariate time

series data contains three modules. These modules are

• Fuzzy MAR: This algorithm produces fuzzy expert rules, by means of fuzzy

linear regression equations. These equations are used for knowledge-base of

the neuro-fuzzy system. They determine which variables must be used in the

second module and the number of time intervals needed for the prediction.

This module can be thought as a fuzzy rule extraction module.

• ANFIS unfolded in time: This module is based on the neuro-fuzzy model

ANFIS, and the knowledge-base is used to forecast the future values of mul-

tivariate time series data. This module uses online learning so that the data

can be supplied continuously. In other words, the data at a time interval

is presented to the network, and the parameters are updated according to

the error obtained for that time interval. This method of learning is more

convenient for temporal data.

41

• System Interface: This module provides necessary input to the AN-

FIS unfolded in time by using the output produced by the Fuzzy MAR

Algorithm.

The neuro-fuzzy approach can be applied to many areas where temporal data

is to be forecasted. The data set must be multivariate. It is quite appropriate to

have more than one dependent variable in the same data set. Some examples for

application areas are presented in the next chapter.

42

CHAPTER 5

EXPERIMENTAL RESULTS

In this chapter, the experimental results for the proposed system are presented.

The experiments on Fuzzy MAR Algorithm are given in the first section. The

experiments are performed on one synthetic data set produced by an artificial

function and 13 real data sets which are taken from the book by Reinsel [36].

Each of the series variable in the system is described by a fuzzy equation by

using a number of lags of a number of variables. Fuzzy outputs are obtained

by using fuzzy equations and are compared with the expected output. Since

the main purpose to use Fuzzy MAR Algorithm is extracting the rules to be

stored in ANFIS unfolded in time, necessary information is obtained from the

fuzzy equations and is used in neuro-fuzzy system.

The second section presents experiments performed on ANFIS unfolded in time.

The first set of experiments is performed by using Gas Furnace data which is a

benchmark problem in order to compare the performance of the proposed system

with similar neuro-fuzzy systems. The rest of the section is dedicated to the

experiments performed to test the Fuzzy MAR Algorithm.

5.1 Fuzzy MAR Algorithm

The synthetic data is prepared and tested in order to show how the Fuzzy MAR

Algorithm produces fuzzy equations and how it treats noise in the data. Then,

real data sets are tested in order to obtain fuzzy equations for time series variables

43

in multivariate systems. In order to evaluate the accuracy of the model obtained,

Accuracy Rate is used. Accuracy Rate is defined as the ratio of the number of

expected outputs found in the intervals of obtained outputs to the total number

of outputs.

5.1.1 Synthetic Data

A univariate time series is created and tested in a set of experiments. The time

series is similar to Fibonacci series. The initial values of the series are given

as x0 = 0.5 and x1 = 0.6. The function is 0.67 ∗ xt−1 + 0.47 ∗ xt−2. By using

this equation, synthetic data containing 200 time instances shown in Table A.1 is

produced. Fuzzy MAR algorithm is tested and a fuzzy equation is obtained. This

equation can be seen in Table 5.1. The fuzzy coefficients are (0.67,0) and (0.47,0)

in which the width values are zero. It can be said that if the data is without noise

Fuzzy MAR finds the exact crisp coefficients in the equation defining the data.

Since the problem is an LP problem which finds a fuzzy equation with minimum

modified BIC value, the result obtained is as expected.

Table 5.1: Fuzzy equations for Synthetic Data

Variable Fuzzy Equation

x (0.67, 0)xt−1 + (0.47, 0)xt−2

(noisy)x (0.2734, 0)xt−1 + (0.9058, 0.03307)xt−2

As the second step, uniform random noise is added to the data. The aim

of the procedure is to describe how the noise affects the results. At the end

of the processing, the same variables are found for 2 lags as in the minimum

modified BIC value. The fuzzy equation obtained by using the synthetic data

with random noise is also shown in Table 5.1. On the other hand, the coefficients

become different since the algorithm tries to fit an equation to a modified data

set. By using the fuzzy equation, it is found that 152 samples are within the

interval numbers, where 46 is out of the interval, which yields an accuracy rate

of 0.767677. Test results for different lags are listed in 5.2.

44

Table 5.2: Accuracy Rates for Synthetic Noisy Data

Lag Equation Rate

3 (0.3737, 0.0005141)x0,t−1 + (0.6151, 0)x0,t−2 0.741117+(0.1952, 0.03075)x0,t−3

4 (0.0486, 0)x0,t−1 + (0.3169, 0)x0,t−2 0.688776+(0.5621, 0.02907)x0,t−3 + (0.3965, 0)x0,t−4

5 (0.2287, 0)x0,t−1 + (0.2018, 0.02408)x0,t−2 0.600000+(0.1387, 0)x0,t−3 + (0.5234, 0)x0,t−4

+(0.2604, 0)x0,t−5

6 (0.1289, 0)x0,t−1 + (0.2076, 0)x0,t−2 0.587629+(0.1202, 0.0208)x0,t−3 + (0.4263, 0)x0,t−4

+(0.2495, 0.004869)x0,t−5 + (0.2979, 0)x0,t−6

7 (0.1289, 0)x0,t−1 + (0.2076, 0)x0,t−2 0.585492+(0.1202, 0.0208)x0,t−3 + (0.4263, 0)x0,t−4

+(0.2495, 0.004869)x0,t−5 + (0.2979, 0)x0,t−6

8 (0.07262, 0)x0,t−1 + (0.1469, 0)x0,t−2 0.562500+(0.2004, 0)x0,t−3 + (0.2313, 0)x0,t−4

+(0.1674, 0.02878)x0,t−5 + (0.1899, 0)x0,t−6

+(0.09604, 0)x0,t−7 + (0.5001, 0)x0,t−8

9 (0.1098, 0)x0,t−1 + (0.2555, 0)x0,t−2 0.518325+(0.2709, 0)x0,t−3 + (0.1365, 0)x0,t−4

+(0.08758, 0.02707)x0,t−5

+(0.2556, 0)x0,t−8 + (0.4969, 0)x0,t−9

10 (0.08275, 0)x0,t−1 + (0.2289, 0)x0,t−2 0.505263+(0.2731, 0.004673)x0,t−3 + (0.2052, 0)x0,t−4

+(0.1687, 0)x0,t−5 + (0.0911, 0.02803)x0,t−8

+(0.461, 0)x0,t−9 + (0.1029, 0)x0,t−10

45

Table 5.3: Expected and Obtained Output for Variable x (Synthetical Data withRandom Noise)

Expected ExpectedTime Output Obtained Result Time Output Obtained Result

2 0.626499 (0.600421,0.633487) 52 71.508 (68.4544,72.3598)3 0.71182 (0.694942,0.734622) 53 76.9886 (73.5463,77.6378)4 0.803544 (0.741396,0.782828) 54 83.3371 (83.4582,88.1872)5 0.839407 (0.840938,0.888013) 55 91.7586 (89.9771,95.0686)6 0.920627 (0.930797,0.983938) 56 101.32 (97.8203,103.332)7 1.06757 (0.984302,1.03981) 57 111.297 (107.784,113.853)8 1.15198 (1.09536,1.15625) 58 125.542 (118.857,125.558)9 1.21333 (1.24669,1.31729) 59 134.527 (131.459,138.819)

10 1.39984 (1.33713,1.41331) 60 150.781 (146.348,154.65)11 1.50858 (1.44166,1.5219) 61 159.161 (158.634,167.53)12 1.63537 (1.63417,1.72675) 62 181.497 (175.111,185.083)13 1.85463 (1.76374,1.86351) 63 196.2 (188.531,199.057)14 1.96828 (1.93434,2.04249) 64 214.86 (212.045,224.048)15 2.25586 (2.15678,2.27944) 65 236.461 (229.978,242.954)16 2.45773 (2.3346,2.46477) 66 254.658 (252.17,266.38)17 2.61255 (2.64078,2.78996) 67 291.442 (275.998,291.636)18 2.85668 (2.85929,3.02182) 68 309.519 (301.936,318.777)19 3.21187 (3.06115,3.23392) 69 350.674 (338.982,358.256)20 3.56928 (3.37132,3.56024) 70 366.109 (366.011,386.48)21 3.85565 (3.77904,3.99145) 71 405.221 (406.149,429.341)22 4.27612 (4.16927,4.40531) 72 450.816 (430.313,454.525)23 4.57113 (4.53416,4.78914) 73 503.323 (476.915,503.713)24 5.07453 (4.98178,5.26458) 74 540.205 (531.063,560.877)25 5.71471 (5.37689,5.67919) 75 609.191 (586.973,620.26)26 6.00067 (5.99126,6.32685) 76 662.548 (638.023,673.748)27 6.75417 (6.62817,7.0061) 77 720.063 (712.819,753.107)28 7.19211 (7.08374,7.48059) 78 816.444 (775.112,818.928)29 8.06284 (7.86111,8.30778) 79 888.41 (851.659,899.279)30 8.78707 (8.48138,8.95701) 80 973.698 (955.453,1009.45)31 9.68024 (9.43932,9.97254) 81 1071.73 (1041.58,1100.33)32 10.8738 (10.3156,10.8967) 82 1180.6 (1142.82,1207.21)33 11.6741 (11.4214,12.0616) 83 1283.63 (1258.14,1329.02)34 13.1268 (12.6819,13.401) 84 1385.1 (1381.33,1459.4)35 14.3646 (13.7776,14.5497) 85 1504.14 (1498.99,1583.88)36 15.5377 (15.3839,16.252) 86 1639.34 (1620.09,1711.69)37 17.4627 (16.7849,17.7349) 87 1887.43 (1760.95,1860.43)38 19.0148 (18.335,19.3626) 88 2051.09 (1946.78,2055.19)39 20.2024 (20.4394,21.5943) 89 2164.62 (2208.05,2332.87)40 23.1254 (22.1188,23.3763) 90 2432.17 (2381.92,2517.57)41 24.9276 (23.9544,25.2905) 91 2697.98 (2554.15,2697.3)42 27.5841 (26.9982,28.5276) 92 2904.87 (2860.33,3021.18)43 29.3673 (29.2973,30.9459) 93 3193.49 (3148.88,3327.31)44 32.7011 (32.1034,33.9276) 94 3486.68 (3408.36,3600.46)45 37.1032 (34.5711,36.5133) 95 3972.9 (3740.41,3951.61)46 38.8759 (38.6843,40.8469) 96 4377.22 (4129.23,4359.82)47 42.9389 (43.011,45.4647) 97 4689.49 (4664.13,4926.87)48 47.1679 (45.6689,48.2399) 98 5249.06 (5102.38,5391.86)49 52.2348 (50.3712,53.2109) 99 5592.7 (5527.9,5838.03)50 59.0535 (55.4474,58.5667) 100 6182.63 (6110.23,6457.36)51 61.868 (61.7337,65.1882) 101 6981.92 (6571.43,6941.29)

46

Table 5.3: Expected and Obtained Output for Variable x(Synthetical Data withRandom Noise) (cont’d)

Expected ExpectedTime Output Obtained Result Time Output Obtained Result

102 7498.56 (7304.82,7713.69) 151 728976 (703252,742349)103 8395.38 (8143.66,8605.4) 152 796504 (797158,842461)104 9088.85 (8839.75,9335.66) 153 885605 (853987,902196)105 10120.2 (9812.06,10367.3) 154 984722 (937283,989958)106 10970.9 (10699.3,11300.3) 155 1.09348e+06 (1.04214e+06,1.10071e+06)107 11759.3 (11831.9,12501.2) 156 1.16042e+06 (1.15838e+06,1.22351e+06)108 12962.7 (12789.9,13515.5) 157 1.3188e+06 (1.27161e+06,1.34392e+06)109 14514.6 (13807,14584.7) 158 1.40854e+06 (1.37333e+06,1.45007e+06)110 15410.7 (15281.7,16138.9) 159 1.56598e+06 (1.53609e+06,1.62331e+06)111 16955.8 (16881.1,17841) 160 1.7481e+06 (1.65746e+06,1.7061e+06)112 19036.9 (18085.6,19104.8) 161 1.84777e+06 (1.84466e+06,1.94822e+06)113 21519 (20003,21124.4) 162 2.04278e+06 (2.03086e+06,2.14647e+06)114 23443.1 (22498,23756.9) 163 2.30429e+06 (2.17116e+06,2.29336e+06)115 25853.2 (25190.3,26613.4) 164 2.50096e+06 (2.41286e+06,2.54795e+06)116 27251.6 (27528.5,29078.9) 165 2.64121e+06 (2.69486e+06,2.84725e+06)117 31263.7 (30014.3,31724.1) 166 2.96943e+06 (2.90485e+06,3.07025e+06)118 34077.5 (32331.7,34133.9) 167 3.35508e+06 (3.11698e+06,3.29166e+06)119 37706.7 (36602.5,38670.1) 168 3.54842e+06 (3.50888e+06,3.70526e+06)120 39193.2 (40050.5,42304.2) 169 3.8777e+06 (3.89833e+06,4.12021e+06)121 43708 (43624.4,46118.1) 170 4.31906e+06 (4.15709e+06,4.39176e+06)122 48871.9 (46156.1,48748.1) 171 4.66011e+06 (4.56514e+06,4.82159e+06)123 52861 (51508.2,54398.8) 172 5.08252e+06 (5.04358e+06,5.32922e+06)124 59003.2 (57105.7,60337.8) 173 5.58839e+06 (5.45673e+06,5.76492e+06)125 65590.1 (62266.5,65762.4) 174 6.29393e+06 (5.96369e+06,6.29982e+06)126 69383.8 (69428.1,73330.2) 175 7.05541e+06 (6.59809e+06,6.96767e+06)127 77726 (76214.1,80551.8) 176 7.59875e+06 (7.42205e+06,7.83829e+06)128 84286.2 (81805.8,86394.4) 177 8.4477e+06 (8.23519e+06,8.70178e+06)129 90871.8 (90880.2,96020.5) 178 9.34401e+06 (8.9415e+06,9.44403e+06)130 104816 (98406.1,103980) 179 1.0296e+07 (9.92748e+06,1.04862e+07)131 114445 (107966,113976) 180 1.12216e+07 (1.097e+07,1.1588e+07)132 126951 (122768,129700) 181 1.2344e+07 (1.2054e+07,1.27349e+07)133 134938 (134591,142160) 182 1.31806e+07 (1.31686e+07,1.39107e+07)134 150247 (147690,156086) 183 1.42902e+07 (1.4377e+07,1.51933e+07)135 164327 (158846,167770) 184 1.59208e+07 (1.54105e+07,1.62821e+07)136 177082 (176057,185993) 185 1.72102e+07 (1.68247e+07,1.77698e+07)137 194626 (191833,202700) 186 1.88777e+07 (1.86003e+07,1.96532e+07)138 220525 (207761,219472) 187 2.15618e+07 (2.01816e+07,2.13198e+07)139 238621 (230154,243025) 188 2.27424e+07 (2.23707e+07,2.36192e+07)140 267363 (257705,272289) 189 2.57476e+07 (2.50361e+07,2.6462e+07)141 293033 (281357,297138) 190 2.80575e+07 (2.68881e+07,2.83921e+07)142 317718 (313460,331141) 191 3.00922e+07 (3.01424e+07,3.18452e+07)143 342975 (342612,361991) 192 3.47788e+07 (3.27147e+07,3.45703e+07)144 387064 (371062,392073) 193 3.81566e+07 (3.57718e+07,3.77619e+07)145 416289 (405158,427840) 194 3.96356e+07 (4.07856e+07,4.30857e+07)146 451332 (451628,477226) 195 4.52894e+07 (4.4138e+07,4.66614e+07)147 488801 (486715,514245) 196 4.77814e+07 (4.69746e+07,4.95958e+07)148 549516 (527543,557391) 197 5.39414e+07 (5.25903e+07,5.55854e+07)149 591191 (576844,609170) 198 5.97608e+07 (5.64494e+07,5.96093e+07)150 685016 (641228,677570) 199 6.44507e+07 (6.34166e+07,6.69839e+07)

47

0

1e+07

2e+07

3e+07

4e+07

5e+07

6e+07

7e+07

0 1e+07 2e+07 3e+07 4e+07 5e+07 6e+07

x0,t

x0,t-1

Fuzzy Coefficient: (0.67, 0)

0

1e+07

2e+07

3e+07

4e+07

5e+07

6e+07

7e+07

0 1e+07 2e+07 3e+07 4e+07 5e+07 6e+07

x0,t

x0,t-2


Figure 5.1: Linearity Figures for Synthetic Data (xt = 0.67xt−1 + 0.47xt−2)

0

1e+07

2e+07

3e+07

4e+07

5e+07

6e+07

7e+07

0 1e+07 2e+07 3e+07 4e+07 5e+07 6e+07

x0,t

x0,t-1


0

1e+07

2e+07

3e+07

4e+07

5e+07

6e+07

7e+07

0 1e+07 2e+07 3e+07 4e+07 5e+07 6e+07

x0,t

x0,t-2

Fuzzy Coefficient: (0.905835, 0.0330666)

Figure 5.2: Linearity Figures for Synthetic Data (xt = 0.67xt−1 + 0.47xt−2) withuniform random noise

48

5.1.2 Real Data

All of the thirteen data sets are used in the experiments. The data sets are

processed and fuzzy equations are obtained for each of the variables in each of

the data sets. These fuzzy equations are converted to the fuzzy rules that can be

used in any fuzzy expert system. During the experiments, the fuzzy equations

are also used in order to compute the output value. The outputs obtained are

interval numbers and can be seen in Appendices. In this section, the results

of each experiment is presented and discussed. The expected output value is

compared with the obtained value and the rate in which the expected value is

included in the interval (accuracy rate) is computed and listed in Table 5.52.

5.1.2.1 AAA Corporate Bonds and Commercial Paper Interest Rates

Data

Table 5.4: Fuzzy equations for AAA CP Data


x0 (0.9922, 0)x0,t−1 + (0, 0.0441)x0,t−2

x1 (0.5244, 0)x1,t−1 + (0, 0.1173)x1,t−2 + (0.3854, 0.02304)x0,t−1

There exists two time series in this data set which are AAA Bond Rate (x0)

and Commercial Paper Rate (x1). The data is collected quarterly between 1953

and 1970 and can be seen in Table A.3.

Fuzzy MAR Algorithm produces two fuzzy equations each for one variable.

The fuzzy equations obtained for both variables can be seen in Table 5.4. The

program is executed for 2 lags.

Linearity between the output variables and the defining variables can be seen

in Figure 5.3. x0 is defined by using x0,t−1 and x0,t−2. There is an obvious linear

relationship between x0 and the right-hand side variables (x0,t−1, x0,t−2). The

variable x1 is defined by three variables and the figure between x1 and x1,t−1

seems to be the most linear one. The width of the coefficient is zero which obeys

the minimization constraint.

49

The accuracy of the model is 0.7577143 and 0.685724 for the variables x0 and

x1 respectively as can be seen in Table 5.52. The model for the variable x0 is

more accurate than the model for x1. The output results for both of the variables

are presented in Table 5.5 and Table 5.6 for the variables x0 and x1 respectively.

The estimated output has width around 0.36 for the variable x0 and 1.15 for the

variable x1. The difference between the width values also shows that the model

for the variable x0 is more accurate than the model for the variable x1.

50

Table 5.5: Expected and Obtained Output for Variable x0 (AAA-CorporateBonds)

Time Expected Output Obtained Result Time Expected Output Obtained Result

2 3.27 (3.16181,3.43256) 37 4.297 (4.18127,4.5702)3 3.133 (3.09806,3.39113) 38 4.337 (4.06915,4.45808)4 2.957 (2.96446,3.25285) 39 4.257 (4.11382,4.49279)5 2.877 (2.79587,3.07218) 40 4.197 (4.03268,4.41517)6 2.883 (2.72425,2.98504) 41 4.22 (3.97667,4.35211)7 2.887 (2.73374,2.98747) 42 4.287 (4.00214,4.37229)8 2.98 (2.73744,2.9917) 43 4.333 (4.06761,4.43978)9 3.033 (2.82954,3.08415) 44 4.37 (4.11029,4.48838)10 3.1 (2.87803,3.14084) 45 4.407 (4.14498,4.52712)11 3.117 (2.94217,3.20966) 46 4.41 (4.18006,4.56546)12 3.097 (2.95608,3.22948) 47 4.43 (4.1814,4.57007)13 3.263 (2.93549,3.21039) 48 4.42 (4.20112,4.59005)14 3.423 (3.10108,3.37422) 49 4.443 (4.19031,4.58101)15 3.677 (3.25252,3.54029) 50 4.497 (4.21357,4.60339)16 3.7 (3.49749,3.79937) 51 4.613 (4.26614,4.65798)17 3.773 (3.50911,3.8334) 52 4.813 (4.37886,4.77546)18 4.07 (3.58053,3.90684) 53 5.003 (4.57219,4.97902)19 3.997 (3.872,4.20475) 54 5.32 (4.75189,5.17637)20 3.607 (3.78647,4.14542) 55 5.383 (5.05805,5.49928)21 3.58 (3.40272,3.75523) 56 5.12 (5.10658,5.57577)22 3.87 (3.39313,3.71124) 57 5.263 (4.84285,5.31759)23 4.093 (3.68207,3.9978) 58 5.617 (4.99634,5.44788)24 4.13 (3.89055,4.23185) 59 6.027 (5.34128,5.80544)25 4.353 (3.91743,4.2784) 60 6.127 (5.73248,6.22786)26 4.473 (4.13706,4.5013) 61 6.253 (5.81363,6.34517)27 4.57 (4.2463,4.6302) 62 6.077 (5.93424,6.4746)28 4.553 (4.33725,4.73174) 63 6.243 (5.75405,6.30552)29 4.453 (4.3161,4.71915) 64 6.7 (5.92652,6.46247)30 4.313 (4.21763,4.61917) 65 6.887 (6.37265,6.92324)31 4.32 (4.08313,4.47585) 66 7.063 (6.53805,7.12894)32 4.27 (4.09625,4.47662) 67 7.467 (6.70443,7.31182)33 4.283 (4.04633,4.42732) 68 7.893 (7.09753,7.72044)34 4.437 (4.06143,4.43802) 69 8.14 (7.50241,8.16094)35 4.41 (4.21366,4.59139) 70 8.22 (7.7287,8.42481)36 4.41 (4.18008,4.57139) 71 7.907 (7.79719,8.51508)

51

Table 5.6: Expected and Obtained Output for Variable x1(AAA-CorporateBonds)


2 2.747 (2.30671,3.00565) 37 3.203 (2.94012,3.86037)3 2.373 (2.31782,3.08374) 38 3.333 (2.85638,3.81506)4 2.037 (2.05751,2.84621) 39 3.263 (2.94373,3.89487)5 1.633 (1.8614,2.55427) 40 3.31 (2.86279,3.84074)6 1.363 (1.65997,2.27033) 41 3.317 (2.8739,3.83267)7 1.31 (1.56794,2.08382) 42 3.697 (2.8804,3.85124)8 1.613 (1.57326,2.02599) 43 3.907 (3.10312,4.0787)9 1.967 (1.77207,2.21665) 44 3.95 (3.18534,4.25217)10 2.327 (1.94137,2.45947) 45 3.933 (3.19667,4.31446)11 2.833 (2.11291,2.71713) 46 3.91 (3.19612,4.3257)12 3 (2.34218,3.03163) 47 4.063 (3.18714,4.31287)13 3.263 (2.36316,3.17038) 48 4.3 (3.27732,4.39858)14 3.35 (2.54164,3.39568) 49 4.38 (3.38003,4.53671)15 3.63 (2.6144,3.5375) 50 4.38 (3.40252,4.61586)16 3.63 (2.84307,3.79828) 51 4.47 (3.4127,4.64729)17 3.683 (2.81856,3.84051) 52 4.97 (3.50193,4.74187)18 3.953 (2.87281,3.89812) 53 5.427 (3.82604,5.09631)19 3.993 (3.1158,4.16723) 54 5.79 (4.0759,5.4722)20 2.917 (3.07866,4.19005) 55 6 (4.32752,5.84562)21 1.817 (2.3684,3.47121) 56 5.45 (4.4179,6.02405)22 2.13 (1.908,2.75717) 57 4.717 (4.00955,5.65285)23 3.213 (2.30623,2.91074) 58 4.973 (3.74151,5.26238)24 3.303 (2.91823,3.60644) 59 5.303 (4.09,5.45524)25 3.603 (2.85182,3.79577) 60 5.58 (4.38159,5.82578)26 4.193 (3.07939,4.05472) 61 6.08 (4.52438,6.05058)27 4.76 (3.39707,4.4483) 62 5.963 (4.79974,6.39672)28 4.678 (3.66035,4.85445) 63 5.963 (4.61597,6.32212)29 4.073 (3.54469,4.871) 64 6.657 (4.68985,6.3762)30 3.373 (3.20082,4.50329) 65 7.54 (5.21938,6.92679)31 3.27 (2.85398,4.00808) 66 8.487 (5.66878,7.54759)32 3.013 (2.8846,3.87483) 67 8.62 (6.12558,8.21963)33 2.86 (2.74379,3.70756) 68 8.553 (6.23066,8.56545)34 2.897 (2.69842,3.6025) 69 8.167 (6.3343,8.71992)35 3.057 (2.79157,3.66686) 70 7.837 (6.22925,8.61054)36 3.243 (2.86135,3.74407) 71 7.293 (6.13047,8.42489)

52

2

3

4

5

6

7

8

9

2 3 4 5 6 7 8 9

x0,t

x0,t-1


2

3

4

5

6

7

8

9

2 3 4 5 6 7 8 9

x0,t

x0,t-2

Fuzzy Coefficient: (0, 0.0440965)

1

2

3

4

5

6

7

8

9

2 3 4 5 6 7 8 9

x1,t

x0,t-1


1

2

3

4

5

6

7

8

9

1 2 3 4 5 6 7 8 9

x1,t

x1,t-1


1

2

3

4

5

6

7

8

9

1 2 3 4 5 6 7 8 9

x1,t

x1,t-2


Figure 5.3: Linearity Figures for AAA corporate bonds

53

5.1.2.2 Monthly Agriculture Data

Table 5.7: Fuzzy equations for Agriculture Data


x0 (0.8611, 0)x0,t−1 + (0.1203, 0.01958)x0,t−2

+(9.564e − 05, 0)x1,t−1 + (0.0003082, 4.306e − 05)x1,t−2

x1 (8.054, 1.575)x2,t−1 + (0.2305, 0.006622)x1,t−1

+(0.1482, 0.111)x1,t−2 + (32.72, 14.66)x0,t−1

x2 (0.7253, 0.03831)x2,t−1 + (0.1085, 0)x2,t−2+(0.003689, 0.0009542)x3,t−2 + (0.001525, 0)x1,t−1

x3 (9.327, 12.09)x0,t−1 + (14.5, 0)x0,t−2 + (0.2195, 0)x1,t−1

+(0.01634, 0)x1,t−2 + (0.1806, 0)x3,t−1 + (0.2703, 0.0295)x3,t−2

Monthly agriculture data contains four time series which are First difference

of the logarithm of exchange rate (x0), price (x1), logarithm of levels of sales (x2)

and logarithm of shipments (x3). The data set can be seen in Table A.4. The

data is collected between February 1978 and December 1992.

Four fuzzy equations produced by the algorithm can be seen in Table 5.7. The

outputs are obtained for 4 lags.

In Figure 5.4, linearity between the output variables and the defining vari-

ables can be seen. The relationship between x0 and x0,t−1, x0,t−2 is linear. On

the other hand, the relationship with x1,t−1 and x1,t−2 is not linear. In the cor-

responding fuzzy equation, the coefficients have center values very close to zero

when compared to the other coefficients in the same equation. The variable x1

has no apparent linear relationship with any lagged variables and the estimation

rates shown in Table 5.52 is smaller than the other variables in the multivari-

ate system. The variables x2,t−1 and x2,t−2 show linear relationship with x2 with

larger coefficient center values compared to the other coefficients in the same fuzzy

equation. Other variables (x1,t−1 and x3,t−2) does not imply a linear relationship.

The variable x3 does not have any linear relationship.

The accuracy rates of the models the variables x0, x1, x2 and x3 are 0.717514,

0.536723, 0.734463 and 0.581921 respectively. Rates for the variables x1 and x3

54

are smaller than the rates for x0 and x2 as shown in Table 5.52. The output

results for all four variables can be seen in Figures 5.8, 5.9, 5.10 and 5.11. The

maximum width values for the variables x0, x1, x2 and x3 are 2.17, 2500, 11

and 1300 respectively. The fuzzy linear equations fitted for x1 and x3 has large

coefficients to fit to as much data as possible.

55

Table 5.8: Expected and Obtained Output for Variable x0 (Agriculture Data)


2 72.1436 (72.3359,75.5583) 62 91.9752 (89.6718,93.625)3 73.6363 (71.1883,74.4762) 63 91.6901 (90.0306,94.0022)4 72.575 (72.2872,75.4766) 64 93.4442 (89.7597,93.7403)5 70.0887 (71.6261,74.9516) 65 94.8545 (91.1182,95.0463)6 68.0279 (69.1703,72.3314) 66 96.1456 (92.6993,96.6997)7 67.902 (67.354,70.5709) 67 95.6553 (94.3351,98.5517)8 66.4023 (66.3109,69.2102) 68 93.8139 (93.7688,97.9338)9 68.2956 (65.2184,68.115) 69 94.8224 (92.0609,96.2429)10 68.9532 (67.3397,70.3616) 70 95.3536 (92.3618,96.3055)11 70.8264 (68.1063,71.2704) 71 95.4012 (93.1027,97.2028)12 71.6424 (69.2768,72.2867) 72 94.5507 (92.6864,96.647)13 72.2849 (70.3709,73.4952) 73 93.1276 (92.0107,95.9578)14 73.3875 (71.0169,74.242) 74 93.1344 (91.1084,95.1114)15 73.8351 (71.9766,75.0454) 75 94.6948 (91.1227,95.1908)16 74.3947 (73.7917,77.3576) 76 95.0941 (91.8658,95.7361)17 73.6612 (74.3405,77.8529) 77 98.045 (92.4197,96.3586)18 74.2412 (74.5723,78.323) 78 98.7482 (95.6776,99.6589)19 74.3801 (74.5836,78.4322) 79 99.9553 (97.7941,102.426)20 75.2657 (72.7985,75.9459) 80 100.402 (97.55,101.818)21 77.8164 (74.2484,77.6407) 81 99.3523 (97.6848,101.865)22 77.1804 (76.4181,79.7872) 82 100.819 (96.9085,101.147)23 77.1146 (76.1977,79.723) 83 101.938 (97.8596,102.019)24 77.8007 (75.9534,79.3611) 84 103.39 (98.8092,102.945)25 79.3713 (76.7709,80.3218) 85 103.404 (100.502,104.76)26 79.7922 (77.5948,80.9784) 86 100.763 (101.037,105.431)27 76.4102 (78.0068,81.4519) 87 101.552 (98.7987,103.317)28 74.8877 (75.0246,78.3884) 88 101.499 (98.3378,102.44)29 74.7106 (74.089,77.4903) 89 99.2322 (98.5743,102.786)30 75.5064 (74.2941,77.8095) 90 100.008 (96.5746,100.747)31 73.7472 (75.2637,78.7506) 91 100.655 (97.1778,101.327)32 73.3667 (74.6181,78.4682) 92 95.8412 (97.9137,102.059)33 74.8438 (73.4793,77.0503) 93 94.1735 (94.2754,98.6335)34 74.962 (74.5496,78.1176) 94 93.6765 (92.1029,96.1493)35 73.7306 (74.0397,77.4857) 95 91.7611 (91.7101,95.8065)36 75.6072 (72.3984,75.6629) 96 87.5968 (89.5387,93.4787)37 76.0088 (73.777,76.9752) 97 85.7529 (85.4852,89.3051)38 76.9838 (74.0027,77.2228) 98 84.6784 (83.0688,86.6033)39 79.1158 (74.6687,77.7689) 99 83.0553 (81.9364,85.3693)40 80.3512 (77.1798,80.5015) 100 83.9655 (80.6613,84.1143)41 82.5629 (78.3285,81.7452) 101 81.9692 (81.3652,84.7903)42 83.8029 (80.0323,83.3485) 102 80.7844 (79.6334,83.0704)43 82.2222 (81.7894,85.3124) 103 80.2596 (78.5065,81.8369)44 82.2338 (80.9872,84.6453) 104 80.1017 (78.3847,81.8328)45 81.0846 (81.1017,84.8488) 105 81.8333 (78.0429,81.4201)46 80.6541 (79.5238,83.0905) 106 82.7806 (79.4978,82.8704)47 81.4489 (78.8245,82.317) 107 80.1827 (80.4458,83.8635)48 84.3711 (79.3237,82.6886) 108 79.782 (78.3398,81.8197)49 87.9434 (82.5804,86.1557) 109 79.215 (77.8926,81.2332)50 88.5295 (86.1635,89.9506) 110 76.563 (78.196,81.7688)51 86.9241 (86.8106,90.6591) 111 75.3233 (75.8044,79.4168)52 91.2189 (85.3783,89.272) 112 75.9641 (73.8547,77.1463)53 92.9265 (88.3817,92.0523) 113 77.3026 (74.2707,77.5614)54 95.9211 (90.3988,94.2019) 114 76.5162 (75.1775,78.3511)55 96.9391 (93.6941,97.7661) 115 74.9921 (74.8145,78.1553)56 96.3476 (94.8631,98.9826) 116 74.6192 (73.0577,76.2112)57 95.6618 (94.6383,98.9828) 117 71.9004 (73.1565,76.3551)58 91.8009 (93.1058,97.0776) 118 69.7991 (71.2054,74.6153)59 90.774 (90.2997,94.4191) 119 69.3949 (68.3481,71.4104)60 91.4701 (89.2006,93.2144) 120 69.7908 (68.038,70.9896)61 91.6144 (89.7034,93.7073) 121 68.916 (69.0666,72.3059)

56

Table 5.8: Expected and Obtained Output for Variable x0 (AgricultureData)(cont’d)


122 67.8466 (67.848,70.9173) 152 69.6079 (70.1658,73.2726)123 68.0343 (66.8626,69.9568) 153 69.1751 (67.9466,70.9274)124 69.088 (66.4511,69.3607) 154 70.0241 (67.7072,70.6631)125 71.0494 (67.6553,70.5488) 155 69.9899 (68.6465,71.7343)126 71.2644 (70.2219,73.4721) 156 68.8145 (68.6331,71.5683)127 71.3566 (69.8807,73.0278) 157 72.6329 (68.5882,71.9748)128 69.5652 (69.8015,72.8041) 158 73.3007 (70.5912,73.5505)129 67.1983 (68.8322,72.1) 159 73.4972 (71.5191,74.6347)130 66.9505 (66.0221,69.0024) 160 74.6456 (71.6139,74.7083)131 68.112 (66.0518,69.0095) 161 74.1126 (72.7203,75.8261)132 68.3782 (67.4229,70.6101) 162 73.6676 (72.7201,75.9484)133 70.3833 (66.7901,69.716) 163 72.2758 (72.4183,75.7196)134 69.8679 (68.8326,71.7379) 164 71.3388 (70.8541,73.9995)135 71.8275 (69.4628,72.793) 165 70.1196 (70.0438,73.2296)136 73.465 (70.4073,73.542) 166 69.4182 (68.548,71.5572)137 71.9767 (71.6865,74.7617) 167 68.2921 (67.9011,70.934)138 73.0268 (70.7637,73.9855) 168 69.0452 (66.6066,69.4858)139 74.2547 (71.3099,74.4191) 169 71.1068 (67.5095,70.4659)140 72.7193 (72.7899,75.9036) 170 70.5836 (69.4549,72.4353)141 73.3739 (72.8879,76.4366) 171 67.7603 (69.341,72.5023)142 72.6713 (73.3965,76.9794) 172 67.6915 (66.5144,69.4938)143 72.6099 (72.0754,75.5122) 173 67.3393 (66.487,69.4949)144 72.5494 (71.3275,74.5088) 174 67.3503 (66.1688,69.0732)145 74.2819 (71.4246,74.6735) 175 66.7731 (66.671,69.8338)146 74.6865 (72.8612,76.0534) 176 67.54 (65.3924,68.2765)147 73.2494 (73.7302,77.0966) 177 68.741 (66.4323,69.3631)148 73.8561 (72.5587,75.9578) 178 68.7429 (67.8331,70.996)149 74.5365 (72.6474,75.9664)150 73.6267 (72.7879,75.9364)151 71.8571 (72.1989,75.4564)

57

Table 5.9: Expected and Obtained Output for Variable x1(Agriculture Data)


2 4233.6 (3549.56,7021.04) 62 4403.4 (3616.75,7709.87)3 5134.3 (3548.11,7302.9) 63 3923.3 (3676.38,7841.8)4 3708.5 (3726.55,7422.98) 64 3966.1 (3563.17,7701.42)5 5487.3 (3355.62,7169.8) 65 5834.4 (3592.43,7678.98)6 2735.8 (3555.87,6955.68) 66 4648.7 (4070.4,8248.43)7 2762.9 (2925.39,6601.94) 67 5069.8 (3988.55,8647.46)8 4899.1 (2826.59,5889.09) 68 3139.6 (3995.4,8365.64)9 5689.2 (3286.68,6343.38) 69 4497.1 (3531.36,7908.51)10 3599.2 (3605.34,7215.75) 70 2635.2 (3780.1,7775.69)11 4075.6 (3167.66,6939.63) 71 2457 (3394.1,7667.63)12 4875.1 (3279.12,6672.2) 72 3493.2 (3277.46,7132.77)13 2768.9 (3483.35,7013.17) 73 4894.7 (3445.8,7230.11)14 8038.8 (3062.14,6764.68) 74 2595.8 (3835.29,7856.97)15 7214.7 (4179.72,7515.17) 75 2683.4 (3379.38,7684.97)16 9728 (4211.7,8724.85) 76 2993.7 (3321.16,7153.53)17 11197.4 (4813.01,9221.76) 77 9208.3 (3404.42,7273.67)18 2791.6 (5290.41,10288) 78 4667.8 (4832.35,8925.64)19 5573.1 (3375.39,8556.92) 79 3096.4 (4044.45,9469.9)20 4900.5 (3659.35,7002.53) 80 3567.4 (3517.65,7936.48)21 5554.3 (3642.09,7626.32) 81 3128.2 (3497.52,7550.42)22 4477.6 (3778.86,7682.39) 82 2189.7 (3376.78,7487.41)23 6170.3 (3541.57,7552.86) 83 3102.9 (3159.51,7194.82)24 3916.6 (3793.56,7544.74) 84 4010.5 (3375.01,7259.48)25 3916.6 (3382.93,7509.02) 85 5448.4 (3632.84,7771.88)26 2781.4 (3301.08,6960.27) 86 1819.4 (4002.75,8369.46)27 4753.9 (3025.52,6667.9) 87 2739.1 (3194.67,7754.24)28 6771.2 (3373.61,6695.81) 88 2304.1 (3253.31,7029.97)29 6520.1 (3881.73,7628.91) 89 3066.5 (3184.34,7155.45)30 10375.9 (4012.46,8254.75) 90 2659.4 (3298.61,7117.17)31 7935.5 (4951.8,9247.36) 91 4841.5 (3184.03,7157.13)32 8073.1 (4514.91,9581.13) 92 3409.6 (3653.94,7577.23)33 5984.4 (4435.97,8945.42) 93 4749 (3303.66,7539.43)34 3824.8 (4028.7,8598.19) 94 3158.4 (3573.48,7486.26)35 3613.3 (3473.94,7556.84) 95 2634.5 (3269.44,7449.25)36 3015.4 (3338.19,6909.82) 96 1214.3 (3042.27,6798.16)37 1441 (3225.79,6795.36) 97 873.4 (2620.54,6114.83)38 3568.7 (2769.71,6154.43) 98 1594.2 (2441.61,5554.13)39 3703 (3208.26,6301.92) 99 2008.5 (2581.98,5602.28)40 1973.4 (3332.25,6950.85) 100 1733.7 (2695.49,5844.95)41 3370.2 (2941.97,6588.91) 101 1402.7 (2666.5,5931.29)42 4375.1 (3238.55,6589.03) 102 3308 (2412.71,5488.74)43 6127.2 (3488.89,7175.22) 103 2724.2 (2734.31,5692.76)44 4026.6 (3819.42,7671.33) 104 2742.9 (2618.89,5954.62)45 3687.5 (3377.64,7572.94) 105 2479 (2597.2,5798.35)46 2401.9 (3172.64,6848.91) 106 2770.8 (2615.09,5889.56)47 4483.9 (2851.55,6416.65) 107 2333.9 (2688.19,5935.96)48 5615.2 (3269.99,6593.78) 108 5212.7 (2524.71,5740.88)49 4702.1 (3693.63,7599.64) 109 5929.7 (3137.02,6278.42)50 4962.1 (3604.99,7859.17) 110 3410.3 (3419.1,7204.24)51 3101.5 (3665.43,7750.05) 111 3964.3 (2849.75,6690.59)52 2686.1 (3225.47,7294.08) 112 2313.7 (2892.27,6161.82)53 5032.2 (3121.42,6888.12) 113 3645.4 (2552.93,5941.37)54 4222.1 (3633.58,7375.42) 114 1829.2 (2780.03,5842.64)55 6372.1 (3545.97,7862.39) 115 3045.7 (2376.19,5671.35)56 2314.2 (3997.16,8183.2) 116 5668.2 (2566.37,5436.2)57 4339.2 (3100.71,7664.92) 117 2868.1 (3227.52,6408.41)58 4868.3 (3466.41,7173.61) 118 2538.3 (2667.31,6322.2)59 5220.4 (3599.25,7653.91) 119 6061.7 (2457.61,5428.16)60 4315 (3694,7847.82) 120 3908.9 (3243.38,6183.62)61 4463.1 (3573.26,7840.98) 121 4595.4 (2904.81,6612.54)

58



122 2938.5 (2980.63,6202.83) 152 2675.4 (2387.13,5367.92)123 2665.8 (2605.04,5920.97) 153 4403.3 (2464.45,5232.49)124 6328.5 (2482.13,5430.5) 154 2245.5 (2879.24,5824.4)125 4239.4 (3476.61,6524.4) 155 7502.6 (2498.59,5834.6)126 2464.3 (3248.02,7171.24) 156 3074.9 (3609.48,6542.51)127 5501.3 (2710.11,6120.09) 157 3153.8 (2807.08,6820.66)128 2977.1 (3342.94,6410.34) 158 2603.9 (2745.3,5897.23)129 3791.1 (2851.81,6503.98) 159 2647.3 (2643.83,5828.8)130 6568.3 (2854.87,5867.57) 160 3548.3 (2606.18,5660.64)131 3006.6 (3521.77,6754.06) 161 4637.4 (2825.17,5932.4)132 2646.1 (2845.05,6678.75) 162 3029.9 (3099.95,6409.38)133 6668.5 (2628.95,5671.52) 163 4131.7 (2799.94,6330.02)134 4633.6 (3553.68,6629.04) 164 2506.6 (2951.04,6093.01)135 3050.6 (3216.21,7131.33) 165 3336.6 (2615.3,5954.72)136 4009.1 (2833.39,6339.19) 166 1869.9 (2707.51,5656.14)137 3377.7 (2997.51,6202.44) 167 3279.6 (2394.38,5485.75)138 2952.1 (2825.32,6171.69) 168 3216 (2652.52,5412.54)139 7446.1 (2695.24,5911.93) 169 4378.4 (2728.58,5834.47)140 8540.6 (3711.28,6930.83) 170 2504 (3036.29,6210.05)141 6546.5 (4116.58,8313.04) 171 4118.2 (2609.87,5981.68)142 3899.4 (3740.51,8181.28) 172 2948.1 (2848.01,5741.23)143 4712.4 (3059.39,7001.17) 173 6108.9 (2642.89,5875.86)144 4081.9 (3108.84,6456.12) 174 2867.5 (3257.84,6241.89)145 5316.6 (3005.12,6526.71) 175 3671.6 (2624.93,6255.63)146 5511.7 (3307.44,6765.05) 176 6018.1 (2674.74,5580.24)147 5235.3 (3455.99,7227.17) 177 2537.8 (3226.05,6354.93)148 2980.1 (3389.44,7164.92) 178 3641.8 (2561.57,6203.05)149 3937.1 (2875.74,6573.17)150 2599.4 (2988.9,6204.16)151 1941.9 (2669.62,6033.92)

59



2 173.447 (144.986,164.026) 62 133.318 (116.787,134.731)3 168.234 (152.832,172.862) 63 133.57 (122.223,139.749)4 158.034 (152.765,173.012) 64 134.475 (124.258,142.901)5 142.597 (148.069,171.06) 65 139.517 (122.979,140.301)6 135.886 (137.295,157.864) 66 153.527 (129.069,146.553)7 135.847 (128.175,149.181) 67 148.209 (137.555,156.197)8 137.075 (123.174,141.185) 68 145.993 (134.584,151.786)9 141.419 (127.585,145.909) 69 145.75 (132.112,150.94)10 139.71 (131.812,150.402) 70 141.232 (135.294,155.163)11 147.23 (126.74,144.374) 71 139.942 (130.56,150.946)12 146.119 (132.232,150.291) 72 133.515 (129.473,150.151)13 147.5 (134.033,152.378) 73 143.225 (123.016,140.771)14 146.926 (129.968,147.244) 74 144.223 (130.87,149.19)15 148.836 (139.538,158.01) 75 141.059 (132.053,152.508)16 157.976 (144.61,166.773) 76 141.573 (127.335,145.603)17 168.506 (151.466,171.913) 77 137.236 (126.702,144.23)18 153.317 (165.022,188.431) 78 134.875 (133.277,150.487)19 148.907 (147.048,172.182) 79 130.561 (121.289,136.246)20 151.046 (141.981,163.545) 80 118.956 (119.623,137.007)21 146.319 (139.818,160.06) 81 115.718 (107.725,121.305)22 144.915 (139.922,161.283) 82 113.043 (107.892,124.239)23 131.688 (136.492,157.528) 83 117.021 (113.041,135.303)24 134.548 (132.806,154.903) 84 116.166 (110.901,129.298)25 130.476 (124.772,143.51) 85 118.332 (111.602,129.564)26 125.995 (122.729,141.466) 86 118.103 (112.996,129.593)27 127.511 (123.762,146.542) 87 114.01 (110.67,129.435)28 129.155 (122.737,142.43) 88 113.262 (105.671,121.635)29 146.921 (124.136,141.88) 89 113.372 (103.803,119.537)30 157.902 (139.273,160.569) 90 103.213 (100.777,113.6)31 157.558 (152.54,173.222) 91 100.882 (92.6481,104.315)32 155.69 (155.011,179.322) 92 97.1817 (94.6443,107.09)33 160.09 (148.955,169.671) 93 105.416 (88.4475,99.7598)34 160.745 (153.463,177.918) 94 107.143 (98.1604,111.789)35 162.93 (149.754,173.322) 95 104.651 (106.929,127.052)36 162.224 (149.227,171.722) 96 103.245 (97.1079,111.806)37 148.543 (151.118,175.69) 97 100.698 (98.061,115.692)38 149.036 (136.344,157.845) 98 102.41 (90.142,103.741)39 145.377 (138.53,160.129) 99 106 (89.4445,101.293)40 140.685 (135.012,155.449) 100 106.106 (90.3797,100.864)41 141.963 (130.676,152.095) 101 85.5131 (90.9396,101.792)42 134.408 (128.293,146.403) 102 74.5217 (75.9705,85.016)43 123.523 (125.563,143.644) 103 67.4044 (68.0005,75.4425)44 117.825 (117.347,132.786) 104 67.2016 (62.536,70.4852)45 113.163 (111.819,128.756) 105 74.1483 (63.6341,72.9504)46 111.171 (110.773,129.65) 106 74.2227 (71.8212,84.3493)47 108.982 (104.706,121.874) 107 69.6517 (71.3237,82.638)48 115.185 (104.982,121.157) 108 68.3168 (65.8303,75.6115)49 116.602 (112.461,130.319) 109 72.1344 (70.699,81.6974)50 120.554 (110.44,126.819) 110 74.5829 (72.9223,83.2723)51 119.918 (115.481,133.403) 111 79.922 (74.1931,86.8366)52 116.923 (115.759,135.811) 112 79.6116 (81.0217,95.5051)53 112.536 (111.799,130.785) 113 74.3961 (80.731,96.4893)54 105.19 (108.752,125.108) 114 69.3642 (73.1384,84.3045)55 102.356 (100.549,115.333) 115 71.3597 (69.3151,82.1683)56 93.5086 (98.8543,111.865) 116 76.8492 (68.6448,79.3125)57 105.19 (86.7039,99.3332) 117 79.6545 (77.5282,89.2415)58 106.548 (101.49,118.248) 118 80.6142 (78.6292,92.5459)59 108.831 (104.025,120.556) 119 83.174 (77.0414,89.602)60 117.516 (106.836,123.933) 120 83.9695 (83.8194,96.2559)61 123.907 (110.796,127.949) 121 86.7493 (83.0776,96.7738)

60



122 85.0662 (84.6181,97.4785) 152 82.7815 (82.8587,96.7665)123 84.507 (81.7601,94.8573) 153 84.0966 (77.47,88.8189)124 110.075 (85.524,101.891) 154 87.6158 (80.0172,91.0296)125 120.482 (105.15,121.059) 155 89.916 (87.2057,104.015)126 110.185 (110.223,125.768) 156 92.1502 (90.9279,103.545)127 112.859 (102.946,118.661) 157 94.6644 (87.7094,101.72)128 111.83 (106.907,121.857) 158 95.6897 (92.0608,107.842)129 105.263 (105.446,122.275) 159 90.9871 (93.928,110.995)130 108.257 (102.049,118.363) 160 90.2062 (85.8892,99.1161)131 107.692 (107.059,123.205) 161 91.3006 (85.7128,98.5281)132 106.498 (101.776,118.028) 162 95.525 (86.4767,98.284)133 106.727 (101.794,118.965) 163 93.8846 (90.3342,104.76)134 103.295 (106.565,122.781) 164 94.5017 (91.988,106.736)135 105.124 (104.046,122.036) 165 92.7835 (87.5252,100.764)136 101.86 (100.555,117.317) 166 92.321 (88.0647,101.442)137 95.7447 (99.9727,116.484) 167 95.0142 (86.2233,100.194)138 91.0714 (99.7317,119.458) 168 98.7591 (87.6815,100.119)139 91.637 (85.7035,98.4381) 169 100.709 (91.5591,105.058)140 94.8582 (91.6278,103.839) 170 94.4423 (94.3058,107.415)141 97.6043 (97.2667,110.908) 171 94.053 (88.2277,101.467)142 97.3451 (98.089,113.077) 172 93.749 (90.0348,103.448)143 92.2541 (102.722,123.657) 173 87.1503 (88.466,102.182)144 93.5315 (99.9442,120.146) 174 83.2847 (88.1467,100.957)145 96.3222 (96.7842,115.323) 175 83.4066 (81.9153,95.8836)146 104.294 (96.7305,112.696) 176 80.6692 (80.7467,93.2871)147 106.457 (105.512,123.976) 177 81.543 (82.823,95.4081)148 104.987 (105.307,122.447) 178 81.0122 (80.0683,94.291)149 100.437 (100.58,117.251)150 94.4206 (102.116,120.784)151 86.1487 (88.2175,101.378)

61



2 3854.3 (3438.59,5426.16) 62 4405.8 (3880.94,6358.05)3 5701.7 (3420.96,5374.24) 63 3677 (3831.52,6282.14)4 5052.3 (3996.31,6004.77) 64 3559.4 (3738.13,6215.8)5 5550.5 (4050.3,6142.09) 65 3604 (3533.94,6011.04)6 3982.8 (4342.49,6335.83) 66 3062.9 (3945.98,6450.24)7 4097.4 (3574.16,5547.05) 67 4003.5 (3646.16,6184.29)8 4062.5 (3148.79,5026.13) 68 4558.7 (3778.84,6273.18)9 3630.2 (3641.72,5489.55) 69 5010.8 (3686.82,6192.12)10 3551.5 (3736.57,5628.14) 70 5215.9 (4139.1,6701.55)11 3745.2 (3198.07,5080.04) 71 3942.4 (3911.68,6513.65)12 3130.3 (3288.87,5211.5) 72 3848.5 (3669.15,6284.38)13 3779.9 (3432.65,5386.45) 73 4927.4 (3573.07,6092.59)14 5637.2 (2962.7,4895.76) 74 3908.5 (4061.42,6540.98)15 4370.7 (4583.06,6581.11) 75 3500.2 (3634.92,6178.29)16 5500 (4721.6,6840.06) 76 3507.4 (3293.27,5814.27)17 7013.9 (5163.64,7220.91) 77 2421.5 (3287.3,5793.87)18 5320.2 (6082.69,8188.85) 78 3866.4 (4459.66,7038.03)19 4541.8 (4308.23,6517.74) 79 2339.2 (3604.85,6136.16)20 5317.9 (4241,6353.94) 80 3919.1 (3264.83,5910.58)21 5204.2 (4091.06,6179.5) 81 7126 (3276.31,5842.77)22 6290.9 (4395.73,6591.65) 82 4940.8 (4156.56,6790.84)23 4416.2 (4377.69,6551.52) 83 4747.6 (4301.81,7160.78)24 4578.9 (4645.76,6882.12) 84 3945.4 (3943.96,6701.06)25 6877.2 (3753.81,5896.15) 85 5090.7 (3978.9,6759.74)26 5198.9 (4176.8,6366.73) 86 3787 (4343.98,7077.8)27 4112.1 (4199.68,6535.39) 87 3696.7 (3618.85,6356.36)28 5259.4 (4029.11,6183.99) 88 2166.9 (3390.68,6070.36)29 4496.7 (4404.72,6458.65) 89 1969.1 (3023.82,5696.9)30 6412.2 (4499.54,6616.89) 90 2470.8 (2785.44,5313.43)31 4603.6 (5499.18,7590.78) 91 2025.4 (2716.48,5251.57)32 6386 (5177.64,7339.71) 92 2908.4 (3238.63,5818.96)33 5895 (5029.83,7075.98) 93 6241.8 (3034.87,5472.5)34 5245.2 (4904.61,7091.65) 94 3499.8 (4054.85,6504.23)35 6361.4 (4182,6342.93) 95 5092.7 (4012.42,6646.46)36 5301.8 (4150.49,6243.3) 96 3082.6 (3496.85,5922.78)37 5333.3 (4070.23,6274.28) 97 2096.5 (3180.95,5600.14)38 4871.4 (3491.52,5642.76) 98 1237.5 (2365.5,4621.49)39 5574.3 (3860.1,6036.79) 99 1426.4 (2101.78,4273.6)40 3789.1 (3948.27,6149.27) 100 1306.2 (2020.64,4102.52)41 4077.5 (3445.25,5717.6) 101 907.3 (1964.82,4079.86)42 3130 (3357.53,5578.05) 102 1458.8 (1805.43,3865.09)43 4143.5 (3527.91,5795.44) 103 2183.3 (2196.06,4203.53)44 5347.4 (3906.07,6079.47) 104 3587.3 (2346.96,4374.28)45 4531.6 (3912.16,6145.63) 105 2948.3 (2762.36,4828.6)46 4099.8 (3949.37,6226.08) 106 2328.5 (2920.39,5111.35)47 4732.5 (3371.75,5589.92) 107 3019.9 (2736.81,4912.98)48 3900.6 (3809.55,6021.45) 108 2526.9 (2642.19,4718.96)49 4550 (4097.45,6417.37) 109 3630.3 (3307.83,5415.7)50 5692.2 (3864.93,6222.17) 110 4380 (3588.51,5653.58)51 5253.7 (4319.91,6729.64) 111 5060.5 (3447.46,5513.49)52 4051.7 (4124.61,6562.9) 112 2863.7 (3796.3,5876.58)53 3523.2 (3645.22,6161.52) 113 3949.7 (3190.51,5326.44)54 2707.6 (3826.03,6312.71) 114 2724.4 (3128.49,5167.18)55 2863 (3428.47,5956.4) 115 3051.9 (2813.47,4897.22)56 4557 (3759.39,6263.82) 116 4093.8 (2807.68,4782.27)57 4384.2 (3263.57,5762.87) 117 3344.7 (3649.1,5633.99)58 4588.4 (4011.82,6594.47) 118 3176.9 (3195.11,5175.71)59 4269.2 (4157.02,6636.1) 119 3805 (2832.67,4718.25)60 4427.3 (4181.34,6647.62) 120 3255.5 (3644.28,5510.18)61 3830.5 (3923.23,6387.51) 121 3447.1 (3274.4,5186.94)

62



122 5182.9 (3300.34,5159.3) 152 2393.7 (2733.15,4697.04)123 3916.1 (3297.74,5142.14) 153 5289.8 (2532.25,4370.62)124 3307.5 (3384.06,5335.42) 154 3001.1 (3359.86,5174.24)125 3810.2 (3768.36,5670.45) 155 3641.3 (3190.02,5195.8)126 3301.7 (3323.72,5237.35) 156 4468.3 (3885.47,5755.4)127 4327.8 (2957.07,4905.56) 157 5100.5 (3305.79,5185.06)128 4321.7 (3660.35,5581.07) 158 3277.3 (3536.5,5556.91)129 4113.5 (3408.23,5346.16) 159 3093 (3293.64,5367.51)130 4191.3 (3487.17,5367.49) 160 2520.9 (2830.97,4802.02)131 4721.1 (4040.29,5902.34) 161 3723.5 (2881.42,4869.37)132 4212.1 (3411.51,5306.24) 162 3958.2 (3232.69,5174)133 5279.2 (3325.99,5258.41) 163 3142.7 (3223.16,5224.65)134 4562.2 (4271.39,6222.28) 164 3284.2 (3345.39,5327.07)135 4562 (4048.47,6049.86) 165 3613.3 (2818.27,4729.17)136 6491.9 (3482.18,5488.66) 166 2702 (3057.18,4946.95)137 3016 (4039.04,6085.11) 167 3107.8 (2647.78,4539.99)138 2719.1 (3781.07,5905.01) 168 2825.5 (2779.99,4591.2)139 3339 (2762.17,4706.42) 169 3145.2 (2817.39,4670.76)140 3934.3 (3793.86,5750.3) 170 3252.1 (3066.45,4953.02)141 7061.2 (4386.33,6342.21) 171 3422.9 (2801.67,4694.45)142 6881.2 (4650.46,6657.3) 172 3212.9 (3182.16,5012.96)143 5958.4 (4768.85,6943.18) 173 3975.1 (2914.16,4753.38)144 4498.3 (4684.05,6846.27) 174 3222 (3675.84,5494.15)145 5487.1 (4072.33,6178.64) 175 3355.3 (3058.45,4922)146 4706.7 (4154.29,6216.36) 176 4178.3 (3026.46,4831.61)147 4519.7 (4338.48,6468.68) 177 4473.7 (3724.84,5556.41)148 5749 (4069.14,6118.54) 178 3625.5 (3258.63,5167.8)149 3104.5 (3724.07,5777.11)150 3828.4 (3722.71,5864.73)151 2622.8 (2950.98,4914.97)

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65 70 75 80 85 90 95 100 105

x0,t

x0,t-2


65

70

75

80

85

90

95

100

105

0 2000 4000 6000 8000 10000 12000

x0,t

x1,t-1

Fuzzy Coefficient: (9.5641e-05, 0)

65

70

75

80

85

90

95

100

105

0 2000 4000 6000 8000 10000 12000

x0,t

x1,t-2

Fuzzy Coefficient: (0.000308249, 4.3061e-05)

0

2000

4000

6000

8000

10000

12000

65 70 75 80 85 90 95 100 105

x1,t

x0,t-1


0

2000

4000

6000

8000

10000

12000

0 2000 4000 6000 8000 10000 12000

x1,t

x1,t-1


0

2000

4000

6000

8000

10000

12000

0 2000 4000 6000 8000 10000 12000

x1,t

x1,t-2


0

2000

4000

6000

8000

10000

12000

60 80 100 120 140 160 180

x1,t

x2,t-1


Figure 5.4: Linearity Figures for Agriculture data

64

60

80

100

120

140

160

180

0 2000 4000 6000 8000 10000 12000

x2,t

x1,t-1


60

80

100

120

140

160

180

60 80 100 120 140 160 180

x2,t

x2,t-1


60

80

100

120

140

160

180

60 80 100 120 140 160 180

x2,t

x2,t-2


0

1000

2000

3000

4000

5000

6000

7000

8000

65 70 75 80 85 90 95 100 105

x3,t

x0,t-1


0

1000

2000

3000

4000

5000

6000

7000

8000

65 70 75 80 85 90 95 100 105

x3,t

x0,t-2


0

1000

2000

3000

4000

5000

6000

7000

8000

0 2000 4000 6000 8000 10000 12000

x3,t

x1,t-1


0

1000

2000

3000

4000

5000

6000

7000

8000

0 2000 4000 6000 8000 10000 12000

x3,t

x1,t-2


0

1000

2000

3000

4000

5000

6000

7000

8000

0 1000 2000 3000 4000 5000 6000 7000 8000

x3,t

x3,t-1


Figure 5.4: Linearity Figures for Agriculture Data(cont’d)

65

60

80

100

120

140

160

180

0 1000 2000 3000 4000 5000 6000 7000 8000

x2,t

x3,t-2


0

1000

2000

3000

4000

5000

6000

7000

8000

0 1000 2000 3000 4000 5000 6000 7000 8000

x3,t

x3,t-2


Figure 5.4: Linearity Figures for Agriculture Data(cont’d)

66

5.1.2.3 Monthly Flour Price Indices Data

Table 5.12: Fuzzy equations for Flour Price Indices Data


x0 (0, 0.006193)x1,t−1 + (0, 0.02697)x1,t−2+(0.3482, 0.02206)x2,t−1 + (0.6702, 0)x0,t−1

x1 (0.4229, 0.04741)x2,t−1 + (0.5725, 0.003683)x0,t−1

x2 (1.041, 0.07212)x2,t−1

This data set contains monthly flour price indices for three US cities which

are Buffalo (x0), Minneapolis (x1) and Kansas City (x2). The data belongs to

the time interval January 1972 and November 1980. The data set can be seen in

A.6.

The fuzzy equations obtained for three variables using 2 lags can be seen in

5.12. Among all variables, x2,t−1 has the largest coefficient center value for the

variable x2.

In Figure 5.5, linearity between the output variables and the defining variables

can be seen. All of the variables show the same linear characteristics. There is no

visible difference between the figures. The accuracy rates are at the same level as

shown in Table 5.52 which are 0.693878, 0.663265 and 0.683673 for the variables

x0, x1 and x2 respectively.

The output results for three variables are given in Figures 5.13, 5.14 and 5.15.

The widths of the numbers are approximately the same. The accuracy rates are

the average. The expected outputs that are not covered are close to the obtained

results.

67

Table 5.13: Expected and Obtained Output for Variable x0 (Flour Price Data)


2 112.7 (109.867,122.059) 51 135.6 (135.319,150.464)3 114.7 (109.469,122.011) 52 135.4 (127.49,142.065)4 123.4 (111.305,123.9) 53 134.5 (126.949,141.014)5 123.6 (122.895,136.504) 54 136.1 (125.901,139.811)6 116.3 (122.011,136.125) 55 135.6 (127.263,141.151)7 118.5 (113.846,127.403) 56 122.8 (125.195,138.928)8 119.8 (115.91,129.124) 57 119 (114.784,128.005)9 120.3 (118.381,131.946) 58 108.5 (110.372,122.728)10 127.4 (117.973,131.492) 59 113.3 (100.878,112.578)11 125.1 (126.788,140.967) 60 114.8 (106.992,118.538)12 127.6 (123.839,138.327) 61 120.9 (108.213,120.093)13 129 (125.541,139.906) 62 123.7 (113.232,125.409)14 124.6 (129.864,144.701) 63 127.8 (116.346,129.042)15 134.1 (125.247,139.73) 64 125.4 (120.476,133.541)16 146.5 (133.756,148.341) 65 131.5 (118.293,131.393)17 171.2 (147.264,163.221) 66 127.7 (125.309,138.751)18 178.6 (173.062,191.698) 67 131.2 (120.79,134.277)19 172.2 (179.493,199.71) 68 145.2 (124.882,138.427)20 171.5 (172.612,192.86) 69 141.9 (136.108,150.422)21 163.6 (171.285,191.091) 70 139.3 (134.336,149.581)22 185.6 (165.298,184.888) 71 141.1 (131.331,146.107)23 198.8 (182.535,202.778) 72 135.9 (134.833,149.683)24 195.7 (196.938,219.228) 73 136.5 (131.182,146.086)25 190.3 (190.091,212.589) 74 137.2 (131.858,146.492)26 207.9 (187.33,209.366) 75 143.8 (132.804,147.46)27 212.8 (205.169,228.138) 76 138.7 (138.617,153.627)28 199.9 (210.056,234.414) 77 133.9 (134.214,149.453)29 185.3 (197.252,221.155) 78 137.7 (129.714,144.456)30 183 (182.725,205.042) 79 143.8 (133.717,148.4)31 173.5 (178.107,199.196) 80 140.8 (140.311,155.623)32 172.2 (166.717,186.502) 81 153.4 (137.564,153.096)33 165.3 (164.977,183.993) 82 157.5 (149.925,166.048)34 159.9 (157.591,176.073) 83 179.5 (153.963,170.988)35 170.3 (151.208,168.742) 84 177.5 (176.145,194.726)36 172.2 (163.284,181.329) 85 178 (172.462,191.967)37 184.5 (166.618,185.625) 86 176.8 (172.791,192.169)38 185 (179.65,199.732) 87 179.8 (172.129,191.502)39 177.7 (178.926,199.525) 88 174.2 (177.016,196.903)40 169.1 (171.191,191.209) 89 171.1 (171.933,191.694)41 174.7 (161.597,180.502) 90 175.9 (167.843,186.895)42 169.4 (165.273,183.705) 91 172.2 (172.651,191.793)43 177.8 (163.899,182.781) 92 164.7 (166.799,185.722)44 170.1 (170.722,189.723) 93 175.7 (160.062,178.368)45 167.1 (162.294,181.216) 94 177.4 (170.618,189.101)46 171.4 (157.714,175.786) 95 187.5 (170.445,189.394)47 172.3 (162.006,180.112) 96 190.7 (178.107,197.289)48 152.6 (161.748,180.044) 97 190.4 (181.193,201.138)49 144.1 (143.339,160.629) 98 192.4 (183.165,203.36)50 143.5 (136.837,152.744) 99 192.9 (185.204,205.533)

68

Table 5.14: Expected and Obtained Output for Variable x1(Flour Price Data)


2 111.8 (107.59,119.291) 51 129.7 (132.013,145.794)3 113.3 (107.473,119.254) 52 128.4 (124.478,137.433)4 124.5 (109.173,121.111) 53 126.9 (123.651,136.424)5 124.3 (120.881,134.589) 54 128.8 (122.576,135.2)6 116.5 (120.168,133.67) 55 126.5 (123.786,136.498)7 118.6 (112.186,124.667) 56 116.6 (121.549,133.764)8 119.6 (113.888,126.499) 57 113.4 (112.016,123.568)9 119.4 (116.543,129.646) 58 102.8 (107.376,118.274)10 128.6 (116.001,128.9) 59 107.7 (98.3998,108.462)11 126.3 (124.771,138.916) 60 109.4 (104.172,115.037)12 126.8 (122.111,135.898) 61 114.9 (105.438,116.419)13 125.7 (123.495,137.291) 62 117.5 (110.071,121.391)14 120.8 (128.196,142.989) 63 120 (113.279,125.027)15 127.9 (123.704,137.962) 64 117.6 (117.3,129.506)16 147.6 (131.473,146.398) 65 124 (115.335,127.37)17 169.8 (144.872,161.491) 66 119.7 (122.146,135.071)18 177.6 (170.336,190.019) 67 125 (117.882,130.247)19 172.5 (176.986,197.34) 68 141.1 (121.788,134.663)20 170.1 (170.567,190.172) 69 137 (132.154,145.739)21 171.3 (169.005,188.311) 70 132.3 (131.253,145.06)22 189.9 (163.647,182.677) 71 134.8 (128.16,141.54)23 206.9 (179.202,199.162) 72 129.7 (131.699,145.728)24 197.4 (193.807,215.656) 73 128.7 (128.591,142.544)25 195 (187.012,207.568) 74 129.9 (129.083,143.078)26 214.2 (184.617,205.303) 75 137.2 (130.007,144.139)27 219.2 (201.649,224.239) 76 132.8 (135.451,150.058)28 205.6 (206.915,230.166) 77 127.5 (131.611,145.944)29 193.4 (194.846,216.807) 78 131.2 (127.229,141.109)30 185.1 (180.571,200.918) 79 137.1 (130.93,145.227)31 174 (175.283,194.607) 80 135.5 (137.441,152.551)32 173.2 (163.759,181.468) 81 147.1 (135.058,149.976)33 164.5 (161.667,179.026) 82 151.6 (146.769,162.926)34 158.9 (154.476,170.958) 83 173.6 (150.978,167.64)35 169.7 (147.912,163.473) 84 171.6 (172.353,191.415)36 174.4 (159.61,176.708) 85 170.8 (169.188,187.723)37 186.2 (163.432,181.236) 86 172.4 (169.397,187.917)38 184.7 (176.174,195.518) 87 174.9 (168.865,187.414)39 176.4 (175.594,194.725) 88 168.1 (173.951,193.375)40 167.6 (168.063,186.286) 89 164.7 (169.264,188.267)41 170.9 (158.44,175.405) 90 170 (164.947,183.283)42 168.3 (161.287,178.209) 91 164.9 (169.442,188.259)43 176.4 (160.863,178.4) 92 157.9 (163.583,181.424)44 168.6 (166.993,184.933) 93 169.2 (157.139,174.375)45 164.6 (159.046,176.028) 94 168.6 (166.925,185.134)46 170.1 (154.111,170.255) 95 179.8 (166.728,184.655)47 169.4 (158.096,174.661) 96 179 (173.562,191.838)48 149.6 (157.782,174.145) 97 179.2 (176.809,195.469)49 139.5 (140.418,155.082) 98 181.4 (179.116,198.4)50 137.3 (133.969,148.162) 99 181.8 (181.08,200.587)

69

Table 5.15: Expected and Obtained Output for Variable x2(Flour Price Data)


2 115.5 (111.06,127.591) 51 126.1 (130.054,149.412)3 117 (111.932,128.593) 52 124.2 (122.205,140.394)4 135 (113.386,130.263) 53 122.7 (120.363,138.279)5 132.8 (130.83,150.303) 54 123.5 (118.91,136.609)6 122.6 (128.698,147.854) 55 118.3 (119.685,137.499)7 123.8 (118.813,136.497) 56 112.3 (114.646,131.71)8 128.9 (119.976,137.833) 57 105.7 (108.831,125.03)9 126.7 (124.918,143.512) 58 97.7 (102.435,117.682)10 139.3 (122.786,141.062) 59 105.8 (94.6819,108.775)11 135.7 (134.997,155.09) 60 106.9 (102.532,117.793)12 135.6 (131.508,151.082) 61 110 (103.598,119.018)13 146 (131.411,150.971) 62 114.3 (106.602,122.469)14 140.7 (141.49,162.55) 63 118.8 (110.769,127.257)15 147 (136.354,156.649) 64 117.2 (115.13,132.267)16 163.9 (142.459,163.663) 65 126.1 (113.58,130.485)17 194.3 (158.837,182.479) 66 120.5 (122.205,140.394)18 200.8 (188.298,216.325) 67 125.6 (116.778,134.159)19 193.4 (194.597,223.562) 68 132 (121.72,139.837)20 190.3 (187.426,215.323) 69 134.6 (127.922,146.963)21 188 (184.421,211.872) 70 130.3 (130.442,149.858)22 196.1 (182.192,209.311) 71 137 (126.275,145.07)23 215 (190.042,218.329) 72 136.6 (132.768,152.53)24 201.6 (208.358,239.371) 73 137 (132.38,152.084)25 203.4 (195.372,224.452) 74 138.4 (132.768,152.53)26 222.1 (197.117,226.456) 75 142.9 (134.125,154.088)27 228.7 (215.239,247.276) 76 140.4 (138.486,159.098)28 216.1 (221.635,254.624) 77 136 (136.063,156.315)29 200.2 (209.424,240.596) 78 140.1 (131.799,151.416)30 189.6 (194.016,222.894) 79 148.2 (135.772,155.981)31 173.3 (183.743,211.092) 80 146.4 (143.622,164.999)32 169.7 (167.946,192.944) 81 158.5 (141.877,162.995)33 161 (164.458,188.936) 82 163.5 (153.604,176.467)34 151.7 (156.026,179.25) 83 187.1 (158.449,182.034)35 167.1 (147.014,168.896) 84 181.7 (181.32,208.309)36 174.4 (161.938,186.042) 85 181.5 (176.087,202.297)37 189.7 (169.013,194.169) 86 181.9 (175.893,202.074)38 187.4 (183.84,211.204) 87 190.9 (176.281,202.519)39 178.4 (181.611,208.643) 88 186.9 (185.003,212.54)40 165.8 (172.889,198.623) 89 180.1 (181.126,208.086)41 164.9 (160.678,184.594) 90 184.8 (174.536,200.515)42 171.8 (159.806,183.592) 91 174.8 (179.091,205.748)43 175.4 (166.493,191.274) 92 169 (169.4,194.615)44 165.9 (169.982,195.283) 93 178.4 (163.779,188.157)45 157.3 (160.775,184.706) 94 175.3 (172.889,198.623)46 161.4 (152.441,175.131) 95 178.2 (169.885,195.171)47 159.2 (156.414,179.696) 96 182 (172.695,198.4)48 142.8 (154.282,177.246) 97 188.6 (176.378,202.631)49 138.5 (138.389,158.987) 98 190.8 (182.774,209.979)50 134.2 (134.222,154.2) 99 192.2 (184.906,212.428)

70

100

120

140

160

180

200

220

100 120 140 160 180 200 220

x0,t

x0,t-1


100

120

140

160

180

200

220

100 120 140 160 180 200 220

x0,t

x1,t-1


100

120

140

160

180

200

220

100 120 140 160 180 200 220

x0,t

x1,t-2


100

120

140

160

180

200

220

80 100 120 140 160 180 200 220 240

x0,t

x2,t-1


100

120

140

160

180

200

220

100 120 140 160 180 200 220

x1,t

x0,t-1


100

120

140

160

180

200

220

80 100 120 140 160 180 200 220 240

x1,t

x2,t-1


80

100

120

140

160

180

200

220

240

80 100 120 140 160 180 200 220 240

x2,t

x2,t-1


Figure 5.5: Linearity Figures for Monthly Flour Price Indices

71

5.1.2.4 Monthly Forestry Data

Table 5.16: Fuzzy equations for Forestry Data


x0 (0.472, 0)x0,t−1 + (0.5729, 0.05823)x0,t−2 + (11.98, 65.35)x1,t−1

x1 (0.8564, 0.02724)x1,t−1 + (9.235e − 05, 3.565e − 05)x0,t−1+(0.002885, 0)x3,t−1

x2 (0.7309, 0)x2,t−1 + (0.003844, 0)x2,t−2 + (20.12, 11.74)x1,t−1

+(0.2402, 0.1635)x3,t−1 + (0.01071, 0.003766)x0,t−1

x3 (0.2716, 0.04415)x3,t−1 + (0.7173, 0)x3,t−2

Monthly Forestry data contains four variables, which are lumber production

(x0), lumber price (x1), the price that housing starts (x2) and disposable income

(x4). The data can be seen in A.7.

The fuzzy equations obtained by using Fuzzy MAR can also be seen in 5.16.

For the variable x0, x0,t−1 and x0,t−2 have the smaller coefficient width when

compared to xt−1 (65.35). For x1, x1,t−1 has the largest center value (small width

value). For x2, x2,t−1 has the smallest width and largest center among the rest

of the coefficients (excluding x1,t−1 which has the largest width). In Figure 5.6,

linearity between the output variables and the defining variables can be seen.

x3 has obvious linear relationships with both of the lagged variables x3,t−1 and

x3,t−2. The variable x0 has a linear relationship with x0,t−1 and x0,t−2. x1 has

linear relationship with x1,t−1 and x2 has linear relationship with x2,t−1.

The accuracy rates in Table 5.52 for the variables x0 (0.854730) and especially

x3 (0.932432) are better than for x1 and x2 (0.797980 and 0.722973). The output

results for all four variables can be seen in Figures 5.17, 5.18, 5.19 and 5.20.

The outputs for x0 and x2 have larger width values since the accuracy rates are

smaller.

72

Table 5.17: Expected and Obtained Output for Variable x0 (Forest Data)


2 631 (566.572,820.838) 62 671 (592.673,850.859)3 617 (566.638,822.53) 63 619 (596.812,848.449)4 609 (546.595,789.953) 64 589 (570.058,813.343)5 575 (540.279,770.583) 65 567 (532.525,761.454)6 563 (522.474,746.069) 66 505 (511.554,725.108)7 636 (501.425,715.884) 67 577 (472.111,680.306)8 545 (529.873,742.517) 68 443 (476.141,672.32)9 523 (524.528,745.594) 69 350 (451.001,653.002)10 579 (467.639,677.298) 70 466 (339.816,522.109)11 556 (482.64,689.999) 71 460 (347.628,516.817)12 592 (501.946,712.546) 72 549 (403.106,589.214)13 631 (507.831,713.961) 73 615 (440.892,629.072)14 628 (544.777,755.073) 74 606 (515.174,720.267)15 589 (563.578,778.063) 75 586 (542.525,761.107)16 613 (543.926,757.329) 76 615 (529.938,743.838)17 594 (535.634,743.385) 77 609 (534.943,742.567)18 616 (538.133,750.849) 78 599 (547.779,756.963)19 682 (538.188,750.037) 79 700 (540.345,747.915)20 608 (579.94,796.116) 80 584 (584.137,787.433)21 607 (579.559,802.009) 81 538 (580.913,797.093)22 597 (541.572,754.006) 82 636 (498.165,704.105)23 658 (535.361,750.071) 83 583 (517.454,726.089)24 710 (557.083,775.333) 84 623 (540.119,766.924)25 750 (610.923,841.414) 85 634 (525.95,760.712)26 694 (656.026,893.892) 86 598 (553.287,788.891)27 731 (649.393,893.852) 87 664 (542.177,778.525)28 718 (636.945,877.549) 88 710 (555.492,785.949)29 721 (647.515,898.152) 89 743 (610.99,849.577)30 715 (641.066,893.064) 90 716 (648.378,896.883)31 756 (639.276,892.853) 91 761 (651.674,906.336)32 694 (655.683,908.208) 92 656 (659.128,910.372)33 688 (647.523,904.822) 93 663 (633.442,888.187)34 691 (613.176,862.288) 94 651 (581.712,826.661)35 730 (611.722,859.6) 95 702 (579.042,826.296)36 782 (630.872,881.06) 96 787 (597.284,842.291)37 770 (674.126,932.189) 97 778 (662.465,916.312)38 776 (694.185,960.852) 98 729 (701.842,965.889)39 803 (690.932,955.979) 99 728 (674.409,936.61)40 744 (707.126,971.983) 100 708 (647.928,906.357)41 802 (693.443,960.79) 101 759 (633.41,902.899)42 770 (690.093,951.446) 102 742 (641.817,922.126)43 815 (704.489,973.459) 103 764 (655.506,952.863)44 710 (708.949,974.95) 104 629 (660.134,948.179)45 697 (682.115,954.428) 105 621 (609.614,895.634)46 659 (623.347,879.982) 106 608 (535.796,807.467)47 640 (599.319,852.977) 107 622 (524.994,797.079)48 731 (568.851,822.841) 108 728 (524.724,795.752)49 643 (600.612,855.84) 109 730 (581.584,855.158)50 705 (601.97,877.455) 110 735 (636.839,923.358)51 649 (585.353,852.028) 111 728 (640.052,927.235)52 628 (589.152,867.163) 112 669 (639.914,926.229)53 689 (550.659,821.561) 113 733 (609.917,891.861)54 644 (570.418,834.522) 114 688 (609.223,885.604)55 684 (577.186,856.693) 115 737 (620.785,904.959)56 618 (572.546,847.689) 116 663 (621.653,898.377)57 557 (562.795,840.609) 117 646 (612.283,893.89)58 599 (501.797,767.612) 118 654 (566.317,838.954)59 573 (494.115,743.299) 119 642 (563.473,828.915)60 670 (505.927,753.962) 120 742 (563.213,826.437)61 681 (538.347,783.303) 121 665 (602.883,868.017)

73

Table 5.17: Expected and Obtained Output for Variable x0 (ForestData)(cont’d)


122 669 (618.293,894.378) 182 615 (575.608,799.625)123 673 (581.975,845.604) 183 636 (554.975,768.617)124 670 (586.477,849.212) 184 605 (559.786,770.774)125 726 (586.981,850.522) 185 657 (555.76,769.69)126 638 (609.231,878.879) 186 617 (564.182,774.907)127 743 (595.468,874.213) 187 893 (573.652,786.551)128 594 (600.699,866.83) 188 824 (683.894,890.749)129 522 (585.666,860.966) 189 679 (794.052,1031.4)130 633 (478.616,727.703) 190 769 (689.586,920.168)131 599 (496.878,730.405) 191 783 (657.643,870.829)132 640 (539.279,782.503) 192 890 (712.173,931.919)133 425 (542.705,778.114) 193 992 (769.516,991.781)134 493 (470.222,691.088) 194 1039 (873.4,1106.48)135 553 (392.178,586.739) 195 894 (942.096,1201.7)136 566 (454.261,659.76) 196 920 (893.965,1168.56)137 614 (490.455,704.954) 197 920 (836.251,1082.59)138 575 (519.032,736.927) 198 895 (853.716,1093.09)139 626 (527.326,745.922) 199 944 (843.273,1079.32)140 519 (534.564,740.685) 200 783 (854.771,1084.88)141 515 (509.64,723.346) 201 761 (804.926,1038.48)142 569 (451.742,655.288) 202 881 (711.824,926.273)143 492 (476.365,676.518) 203 825 (755.237,971.777)144 552 (468.296,673.562) 204 968 (789.8,1022.2)145 604 (457.016,653.236) 205 878 (824.855,1059.58)146 546 (510.406,718.593) 206 949 (851.772,1113.45)147 559 (508.941,725.275) 207 955 (840.593,1087.79)148 557 (486.159,693.451) 208 798 (881.937,1132.58)149 512 (494.575,696.846) 209 919 (811.337,1061.67)150 485 (473.332,672.86) 210 676 (787.368,1020.09)151 488 (439.638,628.515) 211 1014 (736.573,979.421)152 364 (428.294,611.243) 212 846 (772.59,983.323)153 415 (372.504,552.888) 213 833 (867.101,1117.59)154 366 (330.374,502.147) 214 966 (774.227,1005.82)155 419 (334.297,510.048) 215 912 (829.678,1061.3)156 487 (334.999,502.822) 216 1008 (871.177,1121.88)157 515 (394.163,568.668) 217 1092 (886.323,1136.54)158 490 (440.23,627.908) 218 988 (975.1,1237.17)159 556 (443.376,633.007) 219 1071 (970.332,1239.53)160 547 (458.883,652.4) 220 1086 (953.817,1216.2)161 582 (487.606,691.253) 221 1007 (1000.4,1280.32)162 643 (502.52,697.707) 222 1004 (972.149,1250.6)163 563 (549.323,748.587) 223 1102 (929.962,1199.49)164 513 (544.616,746.931) 224 929 (978.551,1238.22)165 505 (478.829,674.312) 225 994 (947.494,1218.2)166 521 (447.884,641.06) 226 938 (887.905,1141.47)167 515 (449.343,646.423) 227 966 (894.032,1157.36)168 561 (452.739,656.597) 228 1124 (877.779,1136.17)169 550 (470.454,675.835) 229 1085 (967.428,1227.45)170 584 (488.564,700.217) 230 1081 (1030.75,1308.16)171 594 (498.389,710.134) 231 1127 (1010.64,1278.99)172 570 (519.039,738.585) 232 1033 (1030.92,1297.26)173 598 (513.936,732.007) 233 1042 (1010.75,1281.18)174 551 (516.304,727.894) 234 1056 (970.918,1219.86)175 586 (511.42,719.2) 235 1089 (984.527,1228.72)176 536 (502.869,707.332) 236 1028 (1006.72,1253.99)177 471 (496.756,706.588) 237 971 (992.198,1249.95)178 577 (438.267,647.362) 238 1067 (933.751,1184.77)179 610 (454.153,657.365) 239 945 (950.209,1193.4)180 661 (523.788,740.559) 240 1047 (942.326,1195.99)181 626 (565.51,784.49) 241 1064 (928.872,1165.4)

74

Table 5.17: Expected and Obtained Output for Variable x0 (cont’d)


242 1123 (990.615,1236.03) 270 1039 (968.862,1214.41)243 1153 (1028.37,1272.93) 271 1169 (979.658,1225.76)244 988 (1070.73,1327.46) 272 992 (1036.36,1280.09)245 1049 (1006.21,1271.54) 273 903 (1018.95,1279.73)246 1038 (952.513,1192.66) 274 1250 (884.388,1128.13)247 1054 (979.704,1224.54) 275 1083 (1000.37,1238.61)248 990 (978.696,1229.35) 276 1177 (1095.36,1385.72)249 904 (959.612,1205.06) 277 1180 (1051.35,1328.21)250 1065 (886.169,1123.95) 278 1067 (1100.42,1389.99)251 1006 (918.256,1145.14) 279 1170 (1049.79,1336.87)252 1080 (971.614,1221.31) 280 1228 (1044.19,1308.46)253 1121 (974.962,1220.75) 281 1115 (1125.84,1399)254 1091 (1030.29,1289.86) 282 1143 (1100.27,1385.31)255 1142 (1036.21,1303.05) 283 1215 (1051.67,1332.49)256 1239 (1046.33,1306.03) 284 1108 (1103.01,1379.88)257 1114 (1118.32,1384.06) 285 1076 (1088.74,1376.02)258 990 (1111.23,1383.41) 286 1115 (1013.92,1300.11)259 1147 (990.787,1242.5) 287 1169 (1009.88,1306.98)260 947 (1004.93,1232.74) 288 1151 (1050.99,1363.53)261 867 (990.759,1238.28) 289 1125 (1061.98,1401.16)262 950 (850.375,1073.84) 290 1088 (1040.99,1376.73)263 924 (847.995,1063.08) 291 1244 (1015.73,1334.77)264 1024 (879.366,1101.85) 292 1264 (1077.24,1375.01)265 1102 (911.27,1135.41) 293 1237 (1168.81,1480.24)266 1071 (996.525,1239.72) 294 1260 (1164.23,1483.2)267 1123 (1021.89,1274.53) 295 1322 (1159.62,1479.3)268 1021 (1025.07,1287.34) 296 1092 (1195.85,1530.13)269 1049 (1002.63,1273.57) 297 1179 (1114.23,1467.92)

75

Table 5.18: Expected and Obtained Output for Variable x1(Forest Data)


1 1.3796 (1.32795,1.44784) 61 1.3784 (1.29994,1.42201)2 1.3662 (1.33739,1.45991) 62 1.3185 (1.31178,1.43545)3 1.2997 (1.32546,1.44489) 63 1.2635 (1.25959,1.37928)4 1.2123 (1.26971,1.38452) 64 1.2 (1.21054,1.32352)5 1.1681 (1.19761,1.30708) 65 1.1091 (1.15223,1.25961)6 1.1285 (1.16026,1.2649) 66 1.0877 (1.07109,1.17195)7 1.1253 (1.12659,1.22822) 67 1.051 (1.04981,1.14509)8 1.1247 (1.12827,1.23493) 68 1.0314 (1.02202,1.12043)9 1.1185 (1.12225,1.22239) 69 1 (0.996307,1.08409)10 1.1205 (1.11497,1.2132) 70 0.9826 (0.965081,1.04452)11 1.0954 (1.11912,1.22146) 71 1.0087 (0.954934,1.0417)12 1.0817 (1.08828,1.18761) 72 1.0299 (0.977433,1.06519)13 1.0815 (1.07835,1.17951) 73 1.08 (1.00073,1.09599)14 1.0788 (1.09179,1.19571) 74 1.1244 (1.04718,1.14988)15 1.0732 (1.0882,1.19175) 75 1.0966 (1.08935,1.19383)16 1.0647 (1.08037,1.18085) 76 1.0664 (1.06069,1.16223)17 1.0813 (1.07598,1.17771) 77 1.0525 (1.03414,1.13609)18 1.0916 (1.09017,1.19144) 78 1.0455 (1.02262,1.12339)19 1.1051 (1.0995,1.2029) 79 1.0217 (1.01613,1.1158)20 1.0943 (1.11384,1.22268) 80 1.0303 (1.00223,1.10781)21 1.0836 (1.10061,1.20359) 81 1.0553 (1.00311,1.10089)22 1.1019 (1.09211,1.19443) 82 1.1169 (1.02086,1.11672)23 1.1379 (1.10772,1.21033) 83 1.1686 (1.07835,1.18456)24 1.1772 (1.13926,1.24818) 84 1.2767 (1.11883,1.22408)25 1.1873 (1.17486,1.28962) 85 1.2475 (1.21086,1.32484)26 1.2021 (1.18516,1.30333) 86 1.2434 (1.18615,1.29933)27 1.2225 (1.19405,1.30903) 87 1.2304 (1.18045,1.29084)28 1.2663 (1.21574,1.33448) 88 1.2338 (1.1725,1.28688)29 1.2883 (1.24889,1.36908) 89 1.2687 (1.17767,1.29553)30 1.2977 (1.26697,1.38857) 90 1.2864 (1.20889,1.33099)31 1.295 (1.27546,1.39715) 91 1.2843 (1.22131,1.34246)32 1.295 (1.27565,1.40011) 92 1.271 (1.22183,1.34607)33 1.2876 (1.27263,1.39267) 93 1.2896 (1.20573,1.32175)34 1.2835 (1.26594,1.38516) 94 1.301 (1.22169,1.33923)35 1.2985 (1.26107,1.38027) 95 1.2945 (1.22908,1.34638)36 1.324 (1.2749,1.3977) 96 1.3167 (1.22406,1.34465)37 1.3435 (1.29923,1.42713) 97 1.319 (1.24773,1.37559)38 1.3418 (1.3151,1.44321) 98 1.3129 (1.24813,1.37548)39 1.335 (1.31417,1.44261) 99 1.3277 (1.23993,1.36345)40 1.33 (1.30797,1.43797) 100 1.4132 (1.253,1.37725)41 1.3367 (1.30189,1.42741) 101 1.5138 (1.32388,1.45137)42 1.3433 (1.31189,1.44191) 102 1.5988 (1.41107,1.54768)43 1.3491 (1.31576,1.44385) 103 1.5427 (1.48032,1.62034)44 1.3573 (1.32632,1.45795) 104 1.5076 (1.43439,1.57292)45 1.3309 (1.32866,1.45324) 105 1.5181 (1.3978,1.52479)46 1.3197 (1.3039,1.42611) 106 1.5284 (1.40559,1.53258)47 1.3561 (1.29045,1.40934) 107 1.5319 (1.41247,1.5391)48 1.3825 (1.31725,1.43678) 108 1.5389 (1.41578,1.5436)49 1.4564 (1.34116,1.46861) 109 1.5435 (1.43639,1.57215)50 1.4674 (1.39626,1.52146) 110 1.5468 (1.43949,1.57564)51 1.4989 (1.40681,1.53703) 111 1.5357 (1.44169,1.57838)52 1.4944 (1.4271,1.55505) 112 1.5085 (1.4313,1.56689)53 1.4611 (1.42403,1.55023) 113 1.5185 (1.40522,1.53512)54 1.5246 (1.39265,1.52139) 114 1.5211 (1.40933,1.54433)55 1.5313 (1.44559,1.57458) 115 1.5042 (1.40829,1.54023)56 1.5161 (1.45626,1.58847) 116 1.4979 (1.39645,1.53096)57 1.4831 (1.44037,1.56704) 117 1.4952 (1.38672,1.51561)58 1.4102 (1.40633,1.52686) 118 1.4553 (1.38296,1.51049)59 1.364 (1.34261,1.46216) 119 1.4312 (1.34885,1.47478)60 1.3636 (1.29806,1.41324) 120 1.4565 (1.32693,1.45069)

76



121 1.4512 (1.35279,1.48506) 181 1.125 (1.06088,1.16968)122 1.4245 (1.34232,1.46881) 182 1.0768 (1.05318,1.15911)123 1.4141 (1.31904,1.44436) 183 1.0663 (1.01245,1.11498)124 1.4167 (1.31017,1.4352) 184 1.0701 (1.00546,1.10891)125 1.4661 (1.31161,1.43658) 185 1.0732 (1.00696,1.10841)126 1.4858 (1.35521,1.48686) 186 1.0435 (1.01326,1.11858)127 1.4677 (1.36479,1.49124) 187 1.0329 (0.987224,1.08808)128 1.4443 (1.35463,1.48758) 188 1.0203 (0.993607,1.11356)129 1.3765 (1.32649,1.44754) 189 1.03 (0.979282,1.09363)130 1.3216 (1.26543,1.37765) 190 1.0261 (0.979953,1.08449)131 1.2969 (1.22538,1.34253) 191 0.9961 (0.982055,1.0928)132 1.2674 (1.20047,1.31384) 192 1.0029 (0.957535,1.06764)133 1.1196 (1.17695,1.29164) 193 0.9903 (0.968791,1.0869)134 1.1099 (1.0412,1.13251) 194 1.1024 (0.966145,1.09084)135 1.133 (1.03593,1.13155) 195 1.1752 (1.06296,1.19711)136 1.1484 (1.05812,1.15929) 196 1.0882 (1.11324,1.24102)137 1.1628 (1.07083,1.17376) 197 1.0117 (1.04278,1.16767)138 1.1254 (1.0846,1.19174) 198 0.9863 (0.979603,1.10033)139 1.0647 (1.05182,1.15414) 199 0.9631 (0.957588,1.07515)140 1.0773 (1.00414,1.10679) 200 0.9458 (0.940931,1.06072)141 1.0949 (1.00849,1.10419) 201 0.9431 (0.917534,1.0249)142 1.0725 (1.02292,1.1193) 202 0.9787 (0.914147,1.0198)143 1.0635 (1.00587,1.10488) 203 0.9931 (0.950974,1.06712)144 1.0629 (0.993133,1.08616) 204 1.0608 (0.961604,1.07454)145 1.101 (0.995402,1.09268) 205 1.1396 (1.02802,1.15484)146 1.117 (1.02888,1.13194) 206 1.109 (1.09012,1.21482)147 1.0995 (1.03844,1.13823) 207 1.0721 (1.06797,1.19607)148 1.0495 (1.02471,1.12447) 208 1.0644 (1.03734,1.16386)149 1.0303 (0.983587,1.08049) 209 1.0695 (1.02267,1.13757)150 0.9889 (0.956658,1.0493) 210 1.0392 (1.0339,1.1577)151 0.9676 (0.929586,1.01805) 211 1.01 (0.995032,1.09986)152 0.9453 (0.912139,0.999657) 212 1.013 (0.989315,1.11665)153 0.9899 (0.886468,0.963929) 213 1.0181 (0.982081,1.0976)154 0.9749 (0.925901,1.00943) 214 1.0299 (0.986133,1.101)155 0.9579 (0.909726,0.988942) 215 1.0574 (1.00286,1.12786)156 0.9618 (0.898737,0.980807) 216 1.1018 (1.02278,1.14543)157 1.002 (0.906226,0.993357) 217 1.107 (1.06475,1.19666)158 0.992 (0.942106,1.03342) 218 1.0867 (1.06955,1.20773)159 1.044 (0.932039,1.02103) 219 1.1272 (1.0492,1.17886)160 1.0627 (0.972403,1.06893) 220 1.1874 (1.08697,1.22476)161 1.006 (0.993536,1.09044) 221 1.1628 (1.13755,1.27968)162 1.006 (0.948447,1.04476) 222 1.1649 (1.11255,1.24771)163 0.975 (0.952031,1.05269) 223 1.0922 (1.11421,1.24927)164 0.994 (0.921633,1.0149) 224 1.0893 (1.06051,1.1986)165 1.0209 (0.934691,1.02543) 225 1.1123 (1.0476,1.17319)166 1.0579 (0.956611,1.04825) 226 1.1291 (1.07148,1.20297)167 1.0955 (0.988575,1.08337) 227 1.1412 (1.08199,1.21039)168 1.1125 (1.019,1.11541) 228 1.1287 (1.0941,1.22516)169 1.1195 (1.03598,1.1366) 229 1.121 (1.09296,1.23461)170 1.13 (1.04222,1.14243) 230 1.0864 (1.08276,1.22121)171 1.1594 (1.04635,1.14956) 231 1.0746 (1.05385,1.19012)172 1.1392 (1.07695,1.18248) 232 1.0649 (1.04642,1.18533)173 1.111 (1.05962,1.16234) 233 0.9843 (1.03254,1.16422)174 1.0569 (1.03699,1.14017) 234 0.9399 (0.966238,1.09417)175 1.0734 (0.989854,1.08673) 235 0.951 (0.930097,1.05661)176 1.0833 (1.00615,1.10642) 236 1.0018 (0.942263,1.07173)177 1.1222 (1.01279,1.11003) 237 1.0046 (0.980224,1.10811)178 1.1351 (1.04124,1.13597) 238 0.9955 (0.979254,1.10323)179 1.1444 (1.058,1.16099) 239 0.9901 (0.975926,1.10625)180 1.1319 (1.06836,1.17422) 240 0.9677 (0.965105,1.08644)

77



241 0.9448 (0.951959,1.07934) 270 0.9483 (0.950321,1.07792)242 0.9231 (0.932171,1.05952) 271 0.939 (0.932764,1.05852)243 0.9637 (0.916943,1.04731) 272 0.9536 (0.932485,1.067)244 1.0027 (0.952675,1.0874) 273 0.981 (0.934198,1.05689)245 0.9571 (0.975833,1.10092) 274 1.0182 (0.953406,1.07125)246 0.9386 (0.942101,1.06905) 275 1.1078 (1.0039,1.14851)247 0.9929 (0.92557,1.05073) 276 1.1533 (1.06901,1.2066)248 0.9388 (0.972148,1.1014) 277 1.1668 (1.11208,1.25885)249 0.9372 (0.923695,1.04544) 278 1.1451 (1.12362,1.27134)250 0.9304 (0.917439,1.03296) 279 1.0712 (1.09875,1.23723)251 0.9615 (0.919998,1.04663) 280 1.0475 (1.0428,1.1846)252 0.9842 (0.943174,1.0673) 281 1.0867 (1.02653,1.17117)253 1.0237 (0.966576,1.09721) 282 1.1551 (1.05315,1.19186)254 1.0428 (1.00178,1.13749) 283 1.0999 (1.11133,1.25577)255 1.0149 (1.01764,1.15226) 284 1.1154 (1.07015,1.21672)256 1.0157 (0.995243,1.13197) 285 1.2024 (1.07717,1.21695)257 0.9785 (1.00137,1.14506) 286 1.3144 (1.14696,1.2892)258 0.9333 (0.961547,1.0943) 287 1.3978 (1.2423,1.39343)259 0.8609 (0.915725,1.03717) 288 1.5535 (1.31422,1.47374)260 0.8718 (0.862119,0.990813) 289 1.5432 (1.44249,1.60921)261 0.866 (0.860864,0.975891) 290 1.4386 (1.4328,1.5971)262 0.8731 (0.853364,0.962371) 291 1.3088 (1.34376,1.49973)263 0.8558 (0.86527,0.980582) 292 1.2743 (1.24489,1.40491)264 0.8916 (0.848356,0.960872) 293 1.3142 (1.21738,1.37694)265 0.9483 (0.884864,1.00646) 294 1.3437 (1.25022,1.41003)266 0.9511 (0.936386,1.06663) 295 1.4349 (1.27589,1.43895)267 1.0524 (0.936772,1.06496) 296 1.5282 (1.35502,1.52747)268 1.0724 (1.02396,1.16138) 297 1.5938 (1.4199,1.58103)269 0.969 (1.03462,1.16585)

78



2 155.5 (137.034,192.098) 62 149 (137.019,189.284)3 147.3 (134.266,188.886) 63 147.6 (128.636,179.197)4 125.2 (127.61,180.583) 64 126.6 (126.736,175.557)5 124.9 (110.661,161.614) 65 111.1 (110.537,157.191)6 129.3 (109.783,159.581) 66 98.3 (98.0931,141.935)7 123.4 (112.578,161.338) 67 96.7 (88.0679,130.938)8 94.6 (108.768,158.025) 68 75.1 (87.0036,129.391)9 84.1 (87.0497,135.566) 69 55.1 (70.0661,110.772)10 66.4 (79.0364,127.14) 70 56.1 (54.4588,93.7366)11 74.3 (66.4473,114.942) 71 54.7 (55.7116,95.194)12 114.7 (71.5509,118.294) 72 80.2 (54.9012,95.0872)13 128.4 (101.227,147.852) 73 97.9 (74.3466,115.777)14 125 (111.968,160.172) 74 116.1 (88.2903,131.527)15 135.2 (109.461,157.445) 75 110.3 (102.126,147.002)16 140.8 (116.559,164.008) 76 119.3 (97.4658,141.032)17 128.7 (120.821,168.398) 77 117.3 (103.885,146.602)18 130.9 (112.046,160.04) 78 111.9 (102.309,144.694)19 140.9 (113.834,162.184) 79 123.6 (98.2238,140.355)20 126.9 (121.707,170.803) 80 96.9 (107.259,149.604)21 121.4 (110.907,159.183) 81 76.1 (87.0651,128.776)22 110.6 (106.748,154.814) 82 72.5 (71.6404,113.549)23 102.2 (98.9442,147.479) 83 89.9 (70.1492,114.341)24 167.9 (93.4416,143.081) 84 118.4 (82.934,128.011)25 201.1 (142.119,193.08) 85 137.2 (105.017,152.948)26 198.5 (166.99,218.45) 86 147.9 (118.669,165.87)27 193.8 (164.947,216.31) 87 154.2 (126.271,173.075)28 194.3 (162.001,214.426) 88 136.6 (131.241,178.133)29 204.5 (162.561,215.64) 89 145.9 (118.744,166.034)30 173.8 (170.214,223.794) 90 151.8 (126.005,174.409)31 179.7 (147.88,201.754) 91 148.4 (130.295,178.829)32 173.7 (152.339,206.47) 92 127.1 (128.12,176.912)33 152.1 (147.559,201.279) 93 107.4 (111.722,159.509)34 149.1 (131.64,185.118) 94 81.3 (97.4501,145.743)35 152.2 (129.307,182.524) 95 112.5 (78.2741,126.588)36 203.9 (131.936,185.707) 96 173.6 (101.21,149.47)37 211.6 (170.315,225.103) 97 182.2 (146.774,196.245)38 225.8 (176.232,231.431) 98 201.3 (153.224,202.569)39 223.1 (186.67,241.891) 99 197.6 (166.816,215.608)40 206.5 (184.827,239.854) 100 189.8 (164.325,213.553)41 228.6 (172.27,226.895) 101 194 (159.217,210.43)42 203 (188.848,244.198) 102 177.7 (163.478,217.538)43 216.5 (170.061,225.349) 103 193.1 (152.168,208.066)44 185.7 (180.276,236.402) 104 154.8 (163.025,217.697)45 150.5 (157.196,212.893) 105 129.2 (133.865,186.714)46 146.6 (130.983,185.72) 106 88.6 (115.028,168.01)47 138 (127.586,181.546) 107 101.3 (85.2278,138.25)48 200 (121.397,175.809) 108 172.1 (94.4698,147.636)49 205 (167.448,222.808) 109 197.5 (147.293,202.419)50 234 (171.317,227.616) 110 211 (166.159,221.314)51 202.6 (192.998,249.786) 111 216 (176.164,231.342)52 202.6 (169.965,226.771) 112 192.2 (179.708,234.483)53 197.2 (169.71,226.461) 113 190.9 (161.69,215.361)54 148.4 (165.716,221.325) 114 180.5 (160.969,214.471)55 147.1 (130.324,187.408) 115 192.1 (153.055,206.207)56 133.3 (129.596,187.461) 116 158.6 (161.676,214.731)57 90.4 (118.931,175.994) 117 119.5 (136.661,188.976)58 84.5 (86.7384,142.202) 118 88.2 (107.8,159.86)59 109.4 (81.7916,135.218) 119 84.5 (84.4556,135.471)60 124.8 (99.273,150.879) 120 152.9 (81.3124,131.529)61 159.8 (111.207,163.16) 121 161 (132.175,183.65)

79



122 189.1 (137.733,188.31) 182 180.7 (142.783,183.539)123 191.8 (158.069,207.894) 183 184 (148.281,187.807)124 164.2 (160.079,209.635) 184 162.1 (150.796,190.293)125 170.3 (139.904,189.437) 185 147.4 (134.622,173.987)126 163.7 (145.044,196.099) 186 148.5 (124.202,164.119)127 169 (139.751,190.405) 187 152.3 (124.445,163.461)128 118.7 (144.148,195.043) 188 126.2 (129.042,169.834)129 91.6 (106.167,155.357) 189 98.9 (109.398,149.378)130 73.1 (85.0782,132.048) 190 105.4 (88.4412,127.653)131 79.9 (71.7423,118.166) 191 95.4 (93.6856,133.51)132 85.1 (76.1311,121.434) 192 145 (86.2361,125.412)133 96.2 (79.9585,124.72) 193 175.8 (123.237,163.327)134 91.7 (85.3326,124.89) 194 170.2 (146.595,187.387)135 116.4 (82.4484,122.167) 195 163.2 (143.918,187.833)136 120.1 (101.084,141.756) 196 160.7 (138.334,182.656)137 129.9 (104.081,145.122) 197 160.7 (135.937,178.434)138 138.3 (111.689,153.329) 198 147.7 (135.293,176.022)139 152.7 (117.293,157.811) 199 173 (125.417,165.414)140 112.9 (127.689,167.138) 200 124.1 (143.999,183.797)141 95.9 (98.0173,136.953) 201 120.5 (107.096,145.284)142 84.5 (85.5611,124.887) 202 115.6 (104.104,142.074)143 71.9 (77.3103,116.343) 203 107.2 (101.654,141.421)144 107.8 (67.423,105.561) 204 151 (95.2773,135.172)145 123 (94.0075,132.511) 205 188.2 (128.875,171.683)146 109.9 (105.907,145.577) 206 186.6 (156.317,200.51)147 105.8 (96.1111,135.673) 207 183.6 (155.506,199.423)148 99.9 (93.0088,132.261) 208 172 (153.03,196.084)149 86.3 (88.2602,126.375) 209 163.8 (143.401,185.162)150 84.1 (77.5992,113.965) 210 154 (138.25,181.057)151 87.2 (75.6384,111.825) 211 154.8 (129.116,169.385)152 64.6 (77.7391,113.453) 212 115.6 (131.748,173.801)153 59.1 (60.1813,94.4205) 213 113 (101.954,142.79)154 47.2 (56.7909,92.4118) 214 105.1 (99.8704,140.792)155 51.3 (47.5809,82.3718) 215 102.8 (95.0934,137.226)156 78.2 (50.7601,85.5624) 216 141.2 (93.2422,135.635)157 84.1 (70.9518,106.403) 217 159.3 (122.33,166.455)158 98.8 (75.9242,112.639) 218 158 (136.219,180.615)159 91.1 (86.4237,122.675) 219 162.9 (134.509,177.914)160 106.8 (81.5738,118.807) 220 152.4 (138.988,183.911)161 96 (93.2763,131.577) 221 143.6 (131.936,178.365)162 106.4 (85.2098,122.436) 222 152 (124.706,169.948)163 110.5 (93.1963,130.896) 223 139.1 (130.811,176.09)164 108.9 (95.4132,131.764) 224 118.8 (121.514,165.939)165 82.9 (94.0752,130.512) 225 85.4 (105.383,148.358)166 91.3 (75.2382,112.254) 226 78.2 (81.5683,125.701)167 96.3 (81.7087,119.757) 227 90.2 (75.9228,120)168 134.6 (85.6577,124.497) 228 128.8 (84.9746,129.603)169 135.8 (114.138,153.755) 229 153.2 (114.232,159.787)170 174.9 (115.173,154.991) 230 140.2 (131.835,176.734)171 173.2 (143.906,183.489) 231 150.2 (122.11,166.166)172 161.6 (143.28,184.27) 232 137 (129.583,173.68)173 176.8 (134.48,174.904) 233 136.8 (119.226,162.33)174 154.9 (145.481,185.359) 234 131.1 (118.417,159.697)175 159.3 (128.764,167.06) 235 135.1 (113.972,154.302)176 136 (132.293,171.312) 236 113 (117.225,158.187)177 108.3 (115.05,154.069) 237 94.2 (101.071,142.688)178 109.1 (94.5871,134.016) 238 100.1 (86.8721,128.118)179 130 (95.9111,136.447) 239 85.8 (91.6699,133.286)180 137.5 (111.517,152.61) 240 117.8 (80.3638,121)181 172.7 (117.328,158.51) 241 129.4 (104.208,145.046)

80



242 131.7 (112.688,152.916) 270 86.6 (87.3425,127.905)243 143.2 (114.626,154.723) 271 101.8 (81.1503,121.171)244 134.7 (123.599,164.916) 272 75.6 (93.0551,133.843)245 122.4 (116.616,157.627) 273 65.6 (72.8482,112.606)246 109.3 (107.651,148.121) 274 71.6 (65.091,104.996)247 130.1 (97.7833,137.671) 275 78.8 (72.1591,115.548)248 96.6 (113.518,154.876) 276 111.6 (77.0429,121.311)249 75 (88.2173,127.826) 277 107.6 (102.077,148.124)250 99.2 (71.6896,110.607) 278 115.2 (99.4187,145.825)251 86.9 (90.3301,130.194) 279 117.8 (103.979,148.971)252 108.5 (81.3038,121.538) 280 106.2 (105.991,149.965)253 119 (97.7577,139.125) 281 109.9 (97.7295,141.594)254 121.1 (106.134,148.753) 282 106 (99.9464,143.939)255 117.8 (107.707,150.744) 283 111.8 (97.8746,143.671)256 111.2 (105.366,147.887) 284 84.5 (102.15,147.25)257 102.8 (101.209,144.476) 285 78.6 (81.6123,126.297)258 93.1 (93.8142,135.051) 286 70.5 (77.6878,124.114)259 94.2 (85.4175,124.51) 287 74.6 (72.9617,122.343)260 81.4 (86.6021,124.896) 288 95.5 (76.9932,128.707)261 57.4 (75.9813,113.144) 289 117.8 (93.4687,148.725)262 52.5 (57.8361,94.469) 290 120.9 (109.589,164.444)263 59.1 (54.8335,92.4071) 291 128.5 (110.801,162.897)264 73.8 (59.2837,96.1311) 292 115.3 (116.361,166.573)265 99.7 (71.0781,109.65) 293 121.8 (106.592,156.139)266 97.7 (91.0833,131.584) 294 118.5 (111.473,161.9)267 103.4 (89.5245,129.838) 295 123.2 (109.49,160.773)268 103.5 (94.8993,138.011) 296 102.3 (114.108,167.998)269 94.7 (94.4496,137.246) 297 98.7 (98.0497,152.463)

81



2 54.4056 (51.2426,56.0144) 62 44.4961 (42.9412,46.9302)3 54.4678 (51.1396,55.9435) 63 44.3231 (42.5261,46.455)4 54.7514 (51.4142,56.2236) 64 42.9481 (41.9987,45.9123)5 55.1765 (51.5233,56.3578) 65 41.381 (41.5619,45.3541)6 55.1229 (51.8235,56.6954) 66 41.3739 (40.2191,43.873)7 55.1894 (52.1162,56.9834) 67 40.8735 (39.0934,42.7466)8 55.0637 (52.0929,56.966) 68 40.2265 (38.9745,42.5835)9 54.7548 (52.112,56.974) 69 40.2548 (38.4684,42.0203)10 54.5178 (51.9516,56.7863) 70 39.4582 (38.0107,41.5651)11 51.4932 (51.6761,56.4899) 71 39.8741 (37.8499,41.3339)12 51.2834 (50.8182,55.365) 72 40.1072 (37.373,40.8938)13 55.231 (48.6009,53.1291) 73 40.513 (37.7244,41.2657)14 54.8207 (49.3481,54.2249) 74 42.5475 (37.9839,41.5611)15 54.4851 (52.0866,56.9271) 75 40.9966 (38.7376,42.4945)16 54.9353 (51.7159,56.5268) 76 39.8995 (39.8444,43.4643)17 55.4553 (51.5776,56.4282) 77 40.0203 (38.4823,42.0054)18 55.2965 (52.0188,56.9154) 78 39.9781 (37.7228,41.2565)19 55.0889 (52.3557,57.2382) 79 40.015 (37.7999,41.3299)20 55.062 (52.1946,57.0588) 80 40.1294 (37.778,41.3112)21 55.2102 (52.0395,56.9014) 81 40 (37.8305,41.3738)22 55.5603 (52.0539,56.9289) 82 40.3005 (37.8831,41.415)23 54.9469 (52.2399,57.1457) 83 40.5159 (37.8586,41.4171)24 54.9683 (52.3515,57.2032) 84 40.56 (38.1232,41.7006)25 54.8496 (51.9164,56.7699) 85 40.1716 (38.2877,41.8691)26 54.7795 (51.9047,56.7478) 86 40.0822 (38.231,41.7781)27 55.7094 (51.8036,56.6405) 87 39.7631 (37.9321,41.4713)28 54.8668 (51.9648,56.8838) 88 39.6769 (37.7954,41.3064)29 54.7506 (52.4402,57.2849) 89 39.8176 (37.5469,41.0503)30 55.1097 (51.8094,56.6437) 90 39.568 (37.517,41.0328)31 55.1462 (51.8077,56.6738) 91 39.4685 (37.5612,41.055)32 55.3185 (52.0736,56.9429) 92 39.771 (37.3595,40.8445)33 55.2461 (52.139,57.0235) 93 39.8208 (37.3569,40.8686)34 54.6753 (52.2461,57.1242) 94 39.3408 (37.5853,41.1014)35 54.3929 (52.0643,56.8921) 95 38.4693 (37.5118,40.9855)36 54.4745 (51.5907,56.3934) 96 38.6225 (36.9693,40.3661)37 54.6081 (51.4066,56.2166) 97 38.2774 (36.379,39.7893)38 54.6557 (51.4956,56.3173) 98 38.1518 (36.4104,39.7902)39 53.932 (51.6022,56.4282) 99 38.4477 (36.1343,39.503)40 54.42 (51.4718,56.2339) 100 38.8372 (36.1115,39.5063)41 54.8204 (51.0636,55.8688) 101 39.1462 (36.4123,39.8416)42 54.893 (51.5047,56.3453) 102 39.0551 (36.762,40.2185)43 56.005 (51.8085,56.6554) 103 38.8247 (36.9629,40.4114)44 56.5211 (52.1134,57.0586) 104 38.8815 (36.8452,40.2733)45 55.781 (53.0285,58.0192) 105 38.7205 (36.6928,40.126)46 55.0841 (53.2304,58.1558) 106 38.4027 (36.697,40.1159)47 54.2877 (52.541,57.4048) 107 38.2667 (36.5092,39.9001)48 53.1982 (51.86,56.6535) 108 41.3186 (36.2503,39.6292)49 52.789 (51.0409,55.7382) 109 41.029 (36.8468,40.4951)50 52.0652 (50.1663,54.8275) 110 40.7453 (38.9702,42.593)51 51.1473 (49.7082,54.3054) 111 40.4729 (38.6979,42.2956)52 51.7884 (48.9802,53.4964) 112 40.4091 (38.4325,42.0061)53 49.2821 (48.4676,53.0404) 113 37.7017 (38.2226,41.7906)54 50.2677 (48.3575,52.709) 114 37.4789 (37.5611,40.8901)55 51.2549 (46.7838,51.2223) 115 37.2688 (35.5683,38.8776)56 51.4129 (47.7153,52.241) 116 37.1609 (35.3607,38.6514)57 50.2975 (48.4594,52.999) 117 36.967 (35.1854,38.4667)58 48.3347 (48.3191,52.7602) 118 36.4539 (35.0639,38.328)59 46.682 (47.0726,51.3404) 119 36.0187 (34.8082,38.027)60 45.5395 (45.2887,49.4106) 120 35.7454 (34.3411,37.5215)61 45.1765 (43.8433,47.8644) 121 35.1534 (33.9668,37.123)

82



122 34.6787 (33.6361,36.7401) 182 29.3939 (27.814,30.4137)123 34.5115 (33.1035,36.1655) 183 29.575 (27.8048,30.4002)124 34.3245 (32.7249,35.7722) 184 29.6132 (27.8113,30.4227)125 34.1382 (32.5625,35.5932) 185 29.8815 (27.9499,30.5647)126 33.5266 (32.386,35.4003) 186 30.1799 (28.0383,30.6768)127 33.1449 (32.1132,35.0736) 187 30.0145 (28.2986,30.9634)128 33.0448 (31.5877,34.5143) 188 30.026 (28.4751,31.1253)129 32.7782 (31.2911,34.2089) 189 30.3208 (28.359,31.0103)130 32.4941 (31.1587,34.0529) 190 30.4014 (28.4343,31.1116)131 31.6237 (30.9028,33.772) 191 30.2488 (28.6641,31.3485)132 31.1429 (30.5011,33.2934) 192 30.0931 (28.6872,31.3581)133 30.7938 (29.7674,32.5172) 193 30.7928 (28.5424,31.1995)134 30.4156 (29.3431,32.0621) 194 31.2106 (28.5898,31.3087)135 30.2909 (29.0067,31.6923) 195 30.5682 (29.1867,31.9426)136 30.0133 (28.707,31.3816) 196 30.6357 (29.3404,32.0395)137 29.7071 (28.5544,31.2045) 197 30.7225 (28.8949,31.5999)138 29.8582 (28.2856,30.9087) 198 30.8825 (28.963,31.6758)139 29.7737 (28.1004,30.7368) 199 30.8134 (29.0617,31.7885)140 29.7629 (28.1895,30.8185) 200 30.8395 (29.1608,31.8815)141 29.7868 (28.1265,30.7545) 201 30.8736 (29.1171,31.8402)142 29.2532 (28.1242,30.7543) 202 31.0494 (29.1436,31.8697)143 28.9365 (28.0199,30.6029) 203 31.6962 (29.208,31.9496)144 28.7165 (27.5652,30.1202) 204 32.4496 (29.4812,32.2799)145 28.35 (27.2879,29.8235) 205 33.0994 (30.1166,32.9818)146 28.2045 (27.0468,29.55) 206 32.82 (30.8048,33.7274)147 28.2173 (26.7508,29.2412) 207 32.6907 (31.2074,34.1053)148 28.3737 (26.6493,29.1408) 208 32.9044 (30.9775,33.864)149 25.4414 (26.6941,29.1994) 209 32.9507 (30.9334,33.8388)150 28.4868 (26.1394,28.3858) 210 32.9628 (31.0972,34.0067)151 28.5015 (24.7285,27.2438) 211 32.7302 (31.1332,34.0437)152 28.4494 (26.9165,29.4331) 212 32.6623 (31.0889,33.979)153 28.2976 (26.9152,29.4272) 213 32.8566 (30.9067,33.7907)154 27.9649 (26.8433,29.3419) 214 32.6497 (30.9021,33.8033)155 28 (26.6587,29.1279) 215 32.7139 (30.9945,33.8774)156 28.1386 (26.428,28.9004) 216 32.6126 (30.8606,33.7492)157 28.4719 (26.4847,28.9693) 217 31.1315 (30.8837,33.7633)158 28.3477 (26.6599,29.174) 218 31.9542 (30.4742,33.223)159 26.097 (26.8708,29.3738) 219 31.7777 (29.5988,32.4203)160 28.2251 (26.2699,28.5742) 220 31.713 (30.1488,32.9547)161 28.2034 (25.1393,27.6315) 221 31.6696 (30.0075,32.8077)162 28.247 (26.6609,29.1512) 222 31.702 (29.9512,32.7476)163 28.1916 (26.6553,29.1494) 223 32.0538 (29.9275,32.7267)164 28.2403 (26.674,29.1632) 224 31.8119 (30.0307,32.861)165 28.2647 (26.6453,29.1389) 225 32.2044 (30.228,33.037)166 28.3962 (26.6858,29.1815) 226 32.1185 (30.1438,32.9874)167 28.2527 (26.7332,29.2405) 227 32.2901 (30.4058,33.2418)168 28.3496 (26.7949,29.2895) 228 32.3823 (30.3832,33.2344)169 28.7171 (26.714,29.2172) 229 31.8261 (30.5273,33.3866)170 26.4613 (26.8671,29.4027) 230 31.8254 (30.4669,33.2771)171 28.4238 (26.6177,28.9542) 231 31.7379 (30.0678,32.8779)172 28.695 (25.4458,27.9556) 232 31.5616 (30.0474,32.8498)173 28.4057 (26.9153,29.449) 233 31.5667 (29.9445,32.7313)174 28.5373 (27.044,29.5522) 234 31.5245 (29.8192,32.6065)175 28.7554 (26.8664,29.3862) 235 31.903 (29.8133,32.5968)176 29.1939 (27.0104,29.5495) 236 31.6602 (29.8691,32.686)177 29.1535 (27.2666,29.8444) 237 31.6395 (30.0854,32.8809)178 29.172 (27.572,30.1462) 238 31.2145 (29.9065,32.7002)179 29.4448 (27.5472,30.123) 239 31.4134 (29.795,32.5512)180 29.4398 (27.6225,30.2224) 240 31.2897 (29.5354,32.3091)181 29.4423 (27.8171,30.4165) 241 30.6776 (29.6499,32.4127)

83



242 30.4761 (29.422,32.1307) 270 30.3686 (28.6725,31.3487)243 30.6041 (28.9371,31.628) 271 30.39 (28.6482,31.3297)244 30.6658 (28.8216,31.5239) 272 30.2663 (28.6958,31.3792)245 30.8795 (28.9275,31.6352) 273 30.799 (28.683,31.3555)246 30.6824 (29.0203,31.7469) 274 30.7907 (28.7155,31.4349)247 30.9078 (29.1288,31.838) 275 30.8922 (29.0957,31.8144)248 30.9184 (29.0387,31.7678) 276 30.8966 (29.1128,31.8405)249 30.9 (29.2028,31.9328) 277 30.9579 (29.1866,31.9147)250 30.577 (29.2062,31.9346) 278 30.7952 (29.2037,31.9372)251 30.8322 (29.1195,31.8194) 279 30.6161 (29.2107,31.9298)252 30.9658 (28.9459,31.6683) 280 30.648 (29.0533,31.7566)253 31.0114 (29.1593,31.8935) 281 30.8275 (28.932,31.6382)254 31.6099 (29.2655,32.0038) 282 30.7864 (28.9957,31.7177)255 30.8609 (29.4344,32.2254) 283 30.9636 (29.1152,31.8335)256 30.8498 (29.6933,32.4183) 284 31.0433 (29.126,31.86)257 30.1923 (29.1535,31.8775) 285 30.8597 (29.2712,32.0123)258 29.7365 (28.996,31.662) 286 30.9551 (29.2866,32.0115)259 28.8767 (28.4207,31.0464) 287 30.8547 (29.1766,31.9099)260 29.2398 (27.8982,30.448) 288 30.9225 (29.2222,31.9466)261 29.8795 (27.364,29.9459) 289 31.0335 (29.1656,31.896)262 30.3345 (27.77,30.4083) 290 30.9599 (29.2395,31.9797)263 29.9548 (28.3323,31.0108) 291 30.9264 (29.3024,32.0361)264 30.3554 (28.5724,31.2173) 292 30.9102 (29.242,31.9727)265 30.3862 (28.3911,31.0714) 293 31.358 (29.2143,31.9436)266 30.3245 (28.6855,31.3685) 294 31.326 (29.3045,32.0733)267 30.4115 (28.6936,31.3711) 295 31.3258 (29.6184,32.3844)268 30.3618 (28.6691,31.3543) 296 31.5223 (29.5954,32.3614)269 30.309 (28.7202,31.4011) 297 31.8209 (29.64,32.4233)

84

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Fuzzy Coefficient: (9.23515e-05, 3.56527e-05)

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Figure 5.6: Linearity Figures for Forestry Data

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Figure 5.6: Linearity Figures for Forestry Data(cont’d)

86

5.1.2.5 Gas Furnace Data

Table 5.21: Fuzzy equations for Gas Furnace Data


x0 (1.121, 0.1912)x0,t−1 + (0, 0.1117)x0,t−4 + (0, 0.004617)x1,t−1

x1 (0.9662, 0.01582)x1,t−1 + (0.03636, 0)x1,t−4 + (0, 0.1352)x0,t−1

Gas Furnace data contains two variables which are gas flow rate (x0) and CO2

concentration (x1). The data can be seen in A.8. The fuzzy equations for the

variables x0 and x1 are shown in the Table 5.21. x0,t−1 is the largest coefficient

center value for the fuzzy equation for x0 and x1,t−1 is the largest coefficient value

for x1. Accuracy rate of the model for x0 (0.482877) is worse than the rate of the

model for x1 (0.708904).


can be seen. The defining variables with largest coefficient center values in the

fuzzy equations have apparent linear relationship with the respective variables.

The output results can be seen in Figures 5.22, 5.23. The widths of the

obtained results for the variable x0 are large and the model does not fit well.

87

Table 5.22: Expected and Obtained Output for Variable x0(Gas Furnace Data)


4 0.373 (0.0802995,0.679532) 64 1.223 (0.680874,1.83173)5 0.441 (0.100191,0.735848) 65 1.257 (0.776529,1.96469)6 0.461 (0.144903,0.843551) 66 1.157 (0.814322,2.00311)7 0.348 (0.147357,0.885924) 67 0.913 (0.721286,1.87201)8 0.127 (0.039906,0.740098) 68 0.62 (0.484007,1.56238)9 -0.18 (-0.172196,0.456852) 69 0.255 (0.208314,1.18135)10 -0.588 (-0.458877,0.055426) 70 -0.28 (-0.119339,0.690894)11 -1.055 (-0.825509,-0.492429) 71 -1.08 (-0.589853,-0.0377368)12 -1.421 (-1.23677,-1.1279) 72 -1.551 (-1.30261,-1.11809)13 -1.52 (-1.54546,-1.63956) 73 -1.799 (-1.70243,-1.77397)14 -1.302 (-1.59653,-1.81039) 74 -1.825 (-1.87781,-2.15445)15 -0.814 (-1.34588,-1.57241) 75 -1.456 (-1.81955,-2.27099)16 -0.475 (-0.856462,-0.968031) 76 -0.944 (-1.43132,-1.83214)17 -0.193 (-0.533981,-0.53068) 77 -0.57 (-0.935081,-1.18079)18 0.088 (-0.296199,-0.136389) 78 -0.431 (-0.588682,-0.688911)19 0.435 (-0.0876585,0.284901) 79 -0.577 (-0.503461,-0.462578)20 0.771 (0.20026,0.774745) 80 -0.96 (-0.69544,-0.597843)21 0.866 (0.484321,1.24379) 81 -1.616 (-1.09,-1.06174)22 0.875 (0.544472,1.39657) 82 -1.875 (-1.71254,-1.90955)23 0.891 (0.519158,1.44206) 83 -1.891 (-1.93421,-2.2684)24 0.987 (0.500655,1.49642) 84 -1.746 (-1.9063,-2.33217)25 1.263 (0.582512,1.62974) 85 -1.474 (-1.70286,-2.21061)26 1.775 (0.839908,1.99097) 86 -1.201 (-1.42479,-1.87902)27 1.976 (1.31589,2.66258) 87 -0.927 (-1.17293,-1.51898)28 1.934 (1.4934,2.9356) 88 -0.524 (-0.936279,-1.14149)29 1.866 (1.42583,2.90902) 89 0.04 (-0.592054,-0.582436)30 1.832 (1.30913,2.87331) 90 0.788 (-0.0968984,0.186554)31 1.767 (1.25877,2.84746) 91 0.943 (0.569643,1.19657)32 1.608 (1.20535,2.75519) 92 0.93 (0.671941,1.44169)33 1.265 (1.06653,2.53763) 93 1.006 (0.601476,1.48302)34 0.79 (0.751955,2.08341) 94 1.137 (0.594574,1.66027)35 0.36 (0.317681,1.45302) 95 1.198 (0.705957,1.84251)36 0.115 (-0.0647249,0.871626) 96 1.054 (0.769189,1.916)37 0.088 (-0.25646,0.51422) 97 0.595 (0.629154,1.73327)38 0.331 (-0.232657,0.429899) 98 -0.08 (0.190631,1.143)39 0.645 (0.0366357,0.705265) 99 -0.314 (-0.441317,0.262006)40 0.96 (0.3508,1.0949) 100 -0.288 (-0.642285,-0.061512)41 1.409 (0.643391,1.50835) 101 -0.153 (-0.569618,-0.0759028)42 2.67 (1.03315,2.12497) 102 -0.109 (-0.372428,0.0294946)43 2.834 (2.17116,3.81336) 103 -0.187 (-0.308161,0.0638489)44 2.812 (2.29073,4.06138) 104 -0.255 (-0.386339,-0.0328011)45 2.483 (2.22567,4.07713) 105 -0.229 (-0.466474,-0.105081)46 1.929 (1.78685,3.77853) 106 -0.007 (-0.448144,-0.065135)47 1.485 (1.25956,3.06408) 107 0.254 (-0.233534,0.217844)48 1.214 (0.854841,2.47363) 108 0.33 (0.0162109,0.553103)49 1.239 (0.640605,2.08044) 109 0.102 (0.0839522,0.655707)50 1.608 (0.723877,2.05321) 110 -0.423 (-0.152781,0.381403)51 1.905 (1.11232,2.49184) 111 -1.139 (-0.667636,-0.280472)52 2.023 (1.41451,2.85534) 112 -2.275 (-1.34029,-1.21265)53 1.815 (1.51955,3.01478) 113 -2.594 (-2.37033,-2.72884)54 0.535 (1.28453,2.78359) 114 -2.716 (-2.61052,-3.20365)55 0.122 (0.0633866,1.13576) 115 -2.51 (-2.64949,-3.43813)56 0.009 (-0.330464,0.603914) 116 -1.79 (-2.33943,-3.28647)57 0.164 (-0.41227,0.432442) 117 -1.346 (-1.64191,-2.37018)58 0.671 (-0.129374,0.496962) 118 -1.081 (-1.22203,-1.79488)59 1.019 (0.382035,1.12194) 119 -0.91 (-1.0024,-1.42054)60 1.146 (0.712598,1.57138) 120 -0.876 (-0.922949,-1.11672)61 1.155 (0.809183,1.75945) 121 -0.885 (-0.938167,-1.02529)62 1.112 (0.760457,1.82835) 122 -0.8 (-0.971516,-1.01212)63 1.121 (0.683462,1.80897) 123 -0.544 (-0.907912,-0.885202)

88

Table 5.22: Expected and Obtained Output for Variable x0(Gas FurnaceData)(cont’d)


124 -0.416 (-0.670514,-0.548803) 184 0.063 (-0.42835,-0.324758)125 -0.271 (-0.548219,-0.3842) 185 0.084 (-0.0650644,0.206272)126 0 (-0.421082,-0.186335) 186 0 (-0.0824782,0.270755)127 0.403 (-0.196385,0.196385) 187 0.001 (-0.220541,0.220541)128 0.841 (0.165771,0.73751) 188 0.209 (-0.260947,0.263188)129 1.285 (0.558101,1.32691) 189 0.556 (-0.0662538,0.534705)130 1.607 (0.943319,1.93687) 190 0.782 (0.268913,0.977301)131 1.746 (1.20085,2.40107) 191 0.858 (0.47934,1.27343)132 1.683 (1.28529,2.62818) 192 0.918 (0.526752,1.39636)133 1.485 (1.18267,2.58959) 193 0.862 (0.545613,1.51198)134 0.993 (0.967275,2.36119) 194 0.416 (0.471547,1.46053)135 0.648 (0.499953,1.72575) 195 -0.336 (0.0507888,0.88163)136 0.577 (0.189067,1.26336) 196 -0.959 (-0.652161,-0.100947)137 0.577 (0.14657,1.14671) 197 -1.813 (-1.22317,-0.926329)138 0.632 (0.200602,1.09268) 198 -2.378 (-1.96672,-2.09692)139 0.747 (0.287954,1.12861) 199 -2.499 (-2.41484,-2.91519)140 0.9 (0.400012,1.27431) 200 -2.473 (-2.46281,-3.13843)141 0.993 (0.539462,1.47779) 201 -2.33 (-2.35295,-3.19002)142 0.968 (0.618381,1.60732) 202 -2.053 (-2.16799,-3.05445)143 0.79 (0.581374,1.58829) 203 -1.739 (-1.90392,-2.69766)144 0.399 (0.399749,1.37095) 204 -1.261 (-1.61725,-2.28053)145 -0.161 (0.0268325,0.867483) 205 -0.569 (-1.19075,-1.63565)146 -0.553 (-0.490459,0.129595) 206 -0.137 (-0.578906,-0.696446)147 -0.603 (-0.834035,-0.405455) 207 -0.024 (-0.21103,-0.0960407)148 -0.424 (-0.837762,-0.513797) 208 -0.05 (-0.157074,0.103281)149 -0.194 (-0.612517,-0.337833) 209 -0.135 (-0.255305,0.143235)150 -0.049 (-0.360015,-0.0748153) 210 -0.276 (-0.376106,0.0735178)151 0.06 (-0.223801,0.113973) 211 -0.534 (-0.514253,-0.104371)152 0.161 (-0.145706,0.280189) 212 -0.871 (-0.745632,-0.451272)153 0.301 (-0.0784353,0.439299) 213 -1.243 (-1.04616,-0.90609)154 0.517 (0.0359667,0.638692) 214 -1.439 (-1.37436,-1.41169)155 0.566 (0.22642,0.93238) 215 -1.422 (-1.52773,-1.69763)156 0.56 (0.262533,1.00609) 216 -1.175 (-1.47567,-1.71159)157 0.573 (0.242242,1.01294) 217 -0.813 (-1.2096,-1.42403)158 0.592 (0.231123,1.05319) 218 -0.634 (-0.854445,-0.967807)159 0.671 (0.245619,1.08128) 219 -0.582 (-0.693649,-0.727394)160 0.933 (0.32157,1.1824) 220 -0.625 (-0.674286,-0.630204)161 1.337 (0.565044,1.52617) 221 -0.713 (-0.755153,-0.645717)162 1.46 (0.938458,2.05828) 222 -0.848 (-0.8551,-0.743013)163 1.353 (1.04489,2.22754) 223 -1.039 (-0.983626,-0.917074)164 0.772 (0.917089,2.11551) 224 -1.346 (-1.15406,-1.17475)165 0.218 (0.334206,1.39615) 225 -1.628 (-1.42775,-1.58916)166 -0.237 (-0.191269,0.679893) 226 -1.619 (-1.67388,-1.9751)167 -0.714 (-0.599491,0.0682808) 227 -1.149 (-1.64372,-1.98509)168 -1.099 (-0.977526,-0.622828) 228 -0.488 (-1.17347,-1.40189)169 -1.269 (-1.27537,-1.18792) 229 -0.16 (-0.530309,-0.563491)170 -1.175 (-1.38672,-1.4576) 230 -0.007 (-0.228732,-0.129891)171 -0.676 (-1.25161,-1.38203) 231 -0.092 (-0.141779,0.126089)172 0.033 (-0.750301,-0.76488) 232 -0.62 (-0.295545,0.0893367)173 0.556 (-0.0768815,0.150847) 233 -1.086 (-0.820673,-0.56899)174 0.643 (0.392769,0.853444) 234 -1.525 (-1.26539,-1.16876)175 0.484 (0.415133,1.02608) 235 -1.858 (-1.6612,-1.75693)176 0.109 (0.188143,0.896691) 236 -2.029 (-1.9076,-2.2569)177 -0.31 (-0.212852,0.457164) 237 -2.024 (-2.01543,-2.53236)178 -0.697 (-0.606971,-0.0878603) 238 -1.961 (-1.96636,-2.57021)179 -1.047 (-0.943867,-0.618383) 239 -1.952 (-1.87569,-2.51968)180 -1.218 (-1.22594,-1.1208) 240 -1.794 (-1.85191,-2.52328)181 -1.183 (-1.33901,-1.391) 241 -1.302 (-1.70837,-2.31268)182 -0.873 (-1.26648,-1.38508) 242 -1.03 (-1.26038,-1.65791)183 -0.336 (-0.942923,-1.01381) 243 -0.918 (-1.00994,-1.2987)

89

Table 5.22: Expected and Obtained Output for Variable x0(Gas FurnaceData)(cont’d)


244 -0.798 (-0.924398,-1.1332) 270 -0.947 (-0.99146,-0.685101)245 -0.867 (-0.867808,-0.920823) 271 -1.029 (-1.12519,-0.997405)246 -1.047 (-0.961404,-0.981883) 272 -0.928 (-1.14172,-1.16467)247 -1.123 (-1.13846,-1.20827) 273 -0.645 (-1.01914,-1.06088)248 -0.876 (-1.21974,-1.29734) 274 -0.424 (-0.737539,-0.708158)249 -0.395 (-0.98059,-0.982869) 275 -0.276 (-0.527568,-0.422783)250 0.185 (-0.510603,-0.374746) 276 -0.158 (-0.404508,-0.214116)251 0.662 (0.0374833,0.377174) 277 -0.033 (-0.328278,-0.0258616)252 0.709 (0.452826,1.03098) 278 0.102 (-0.23677,0.162804)253 0.605 (0.442788,1.14636) 279 0.251 (-0.127351,0.355973)254 0.501 (0.283178,1.07286) 280 0.28 (-0.00018232,0.562772)255 0.603 (0.136919,0.986018) 281 0 (0.0160441,0.611546)256 0.943 (0.232023,1.11954) 282 -0.493 (-0.257461,0.257461)257 1.223 (0.564302,1.54933) 283 -0.759 (-0.730061,-0.374945)258 1.249 (0.840807,1.90041) 284 -0.824 (-0.979636,-0.72158)259 0.824 (0.855429,1.94407) 285 -0.74 (-1.00878,-0.838126)260 0.102 (0.422396,1.42451) 286 -0.528 (-0.87748,-0.78115)261 0.025 (-0.277703,0.506325) 287 -0.204 (-0.656707,-0.526748)262 0.382 (-0.34895,0.404985) 288 0.034 (-0.356122,-0.101122)263 0.922 (0.0322153,0.823996) 289 0.204 (-0.148889,0.225097)264 1.032 (0.614815,1.45175) 290 0.253 (-0.0191631,0.476407)265 0.866 (0.716432,1.59668) 291 0.195 (-0.0125751,0.579647)266 0.527 (0.513018,1.42803) 292 0.131 (-0.0926109,0.529682)267 0.093 (0.132508,1.04871) 293 0.017 (-0.170167,0.463789)268 -0.458 (-0.280431,0.488881) 294 -0.182 (-0.2793,0.317404)269 -0.748 (-0.766221,-0.260337) 295 -0.262 (-0.455493,0.0475593)

90

Table 5.23: Expected and Obtained Output for Variable x1(Gas Furnace Data)


4 53.4 (52.7565,54.5409) 64 50.1 (49.716,51.6169)5 53.1 (52.6496,54.44) 65 49.8 (49.3257,51.2415)6 52.7 (52.3517,54.151) 66 49.6 (49.0215,50.9369)7 52.4 (51.9688,53.7608) 67 49.4 (48.8194,50.7015)8 52.2 (51.6953,53.4473) 68 49.3 (48.6478,50.4576)9 52 (51.5242,53.2101) 69 49.2 (48.5815,50.3089)10 52 (51.3611,52.9577) 70 49.3 (48.5285,50.1541)11 52.4 (51.4053,52.8916) 71 49.7 (48.6886,50.1727)12 53 (51.8413,53.2141) 72 50.3 (49.1732,50.4537)13 54 (52.4538,53.7465) 73 51.3 (49.8035,50.9757)14 54.9 (53.4176,54.7152) 74 52.8 (50.791,51.9278)15 56 (54.258,55.643) 75 54.4 (52.2347,53.4119)16 56.8 (55.2593,56.8111) 76 56 (53.7273,55.0549)17 56.8 (56.0101,57.6789) 77 56.9 (55.2151,56.7317)18 56.4 (56.0048,57.7497) 78 57.5 (56.0744,57.7206)19 55.7 (55.6266,57.4349) 79 57.3 (56.684,58.3868)20 55 (54.9435,56.8234) 80 56.6 (56.5719,58.2288)21 54.3 (54.2328,56.1814) 81 56 (55.9911,57.5223)22 53.2 (53.5402,55.4923) 82 55.4 (55.5313,56.8663)23 52.3 (52.4681,54.3878) 83 55.4 (54.9888,56.2348)24 51.6 (51.5851,53.4807) 84 56.4 (54.9655,56.2072)25 51.2 (50.8814,52.7808) 85 57.2 (55.8745,57.187)26 50.8 (50.4239,52.3853) 86 58 (56.5762,57.9876)27 50.5 (49.9418,52.0289) 87 58.4 (57.2997,58.8101)28 50 (49.6041,51.7361) 88 58.4 (57.6791,59.2763)29 49.2 (49.12,51.2248) 89 58.1 (57.6538,59.3599)30 48.4 (48.3544,50.4154) 90 57.7 (57.3215,59.1706)31 47.9 (47.5877,49.6143) 91 57 (56.8548,58.8934)32 47.6 (47.1031,49.0963) 92 56 (56.1685,58.2269)33 47.5 (46.8104,48.7511) 93 54.7 (55.209,57.2322)34 47.5 (46.7327,48.5775) 94 53.2 (53.9487,55.9513)35 47.6 (46.7787,48.4951) 95 52.1 (52.4799,54.4705)36 48.1 (46.9209,48.5243) 96 51.6 (51.3899,53.3621)37 49 (47.4256,48.9786) 97 51 (50.8869,52.8044)38 50 (48.2846,49.8587) 98 50.5 (50.3241,52.0986)39 51.1 (49.2058,50.8773) 99 50.4 (49.9002,51.4763)40 51.8 (50.227,52.0181) 100 51 (49.8186,51.3283)41 51.9 (50.8824,52.7808) 101 51.8 (50.3635,51.8993)42 51.7 (50.9531,52.9761) 102 52.4 (51.0874,52.6849)43 51.2 (50.6326,52.9901) 103 53 (51.648,53.2765)44 50 (50.1607,52.5467) 104 53.4 (52.2506,53.877)45 48.3 (49.0268,51.3689) 105 53.6 (52.6691,54.2897)46 47 (47.4483,49.6477) 106 53.7 (52.8775,54.5114)47 45.8 (46.2695,48.278) 107 53.8 (52.9643,54.6614)48 45.6 (45.1454,46.9959) 108 53.8 (53.0386,54.8095)49 46 (44.9302,46.7011) 109 53.8 (53.0356,54.827)50 46.9 (45.2597,47.05) 110 53.3 (53.0701,54.7998)51 47.8 (46.0215,47.9401) 111 53 (52.6695,54.2415)52 48.2 (46.8295,48.8568) 112 52.9 (52.4811,53.8501)53 48.3 (47.2082,49.2801) 113 53.4 (52.5396,53.5984)54 47.9 (47.3641,49.3829) 114 54.6 (53.0398,54.0281)55 47.2 (47.1897,48.8498) 115 56.4 (54.1858,55.1791)56 47.2 (46.5948,48.1211) 116 58 (55.865,56.971)57 48.1 (46.6137,48.1095) 117 59.4 (57.3065,58.6578)58 49.4 (47.4335,48.9997) 118 60.2 (58.6207,60.1363)59 50.6 (48.5751,50.3195) 119 60 (59.4107,61.0231)60 51.5 (49.6685,51.5449) 120 59.4 (59.2557,60.908)61 51.6 (50.5394,52.4787) 121 58.4 (58.7317,60.3743)62 51.2 (50.6805,52.6253) 122 57.6 (57.8116,59.4201)63 50.5 (50.3498,52.2703) 123 56.9 (57.0325,58.6387)

91

Table 5.23: Expected and Obtained Output for Variable x1 (Gas FurnaceData)(cont’d)


124 56.4 (56.3108,57.9641) 184 55.4 (53.8387,55.4754)125 56 (55.782,57.454) 185 55.9 (54.5705,56.3404)126 55.7 (55.3531,57.0517) 186 55.9 (55.072,56.8633)127 55.3 (55.0059,56.7683) 187 55.2 (55.1124,56.8811)128 55 (54.5531,56.4117) 188 54.4 (54.4761,56.2228)129 54.4 (54.1943,56.1618) 189 53.7 (53.7058,55.4835)130 53.7 (53.5531,55.6217) 190 53.6 (52.9936,54.843)131 52.8 (52.8298,54.9632) 191 53.6 (52.8426,54.7499)132 51.6 (51.9447,54.0873) 192 53.2 (52.8032,54.7311)133 50.6 (50.7909,52.8785) 193 52.5 (52.3895,54.3209)134 49.4 (49.8419,51.8442) 194 52 (51.7282,53.6223)135 48.8 (48.7352,50.5666) 195 51.4 (51.3133,53.071)136 48.5 (48.1679,49.8871) 196 51 (50.8301,52.3656)137 48.7 (47.856,49.5465) 197 50.9 (50.5087,51.8631)138 49.2 (48.0025,49.6993) 198 52.4 (50.5109,51.6313)139 49.8 (48.4484,50.1759) 199 53.5 (51.9911,53.0062)140 50.4 (48.9922,50.7698) 200 55.6 (53.0383,54.0555)141 50.7 (49.5491,51.387) 201 58 (55.027,56.1177)142 50.9 (49.8398,51.7123) 202 59.5 (57.3432,58.5484)143 50.7 (50.0551,51.9272) 203 60 (58.7713,60.0989)144 50.5 (49.9109,51.7285) 204 60.4 (59.2804,60.7087)145 50.4 (49.7845,51.4902) 205 60.5 (59.6833,61.2534)146 50.2 (49.7725,51.3236) 206 60.2 (59.7393,61.4997)147 50.4 (49.6281,51.0669) 207 59.7 (59.414,61.2816)148 51.2 (49.8176,51.2493) 208 59 (58.938,60.8204)149 52.3 (50.5501,52.0555) 209 57.6 (58.2799,60.1331)150 53.2 (51.5572,53.1595) 210 56.4 (56.9499,58.7359)151 53.9 (52.4002,54.0702) 211 55.2 (55.8103,57.5202)152 54.1 (53.0799,54.8015) 212 54.5 (54.6793,56.2814)153 54 (53.2963,55.0515) 213 54.1 (54.0086,55.4975)154 53.6 (53.2151,55.005) 214 54.1 (53.6351,55.0108)155 53.2 (52.8312,54.6668) 215 54.4 (53.618,54.9407)156 53 (52.4516,54.2879) 216 55.5 (53.8754,55.2122)157 52.8 (52.2587,54.087) 217 56.2 (54.8729,56.3112)158 52.3 (52.0524,53.8778) 218 57 (55.4892,57.0476)159 51.9 (51.5601,53.3748) 219 57.3 (56.2363,57.8683)160 51.6 (51.1619,52.9854) 220 57.4 (56.5543,58.21)161 51.6 (50.8341,52.7189) 221 57 (56.6806,58.3278)162 51.4 (50.7614,52.7554) 222 56.4 (56.3415,57.9522)163 51.2 (50.5401,52.5611) 223 55.9 (55.8004,57.3556)164 50.7 (50.3536,52.3393) 224 55.5 (55.3546,56.8424)165 50 (49.9569,51.7697) 225 55.3 (55.0014,56.3936)166 49.4 (49.3592,51.0001) 226 55.2 (54.8277,56.1372)167 49.3 (48.8432,50.3422) 227 55.4 (54.7132,56.0221)168 49.7 (48.7945,50.1613) 228 56 (54.8252,56.2675)169 50.6 (49.2012,50.4766) 229 56.5 (55.2989,56.9387)170 51.8 (50.0578,51.3157) 230 57.1 (55.7261,57.4705)171 53 (51.1819,52.5032) 231 57.3 (56.2829,58.0876)172 54 (52.2695,53.7636) 232 56.8 (56.5063,58.2944)173 55.3 (53.1568,54.8742) 233 55.6 (56.1206,57.7502)174 55.9 (54.3652,56.2652) 234 55 (55.065,56.5305)175 55.9 (54.9673,56.9098) 235 54.1 (54.5613,55.8893)176 54.6 (55.0252,56.9247) 236 54.3 (53.7328,54.9422)177 53.5 (53.8876,55.6446) 237 55.3 (53.9024,55.0719)178 52.4 (52.9206,54.5295) 238 56.4 (54.8303,56.0328)179 52.1 (51.9275,53.397) 239 57.2 (55.8345,57.0888)180 52.3 (51.6424,53.0078) 240 57.8 (56.6008,57.883)181 53 (51.8156,53.1411) 241 58.3 (57.1861,58.5299)182 53.8 (52.4362,53.7933) 242 58.6 (57.6348,59.1274)183 54.6 (53.1437,54.6099) 243 58.8 (57.9122,59.4879)

92

Table 5.23: Expected and Obtained Output for Variable x1 (Gas FurnaceData)(cont’d)


244 58.8 (58.109,59.7212) 270 50.8 (50.9549,52.379)245 58.6 (58.1109,59.7556) 271 51.2 (50.3897,51.741)246 58 (57.9411,59.5608) 272 52 (50.7192,52.0609)247 57.4 (57.4025,58.9545) 273 52.8 (51.4149,52.8093)248 57 (56.8425,58.355) 274 53.8 (52.1152,53.6114)249 56.4 (56.4217,57.9883) 275 54.5 (53.0503,54.6378)250 56.3 (55.7646,57.4423) 276 54.9 (53.7246,55.3744)251 56.4 (55.5694,57.4007) 277 54.9 (54.1179,55.8122)252 56.4 (55.5854,57.5488) 278 54.8 (54.1374,55.8655)253 56 (55.5572,57.5333) 279 54.4 (54.0495,55.811)254 55.2 (55.1875,57.1228) 280 53.7 (53.6638,55.4528)255 54 (54.4449,56.3268) 281 53.3 (52.9946,54.7693)256 53 (53.2906,55.1621) 282 52.8 (52.6486,54.335)257 52 (52.2797,54.2115) 283 52.6 (52.2255,53.7628)258 51.6 (51.2624,53.2382) 284 52.6 (52.046,53.505)259 51.6 (50.8351,52.8053) 285 53 (52.0402,53.4817)260 51.1 (50.8561,52.7115) 286 54.3 (52.3908,53.8677)261 50.4 (50.4422,52.0865) 287 56 (53.5904,55.1657)262 50 (49.7728,51.3741) 288 57 (55.1623,56.879)263 50 (49.3443,51.0296) 289 58 (56.0951,57.9077)264 52 (49.2532,51.0844) 290 58.6 (57.0698,58.96)265 54 (51.1137,53.0379) 291 58.5 (57.6952,59.6177)266 55.1 (53.0224,54.965) 292 58.3 (57.6443,59.548)267 54.5 (54.1136,55.9994) 293 57.8 (57.4993,59.3793)268 52.8 (53.6747,55.4242) 294 57.3 (57.0613,58.8947)269 51.4 (52.2063,53.753) 295 57 (56.6094,58.3731)

93

-3

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3

x0,t

x0,t-1


-3

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3

x0,t

x0,t-4


-3

-2

-1

0

1

2

3

44 46 48 50 52 54 56 58 60 62

x0,t

x1,t-1


44

46

48

50

52

54

56

58

60

62

-3 -2 -1 0 1 2 3

x1,t

x0,t-1


44

46

48

50

52

54

56

58

60

62

44 46 48 50 52 54 56 58 60 62

x1,t

x1,t-1


44

46

48

50

52

54

56

58

60

62

44 46 48 50 52 54 56 58 60 62

x1,t

x1,t-4


Figure 5.7: Linearity Figures for Gas Furnace Data

94

5.1.2.6 Grain Price Data

Table 5.24: Fuzzy equations for Grain Price Data


x0 (1.021, 0.01166)x0,t−1 + (0, 0.1516)x1,t−1

x1 (0.4969, 0)x1,t−1 + (0.3486, 0)x1,t−2 + (0.02644, 0.008703)x0,t−1

x2 (0.4533, 0.04607)x2,t−1 + (0.2456, 0)x1,t−1 + (0.07505, 0.006236)x0,t−1

x3 (1.036, 0)x3,t−1 + (0, 0.08444)x3,t−2

Monthly US Grain Price Data contains the price in dollars per 100-pound

sack for wheat flour (x0) and per bushel for corn (x1), wheat (x2) and rye(x3).

The data is obtained between January 1961 and October 1972 and it can be seen

in Table A.9.


can be seen. In the figure, x0 has an obvious linear relationship with x0,t−1. As it

can be seen in the fuzzy equation in Table 5.24, the coefficient width for x0,t−1 is

close to zero and the center is larger than x1,t−1. In the equation for the variable

x3, the largest coefficient center is for the lagged variable x3,t−1.

The variables x0 (0.829787) and x3 (0.842857) are modeled more accurately

than x1 (0.721429) and x2 (0.659574) as can be seen in Table 5.52. The output

results for all four variables can be seen in Figures 5.25, 5.26, 5.27, 5.28. The

values out of the intervals are very close to them.

95

Table 5.25: Expected and Obtained Output for Variable x0 (Grain Prices Data)


1 5.46 (5.40458,5.87604) 72 7.35 (7.38044,8.02427)2 5.5 (5.28835,5.74968) 73 7.48 (7.11893,7.73995)3 5.53 (5.32709,5.7918) 74 7.55 (7.24484,7.87684)4 5.63 (5.35615,5.82339) 75 7.27 (7.31264,7.95056)5 5.65 (5.453,5.92869) 76 7.43 (7.04145,7.6557)6 5.7 (5.47237,5.94976) 77 7.34 (7.19641,7.82419)7 5.68 (5.5208,6.00241) 78 7.4 (7.10924,7.72942)8 5.75 (5.50143,5.98135) 79 7.29 (7.16736,7.7926)9 5.66 (5.56923,6.05506) 80 7.27 (7.06082,7.67676)10 5.7 (5.48206,5.96029) 81 7.24 (7.04145,7.6557)11 5.75 (5.5208,6.00241) 82 7.18 (7.01239,7.62411)12 5.75 (5.56923,6.05506) 83 7.21 (6.95427,7.56093)13 5.78 (5.56923,6.05506) 84 7.18 (6.98333,7.59252)14 5.72 (5.59829,6.08665) 85 7.04 (6.95427,7.56093)15 5.8 (5.54017,6.02347) 86 6.94 (6.81868,7.4135)16 5.95 (5.61766,6.10771) 87 6.76 (6.72182,7.3082)17 6.09 (5.76294,6.26567) 88 6.78 (6.54748,7.11865)18 6.25 (5.89854,6.4131) 89 6.77 (6.56685,7.13971)19 6.37 (6.05351,6.58159) 90 6.73 (6.55716,7.12918)20 6.45 (6.16974,6.70795) 91 6.72 (6.51842,7.08705)21 6.44 (6.24722,6.7922) 92 6.85 (6.50874,7.07652)22 6.46 (6.23754,6.78167) 93 6.89 (6.63465,7.21342)23 6.39 (6.25691,6.80273) 94 6.9 (6.67339,7.25554)24 6.03 (6.18911,6.72902) 95 6.85 (6.68308,7.26607)25 6.05 (5.84043,6.34992) 96 6.85 (6.63465,7.21342)26 6.02 (5.8598,6.37098) 97 6.83 (6.63465,7.21342)27 6.15 (5.83074,6.33939) 98 6.76 (6.61528,7.19236)28 6.09 (5.95666,6.47628) 99 6.75 (6.54748,7.11865)29 6.05 (5.89854,6.4131) 100 6.75 (6.53779,7.10812)30 5.7 (5.8598,6.37098) 101 6.75 (6.53779,7.10812)31 5.28 (5.5208,6.00241) 102 6.92 (6.53779,7.10812)32 5.76 (5.11401,5.56013) 103 6.86 (6.70245,7.28713)33 5.98 (5.57892,6.06559) 104 6.78 (6.64433,7.22395)34 6.41 (5.792,6.29726) 105 6.89 (6.56685,7.13971)35 6.83 (6.20848,6.75008) 106 6.95 (6.67339,7.25554)36 6.45 (6.61528,7.19236) 107 7.14 (6.73151,7.31873)37 6.9 (6.24722,6.7922) 108 7.14 (6.91553,7.51881)38 6.64 (6.68308,7.26607) 109 7.17 (6.91553,7.51881)39 6.68 (6.43125,6.99228) 110 7.19 (6.94459,7.5504)40 6.7 (6.46999,7.0344) 111 7.19 (6.96396,7.57146)41 6.53 (6.48936,7.05546) 112 7.19 (6.96396,7.57146)42 7.09 (6.32471,6.87644) 113 7.2 (6.96396,7.57146)43 6.7 (6.8671,7.46615) 114 7.16 (6.97365,7.58199)44 6.73 (6.48936,7.05546) 115 7.21 (6.9349,7.53987)45 6.84 (6.51842,7.08705) 116 7.2 (6.98333,7.59252)46 6.9 (6.62496,7.20289) 117 7.35 (6.97365,7.58199)47 6.9 (6.68308,7.26607) 118 7.43 (7.11893,7.73995)48 6.68 (6.68308,7.26607) 119 7.5 (7.19641,7.82419)49 6.7 (6.46999,7.0344) 120 7.45 (7.26421,7.89791)50 6.7 (6.48936,7.05546) 121 7.45 (7.21579,7.84525)51 6.7 (6.48936,7.05546) 122 7.45 (7.21579,7.84525)52 6.7 (6.48936,7.05546) 123 7.51 (7.21579,7.84525)53 6.9 (6.48936,7.05546) 124 7.58 (7.2739,7.90844)54 7.07 (6.68308,7.26607) 125 7.55 (7.3417,7.98215)55 7.1 (6.84773,7.44509) 126 7.52 (7.31264,7.95056)56 7.1 (6.87679,7.47668) 127 7.48 (7.28359,7.91897)57 7.2 (6.87679,7.47668) 128 7.48 (7.24484,7.87684)58 7.28 (6.97365,7.58199) 129 7.48 (7.24484,7.87684)59 7.21 (7.05113,7.66623) 130 7.48 (7.24484,7.87684)60 7.22 (6.98333,7.59252) 131 7.48 (7.24484,7.87684)61 7.22 (6.99302,7.60305) 132 7.28 (7.24484,7.87684)62 7.22 (6.99302,7.60305) 133 7.15 (7.05113,7.66623)63 7.19 (6.99302,7.60305) 134 7.15 (6.92522,7.52934)64 7.2 (6.96396,7.57146) 135 7.15 (6.92522,7.52934)65 7.63 (6.97365,7.58199) 136 7.07 (6.92522,7.52934)66 8.07 (7.39013,8.0348) 137 7.07 (6.84773,7.44509)67 8.09 (7.81629,8.49815) 138 7.07 (6.84773,7.44509)68 7.92 (7.83567,8.51921) 139 7.66 (6.84773,7.44509)69 7.72 (7.67101,8.34019) 140 8.39 (7.41918,8.06639)70 7.73 (7.4773,8.12958) 141 8.57 (8.12623,8.83512)71 7.62 (7.48698,8.14011)

96

Table 5.26: Expected and Obtained Output for Variable x1 (Grain Price Data)


2 1.18 (28.728,84.4087) 72 1.44 (30.9995,91.0828)3 1.09 (28.4341,83.545) 73 1.39 (32.5495,95.637)4 1.15 (30.9461,90.9258) 74 1.42 (32.1487,94.4592)5 1.13 (28.6746,84.2516) 75 1.38 (31.0797,91.3184)6 1.18 (30.0642,88.3347) 76 1.39 (32.2823,94.8518)7 1.15 (28.5944,84.0161) 77 1.38 (31.7478,93.2814)8 1.11 (34.6874,101.919) 78 1.33 (31.534,92.6532)9 1.11 (31.2133,91.711) 79 1.21 (30.9995,91.0828)10 1.14 (32.2021,94.6162) 80 1.2 (31.828,93.517)11 1.12 (34.6874,101.919) 81 1.2 (29.3694,86.2931)12 1.09 (35.2753,103.646) 82 1.08 (31.3469,92.1036)13 1.11 (34.6072,101.683) 83 1.18 (30.4651,89.5124)14 1.11 (34.6874,101.919) 84 1.15 (29.8771,87.785)15 1.12 (34.5538,101.526) 85 1.17 (30.6788,90.1406)16 1.16 (33.4848,98.3852) 86 1.14 (30.8392,90.6117)17 1.15 (32.8167,96.4222) 87 1.16 (31.8012,93.4384)18 1.12 (32.8702,96.5792) 88 1.21 (31.5073,92.5747)19 1.14 (34.1529,100.348) 89 1.16 (30.5987,89.905)20 1.13 (30.0642,88.3347) 90 1.15 (30.1978,88.7273)21 1.12 (30.5987,89.905) 91 1.08 (28.6746,84.2516)22 1.1 (29.9306,87.9421) 92 1.08 (30.0642,88.3347)23 1.17 (30.6788,90.1406) 93 1.1 (27.953,82.1316)24 1.21 (31.1064,91.3969) 94 1.17 (29.9039,87.8635)25 1.21 (32.3357,95.0088) 95 1.15 (30.5987,89.905)26 1.2 (33.6185,98.7778) 96 1.19 (31.24,91.7895)27 1.21 (32.8167,96.4222) 97 1.17 (31.2133,91.711)28 1.23 (31.9349,93.831) 98 1.17 (32.2021,94.6162)29 1.3 (32.9504,96.8148) 99 1.22 (32.0952,94.3022)30 1.34 (31.9616,93.9096) 100 1.31 (32.4693,95.4014)31 1.34 (32.603,95.794) 101 1.32 (32.4693,95.4014)32 1.36 (32.2021,94.6162) 102 1.3 (33.1374,97.3644)33 1.22 (31.6676,93.0458) 103 1.29 (32.2021,94.6162)34 1.18 (36.2107,106.394) 104 1.27 (31.6676,93.0458)35 1.23 (38.2951,112.519) 105 1.27 (27.6591,81.2679)36 1.26 (37.9477,111.498) 106 1.16 (28.1935,82.8383)37 1.23 (38.0279,111.734) 107 1.18 (29.5297,86.7643)38 1.24 (39.0968,114.874) 108 1.26 (29.797,87.5495)39 1.26 (36.7719,108.043) 109 1.26 (30.3314,89.1199)40 1.29 (34.821,102.311) 110 1.25 (30.3314,89.1199)41 1.26 (34.2064,100.505) 111 1.29 (30.3314,89.1199)42 1.23 (33.7521,99.1704) 112 1.32 (30.5987,89.905)43 1.25 (33.8055,99.3274) 113 1.36 (31.1332,91.4754)44 130 (33.2711,97.757) 114 1.39 (31.1332,91.4754)45 1.25 (32.3357,95.0088) 115 1.37 (32.4693,95.4014)46 1.19 (34.2064,100.505) 116 1.45 (28.9953,85.1939)47 1.28 (33.0305,97.0503) 117 1.44 (28.4608,83.6235)48 1.29 (32.9504,96.8148) 118 1.42 (28.9953,85.1939)49 1.31 (31.6676,93.0458) 119 1.33 (28.9953,85.1939)50 1.34 (31.6409,92.9673) 120 1.39 (30.5987,89.905)51 1.35 (30.9995,91.0828) 121 1.4 (31.1332,91.4754)52 1.37 (31.8012,93.4384) 122 1.36 (30.8659,90.6902)53 1.36 (31.2668,91.868) 123 1.33 (31.1332,91.4754)54 1.33 (31.3469,92.1036) 124 1.31 (30.3314,89.1199)55 1.29 (29.3961,86.3717) 125 1.38 (30.5987,89.905)56 1.31 (26.7505,78.5982) 126 1.3 (31.6676,93.0458)57 1.27 (29.0754,85.4294) 127 1.29 (31.1332,91.4754)58 1.17 (29.6901,87.2354) 128 1.17 (27.6591,81.2679)59 1.24 (29.5297,86.7643) 129 1.12 (23.3833,68.7047)60 1.32 (30.0642,88.3347) 130 1.09 (25.2539,74.2011)61 1.32 (31.534,92.6532) 131 1.26 (25.7884,75.7715)62 1.26 (33.2176,97.6) 132 1.23 (23.9177,70.2751)63 1.3 (32.7366,96.1866) 133 1.21 (26.1893,76.9493)64 1.32 (30.8659,90.6902) 134 1.23 (27.5254,80.8753)65 1.31 (33.8857,99.563) 135 1.25 (28.0599,82.4457)66 1.42 (30.8125,90.5332) 136 1.28 (27.5254,80.8753)67 1.45 (30.1978,88.7273) 137 1.28 (27.7927,81.6605)68 1.46 (34.2064,100.505) 138 1.3 (28.3272,83.2309)69 1.4 (33.3512,97.9926) 139 1.29 (26.4565,77.7345)70 1.35 (33.2176,97.6) 140 1.39 (26.1893,76.9493)71 1.44 (30.9461,90.9258) 141 1.31 (25.6548,75.3789)

97

Table 5.27: Expected and Obtained Output for Variable x2 (Grain Price Data)


1 1.58 (1.35755,1.54758) 72 1.8 (1.62349,1.88299)2 1.56 (1.32796,1.51391) 73 1.68 (1.58791,1.83821)3 1.39 (1.33743,1.52473) 74 1.81 (1.53117,1.7859)4 1.28 (1.24161,1.42994) 75 1.8 (1.60723,1.86435)5 1.28 (1.20387,1.3956) 76 1.67 (1.57984,1.82743)6 1.36 (1.19557,1.38798) 77 1.57 (1.52588,1.77891)7 1.46 (1.27803,1.47214) 78 1.48 (1.46749,1.71745)8 1.45 (1.30262,1.49605) 79 1.42 (1.4331,1.68511)9 1.47 (1.30899,1.5048) 80 1.45 (1.38021,1.62848)10 1.51 (1.32897,1.52172) 81 1.49 (1.40648,1.65407)11 1.53 (1.35506,1.54917) 82 1.44 (1.41817,1.66473)12 1.52 (1.36381,1.55963) 83 1.5 (1.38584,1.63036)13 1.52 (1.35946,1.55528) 84 1.52 (1.42241,1.66795)14 1.54 (1.36062,1.55746) 85 1.56 (1.43119,1.6757)15 1.59 (1.35928,1.55408) 86 1.51 (1.44752,1.68727)16 1.65 (1.38471,1.58223) 87 1.46 (1.41442,1.65076)17 1.63 (1.4245,1.62713) 88 1.47 (1.37185,1.60206)18 1.62 (1.43267,1.64007) 89 1.26 (1.37545,1.60634)19 1.57 (1.41178,1.62463) 90 1.29 (1.26232,1.49288)20 1.59 (1.39905,1.61598) 91 1.19 (1.28338,1.51257)21 1.55 (1.40984,1.62949) 92 1.18 (1.22005,1.4489)22 1.59 (1.39457,1.61388) 93 1.28 (1.23681,1.47009)23 1.58 (1.41824,1.63824) 94 1.36 (1.29286,1.52751)24 1.6 (1.41669,1.63431) 95 1.34 (1.33678,1.57176)25 1.63 (1.41046,1.61581) 96 1.43 (1.32346,1.55674)26 1.64 (1.42118,1.62721) 97 1.4 (1.37386,1.60713)27 1.68 (1.41823,1.62324) 98 1.35 (1.35716,1.58976)28 1.63 (1.45323,1.66267) 99 1.36 (1.33048,1.5607)29 1.38 (1.41827,1.62567) 100 1.4 (1.33468,1.56456)30 1.3 (1.29783,1.50386) 101 1.33 (1.35858,1.58846)31 1.3 (1.23243,1.42654) 102 1.3 (1.31829,1.54817)32 1.47 (1.20042,1.38023) 103 1.28 (1.31168,1.54734)33 1.64 (1.34576,1.54192) 104 1.33 (1.27151,1.50513)34 1.69 (1.45731,1.66096) 105 1.33 (1.29399,1.52488)35 1.69 (1.50859,1.72688) 106 1.42 (1.31023,1.54487)36 1.78 (1.53761,1.7702) 107 1.48 (1.35996,1.59664)37 1.75 (1.56275,1.78241) 108 1.5 (1.40563,1.64878)38 1.48 (1.56337,1.79835) 109 1.52 (1.41538,1.65854)39 1.66 (1.40121,1.62734) 110 1.54 (1.42717,1.67135)40 1.54 (1.48768,1.71517) 111 1.54 (1.44004,1.6849)41 1.43 (1.42752,1.65569) 112 1.55 (1.44355,1.68841)42 1.46 (1.36268,1.58506) 113 1.4 (1.44843,1.69329)43 1.44 (1.4118,1.65325) 114 1.43 (1.38472,1.62992)44 1.52 (1.36944,1.59761) 115 1.45 (1.37381,1.61765)45 1.5 (1.42279,1.65198) 116 1.68 (1.38345,1.62899)46 1.56 (1.41277,1.6457) 117 1.67 (1.49847,1.74367)47 1.59 (1.44558,1.68056) 118 1.74 (1.50377,1.75408)48 1.57 (1.45178,1.68676) 119 1.63 (1.55388,1.80691)49 1.58 (1.42692,1.65441) 120 1.72 (1.50848,1.7639)50 1.56 (1.42894,1.65712) 121 1.74 (1.54724,1.80095)51 1.56 (1.42446,1.65263) 122 1.68 (1.55875,1.81246)52 1.46 (1.42095,1.64912) 123 1.66 (1.52421,1.77793)53 1.45 (1.37269,1.60086) 124 1.66 (1.52028,1.77604)54 1.43 (1.36856,1.60355) 125 1.63 (1.53206,1.7902)55 1.53 (1.35296,1.59373) 126 1.44 (1.51188,1.76899)56 1.54 (1.41905,1.66084) 127 1.31 (1.39433,1.65043)57 1.53 (1.42796,1.66976) 128 1.33 (1.30011,1.55484)58 1.66 (1.42882,1.67402) 129 1.39 (1.32216,1.57689)59 1.74 (1.50117,1.74909) 130 1.49 (1.35494,1.60967)60 1.71 (1.5451,1.79064) 131 1.6 (1.39142,1.64616)61 1.75 (1.54221,1.78809) 132 1.58 (1.46001,1.71474)62 1.67 (1.55856,1.80444) 133 1.52 (1.44546,1.69339)63 1.65 (1.50724,1.75312) 134 1.58 (1.41089,1.65438)64 1.67 (1.5153,1.76016) 135 1.68 (1.43664,1.68014)65 1.77 (1.50554,1.75073) 136 1.67 (1.48718,1.73067)66 1.9 (1.57945,1.83929) 137 1.36 (1.48039,1.72116)67 1.89 (1.69905,1.97388) 138 1.47 (1.31688,1.55765)68 1.92 (1.68991,1.96542) 139 1.6 (1.36878,1.60955)69 1.69 (1.69213,1.96185) 140 1.86 (1.46871,1.72957)70 1.75 (1.55145,1.81435) 141 1.92 (1.65032,1.93605)71 1.83 (1.58174,1.84499)

98

Table 5.28: Expected and Obtained Output for Variable x3 (Grain Prices Data)


2 1.158 (1.01101,1.19256) 72 1.203 (1.1633,1.35921)3 1.073 (1.10928,1.28897) 73 1.163 (1.14287,1.34858)4 1.125 (1.01332,1.20889) 74 1.208 (1.10271,1.30589)5 1.07 (1.07434,1.25556) 75 1.188 (1.15269,1.34911)6 1.298 (1.013,1.203) 76 1.18 (1.12818,1.3322)7 1.168 (1.25374,1.43445) 77 1.16 (1.12159,1.32223)8 1.205 (1.09987,1.31909) 78 1.191 (1.10155,1.30084)9 1.298 (1.14916,1.34643) 79 1.099 (1.13534,1.33125)10 1.32 (1.24234,1.44585) 80 1.173 (1.03746,1.2386)11 1.295 (1.25727,1.47649) 81 1.14 (1.12185,1.30746)12 1.298 (1.22952,1.45246) 82 1.118 (1.08143,1.27954)13 1.293 (1.23474,1.45345) 83 1.148 (1.06144,1.25397)14 1.253 (1.22931,1.44853) 84 1.154 (1.09436,1.28318)15 1.228 (1.18831,1.40669) 85 1.19 (1.09804,1.29192)16 1.23 (1.1658,1.37742) 86 1.179 (1.13481,1.32971)17 1.278 (1.16998,1.37738) 87 1.145 (1.12038,1.32136)18 1.125 (1.21952,1.42725) 88 1.13 (1.0861,1.28522)19 1.145 (1.05703,1.27287) 89 1.073 (1.07344,1.26682)20 1.12 (1.09066,1.28066) 90 1.125 (1.01568,1.20653)21 1.148 (1.06309,1.25646) 91 1.046 (1.07434,1.25556)22 1.164 (1.09419,1.28335) 92 1.119 (0.988147,1.17815)23 1.21 (1.10839,1.30228) 93 1.145 (1.07041,1.24707)24 1.258 (1.15468,1.35126) 94 1.169 (1.09117,1.28016)25 1.228 (1.2005,1.40485) 95 1.168 (1.11383,1.3072)26 1.195 (1.16538,1.37784) 96 1.205 (1.11076,1.3082)27 1.233 (1.13374,1.34114) 97 1.201 (1.14916,1.34643)28 1.196 (1.17588,1.3777) 98 1.215 (1.1419,1.34541)29 1.22 (1.13435,1.34259) 99 1.215 (1.15673,1.35957)30 1.205 (1.16233,1.36432) 100 1.24 (1.15555,1.36075)31 1.185 (1.14477,1.35082) 101 1.205 (1.18144,1.38664)32 1.355 (1.12533,1.32884) 102 1.185 (1.14308,1.35251)33 1.433 (1.30305,1.50319) 103 1.035 (1.12533,1.32884)34 1.42 (1.36947,1.59831) 104 1.055 (0.971689,1.17182)35 1.423 (1.34942,1.59144) 105 1.105 (1.00507,1.17987)36 1.463 (1.35362,1.59345) 106 1.115 (1.05515,1.23333)37 1.376 (1.39479,1.63512) 107 1.135 (1.06129,1.24791)38 1.303 (1.30132,1.54841) 108 1.135 (1.08115,1.26946)39 1.28 (1.23308,1.46547) 109 1.135 (1.07946,1.27115)40 1.263 (1.21543,1.43549) 110 1.145 (1.07946,1.27115)41 1.265 (1.19976,1.41594) 111 1.165 (1.08982,1.28151)42 1.245 (1.20327,1.41658) 112 1.165 (1.10968,1.30306)43 1.21 (1.18239,1.39604) 113 1.215 (1.10799,1.30475)44 1.28 (1.14784,1.3581) 114 1.085 (1.15977,1.35653)45 1.236 (1.22328,1.42764) 115 1.065 (1.02093,1.22613)46 1.233 (1.17181,1.38798) 116 1.085 (1.0112,1.19444)47 1.185 (1.17241,1.38116) 117 1.085 (1.0336,1.21347)48 1.184 (1.12296,1.3312) 118 1.145 (1.03191,1.21515)49 1.16 (1.12598,1.32611) 119 1.165 (1.09404,1.27729)50 1.19 (1.10121,1.30118) 120 1.155 (1.10968,1.30306)51 1.17 (1.1343,1.33022) 121 1.165 (1.09764,1.2944)52 1.173 (1.11106,1.31204) 122 1.135 (1.10884,1.30391)53 1.1 (1.11586,1.31346) 123 1.145 (1.07693,1.27369)54 1.001 (1.04001,1.23812) 124 1.185 (1.08982,1.28151)55 1.088 (0.94366,1.12944) 125 1.165 (1.13039,1.32377)56 1.111 (1.04211,1.21117) 126 1.035 (1.10631,1.30644)57 1.105 (1.05858,1.24233) 127 0.875 (0.973378,1.17013)58 1.125 (1.05042,1.23806) 128 0.945 (0.818674,0.993475)59 1.18 (1.07164,1.25826) 129 0.965 (0.904671,1.05245)60 1.243 (1.12691,1.31691) 130 0.895 (0.91947,1.07907)61 1.225 (1.1875,1.38679) 131 0.98 (0.845295,1.00827)62 1.155 (1.16354,1.37347) 132 1.03 (0.939225,1.09038)63 1.268 (1.09257,1.29946) 133 1.05 (0.983823,1.14933)64 1.153 (1.2155,1.41056) 134 1.03 (1.00031,1.17427)65 1.13 (1.08687,1.30102) 135 1.04 (0.977912,1.15525)66 1.28 (1.07277,1.26749) 136 1.06 (0.989956,1.16391)67 1.248 (1.23003,1.42088) 137 0.99 (1.00982,1.18547)68 1.243 (1.18423,1.40041) 138 0.98 (0.935647,1.11467)69 1.158 (1.18176,1.39253) 139 0.96 (0.931203,1.0984)70 1.16 (1.09416,1.30409) 140 0.99 (0.911337,1.07685)71 1.218 (1.10341,1.29898) 141 1.01 (0.944091,1.10623)

99

5

5.5

6

6.5

7

7.5

8

8.5

9

5 5.5 6 6.5 7 7.5 8 8.5

x0,t

x0,t-1


5

5.5

6

6.5

7

7.5

8

8.5

9

1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5

x0,t

x1,t-1


1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

5 5.5 6 6.5 7 7.5 8 8.5

x1,t

x0,t-1


1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5

x1,t

x1,t-1


1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5

x1,t

x1,t-2


1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

5 5.5 6 6.5 7 7.5 8 8.5

x2,t

x0,t-1


1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5

x2,t

x1,t-1


1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

x2,t

x2,t-1


Figure 5.8: Linearity Figures for Monthly Grain Price Data

100

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

x3,t

x3,t-1


0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

x3,t

x3,t-2


Figure 5.8: Linearity Figures for Monthly Grain Price Data(cont’d)

101

5.1.2.7 Housing Starts and Sold Data

Table 5.29: Fuzzy equations for Housing Data


x0 (1.528, 0.1255)x1,t−1 + (0, 0.1022)x1,t−2 + (0.09332, 0)x0,t−1

x1 (1.007, 0.1316)x1,t−1 + (0.01176, 0)x1,t−2

Monthly US housing starts and sold data contains two variables, which are

the prices that housing starts (x0) and the prices that housing sold (x1). The

data is collected for the period January 1965 and December 1974 and it can be

seen in Table A.10.


can be seen. Lagged variable x1,t−1 has the largest coefficient center value both in

the equation for x0 and x1 in the fuzzy equation in Table 5.29. This is reflected

to the linearity figures.

The output results can be seen in Figures 5.30 and 5.31. The outputs obtained

for both of the variables show large width values. The accuracy rates are 0.644068

and 0.686441 for x0 and x1 respectively which are low values.

102

Table 5.30: Expected and Obtained Output for Variable x0 (Housing Data)


2 82.2 (62.2098,81.0201) 61 41.4 (47.9239,62.1803)3 100.9 (77.481,99.7764) 62 61.9 (41.0468,55.2754)4 98.4 (72.6978,95.8297) 63 73.8 (53.2854,68.2481)5 97.4 (79.8837,103.452) 64 74.8 (62.0925,79.992)6 96.5 (83.4854,108.829) 65 83 (62.9745,82.3517)7 88.8 (74.6824,99.134) 66 75.5 (65.0395,84.8721)8 80.9 (84.3915,109.373) 67 77.3 (64.2374,84.2745)9 85.8 (68.9182,92.8215) 68 76 (70.0135,91.0543)10 72.4 (64.7897,85.6446) 69 79.4 (65.2772,86.383)11 61.2 (61.144,80.6792) 70 67.4 (64.4991,84.7406)12 46.6 (53.2931,71.1648) 71 69 (57.8733,76.9067)13 50.4 (59.452,77.5559) 72 54.9 (54.2254,71.6882)14 83.2 (60.6975,80.0747) 73 58.3 (64.4327,83.2894)15 94.3 (77.6766,99.7675) 74 91.6 (69.5403,91.0364)16 84.7 (72.0819,95.2138) 75 116 (90.4655,116.041)17 79.8 (71.595,93.909) 76 115.6 (91.4135,119.648)18 69.1 (58.5193,78.575) 77 116.9 (85.768,112.998)19 69.4 (58.4409,76.6564) 78 107.7 (87.7003,114.364)20 59.4 (52.8607,70.0725) 79 111.7 (93.7497,121.872)21 53.5 (42.5221,57.1596) 80 102.1 (90.8077,119.451)22 50.2 (45.4913,59.1994) 81 102.9 (73.2918,98.5148)23 38 (37.9686,50.831) 82 92.9 (77.3973,100.669)24 40.2 (33.1351,44.2225) 83 80.4 (73.4556,96.6339)25 40.3 (42.0594,54.0389) 84 76.2 (64.0813,85.3451)26 66.6 (45.6615,59.6205) 85 76.3 (74.117,95.9105)27 79.8 (60.4275,77.2582) 86 111.4 (77.6169,101.594)28 87.3 (64.9454,84.369) 87 119.8 (88.998,115.094)29 87.6 (72.3488,93.6405) 88 135.2 (96.1787,124.757)30 82.3 (69.0615,90.8737) 89 131.9 (95.7026,125.052)31 83.7 (67.3694,88.5217) 90 119.1 (94.0948,122.989)32 78.2 (69.0043,90.2031) 91 131.3 (93.0026,121.692)33 81.7 (62.7806,83.1802) 92 120.5 (106.759,137.707)34 69.1 (66.3203,86.4038) 93 117 (89.409,119.437)35 47 (49.5175,67.2498) 94 97.4 (95.8152,124.598)36 45.2 (44.3735,59.1039) 95 73.2 (73.9485,100.036)37 55.4 (50.1204,65.2411) 96 77.1 (67.5133,89.7344)38 79.3 (61.8797,79.8257) 97 73.6 (78.1005,101.26)39 98 (67.4983,87.8328) 98 105.1 (82.6664,108.261)40 86.8 (68.9367,89.8846) 99 120.5 (96.4134,124.833)41 81.4 (63.6854,83.8805) 100 131.6 (92.8265,122.129)42 86.4 (60.6841,79.7639) 101 114.8 (95.5711,124.511)43 82.5 (65.5613,84.985) 102 114.7 (88.2929,116.434)44 80.1 (69.0968,89.8866) 103 106.8 (78.8782,104.445)45 85.6 (60.1538,80.0515) 104 84.5 (77.4546,101.339)46 64.8 (59.8784,78.2984) 105 86 (64.2594,85.9322)47 53.8 (46.8233,63.0314) 106 70.5 (59.6091,78.6425)48 51.3 (46.6146,61.187) 107 46.8 (52.9634,70.1752)49 47.9 (49.1854,64.2596) 108 43.3 (39.9442,54.3308)50 71.9 (57.0759,74.0646) 109 57.6 (51.6519,66.4101)51 85 (62.9083,81.8766) 110 77.3 (60.5807,78.4802)52 91.3 (62.4221,81.7528) 111 102.3 (77.2282,99.1147)53 82.7 (64.5143,83.8915) 112 96.4 (78.4367,102.572)54 73.5 (65.0115,84.8441) 113 99.6 (80.6902,105.328)55 69.5 (57.0405,75.823) 114 90.9 (70.97,94.2599)56 71.5 (58.5805,76.5915) 115 79.8 (69.4717,91.0794)57 68 (48.8505,65.3096) 116 73.4 (62.9299,83.3295)58 55.1 (47.8375,62.6144) 117 69.5 (57.1334,75.7114)59 42.8 (45.3339,59.6554) 118 57.9 (48.7661,65.0207)60 33.4 (40.0821,53.4464) 119 41 (44.0909,58.3659)

103

Table 5.31: Expected and Obtained Output for Variable x1 (Housing Data)


2 53 (38.9859,50.5639) 61 29 (30.1094,39.0561)3 49 (46.9394,60.8857) 62 36 (25.8006,33.4316)4 54 (43.5418,56.4355) 63 42 (31.8729,41.3459)5 57 (47.8741,62.0835) 64 43 (37.2106,48.2624)6 51 (50.5606,65.5594) 65 44 (38.1571,49.472)7 58 (45.3406,58.7606) 66 44 (39.0447,50.6227)8 48 (51.4012,66.6631) 67 48 (39.0565,50.6345)9 44 (42.7247,55.3553) 68 45 (42.56,55.1906)10 42 (39.1035,50.6816) 69 44 (39.9794,51.8206)11 37 (37.3047,48.3565) 70 40 (39.0682,50.6463)12 42 (32.9018,42.6378) 71 37 (35.5529,46.0784)13 43 (37.2224,48.2741) 72 45 (32.8782,42.6143)14 53 (38.1571,49.472) 73 49 (39.85,51.6912)15 49 (46.9276,60.8739) 74 62 (43.4476,56.3414)16 49 (43.5418,56.4355) 75 62 (54.8812,71.1957)17 40 (43.4947,56.3884) 76 58 (55.0341,71.3486)18 40 (35.6118,46.1373) 77 59 (51.5306,66.7925)19 36 (35.5059,46.0314) 78 64 (52.3594,67.8845)20 29 (32.0024,41.4753) 79 62 (56.7506,73.5914)21 31 (25.8241,33.4551) 80 50 (55.0576,71.3722)22 26 (27.4935,35.6508) 81 52 (44.5235,57.6804)23 23 (23.1376,29.9792) 82 50 (46.1341,59.8173)24 29 (20.4512,26.5033) 83 44 (44.4059,57.5627)25 32 (25.6712,33.3022) 84 51 (39.1271,50.7051)26 41 (28.3694,36.7898) 85 54 (45.1876,58.6076)27 44 (36.2876,47.0763) 86 60 (47.8976,62.1071)28 49 (39.0212,50.5992) 87 65 (53.1882,68.9765)29 47 (43.4359,56.3296) 88 64 (57.6382,74.7422)30 46 (41.7429,54.1104) 89 63 (56.8212,73.662)31 47 (40.8435,52.9478) 90 63 (55.9335,72.5112)32 43 (41.7076,54.0751) 91 72 (55.9218,72.4994)33 45 (38.2159,49.5308) 92 61 (63.8047,82.7506)34 34 (39.9206,51.7618) 93 65 (54.2759,70.3273)35 31 (30.3094,39.2561) 94 51 (57.65,74.7539)36 35 (27.5524,35.7096) 95 47 (45.4347,58.8547)37 43 (31.0206,40.2304) 96 54 (41.7665,54.1339)38 46 (38.0747,49.3896) 97 58 (47.8506,62.06)39 46 (40.7965,52.9008) 98 66 (51.4365,66.6984)40 43 (40.8318,52.9361) 99 63 (58.4906,75.8576)41 41 (38.2041,49.519) 100 64 (55.9571,72.5347)42 44 (36.4171,47.2057) 101 60 (56.7976,73.6384)43 47 (39.0212,50.5992) 102 53 (53.3059,69.0941)44 41 (41.6841,54.0516) 103 52 (47.1276,61.0739)45 40 (36.4641,47.2527) 104 44 (46.1694,59.8525)46 32 (35.5176,46.0431) 105 40 (39.1506,50.7286)47 32 (28.4988,36.9192) 106 36 (35.5529,46.0784)48 34 (28.4047,36.8251) 107 28 (32.0024,41.4753)49 40 (30.1565,39.1031) 108 36 (24.9482,32.3161)50 43 (35.4353,45.9608) 109 42 (31.8612,41.3341)51 42 (38.1335,49.4484) 110 53 (37.2106,48.2624)52 43 (37.2929,48.3447) 111 53 (46.9159,60.8622)53 44 (38.1571,49.472) 112 55 (47.0453,60.9916)54 39 (39.0447,50.6227) 113 48 (48.7971,63.2696)55 40 (34.6771,44.9394) 114 47 (42.6894,55.32)56 33 (35.4941,46.0196) 115 43 (41.7312,54.0986)57 32 (29.3747,38.0582) 116 39 (38.2159,49.5308)58 31 (28.4165,36.8369) 117 33 (34.6653,44.9276)59 28 (27.5288,35.6861) 118 30 (29.3629,38.0465)60 34 (24.8894,32.2573) 119 23 (26.6647,34.5588)

104

20

40

60

80

100

120

140

20 40 60 80 100 120 140

x0,t

x0,t-1


20

40

60

80

100

120

140

20 25 30 35 40 45 50 55 60 65 70 75

x0,t

x1,t-1


20

40

60

80

100

120

140

20 25 30 35 40 45 50 55 60 65 70 75

x0,t

x1,t-2


20

25

30

35

40

45

50

55

60

65

70

75

20 25 30 35 40 45 50 55 60 65 70 75

x1,t

x1,t-1


20

25

30

35

40

45

50

55

60

65

70

75

20 25 30 35 40 45 50 55 60 65 70 75

x1,t

x1,t-2


Figure 5.9: Linearity Figures for Monthly Housing Data

105

5.1.2.8 Monthly Interest Rates Data

Table 5.32: Fuzzy equations for Interest Rates Data


x0 (1.024, 0)x0,t−1 + (0, 0.1188)x1,t−1

x1 (0.7448, 0)x1,t−1 + (0.2335, 0)x1,t−2 + (0, 0.09282)x2,t−1

x2 (0.534, 0.09717)x2,t−1 + (0.5283, 0)x1,t−1

Monthly Interest Rate data contains three series which are Federal Funds Rate

(x0), 90-Day Treasury Bill Rate (x1), and the One-Year Treasury Bill Rate (x2).

The data can be seen in Table A.5.

The fuzzy equations are given in Table 5.32. The equations are obtained for

2 lags. The largest coefficient center value is for the lagged variable x0,t−1 in the

fuzzy equation for x0.

In Figure 5.10, the linearity relationship between dependent and independent

variables can be seen. There exist linearity between all of the dependent and

defining variables.

Fuzzy MAR yields the model of the variable x0 with the maximum accuracy

which is 0.852941. The models for both x1 and x2 provides 0.844538 accuracy.

The output results can be seen in Figures 5.33, 5.34 and 5.35. It is observed from

the output results that the widths of the observed outputs are changing between

1 and 1.5 to cover as much expected output as possible.

106

Table 5.33: Expected and Obtained Output for Variable x0 (Interest Rate Data)


2 3.84 (3.5937,4.53471) 62 4.04 (3.60988,4.539)3 3.92 (3.53784,4.32439) 63 4.09 (3.66892,4.6028)4 3.85 (3.62925,4.39678) 64 4.1 (3.72011,4.65399)5 3.32 (3.55046,4.33225) 65 4.04 (3.7351,4.65947)6 3.23 (3.1065,3.69106) 66 4.09 (3.68437,4.58736)7 2.98 (3.03337,3.57991) 67 4.12 (3.7308,4.64329)8 2.6 (2.77744,3.32398) 68 4.01 (3.76151,4.67401)9 2.47 (2.36703,2.95635) 69 4.08 (3.6394,4.5709)10 2.44 (2.25534,2.80188) 70 4.1 (3.69799,4.65563)11 1.98 (2.21631,2.77949) 71 4.32 (3.71134,4.68324)12 1.85 (1.75965,2.29431) 72 4.42 (3.9021,4.94291)13 2.14 (1.62775,2.16004) 73 4.6 (3.97952,5.07024)14 2.02 (1.90325,2.47831) 74 4.65 (4.15666,5.26164)15 1.5 (1.78397,2.3519) 75 4.67 (4.21498,5.30569)16 1.98 (1.26351,1.80768) 76 4.9 (4.23189,5.32973)17 1.73 (1.7549,2.29907) 77 5.17 (4.46497,5.56757)18 1.56 (1.49421,2.04789) 78 5.3 (4.75801,5.82734)19 2 (1.33087,1.86316) 79 5.53 (4.85545,5.99607)20 1.88 (1.76349,2.33142) 80 5.4 (5.0719,6.25054)21 2.26 (1.65371,2.19551) 81 5.53 (4.8901,6.16616)22 2.62 (2.04035,2.5869) 82 5.77 (5.02556,6.29687)23 2.33 (2.38751,2.97683) 83 5.4 (5.27482,6.539)24 2.14 (2.07637,2.6942) 84 4.94 (4.93882,6.11745)25 2.37 (1.8676,2.51395) 85 5 (4.49642,5.61802)26 2.7 (2.10187,2.7506) 86 4.53 (4.57685,5.66043)27 2.69 (2.44089,3.08724) 87 4.05 (4.13134,5.14364)28 2.29 (2.42947,3.07819) 88 3.94 (3.68985,4.60235)29 2.68 (2.02592,2.66276) 89 3.98 (3.60576,4.46122)30 2.71 (2.41923,3.06795) 90 3.79 (3.65384,4.49504)31 2.93 (2.42737,3.12124) 91 3.89 (3.37972,4.38014)32 2.9 (2.66447,3.33458) 92 4 (3.47497,4.48964)33 2.9 (2.63851,3.29912) 93 3.88 (3.56976,4.62007)34 2.94 (2.64326,3.29436) 94 4.12 (3.43027,4.51386)35 2.93 (2.67352,3.34601) 95 4.51 (3.65577,4.77975)36 2.91 (2.65853,3.34052) 96 4.6 (4.02651,5.20752)37 3 (2.6333,3.3248) 97 4.72 (4.11508,5.30322)38 2.98 (2.72425,3.41812) 98 5.05 (4.2403,5.42369)39 2.9 (2.70734,3.39408) 99 5.76 (4.55556,5.7841)40 3 (2.62425,3.31337) 100 6.12 (5.25746,6.53589)41 2.99 (2.72425,3.41812) 101 6.07 (5.59273,6.93771)42 3.02 (2.70569,3.4162) 102 6.02 (5.55818,6.86988)43 3.49 (2.71383,3.46949) 103 6.03 (5.53194,6.79375)44 3.48 (3.17835,3.96727) 104 5.78 (5.56832,6.77785)45 3.5 (3.16098,3.96417) 105 5.92 (5.3005,6.53379)46 3.48 (3.17314,3.99296) 106 5.81 (5.42482,6.69613)47 3.38 (3.14435,3.9808) 107 6.02 (5.30033,6.5954)48 3.48 (3.04198,3.87843) 108 6.3 (5.45471,6.87098)49 3.48 (3.14435,3.9808) 109 6.64 (5.71997,7.17901)50 3.43 (3.14316,3.98199) 110 6.79 (6.07041,7.5247)51 3.47 (3.09079,3.93199) 111 7.41 (6.23585,7.66638)52 3.5 (3.14005,3.96462) 112 8.67 (6.85987,8.31178)53 3.5 (3.16958,3.99652) 113 8.9 (8.15809,9.59336)54 3.42 (3.16958,3.99652) 114 8.61 (8.34602,9.87634)55 3.5 (3.08768,3.91462) 115 9.19 (7.9826,9.646)56 3.45 (3.1672,3.9989) 116 9.15 (8.57874,10.2374)57 3.36 (3.11245,3.95128) 117 9 (8.52472,10.2095)58 3.52 (3.01556,3.86389) 118 8.85 (8.38186,10.0453)59 3.85 (3.17104,4.03601) 119 8.97 (8.19978,9.92021)60 3.9 (3.48511,4.3976) 120 8.98 (8.25372,10.112)61 3.98 (3.53986,4.44522) 121 8.98 (8.25801,10.1281)

107

Table 5.33: Expected and Obtained Output for Variable x0 (Interest RateData)(cont’d)


122 7.76 (8.34594,10.0402) 181 6.24 (6.55541,8.04296)123 8.1 (7.15639,8.73187) 182 5.54 (5.73459,7.04154)124 7.94 (7.51872,9.06568) 183 5.49 (5.01917,6.32374)125 7.6 (7.31571,8.94109) 184 5.22 (4.95372,6.28682)126 7.21 (6.98666,8.57401) 185 5.55 (4.72246,5.96526)127 6.61 (6.61473,8.14743) 186 6.1 (5.04723,6.31616)128 6.29 (6.00525,7.52844) 187 6.14 (5.51641,6.97307)129 6.2 (5.71092,7.16758) 188 6.24 (5.52053,7.05085)130 5.6 (5.64492,7.04931) 189 5.82 (5.62528,7.15085)131 4.9 (5.10554,6.36022) 190 5.22 (5.24997,6.66623)132 4.14 (4.43764,5.59489) 191 5.2 (4.69276,5.99496)133 3.72 (3.7107,4.76577) 192 4.87 (4.67704,5.96974)134 3.71 (3.36866,4.24788) 193 4.77 (4.40693,5.56418)135 4.15 (3.39644,4.19962) 194 4.84 (4.30337,5.463)136 4.63 (3.78985,4.70709) 195 4.82 (4.36077,5.54892)137 4.91 (4.24797,5.23175) 196 5.29 (4.35693,5.51181)138 5.31 (4.46214,5.59087) 197 5.48 (4.79769,6.03336)139 5.57 (4.7944,6.07759) 198 5.31 (4.96725,6.25282)140 5.55 (5.11523,6.28911) 199 5.29 (4.8146,6.0574)141 5.2 (5.12445,6.23893) 200 5.25 (4.80482,6.02623)142 4.91 (4.79348,5.8533) 201 5.03 (4.771,5.97815)143 4.14 (4.52511,5.5279) 202 4.95 (4.56479,5.73392)144 3.5 (3.76179,4.71468) 203 4.65 (4.50309,5.63182)145 3.29 (3.18146,3.98464) 204 4.61 (4.2435,5.27718)146 3.83 (2.98786,3.74827) 205 4.68 (4.17047,5.26831)147 4.17 (3.4777,4.36406) 206 4.69 (4.23619,5.34591)148 4.27 (3.82815,4.70975) 207 4.73 (4.25474,5.34783)149 4.46 (3.9329,4.80974) 208 5.35 (4.30282,5.38165)150 4.55 (4.10127,5.03039) 209 5.39 (4.88763,6.06626)151 4.8 (4.18508,5.13084) 210 5.42 (4.92145,6.11434)152 4.87 (4.43626,5.39153) 211 5.9 (4.93196,6.16525)153 5.04 (4.43188,5.53923) 212 6.14 (5.38771,6.69229)154 5.06 (4.59641,5.72277) 213 6.47 (5.59538,6.976)155 5.33 (4.61213,5.748) 214 6.51 (5.89163,7.35542)156 5.94 (4.85408,6.05886) 215 6.56 (5.93971,7.38924)157 6.58 (5.43816,6.72373) 216 6.7 (5.99446,7.43686)158 7.09 (6.07077,7.40149) 217 6.78 (6.09382,7.62414)159 7.12 (6.53466,7.98181) 218 6.79 (6.17453,7.70723)160 7.84 (6.54517,8.03272) 219 6.89 (6.20377,7.69846)161 8.49 (7.27037,8.78169) 220 7.36 (6.30615,7.80083)162 10.4 (7.83718,9.54573) 221 7.6 (6.77304,8.29624)163 10.5 (9.69507,11.5985) 222 7.81 (6.98072,8.57995)164 10.78 (9.71903,11.7793) 223 8.04 (7.16243,8.82821)165 10.01 (10.0508,12.0208) 224 8.45 (7.38957,9.07198)166 10.03 (9.38968,11.1054) 225 8.96 (7.71781,9.58319)167 9.95 (9.33768,11.1983) 226 9.76 (8.22328,10.1219)168 9.65 (9.30093,11.0713) 227 10.03 (8.96504,11.0181)169 8.97 (8.95579,10.8022) 228 10.07 (9.18916,11.3468)170 9.35 (8.33689,10.0288) 229 10.06 (9.19803,11.4199)171 10.51 (8.6261,10.5176) 230 10.09 (9.19136,11.4061)172 11.31 (9.76966,11.7491) 231 10.01 (9.20306,11.4558)173 11.93 (10.6005,12.5562) 232 10.24 (9.12354,11.3715)174 12.92 (11.2744,13.1517) 233 10.29 (9.34118,11.6248)175 12.01 (12.3295,14.1236) 234 10.47 (9.45771,11.6106)176 11.34 (11.2304,13.3596) 235 10.94 (9.62059,11.8163)177 10.06 (10.6514,12.5667) 236 11.43 (10.0685,12.3307)178 9.45 (9.41235,11.1851) 237 13.77 (10.4822,12.9202)179 8.53 (8.78669,10.5618) 238 13.18 (12.7066,15.4869)180 7.13 (7.88288,9.58192) 239 13.78 (12.0919,14.8936)

108

Table 5.34: Expected and Obtained Output for Variable x1(Interest Rate Data)


2 3.31 (3.55185,4.37795) 62 3.93 (3.43031,4.17287)3 3.23 (3.04816,3.73132) 63 3.93 (3.46669,4.21297)4 3.29 (2.82291,3.53391) 64 3.89 (3.47322,4.21578)5 2.46 (2.83221,3.57662) 65 3.8 (3.44714,4.18228)6 2.3 (2.31251,2.888) 66 3.84 (3.37727,4.09941)7 2.3 (2.00607,2.56856) 67 3.84 (3.38605,4.10819)8 2.48 (1.98821,2.51171) 68 3.92 (3.38889,4.12402)9 2.3 (2.11856,2.64949) 69 4.03 (3.43641,4.19567)10 2.37 (2.02095,2.56302) 70 4.09 (3.53051,4.30277)11 2.25 (2.0357,2.56849) 71 4.38 (3.59438,4.37964)12 2.24 (1.98402,2.47411) 72 4.59 (3.79375,4.64026)13 2.42 (1.94949,2.43772) 73 4.65 (4.00579,4.87644)14 2.39 (2.07007,2.58058) 74 4.59 (4.08836,4.98129)15 2.29 (2.08882,2.60119) 75 4.62 (4.05768,4.95061)16 2.29 (2.0092,2.51785) 76 4.64 (4.07066,4.95431)17 2.33 (1.98771,2.49265) 77 4.5 (4.08421,4.98456)18 2.24 (2.01007,2.52987) 78 4.8 (3.9911,4.87846)19 2.39 (1.95331,2.47125) 79 4.96 (4.167,5.08406)20 2.28 (2.03288,2.57309) 80 5.37 (4.31908,5.3104)21 2.3 (1.98876,2.5234) 81 5.35 (4.6191,5.69581)22 2.48 (1.97611,2.51447) 82 5.32 (4.72591,5.75064)23 2.6 (2.11484,2.6532) 83 4.96 (4.70168,5.72084)24 2.72 (2.23974,2.79109) 84 4.72 (4.47204,5.40024)25 2.73 (2.33671,2.9289) 85 4.56 (4.24544,5.10124)26 2.72 (2.37032,2.96622) 86 4.26 (4.07396,4.92233)27 2.73 (2.38656,2.93976) 87 3.84 (3.84937,4.62535)28 2.68 (2.39909,2.93745) 88 3.6 (3.49251,4.21651)29 2.73 (2.36326,2.90347) 89 3.54 (3.21757,3.93785)30 2.92 (2.39068,2.92718) 90 4.21 (3.09086,3.86312)31 2.82 (2.51788,3.10635) 91 4.27 (3.50717,4.4168)32 2.78 (2.49425,3.06974) 92 4.42 (3.69435,4.63184)33 2.74 (2.45133,3.00639) 93 4.56 (3.81544,4.7622)34 2.83 (2.42055,2.95891) 94 4.73 (3.94359,4.91263)35 2.87 (2.47453,3.02031) 95 4.97 (4.08711,5.08771)36 2.91 (2.52533,3.07111) 96 5 (4.28698,5.32471)37 2.92 (2.55889,3.11581) 97 4.98 (4.39227,5.37616)38 2.89 (2.57568,3.1326) 98 5.17 (4.38995,5.3627)39 2.9 (2.55845,3.1098) 99 5.38 (4.51008,5.51625)40 2.92 (2.55333,3.11582) 100 5.66 (4.7062,5.72165)41 2.99 (2.56777,3.13583) 101 5.52 (4.92942,6.01356)42 3.18 (2.61994,3.19728) 102 5.31 (4.90538,5.95981)43 3.32 (2.75087,3.38205) 103 5.09 (4.7432,5.7438)44 3.38 (2.89022,3.53996) 104 5.19 (4.5526,5.50865)45 3.45 (2.96109,3.62382) 105 5.35 (4.57293,5.53455)46 3.52 (3.02352,3.69368) 106 5.45 (4.70152,5.69098)47 3.52 (3.08643,3.76773) 107 5.96 (4.79665,5.81952)48 3.52 (3.10091,3.78593) 108 6.14 (5.15713,6.2654)49 3.53 (3.10184,3.785) 109 6.12 (5.40098,6.52781)50 3.54 (3.10651,3.79523) 110 6.02 (5.41511,6.56794)51 3.47 (3.10979,3.81151) 111 6.11 (5.33689,6.48786)52 3.48 (3.06278,3.75892) 112 6.04 (5.39543,6.51669)53 3.48 (3.05759,3.74632) 113 6.44 (5.35781,6.49207)54 3.48 (3.06086,3.74772) 114 7 (5.56791,6.84511)55 3.5 (3.06643,3.74215) 115 6.98 (6.04959,7.38434)56 3.53 (3.07854,3.75983) 116 7.09 (6.15708,7.50854)57 3.57 (3.09998,3.79242) 117 7 (6.22784,7.5923)58 3.64 (3.13121,3.83478) 118 7.24 (6.19949,7.53795)59 3.84 (3.18618,3.90275) 119 7.82 (6.3433,7.70961)60 3.81 (3.3422,4.07733) 120 7.87 (6.80717,8.22175)61 3.91 (3.37119,4.09704) 121 7.13 (6.99003,8.38419)

109



122 6.63 (6.49326,7.80202) 181 5.5 (5.74961,6.91357)123 6.51 (5.99916,7.20582) 182 5.49 (5.0417,6.07385)124 6.84 (5.79027,7.0025) 183 5.61 (4.84382,5.90197)125 6.68 (5.95327,7.27503) 184 5.23 (4.86589,6.05398)126 6.45 (5.91579,7.22826) 185 5.34 (4.65637,5.7535)127 6.41 (5.7489,6.97784) 186 6.13 (4.65422,5.74207)128 6.13 (5.67191,6.88785) 187 6.44 (5.19588,6.42852)129 5.91 (5.46889,6.65513) 188 6.42 (5.56293,6.89211)130 5.28 (5.25452,6.41105) 189 5.96 (5.61669,6.9533)131 4.87 (4.81191,5.81251) 190 5.48 (5.33626,6.5392)132 4.44 (4.41052,5.30902) 191 5.44 (4.87137,6.07431)133 3.7 (4.03539,4.8522) 192 4.87 (4.75922,5.90276)134 3.38 (3.43584,4.1487) 193 4.88 (4.39218,5.40206)135 3.86 (3.04702,3.71533) 194 5 (4.2582,5.28479)136 4.14 (3.28433,4.04359) 195 4.86 (4.32299,5.40342)137 4.75 (3.55388,4.41525) 196 5.2 (4.27273,5.30117)138 5.4 (4.00952,4.99898) 197 5.41 (4.45243,5.56256)139 4.94 (4.59798,5.66355) 198 5.23 (4.67522,5.81133)140 4.69 (4.42755,5.45229) 199 5.14 (4.61803,5.69845)141 4.46 (4.1646,5.12807) 200 5.08 (4.52568,5.57269)142 4.22 (3.97577,4.85756) 201 4.92 (4.47298,5.494)143 4.01 (3.76746,4.60098) 202 4.75 (4.36858,5.33205)144 3.38 (3.56338,4.38019) 203 4.35 (4.22225,5.15045)145 3.2 (3.09898,3.80812) 204 4.62 (3.91806,4.77943)146 3.73 (2.79556,3.54925) 205 4.67 (3.99235,4.92055)147 3.71 (3.11392,3.93631) 206 4.6 (4.07778,5.03568)148 3.69 (3.20234,4.06557) 207 4.54 (4.03453,4.998)149 3.91 (3.20041,4.02837) 208 4.96 (3.98186,4.92862)150 3.98 (3.33639,4.21075) 209 5.02 (4.25002,5.25805)151 4.02 (3.42225,4.33189) 210 5.19 (4.39462,5.39893)152 4.66 (3.46839,4.37802) 211 5.49 (4.52039,5.55441)153 4.74 (3.90426,4.91414) 212 5.81 (4.74639,5.85466)154 4.78 (4.1179,5.1185) 213 6.16 (5.0399,6.17788)155 5.07 (4.18401,5.14933) 214 6.1 (5.33909,6.54946)156 5.41 (4.4019,5.38208) 215 6.07 (5.37611,6.58648)157 5.6 (4.69499,5.73086) 216 6.44 (5.33976,6.55013)158 6.09 (4.88339,5.98423) 217 6.45 (5.58234,6.84469)159 6.26 (5.237,6.44922) 218 6.29 (5.6706,6.94408)160 6.36 (5.47986,6.68838) 219 6.29 (5.55748,6.82354)161 7.19 (5.58289,6.81368) 220 6.41 (5.50713,6.79918)162 8.01 (6.18542,7.49418) 221 6.73 (5.5668,6.91826)163 8.67 (6.90452,8.38407) 222 7.01 (5.80994,7.20781)164 8.29 (7.55503,9.09955) 223 7.08 (6.06905,7.51519)165 7.22 (7.4493,8.94742) 224 7.85 (6.19213,7.62712)166 7.83 (6.64721,7.97825) 225 7.99 (6.75596,8.24294)167 7.45 (6.83037,8.20411) 226 8.64 (6.99916,8.56781)168 7.77 (6.72597,8.0273) 227 9.08 (7.44634,9.15422)169 7.12 (6.87558,8.17692) 228 9.35 (7.90351,9.65595)170 7.96 (6.51259,7.72111) 229 9.32 (8.19805,9.96905)171 8.33 (6.90942,8.27201) 230 9.48 (8.25266,9.99582)172 8.23 (7.31241,8.81238) 231 9.46 (8.36575,10.1071)173 7.9 (7.31224,8.83635) 232 9.61 (8.39749,10.1202)174 7.55 (7.04776,8.56258) 233 9.06 (8.50547,10.2263)175 8.96 (6.72118,8.21373) 234 9.24 (8.17355,9.80904)176 8.06 (7.61164,9.26012) 235 9.52 (8.17364,9.82027)177 7.46 (7.30394,8.88559) 236 10.26 (8.39728,10.0977)178 7.47 (6.73328,8.14228) 237 11.7 (8.94603,10.782)179 7.15 (6.62849,7.98181) 238 11.79 (10.0669,12.1516)180 6.26 (6.43891,7.6994) 239 12.04 (10.471,12.5539)

110

Table 5.35: Expected and Obtained Output for Variable x2(Interest Rate Data)


2 3.68 (4.03582,4.90067) 62 4.02 (3.81285,4.59024)3 3.83 (3.35609,4.0713) 63 4 (3.83215,4.61343)4 4.01 (3.37935,4.1237) 64 3.96 (3.82341,4.60081)5 3.1 (3.48967,4.26901) 65 3.89 (3.78481,4.55443)6 3.03 (2.65369,3.25617) 66 3.89 (3.70669,4.4627)7 2.82 (2.53859,3.12747) 67 3.96 (3.72782,4.48384)8 2.86 (2.44686,2.99493) 68 4.09 (3.7584,4.52802)9 2.92 (2.55943,3.11527) 69 4.16 (3.85744,4.65233)10 2.87 (2.49054,3.05804) 70 4.23 (3.94613,4.75462)11 2.64 (2.50568,3.06346) 71 4.56 (4.00841,4.8305)12 2.63 (2.34182,2.8549) 72 4.69 (4.30576,5.19199)13 2.75 (2.33217,2.84331) 73 4.81 (4.47348,5.38498)14 2.76 (2.47968,3.01414) 74 4.81 (4.5576,5.49241)15 2.74 (2.4682,3.0046) 75 4.76 (4.5259,5.46072)16 2.72 (2.40664,2.93915) 76 4.85 (4.51991,5.44501)17 2.8 (2.3979,2.92653) 77 4.78 (4.56979,5.51238)18 2.79 (2.45397,2.99815) 78 4.94 (4.46525,5.39423)19 2.91 (2.40206,2.94429) 79 5.34 (4.69363,5.65371)20 2.88 (2.53372,3.09928) 80 5.8 (4.95287,5.9907)21 2.9 (2.4625,3.02223) 81 5.52 (5.3704,6.49763)22 2.9 (2.48181,3.04542) 82 5.49 (5.23753,6.31034)23 2.97 (2.5769,3.14051) 83 5 (5.20858,6.27556)24 3.19 (2.67087,3.24809) 84 4.61 (4.80436,5.77611)25 3.21 (2.83036,3.45034) 85 4.57 (4.50722,5.40317)26 2.98 (2.84438,3.46824) 86 4.18 (4.40522,5.29339)27 2.9 (2.73864,3.3178) 87 3.9 (4.07638,4.88875)28 2.91 (2.70898,3.27259) 88 3.88 (3.73219,4.49015)29 2.89 (2.68693,3.25248) 89 4.16 (3.59666,4.35073)30 3.17 (2.70461,3.26628) 90 4.9 (3.68726,4.49575)31 3.1 (2.92729,3.54337) 91 5.05 (4.36446,5.31676)32 2.99 (2.84388,3.44636) 92 5.1 (4.46167,5.44313)33 2.9 (2.7747,3.3558) 93 5.22 (4.56276,5.55394)34 2.94 (2.71426,3.27787) 94 5.39 (4.68914,5.70364)35 2.94 (2.77928,3.35066) 95 5.59 (4.8532,5.90074)36 3 (2.80041,3.37179) 96 5.3 (5.06736,6.15376)37 3 (2.84775,3.4308) 97 5.24 (4.95653,5.98658)38 2.97 (2.85303,3.43608) 98 5.42 (4.91976,5.93815)39 3.03 (2.82408,3.4013) 99 5.47 (5.09876,6.15213)40 3.06 (2.85557,3.44445) 100 5.84 (5.23154,6.29463)41 3.11 (2.87924,3.47395) 101 5.68 (5.54108,6.67608)42 3.4 (2.93806,3.54249) 102 5.39 (5.39723,6.50113)43 3.5 (3.16511,3.8259) 103 5.15 (5.15962,6.20716)44 3.57 (3.28275,3.96297) 104 5.18 (4.93856,5.93946)45 3.61 (3.34503,4.03885) 105 5.33 (5.0045,6.01122)46 3.67 (3.39948,4.10108) 106 5.51 (5.15454,6.19042)47 3.69 (3.46267,4.17593) 107 5.97 (5.286,6.35686)48 3.68 (3.4714,4.18855) 108 6.07 (5.75636,6.91662)49 3.71 (3.46704,4.18224) 109 6.21 (5.89513,7.07483)50 3.78 (3.48542,4.20646) 110 6.2 (5.94572,7.15262)51 3.75 (3.52128,4.25592) 111 6.04 (5.88852,7.09348)52 3.71 (3.4712,4.2) 112 6.11 (5.86618,7.04004)53 3.7 (3.45901,4.18004) 113 6.88 (5.85977,7.04724)54 3.64 (3.45464,4.17373) 114 7.19 (6.40743,7.74455)55 3.67 (3.42843,4.13586) 115 7.28 (6.83868,8.23605)56 3.73 (3.4521,4.16536) 116 7.35 (6.86743,8.28229)57 3.79 (3.49416,4.21908) 117 7.21 (6.95612,8.38458)58 3.86 (3.5415,4.27808) 118 7.36 (6.84742,8.24867)59 3.96 (3.60906,4.35924) 119 7.62 (7.03973,8.47014)60 3.91 (3.7584,4.52802) 120 7.51 (7.45971,8.94065)61 4 (3.72071,4.48061) 121 7.05 (7.43808,8.89764)

111



122 6.5 (6.84621,8.21637) 181 5.56 (6.04589,7.26445)123 6.53 (6.34182,7.60509) 182 5.7 (5.33425,6.41483)124 7.12 (6.29153,7.56063) 183 6.4 (5.39012,6.49791)125 7.07 (6.72358,8.10734) 184 5.91 (5.75927,7.00311)126 6.62 (6.61721,7.99126) 185 5.86 (5.34449,6.49309)127 6.55 (6.29914,7.58573) 186 6.64 (5.38076,6.51965)128 6.39 (6.24744,7.52042) 187 7.16 (6.13882,7.4293)129 6.23 (6.02962,7.27151) 188 7.2 (6.52973,7.92127)130 5.39 (5.84351,7.0543) 189 6.48 (6.53664,7.93595)131 4.84 (5.14377,6.19131) 190 6.48 (5.97912,7.2385)132 4.4 (4.68693,5.62758) 191 6.16 (5.72554,6.98492)133 3.84 (4.26757,5.1227) 192 5.44 (5.56463,6.76182)134 3.6 (3.63202,4.37832) 193 5.53 (4.94901,6.00626)135 4.09 (3.35813,4.05778) 194 5.82 (4.9936,6.06835)136 4.64 (3.82574,4.62063) 195 5.54 (5.18367,6.31478)137 5.33 (4.21391,5.11569) 196 5.98 (4.9874,6.06409)138 5.74 (4.83756,5.87344) 197 6.12 (5.35922,6.52142)139 5.52 (5.36004,6.47561) 198 5.82 (5.53131,6.72072)140 5.19 (5.02093,6.09374) 199 5.64 (5.30518,6.43629)141 4.75 (4.74471,5.75338) 200 5.5 (5.17901,6.27513)142 4.49 (4.43101,5.35417) 201 5.19 (5.08616,6.15507)143 4.4 (4.19065,5.06328) 202 5 (4.86622,5.87489)144 3.82 (4.0404,4.89553) 203 4.64 (4.69342,5.66516)145 4.06 (3.45422,4.19664) 204 5 (4.32485,5.22663)146 4.43 (3.46396,4.25302) 205 5.16 (4.62474,5.59648)147 4.65 (3.90558,4.76654) 206 5.19 (4.72104,5.72388)148 4.46 (3.99111,4.89483) 207 5.1 (4.69717,5.70583)149 4.71 (3.89755,4.76434) 208 5.43 (4.62616,5.61733)150 4.9 (4.12297,5.03836) 209 5.41 (4.99219,6.0475)151 4.9 (4.24295,5.19526) 210 5.57 (5.01515,6.06657)152 5.44 (4.26408,5.21639) 211 5.97 (5.17485,6.25737)153 5.39 (4.83806,5.89532) 212 6.13 (5.50806,6.66832)154 5.2 (4.85849,5.90603) 213 6.52 (5.747,6.93836)155 5.28 (4.79663,5.80724) 214 6.52 (6.10226,7.36941)156 5.58 (4.98478,6.01094) 215 6.52 (6.07056,7.33771)157 5.93 (5.29544,6.37991) 216 6.8 (6.05471,7.32186)158 6.53 (5.5487,6.70118) 217 6.86 (6.37248,7.69405)159 6.51 (6.06964,7.33874) 218 6.82 (6.40397,7.73721)160 6.63 (6.15072,7.41593) 219 6.96 (6.30197,7.62743)161 7.05 (6.25596,7.5445) 220 7.28 (6.36313,7.71579)162 7.97 (6.87791,8.24807) 221 7.53 (6.5663,7.98116)163 8.32 (7.71297,9.26193) 222 7.79 (6.84455,8.308)164 8.07 (8.21453,9.83151) 223 7.73 (7.10604,8.62002)165 7.17 (7.90457,9.47297) 224 8.01 (7.11682,8.61913)166 7.4 (6.94617,8.33965) 225 8.45 (7.64591,9.20265)167 7.01 (7.3689,8.80708) 226 9.2 (7.91207,9.55431)168 7.01 (6.9978,8.36018) 227 9.44 (8.58306,10.3711)169 6.51 (7.16685,8.52924) 228 9.54 (8.92034,10.755)170 7.34 (6.60506,7.87027) 229 9.39 (9.10667,10.9608)171 8.08 (7.41137,8.83789) 230 9.38 (9.0253,10.8502)172 8.21 (7.93007,9.50041) 231 9.28 (9.10546,10.9284)173 8.16 (7.93403,9.52963) 232 9.27 (9.05121,10.8548)174 8.04 (7.73785,9.32373) 233 8.81 (9.12609,10.9277)175 8.88 (7.50053,9.06309) 234 8.87 (8.6346,10.3468)176 8.52 (8.61234,10.3382) 235 9.16 (8.7559,10.4798)177 7.59 (7.97962,9.63547) 236 9.89 (9.03049,10.8107)178 7.29 (7.25642,8.73153) 237 11.23 (9.7403,11.6624)179 6.79 (7.13067,8.54747) 238 11.22 (11.0864,13.2689)180 6.27 (6.74321,8.06284) 239 10.92 (11.1295,13.3101)

112

0

2

4

6

8

10

12

14

0 2 4 6 8 10 12 14

x0,t

x0,t-1


0

2

4

6

8

10

12

14

2 3 4 5 6 7 8 9 10 11 12

x0,t

x1,t-1


2

3

4

5

6

7

8

9

10

11

12

13

2 3 4 5 6 7 8 9 10 11 12

x1,t

x1,t-1


2

3

4

5

6

7

8

9

10

11

12

13

2 3 4 5 6 7 8 9 10 11 12

x1,t

x1,t-2


2

3

4

5

6

7

8

9

10

11

12

13

2 3 4 5 6 7 8 9 10 11 12

x1,t

x2,t-1


2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6 7 8 9 10 11 12

x2,t

x1,t-1


2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6 7 8 9 10 11 12

x2,t

x2,t-1


Figure 5.10: Linearity Figures for Interest Rates for the Federal Funds Rate

113

5.1.2.9 Fixed Investment and Changes in Business Inventories Data

Table 5.36: Fuzzy equations for Investment and Inventories Data


x0 (1.015, 0.03844)x0,t−1

x1 (0.01963, 0.05654)x0,t−1 + (0.9724, 0.1635)x1,t−1

Quarterly, seasonally adjusted, US Fixed investment (x0) and changes in busi-

ness inventories (x1) are tested by the algorithm. The data is recorded between

1947 and 1971 and it can be seen in Table A.11.


ables can be seen. There is an obvious linear relationship between x0 and x0,t−1.

On the other hand, x1 has no obvious linearity as x0 with both of the defining

lagged variables in the equation. The accuracy rates given in Table 5.52 (0.806122

both for x0 and x1) show that the models fit well to the time series data. The

output results can be seen in Figures 5.37 and 5.38. The width of the output

obtained for x1 reaches to 11.25 which shows the output widths get larger in order

the model to fit the data well or to cover as much data as possible.

114

Table 5.37: Expected and Obtained Output for Variable x0 (Investment andInventories Data)


2 69.5 (65.9896,71.1874) 51 101.4 (100.156,108.045)3 74.7 (67.8444,73.1882) 52 104.9 (98.9845,106.781)4 77.1 (72.9205,78.6641) 53 101.8 (102.401,110.467)5 77.4 (75.2633,81.1915) 54 98.8 (99.3749,107.202)6 76.6 (75.5562,81.5074) 55 98.6 (96.4464,104.043)7 76.1 (74.7752,80.665) 56 97.7 (96.2512,103.832)8 71.8 (74.2872,80.1384) 57 99.2 (95.3726,102.885)9 68.9 (70.0896,75.6102) 58 101.3 (96.8369,104.464)10 68.5 (67.2587,72.5563) 59 104.6 (98.8868,106.676)11 70.6 (66.8682,72.1351) 60 106.1 (102.108,110.151)12 75.4 (68.9182,74.3466) 61 109.9 (103.572,111.73)13 82.3 (73.6038,79.4013) 62 111.1 (107.282,115.732)14 88.2 (80.3395,86.6674) 63 110.1 (108.453,116.996)15 86.9 (86.0989,92.8805) 64 110.7 (107.477,115.943)16 83.4 (84.8299,91.5116) 65 116 (108.063,116.575)17 80.3 (81.4133,87.8258) 66 118.5 (113.237,122.156)18 79.4 (78.3871,84.5613) 67 122 (115.677,124.788)19 78.6 (77.5085,83.6135) 68 124 (119.094,128.474)20 79.3 (76.7276,82.7711) 69 124 (121.046,130.58)21 80.3 (77.4109,83.5082) 70 124.9 (121.046,130.58)22 75.3 (78.3871,84.5613) 71 126.4 (121.925,131.528)23 80.6 (73.5062,79.296) 72 133.4 (123.389,133.108)24 83.9 (78.68,84.8772) 73 137.9 (130.222,140.479)25 84.2 (81.9013,88.3523) 74 140.1 (134.615,145.218)26 84.4 (82.1942,88.6683) 75 143.8 (136.763,147.535)27 83.8 (82.3894,88.8789) 76 147.5 (140.374,151.431)28 82.8 (81.8037,88.247) 77 146.2 (143.986,155.327)29 84.1 (80.8275,87.194) 78 145 (142.717,153.958)30 87 (82.0966,88.563) 79 139.7 (141.546,152.695)31 88.5 (84.9275,91.6169) 80 136.4 (136.372,147.114)32 92.1 (86.3918,93.1965) 81 139.6 (133.151,143.638)33 96.1 (89.906,96.9875) 82 141.1 (136.274,147.008)34 98.3 (93.8107,101.2) 83 145.5 (137.739,148.588)35 98.8 (95.9583,103.517) 84 148.9 (142.034,153.221)36 96.6 (96.4464,104.043) 85 148.9 (145.353,156.802)37 97.4 (94.2988,101.726) 86 150.7 (145.353,156.802)38 97.6 (95.0797,102.569) 87 155 (147.11,158.697)39 96.6 (95.275,102.779) 88 159.1 (151.308,163.225)40 96.2 (94.2988,101.726) 89 158.4 (155.31,167.543)41 95.3 (93.9083,101.305) 90 158.1 (154.627,166.806)42 96.4 (93.0298,100.357) 91 154.3 (154.334,166.49)43 94.9 (94.1036,101.516) 92 151.8 (150.624,162.488)44 90 (92.6393,99.9361) 93 150 (148.184,159.856)45 87.2 (87.856,94.7761) 94 150.4 (146.427,157.96)46 88 (85.1227,91.8275) 95 149.5 (146.817,158.381)47 93 (85.9037,92.6699) 96 154.3 (145.939,157.434)48 98.3 (90.7846,97.9353) 97 158.4 (150.624,162.488)49 101.6 (95.9583,103.517) 98 162.1 (154.627,166.806)50 102.6 (99.1797,106.992) 99 166 (158.238,170.702)

115

Table 5.38: Expected and Obtained Output for Variable x1(Investment and In-ventories Data)


2 -2.9 (-3.22345,4.12664) 51 8.2 (-4.11105,7.36044)3 2.7 (-4.91131,1.9996) 52 13.5 (2.88948,17.0376)4 4.1 (-0.573634,8.75659) 53 4.9 (7.04726,23.3243)5 5.6 (0.470179,10.5296) 54 3 (0.205464,13.3197)6 6.9 (1.6724,12.2563) 55 -3.9 (-1.22063,10.933)7 5.3 (2.75346,13.672) 56 -3.8 (-6.79441,3.0802)8 -0.3 (1.47773,11.8165) 57 1.9 (-6.6803,3.12524)9 -7.1 (-2.89317,5.12807) 58 6.6 (-2.12515,9.71399)10 -2.5 (-8.28639,-2.81678) 59 6.7 (1.59899,15.2126)11 -7.7 (-4.55085,2.37779) 60 10.6 (1.55806,15.5775)12 4.4 (-8.83447,-3.36882) 61 9.2 (4.65725,20.1217)13 7.7 (0.775594,10.7409) 62 8 (3.38457,18.8209)14 8 (3.19013,15.0148) 63 4.7 (2.36963,17.5492)15 22.1 (3.21499,15.805) 64 7.6 (-0.262705,13.7247)16 13.4 (14.668,31.7218) 65 7 (2.06085,17.0644)17 19.9 (7.76005,21.5731) 66 9.3 (1.37988,16.7866)18 14.6 (13.1321,28.7202) 67 7.1 (3.14798,19.5895)19 7 (8.87835,22.6315) 68 6.1 (1.23928,17.3572)20 7.3 (2.76051,13.9379) 69 8 (0.356583,16.3736)21 -2.7 (2.97733,14.332) 70 7.3 (1.89342,18.5318)22 5.4 (-5.14823,3.04939) 71 7.9 (1.294,17.8052)23 7.2 (1.58815,11.8692) 72 13.4 (1.72394,18.601)24 3.9 (2.84845,14.3174) 73 10.6 (5.91429,25.3815)25 5.1 (0.0573827,10.8204) 74 12.4 (3.48335,22.5438)26 1.9 (1.01694,12.2063) 75 8.8 (4.85809,24.756)27 -5 (-1.5788,8.58671) 76 13.5 (1.80959,20.9486)28 -3.4 (-7.13782,0.703456) 77 17.8 (5.47467,26.5691)29 -4.1 (-5.80672,2.44469) 78 15.1 (9.00077,31.3543)30 -2.7 (-6.42091,1.7486) 79 20.5 (6.86114,28.196)31 1.5 (-5.39556,3.55971) 80 14.6 (11.4247,33.9261)32 5.9 (-2.0537,8.44465) 81 7.5 (6.77418,26.9731)33 8 (1.37241,13.7167) 82 12.2 (0.913115,19.1521)34 7.8 (2.92336,16.4067) 83 13.8 (4.6594,24.6049)35 9.2 (2.68037,16.3471) 84 6.3 (5.79116,26.7575)36 7.5 (3.79433,17.9754) 85 11.8 (-0.400833,18.4974)37 5.5 (2.50047,15.8769) 86 9.2 (4.04792,24.7447)38 4.9 (0.85321,13.666) 87 7.6 (1.87843,21.9285)39 5.4 (0.360508,12.9998) 88 9.8 (0.425507,20.4386)40 2.5 (0.801855,13.4915) 89 12.2 (2.05366,23.2499)41 2.9 (-1.52908,10.167) 90 13.4 (4.02077,25.9226)42 3.7 (-1.17231,10.5528) 91 6.8 (5.00248,27.2628)43 -3 (-0.56583,11.5453) 92 2.9 (-0.195743,19.4766)44 -6.8 (-5.92985,3.82067) 93 4.8 (-3.25802,14.8563)45 -6.2 (-8.82265,-0.868884) 94 6.3 (-1.65474,16.8773)46 0.3 (-8.23396,-0.400627) 95 3.3 (-0.456206,18.6116)47 5.3 (-3.00588,7.04351) 96 7.9 (-2.84957,15.1354)48 5 (0.853864,13.1037) 97 10 (0.694007,20.7261)49 13 (0.415554,13.1667) 98 5 (2.24127,23.4237)50 -0.4 (6.76465,22.505) 99 3.7 (-1.93964,18.0261)

116

60

70

80

90

100

110

120

130

140

150

160

170

60 70 80 90 100 110 120 130 140 150 160 170

x0,t

x0,t-1


-10

-5

0

5

10

15

20

25

60 70 80 90 100 110 120 130 140 150 160 170

x1,t

x0,t-1


-10

-5

0

5

10

15

20

25

-10 -5 0 5 10 15 20 25

x1,t

x1,t-1


Figure 5.11: Linearity Figures for Quarterly Invest-Invent Data

117

5.1.2.10 Annual Sales of Mink Furs and Muskrat Furs Data

Table 5.39: Fuzzy equations for Mink and Muskrat Furs Data


x0 (0.2953, 0)x0,t−1 + (0.3744, 0.02731)x0,t−2 + (0.2582, 0)x1,t−1

x1 (1.005, 0)x1,t−1 + (0, 0.0335)x1,t−2

Natural logarithms of the annual sales of mink furs (x0) and muskrat furs

(x1) by Hudson’s Bay company are tested in this section. The data is recorded

between 1850 and 1911 and can be seen in Table A.12.

The largest center value among the coefficients is in the second equation (equa-

tion for x1) in Table 5.39 and it is for x1,t−1. In Figure 5.12, linearity between

the output variables and the defining variables can be seen. The extracted lagged

variables are not obviously one-to-one linear with the variables to be defined. The

accuracy of the models for x0 and x1 gives the same result. The accuracy rates

in Table 5.52 are 0.516667 and 0.70 for x0 and x1.

The output results can be seen in Figures 5.40 and 5.41. The output obtained

both for x0 and x1 have small width values. Besides, the output values not

included in the model are very close to the intervals obtained.

118

Table 5.40: Expected and Obtained Output for Variable x0 (Mink and MuskratFurs Data)


2 10.121 (9.65989,10.2223) 32 10.7277 (10.251,10.8227)3 10.1327 (9.69584,10.2399) 33 10.7687 (10.386,10.9593)4 10.6543 (9.89067,10.4435) 34 10.8646 (10.4885,11.0745)5 10.8364 (10.0582,10.6117) 35 11.4167 (10.5344,11.1226)6 11.0281 (10.1914,10.7734) 36 11.2451 (10.6579,11.2514)7 11.0341 (10.2365,10.8284) 37 11.0714 (10.5778,11.2014)8 11.2415 (10.3449,10.9473) 38 11.3269 (10.4904,11.1047)9 11.0551 (10.4177,11.0204) 39 10.6152 (10.4805,11.0852)10 10.7084 (10.3805,10.9946) 40 10.48 (10.2471,10.8658)11 10.3448 (10.1203,10.7242) 41 10.2914 (10.0544,10.6343)12 10.8088 (9.9314,10.5163) 42 10.6517 (10.1013,10.6737)13 10.6911 (10.068,10.6331) 43 10.9711 (10.2296,10.7917)14 11.0305 (10.2105,10.8009) 44 10.8359 (10.4871,11.069)15 11.0077 (10.3618,10.9458) 45 10.8452 (10.4638,11.0631)16 10.8475 (10.4219,11.0244) 46 11.1595 (10.4298,11.0217)17 10.9759 (10.2981,10.8994) 47 11.2433 (10.574,11.1664)18 11.2061 (10.3451,10.9377) 48 11.162 (10.6077,11.2173)19 11.2164 (10.5623,11.1618) 49 10.6416 (10.6207,11.2348)20 10.2295 (10.5356,11.1477) 50 10.7359 (10.4929,11.1026)21 10.373 (10.1046,10.7173) 51 10.7751 (10.3634,10.9447)22 10.5781 (9.97177,10.5305) 52 10.9616 (10.4567,11.0431)23 10.7086 (10.2015,10.7681) 53 11.1057 (10.674,11.2625)24 11.0092 (10.3333,10.9111) 54 10.9091 (10.7546,11.3533)25 11.1882 (10.433,11.0179) 55 10.933 (10.6236,11.2302)26 11.2799 (10.5258,11.1272) 56 11.003 (10.5968,11.1927)27 11.278 (10.6428,11.254) 57 10.5756 (10.5177,11.1149)28 11.3415 (10.5996,11.2158) 58 9.9774 (10.2779,10.8789)29 11.0444 (10.6451,11.2611) 59 9.7902 (9.84894,10.4266)30 10.4652 (10.5866,11.2061) 60 9.9891 (9.61279,10.1578)31 10.4957 (10.3009,10.9042) 61 10.4045 (9.84098,10.3758)

119

Table 5.41: Expected and Obtained Output for Variable x1 (Mink and MuskratFurs Data)


2 12.5863 (11.8415,12.6506) 32 13.8444 (13.2648,14.141)3 13.1102 (12.2475,13.0635) 33 13.8824 (13.4639,14.377)4 13.1466 (12.7606,13.6039) 34 13.8953 (13.4949,14.4225)5 12.7531 (12.7796,13.6581) 35 13.6134 (13.5066,14.4367)6 12.4638 (12.3828,13.2636) 36 12.7572 (13.2227,14.1537)7 12.6191 (12.1051,12.9595) 37 12.8483 (12.3712,13.2834)8 12.6556 (12.2709,13.106) 38 12.7509 (12.4915,13.3463)9 12.4461 (12.3024,13.1479) 39 12.3177 (12.3905,13.2514)10 12.0855 (12.0905,12.9385) 40 12.6828 (11.9582,12.8126)11 12.2357 (11.735,12.5689) 41 13.2617 (12.3398,13.1652)12 12.723 (11.8981,12.7078) 42 13.6 (12.9097,13.7595)13 12.7857 (12.383,13.2028) 43 13.7479 (13.2305,14.119)14 13.1417 (12.4297,13.2822) 44 13.3827 (13.3678,14.2791)15 12.9441 (12.7856,13.6423) 45 13.4222 (12.9957,13.9168)16 12.6786 (12.575,13.4555) 46 13.6087 (13.0476,13.9443)17 12.9292 (12.3146,13.1819) 47 13.2208 (13.2338,14.1332)18 13.3344 (12.5755,13.425) 48 13.2515 (12.8376,13.7494)19 12.9096 (12.9745,13.8408) 49 13.461 (12.8814,13.7672)20 12.3556 (12.5338,13.4273) 50 13.5512 (13.091,13.9789)21 13.0036 (11.991,12.856) 51 13.741 (13.1747,14.0766)22 13.4657 (12.6611,13.489) 52 14.3164 (13.3625,14.2705)23 13.5514 (13.1041,13.9753) 53 14.2131 (13.9347,14.8554)24 13.418 (13.1748,14.077) 54 13.7369 (13.8116,14.7708)25 13.1689 (13.0378,13.9457) 55 13.8702 (13.3362,14.2886)26 13.2765 (12.7918,13.6908) 56 13.4518 (13.4862,14.4066)27 12.988 (12.9083,13.7906) 57 12.9177 (13.0611,13.9904)28 13.094 (12.6146,13.5042) 58 12.5151 (12.538,13.4394)29 13.1218 (12.7309,13.6011) 59 12.6188 (12.1511,13.0166)30 13.0775 (12.7553,13.6326) 60 13.5267 (12.2689,13.1074)31 13.628 (12.7098,13.589) 61 13.7784 (13.1783,14.0238)

120

9.6

9.8

10

10.2

10.4

10.6

10.8

11

11.2

11.4

11.6

9.6 9.8 10 10.2 10.4 10.6 10.8 11 11.2 11.4 11.6

x0,t

x0,t-1


9.6

9.8

10

10.2

10.4

10.6

10.8

11

11.2

11.4

11.6

9.6 9.8 10 10.2 10.4 10.6 10.8 11 11.2 11.4 11.6

x0,t

x0,t-2


9.6

9.8

10

10.2

10.4

10.6

10.8

11

11.2

11.4

11.6

12 12.5 13 13.5 14 14.5

x0,t

x1,t-1


12

12.5

13

13.5

14

14.5

12 12.5 13 13.5 14 14.5

x1,t

x1,t-1


12

12.5

13

13.5

14

14.5

12 12.5 13 13.5 14 14.5

x1,t

x1,t-2


Figure 5.12: Linearity Figures for Annual Mink-Muskrat Data

121

5.1.2.11 Power Station Data

Table 5.42: Fuzzy equations for Power Station Data


x0 (1.015, 0)x0,t−1 + (0, 0.02686)x0,t−2 + (0, 0.01001)x2,t−1

x1 (0.9515, 0)x1,t−1 + (0, 0.01655)x1,t−2 + (0.0004147, 0.02323)x2,t−1

x2 (0.995, 0)x2,t−1 + (0, 0.0102)x2,t−2 + (0.04052, 0)x1,t−1

Power station data is taken from a 50 megawatt turbo-alternator and con-

tains in-phase current deviations(x0), out-of-phase current deviations (x1) and

frequency deviations of voltage generated (x2). The data can be seen in Table

A.13.


ables can be seen. There is linearity between x0 and x0,t−1, and it is reflected to

the coefficient in Table A.13 and the accuracy rate shown in Table 5.52. When

the variable x1 is concerned, the lagged variable x1,t−1 is the one with the largest

coefficient center and width zero. In the same way, for the variable x2, the lagged

variable x2,t−1 has largest coefficient center and zero width. The model accuracy

rate is 0.765306 for x0, 0.642857 for x1 and 0.683673 for x2. The output results

can be seen in Figures 5.43, 5.44 and 5.45. It can be said that the outputs not

included in the intervals are very close to the intervals. The widths are changing

between 0.5 and 1.5.

122

Table 5.43: Expected and Obtained Output for Variable x0 (Power Station Data)


2 9.9 (8.923,10.3678) 51 3 (3.11412,4.39914)3 10.7 (9.32207,10.781) 52 2.7 (2.42621,3.66563)4 11.3 (10.1226,11.605) 53 3 (2.14041,3.34224)5 11.6 (10.7103,12.2356) 54 3.3 (2.45306,3.63877)6 11.5 (10.9967,12.5583) 55 3.3 (2.7496,3.95142)7 10.8 (10.8862,12.4659) 56 3.4 (2.74054,3.96048)8 9.7 (10.1801,11.7505) 57 3.6 (2.84307,4.06101)9 8.9 (9.07709,10.6198) 58 3.8 (3.04845,4.26175)10 8.5 (8.29439,9.77805) 59 4.1 (3.24614,4.47018)11 7.6 (7.90675,9.35344) 60 4.6 (3.54636,4.77915)12 6.5 (6.99971,8.43293) 61 5 (4.04996,5.29085)13 5.9 (5.90105,7.29793) 62 5 (4.44165,5.7114)14 5.9 (5.3194,6.6612) 63 4.7 (4.43191,5.72115)15 5.8 (5.33252,6.64809) 64 4.4 (4.12732,5.41656)16 5.8 (5.22498,6.55256) 65 4.7 (3.83178,5.10291)17 6 (5.22166,6.55588) 66 5.2 (4.14243,5.40144)18 6 (5.42473,6.75894) 67 5.2 (4.64503,5.91415)19 5.5 (5.42035,6.76331) 68 5 (4.6296,5.92958)20 5.4 (4.9127,6.25566) 69 5 (4.42654,5.72652)21 6 (4.8296,6.1357) 70 4.9 (4.4269,5.72615)22 6.2 (5.44648,6.73719) 71 4.8 (4.32537,5.62462)23 5.4 (5.63242,6.95737) 72 4.7 (4.22653,5.5204)24 5 (4.81581,6.14949) 73 4.6 (4.13269,5.41118)25 5.8 (4.43117,5.72188) 74 4.3 (4.02884,5.31197)26 6.5 (5.25316,6.52439) 75 4.2 (3.73194,4.99969)27 6.5 (5.93938,7.25959) 76 4.6 (3.63846,4.8901)28 6.9 (5.91558,7.28339) 77 5.2 (4.04727,5.29354)29 7.4 (6.3217,7.68951) 78 5.8 (4.64471,5.91447)30 7.6 (6.82161,8.20491) 79 6.1 (5.23778,6.53976)31 7.2 (7.01125,8.4214) 80 6.6 (5.52726,6.85947)32 6.9 (6.60175,8.01865) 81 7.2 (6.02585,7.37618)33 7.1 (6.31091,7.70031) 82 7.2 (6.62161,7.99879)34 8.1 (6.52403,7.89331) 83 6.8 (6.60849,8.01191)35 9.6 (7.53596,8.91199) 84 7 (6.20537,7.60278)36 10.9 (9.03006,10.4638) 85 7.7 (6.42318,7.7911)37 11 (10.3087,11.825) 86 7.5 (7.13953,8.49617)38 10.5 (10.3743,11.9624) 87 6.4 (6.91767,8.31191)39 10.4 (9.86795,11.4535) 88 5.6 (5.8012,7.19471)40 10.5 (9.78885,11.3295) 89 5 (5.0195,6.35192)41 10.3 (9.90007,11.4213) 90 4 (4.43581,5.71725)42 10 (9.69433,11.221) 91 2.8 (3.43061,4.69183)43 9.2 (9.39711,10.909) 92 2.4 (2.2381,3.44761)44 8.1 (8.59292,10.0887) 93 2.6 (1.86221,3.01126)45 7.3 (7.49157,8.95638) 94 2.1 (2.06701,3.21258)46 6 (6.70586,8.11759) 95 1.6 (1.54598,2.71831)47 4.8 (5.40545,6.77821) 96 1.8 (1.04675,2.20223)48 4.5 (4.21199,5.53494) 97 1.9 (1.26524,2.38986)49 4.4 (3.92863,5.20912) 98 1.8 (1.3614,2.49676)50 3.7 (3.83115,5.10354) 99 1.9 (1.25618,2.39892)

123



2 -10.4 (-9.71318,-7.75456) 51 14.9 (11.9233,14.7615)3 -10.1 (-10.8245,-8.92682) 52 14.9 (12.7589,15.6381)4 -8.3 (-10.5215,-8.65887) 53 14.7 (12.7441,15.653)5 -6.4 (-8.81384,-6.94125) 54 15.4 (12.5538,15.4627)6 -5.5 (-7.04038,-5.09894) 55 16.4 (13.2231,16.1254)7 -4.6 (-6.21777,-4.20881) 56 16 (14.1607,17.0909)8 -3.1 (-5.37177,-3.34232) 57 14.6 (13.7659,16.7245)9 -1.4 (-3.97086,-1.88839) 58 14.1 (12.4518,15.3739)10 -0.2 (-2.37817,-0.246067) 59 14.1 (11.9992,14.875)11 1.2 (-1.27137,0.930924) 60 13.9 (12.0098,14.8644)12 3.2 (0.0317083,2.2923) 61 12.6 (11.8286,14.6647)13 5.4 (1.8978,4.2326) 62 10.4 (10.5927,13.4268)14 6.8 (3.95339,6.36367) 63 8.9 (8.52329,11.3097)15 7.2 (5.24221,7.73923) 64 9.9 (7.13248,9.84606)16 7.7 (5.58595,8.15717) 65 12.1 (8.11106,10.7704)17 8.2 (6.04138,8.65371) 66 13.1 (10.1832,12.8849)18 8.7 (6.50884,9.13772) 67 12.5 (11.1051,13.8657)19 9.1 (6.97859,9.61937) 68 11.3 (10.5131,13.316)20 9.8 (7.35091,10.0082) 69 10.4 (9.38127,12.1643)21 11 (8.02173,10.6691) 70 9.8 (8.53339,11.3)22 11.7 (9.16332,11.8106) 71 9.3 (7.9774,10.7142)23 10.9 (9.80722,12.4988) 72 8.6 (7.51159,10.2285)24 10 (9.03674,11.7469) 73 7.9 (6.86524,9.54242)25 10.6 (8.19365,10.8773) 74 8 (6.19938,8.87663)26 11.7 (8.77714,11.4357) 75 9.1 (6.31752,8.94837)27 11.8 (9.80699,12.4993) 76 11 (7.36249,9.99665)28 9.6 (9.87253,12.6245) 77 12.6 (9.15209,11.8226)29 6.4 (7.77763,10.5329) 78 12.6 (10.6407,13.3788)30 6 (4.77616,7.44468) 79 11 (10.6143,13.4053)31 8 (4.44852,7.01114) 80 9.2 (9.09418,11.8806)32 8.4 (6.36265,8.90274) 81 7.9 (7.40572,10.1438)33 6.8 (6.71699,9.30933) 82 6.7 (6.19858,8.8771)34 6.3 (5.19257,7.78886) 83 5.3 (5.08517,7.70672)35 7.4 (4.74787,7.28192) 84 4.4 (3.7798,6.34771)36 8.1 (5.79821,8.325) 85 4.3 (2.95576,5.45876)37 6.7 (6.44376,9.0116) 86 3.9 (2.9006,5.32271)38 3.8 (5.09783,7.69348) 87 3.2 (2.52167,4.94047)39 1.7 (2.37084,4.90158) 88 2.9 (1.85085,4.27964)40 1.7 (0.441261,2.83422) 89 3.4 (1.57927,3.98025)41 2.9 (0.491979,2.78292) 90 3.8 (2.0691,4.44157)42 4.1 (1.63375,3.92469) 91 3.7 (2.42772,4.84462)43 5.2 (2.76023,5.08159) 92 3.3 (2.32368,4.75845)44 6.7 (3.78699,6.14807) 93 2.6 (1.94018,4.38094)45 7.7 (5.18231,7.60767) 94 2.5 (1.26023,3.72957)46 8 (6.10213,8.59105) 95 3.1 (1.15841,3.64175)47 8.6 (6.36646,8.89777) 96 4 (1.71955,4.2228)48 10.1 (6.90956,9.49726) 97 4.7 (2.57051,5.08433)49 12.2 (8.30175,10.9604) 98 4.5 (3.22165,5.76525)50 14 (10.2659,12.9928) 99 3.6 (3.01749,5.5889)

124



2 47.4 (46.4073,47.3761) 51 52 (52.1824,53.2327)3 47.5 (46.2592,47.228) 52 52 (51.8117,52.8804)4 47.5 (46.3719,47.3386) 53 52 (51.8158,52.8763)5 47.7 (46.4438,47.4126) 54 52 (51.8076,52.8682)6 47.8 (46.7198,47.6886) 55 52.1 (51.836,52.8966)7 47.6 (46.8538,47.8266) 56 52 (51.976,53.0366)8 48.1 (46.6902,47.6651) 57 51.5 (51.8593,52.9219)9 48.1 (47.2505,48.2213) 58 51.5 (51.3061,52.3666)10 48.4 (47.3143,48.2953) 59 51.4 (51.2909,52.3413)11 48.8 (47.6615,48.6425) 60 51 (51.1914,52.2418)12 49.4 (48.1132,49.1003) 61 51.1 (50.7863,51.8346)13 49.6 (48.7872,49.7824) 62 51 (50.8372,51.8774)14 49.9 (49.0692,50.0767) 63 51 (50.6475,51.6897)15 50.5 (49.4224,50.434) 64 50.9 (50.5878,51.6279)16 51.1 (50.0326,51.0503) 65 51.1 (50.5288,51.5689)17 51.1 (50.6437,51.6737) 66 50.8 (50.818,51.8561)18 51 (50.6579,51.7001) 67 51 (50.5579,51.6001)19 51 (50.5786,51.6208) 68 51 (50.7357,51.7718)20 50.5 (50.5959,51.636) 69 51.5 (50.685,51.7252)21 50 (50.1267,51.1669) 70 51.5 (51.1461,52.1862)22 50.1 (49.6829,50.7129) 71 51.5 (51.1167,52.167)23 50 (49.8159,50.8356) 72 51 (51.0964,52.1467)24 50 (49.6829,50.7047) 73 51.5 (50.5705,51.6209)25 50.1 (49.6475,50.6673) 74 51 (51.0448,52.0849)26 50.4 (49.7713,50.7911) 75 51 (50.5462,51.5965)27 50.9 (50.1134,51.1352) 76 51 (50.5959,51.636)28 50.9 (50.6119,51.6398) 77 51.1 (50.6729,51.713)29 50.6 (50.5177,51.5558) 78 51.1 (50.8372,51.8774)30 50.6 (50.0895,51.1276) 79 51 (50.8362,51.8784)31 50.4 (50.0763,51.1083) 80 51.1 (50.6718,51.714)32 50.1 (49.9584,50.9903) 81 51.1 (50.6994,51.7396)33 49.9 (49.6781,50.706) 82 50.8 (50.6457,51.6879)34 49.7 (49.4173,50.4391) 83 50.5 (50.2986,51.3408)35 49.9 (49.2001,50.2178) 84 50.1 (49.9464,50.9825)36 50 (49.4457,50.4593) 85 49 (49.515,50.5449)37 50.1 (49.5715,50.5892) 86 49 (48.4205,49.4422)38 49.7 (49.6133,50.633) 87 49.5 (48.4155,49.4148)39 48.8 (49.0967,50.1185) 88 49.4 (48.8846,49.884)40 48.1 (48.1202,49.1338) 89 49 (48.7679,49.7774)41 48.1 (47.4328,48.4281) 90 49.6 (48.3911,49.3986)42 47.9 (47.4886,48.4696) 91 49.7 (49.0084,50.0078)43 47.9 (47.3382,48.3192) 92 49.9 (49.0978,50.1094)44 48.5 (47.3848,48.3617) 93 50.8 (49.2795,50.2932)45 48.8 (48.0426,49.0195) 94 51.6 (50.1447,51.1624)46 49 (48.3755,49.3647) 95 52.1 (50.9275,51.9636)47 50 (48.5836,49.5789) 96 51.9 (51.4412,52.4935)48 51.1 (49.601,50.6003) 97 51.9 (51.2735,52.3361)49 51.5 (50.7461,51.7659) 98 52 (51.3039,52.3624)50 52.4 (51.218,52.2602) 99 51.6 (51.3953,52.4538)

125

0

2

4

6

8

10

12

0 2 4 6 8 10 12

x0,t

x0,t-1


0

2

4

6

8

10

12

0 2 4 6 8 10 12

x0,t

x0,t-2


0

2

4

6

8

10

12

47 48 49 50 51 52 53

x0,t

x2,t-1


-15

-10

-5

0

5

10

15

20

-15 -10 -5 0 5 10 15 20

x1,t

x1,t-1


-15

-10

-5

0

5

10

15

20

-15 -10 -5 0 5 10 15 20

x1,t

x1,t-2


-15

-10

-5

0

5

10

15

20

47 48 49 50 51 52 53

x1,t

x2,t-1


Figure 5.13: Linearity Figures for Power Station Data

126

47

48

49

50

51

52

53

-15 -10 -5 0 5 10 15 20

x2,t

x1,t-1


47

48

49

50

51

52

53

47 48 49 50 51 52 53

x2,t

x2,t-1


47

48

49

50

51

52

53

47 48 49 50 51 52 53

x2,t

x2,t-2


Figure 5.13: Linearity Figures for Power Station Data(cont’d)

127

5.1.2.12 Production Schedule and Billing Figures Data

Table 5.46: Fuzzy equations for Production and Billing Data


x0 (0.6804, 0.03444)x0,t−1 + (0.3231, 0)x0,t−2

x1 (0.5613, 0.0004929)x1,t−1 + (0, 0.05643)x1,t−2 + (0.8928, 0)x0,t−2

This data set contains two variables which are weekly production figures in

thousands of units (x0) and weekly billing figures in millions of dollars (x1) of a

company. The data is given in Table A.14.


ables can be seen. The figures show that either for x0 or for x1, there is not an

apparent linearity between any lagged variables.

Since there is no obvious linear relationship the model accuracy decreases as

seen in Table 5.52. The accuracy rates for the models are 0.636364 and 0.777778

for x0 and x1 respectively. The output results can be seen in Figures 5.47 and

5.48. The widths are around 1.5 for x0 and 6 for x1.

128

Table 5.47: Expected and Obtained Output for Variable x0 (Production andBilling Data)


2 48.791 (48.17,51.553) 52 49.646 (49.548,53.0859)3 50.114 (47.3849,50.7458) 53 49.415 (48.6636,52.0834)4 52.127 (48.1357,51.5878) 54 48.94 (47.9605,51.3644)5 50.706 (49.8635,53.4542) 55 50.746 (47.5791,50.9502)6 51.1 (49.5961,53.0889) 56 48.912 (48.5921,52.0877)7 50.164 (49.3914,52.9114) 57 49.358 (47.9911,51.3603)8 49.998 (48.9142,52.3696) 58 47.812 (47.6865,51.0865)9 51.269 (48.5045,51.9485) 59 49.362 (46.8321,50.1255)10 48.894 (49.2718,52.8034) 60 50.63 (47.3337,50.7339)11 52.673 (48.1484,51.5164) 61 52.009 (48.6535,52.1411)12 53.406 (49.8219,53.4502) 62 48.907 (49.954,53.5366)13 51.192 (51.5165,55.1953) 63 49.114 (48.396,51.7648)14 50.114 (50.3233,53.8496) 64 52.602 (47.5273,50.9104)15 49.968 (48.9116,52.3636) 65 53.808 (49.8472,53.4706)16 53.321 (48.4689,51.9109) 66 49.9 (51.7532,55.4597)17 51.683 (50.5875,54.2605) 67 48.454 (49.6187,53.056)18 49.843 (50.613,54.1731) 68 49.372 (47.4219,50.7595)19 51.464 (48.8952,52.3286) 69 49.372 (47.5476,50.9485)20 51.671 (49.3477,52.8927) 70 46.517 (47.8442,51.2451)21 50.456 (50.0052,53.5644) 71 47.724 (46.0001,49.2044)22 50.395 (49.2873,52.7628) 72 49.661 (45.8572,49.1446)23 52.591 (48.8553,52.3266) 73 48.638 (47.4983,50.9192)24 51.916 (50.254,53.8766) 74 52.285 (47.4635,50.8138)25 49.967 (50.5276,54.1037) 75 51.087 (49.4886,53.0901)26 51.1 (49.0506,52.4925) 76 48.851 (49.8933,53.4123)27 50.214 (49.1526,52.6726) 77 47.761 (48.0619,51.4269)28 47.217 (48.9465,52.4054) 78 45.251 (46.6353,49.9252)29 48.172 (46.7243,49.9768) 79 49.121 (44.6618,47.7788)30 50.618 (46.3727,49.691) 80 48.933 (46.3504,49.734)31 51.055 (48.2613,51.748) 81 47.942 (47.4795,50.8502)32 52.299 (49.3339,52.8508) 82 48.715 (46.7787,50.0811)33 51.282 (50.2787,53.8812) 83 48.746 (46.9578,50.3134)34 50.533 (50.0237,53.5562) 84 49.058 (47.2276,50.5853)35 50.628 (49.2113,52.6922) 85 51.504 (47.4391,50.8184)36 51.531 (49.0306,52.5181) 86 48.549 (49.1198,52.6676)37 48.488 (49.6446,53.1942) 87 49.037 (48.0015,51.3458)38 46.747 (47.9709,51.3109) 88 49.361 (47.3619,50.7397)39 49.301 (45.863,49.0831) 89 50.291 (47.7289,51.129)40 51.849 (46.9501,50.3461) 90 47.061 (48.4343,51.8985)41 49.635 (49.4212,52.9927) 91 47.768 (46.6485,49.8902)42 50.549 (48.8145,52.2335) 92 47.585 (46.0614,49.3518)43 48.722 (48.6894,52.1714) 93 48.311 (46.1716,49.4494)44 49.824 (47.8047,51.1608) 94 48.049 (46.5814,49.9093)45 51.045 (47.9261,51.3582) 95 45.933 (46.6468,49.9566)46 50.943 (49.0709,52.587) 96 49.883 (45.1954,48.3594)47 50.249 (49.3996,52.9087) 97 46.565 (47.063,50.4991)48 50.538 (48.9183,52.3796) 98 44.731 (46.1962,49.4038)49 50.569 (48.8807,52.362) 99 47.42 (43.9395,47.0207)50 50.671 (48.9941,52.4775) 100 49.694 (45.0837,48.3501)51 51.36 (49.0701,52.5604)

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Table 5.48: Expected and Obtained Output for Variable x1 (Production andBilling Data)


2 100.115 (97.9041,109.63) 52 95.956 (93.6851,105.414)3 101.577 (94.1246,105.951) 53 95.719 (94.2027,105.218)4 100.812 (94.8724,106.271) 54 96.644 (92.5853,103.509)5 96.455 (95.542,107.105) 55 93.025 (92.9112,103.809)6 98.459 (94.939,106.411) 56 92.576 (90.4054,101.404)7 101.856 (95.04,106.022) 57 95.205 (91.9702,102.56)8 98.898 (97.1837,108.395) 58 100.016 (91.8325,102.374)9 99.189 (94.4976,106.09) 59 101.049 (94.7803,105.623)10 99.455 (94.6795,105.938) 60 104.115 (93.7079,105.095)11 98.775 (95.9469,107.239) 61 101.299 (96.7527,108.259)12 103.087 (93.4303,104.751) 62 98.391 (96.1326,107.982)13 100.569 (99.2605,110.509) 63 99.02 (95.8918,107.421)14 105.557 (98.2595,109.992) 64 103.847 (93.6393,104.841)15 109.592 (99.2222,110.676) 65 99.947 (96.4955,107.773)16 105.883 (100.241,112.262) 66 97.395 (97.15,108.968)17 102.97 (97.8032,110.275) 67 104.41 (97.0156,108.391)18 94.108 (99.3723,111.423) 68 111.124 (97.6046,108.699)19 97.793 (93.1046,104.818) 69 111.124 (99.683,111.576)20 99.117 (94.0285,104.745) 70 104.956 (100.124,112.774)21 101.337 (96.0102,107.144) 71 95.525 (96.6648,109.309)22 101.926 (97.3653,108.651) 72 92.163 (89.1752,101.114)23 103.925 (96.4856,108.022) 73 83.464 (88.8996,99.7707)24 103.674 (97.5189,109.124) 74 82.083 (85.9403,96.4235)25 100.041 (99.2259,111.056) 75 89.417 (84.7434,94.2435)26 101.488 (96.6001,108.399) 76 100.043 (92.19,101.542)27 97.475 (95.8765,107.267) 77 111.95 (96.6656,106.855)28 88.786 (94.556,106.105) 78 115.786 (100.747,112.148)29 88.079 (89.1188,100.207) 79 109.128 (101.253,114.001)30 90.521 (86.537,96.6436) 80 102.147 (95.0622,108.237)31 92.665 (88.7989,98.8281) 81 88.77 (94.9781,107.394)32 92.89 (92.0471,102.354) 82 90.126 (87.7025,99.3177)33 98.968 (92.4424,102.992) 83 92.812 (88.333,98.4399)34 103.838 (96.9488,107.529) 84 92.246 (90.4529,100.715)35 110.873 (98.4289,109.7) 85 95.298 (90.0116,100.577)36 110.924 (101.431,113.258) 86 100.044 (92.0336,102.538)37 108.998 (101.147,113.769) 87 102.806 (96.7066,107.56)38 105.282 (100.87,113.496) 88 112.437 (95.3495,106.741)39 105.015 (96.1784,108.583) 89 106.934 (101.03,112.743)40 94.262 (94.684,106.669) 90 106.283 (97.6901,110.484)41 85.9 (90.9491,102.893) 91 102.776 (98.4658,110.638)42 86.393 (89.1413,99.8637) 92 100.181 (93.6523,105.748)43 97.544 (87.913,97.6923) 93 92.541 (93.0261,104.723)44 101.861 (94.9545,104.8) 94 86.056 (88.7247,100.122)45 106.499 (95.1151,106.224) 95 85.623 (86.1673,96.6957)46 104.903 (98.4563,110.057) 96 89.016 (86.0565,95.8526)47 104.643 (98.3896,110.512) 97 96.537 (86.0946,95.8452)48 103.684 (98.2428,110.185) 98 95.561 (93.6472,103.788)49 103.667 (97.1001,109.012) 99 106.057 (89.7133,100.702)50 103.087 (97.4027,109.206) 100 103.59 (94.0171,104.906)51 96.766 (97.1061,108.907)

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Figure 5.14: Linearity Figures for Weekly Production Data

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5.1.2.13 Unemployment and GDP Data

Table 5.49: Fuzzy equations for Unemployment and GDP Data


x0 (0.8161, 0.04231)x0,t−1 + (0.8297, 0.1603)x1,t−1

x1 (0.8817, 0)x1,t−1 + (0.1035, 0)x1,t−2 + (0.004483, 0.002445)x0,t−1

This data set contains two variables which are unemployment (x0) and gross

domestic product (x1) in UK between 1955 and 1969. The data is recorded

quarterly and it can be seen in Table A.15.


ables can be seen. x0 and x0,t−1 has linear relationship, on the other hand, x1,t−1

does not reflect the same relationship. When the fuzzy equation in Table 5.49 is

observed, it can be said that x1,t−1 is an excess (surplus) variable. Although there

is a slight difference between center values of coefficients, the width for x0,t−1 is

much smaller than the width for x1,t−1. For x1, x1,t−1 and x1,t−2 the figures show

the linearity and the coefficient widths are zero whereas for the variable x0,t−1,

the center of the coefficient is much closer to zero and the width is different than

zero.

The accuracy rates for x0 and x1 are 0.517241 and 0.586207 respectively. The

output results can be seen in Figures 5.50 and 5.51. The output width does not

exceed 42 for x0 and 1.33 for x1.

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Table 5.50: Expected and Obtained Output for Variable x0 (Unemployment andGDP Data)


2 201 (216.254,260.328) 31 483 (410.822,479.725)3 199 (210.636,254.022) 32 535 (438.366,510.164)4 207 (209.557,252.998) 33 520 (478.378,554.467)5 215 (215.661,259.737) 34 489 (469.201,545.183)6 240 (222.34,267.327) 35 456 (446.115,519.908)7 245 (241.506,288.522) 36 386 (421.79,493.371)8 295 (245.509,293.012) 37 368 (368.297,434.279)9 293 (284.696,336.667) 38 358 (355.252,420.133)10 279 (283.972,336.168) 39 330 (347.868,412.073)11 287 (272.355,322.991) 40 306 (327.091,389.353)12 331 (278.526,329.829) 41 304 (309.141,369.671)13 396 (313.09,368.363) 42 321 (307.306,367.528)14 432 (362.406,422.708) 43 305 (321.284,383.34)15 462 (390.78,454.374) 44 279 (309.492,370.476)16 454 (414.082,480.257) 45 282 (289.673,348.601)17 446 (408.36,474.082) 46 318 (291.995,351.177)18 426 (403.421,469.066) 47 414 (320.322,382.774)19 402 (388.835,453.213) 48 463 (394.276,464.691)20 363 (371.354,434.224) 49 506 (432.596,507.349)21 342 (342.693,402.991) 50 538 (466.608,545.351)22 325 (326.329,384.795) 51 536 (491.638,573.218)23 312 (313.87,371.231) 52 544 (490.425,571.996)24 291 (304.031,360.398) 53 541 (497.486,580.151)25 293 (288.63,343.628) 54 547 (495.7,578.367)26 304 (290.579,345.939) 55 532 (501.214,584.805)27 330 (298.864,355.045) 56 532 (490.543,573.314)28 357 (319.118,377.563) 57 519 (489.874,572.324)29 401 (339.905,400.583) 58 547 (480.483,562.154)30 447 (374.844,439.671) 59 544 (502.954,587.379)

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Table 5.51: Expected and Obtained Output for Variable x1 (Unemployment andGDP Data)


2 82.3 (81.6773,82.6943) 31 96.5 (96.3907,98.5763)3 83 (81.5258,82.5086) 32 96.16 (96.1087,98.4703)4 82.87 (82.1079,83.0809) 33 99.79 (95.8662,98.4821)5 83.6 (82.0821,83.0942) 34 101.14 (99.0011,101.544)6 83.33 (82.7286,83.7798) 35 102.95 (100.504,102.895)7 83.53 (82.617,83.7905) 36 103.96 (102.172,104.402)8 84.27 (82.7756,83.9735) 37 105.28 (103.108,104.995)9 85.5 (83.5507,84.9931) 38 105.81 (104.339,106.139)10 84.33 (84.7078,86.1404) 39 107.14 (104.923,106.673)11 84.3 (83.7749,85.1391) 40 108.07 (106.093,107.707)12 85.07 (83.6437,85.0469) 41 107.64 (107.002,108.498)13 83.6 (84.4092,86.0276) 42 108.87 (106.715,108.202)14 84.37 (83.3253,85.2615) 43 109.75 (107.79,109.359)15 84.5 (83.9254,86.0376) 44 110.2 (108.66,110.152)16 85.2 (84.1809,86.4398) 45 110.2 (109.095,110.459)17 87.07 (84.7953,87.0151) 46 110.9 (109.148,110.527)18 88.4 (86.5002,88.6809) 47 110.4 (109.839,111.393)19 90.03 (87.8257,89.9086) 48 111 (109.666,111.69)20 92.3 (89.3517,91.3172) 49 112.1 (110.243,112.507)21 92.13 (91.4424,93.2173) 50 112.5 (111.363,113.837)22 93.17 (91.4847,93.1569) 51 113 (111.895,114.525)23 93.5 (92.3494,93.9385) 52 114.3 (112.373,114.993)24 94.77 (92.7216,94.2471) 53 115.1 (113.587,116.247)25 95.37 (93.8327,95.2555) 54 116.4 (114.421,117.066)26 95.03 (94.4973,95.9299) 55 117.8 (115.662,118.337)27 95.23 (94.282,95.7684) 56 116.8 (117.001,119.602)28 95.07 (94.4762,96.0897) 57 117.8 (116.264,118.865)29 96.4 (94.4108,96.1564) 58 119 (117.015,119.553)30 96.97 (95.6567,97.6173) 59 119.6 (118.234,120.909)

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Figure 5.15: Linearity Figures for Quarterly Unemployment and GDP Data

135

Table 5.52: Accuracy Rates for Fuzzy MAR Algorithm Experiments with RealData

Series Variable Rate

x0 0.757143AAA Bonds Interest Rates

x1 0.685714x0 0.717514x1 0.536723

Agriculturex2 0.734463x3 0.581921x0 0.693878

Flour Price Indices x1 0.663265x2 0.683673x0 0.854730x1 0.797980

Forestx2 0.722973x3 0.932432x0 0.482877

Gas Furnacex1 0.708904x0 0.829787x1 0.721429

Grain Price Indicesx2 0.659574x3 0.842857x0 0.644068

Housing Starts and Soldx1 0.686441x0 0.852941

Interest Rates x1 0.844538x2 0.844538x0 0.806122

Investment and Inventoriesx1 0.806122x0 0.516667

Mink-Muskrat Fursx1 0.700000x0 0.765306

Power Station x1 0.642857x2 0.683673x0 0.636364

Production and Billingx1 0.777778x0 0.517241

Umemployment and GDPx1 0.586207

136

5.1.3 Discussion of Experimental Results

Fuzzy MAR Algorithm is tested by using synthetic and real data. Synthetic data

test shows that the algorithm extracts the correct linear relationship between

lagged input and output variables. The width of the fuzzy coefficients produced

by the algorithm is exactly equal to 0 and the output expected is obtained exactly

as expected. This is an obvious situation since the algorithm is based on FLR

which is an LP problem that minimizes the total width of the coefficients. The

approach is very similar to Least Squares Approximation basically. For that

reason, as the data becomes linear the total width of the fuzzy coefficients becomes

smaller.

Uniform random noise is added to the output values in synthetic data exper-

iment. When two lags are used in the experiment, the algorithm yields the same

fuzzy equation. The fuzzy coefficients have different values since the algorithm

tries to obtain the widest equation that keeps all the output values in the interval.

On the other hand, when different number of lags are used, different equations

are obtained. Moreover the rate in which output values are contained in the in-

terval is decreasing as shown in Table 5.2. Since the algorithm aims to obtain the

fuzzy equation which contains as much lagged variables as possible, the resulting

equation gets larger as the algorithm proceeds.

The aim of using Fuzzy MAR Algorithm is to obtain the number of lags and

to extract the fuzzy equations defining each time series variable in a multivariate

system. In order to test the algorithm on real data, thirteen real data sets are

used. The results are as follows:

• The algorithm yields approximately linear relationships between variables

as seen in the related figures.

• The variable with the largest center value and smallest width value has the

most linear relationship with the variable to be defined.

• Since Fuzzy MAR Algorithm aims to find the equation that involves as

many lagged variables as possible by using modified BIC value, the obtained

equation may contain excess variables. A refinement method can be used to

137

limit the number of variables. Another criterion which is based on individual

coefficient center and width values can be developed for that purpose.

5.2 ANFIS unfolded in time

5.2.1 Gas-Furnace Data Experiment

Gas-furnace data is used in some experiments. One of them is a benchmark

experiment in neuro-fuzzy literature. Another experiment is performed to show

the accuracy of the model obtained by using Fuzzy MAR.

5.2.1.1 Gas Furnace Data

Gas-furnace data consists of 296 data pairs. The data has two variables which

are gas flow rate (input X) and the concentration of CO2 in the exhaust gas

(output Y ). The data set consists of measurements sampled at a fixed interval

of 9 seconds. The measured input Xk represents the flow rate of the methane

gas in a gas furnace and the output measurement Yk represents the concentration

of carbon dioxide (CO2) in the gas mixture flowing out of the furnace under a

steady air supply [11].

5.2.1.2 Benchmark Problem

It is stated that the output Y depends on previous values of itself and also the

values of X. Most studies used the inputs Xt−4 and Yt−1 for the output Yt. It is ob-

served that Gas-furnace data in [4] is used as a benchmark problem in the neuro-

fuzzy literature. In order to compare the results for ANFIS unfolded in time,

same data is used in an experiment. The complete set of Gas-furnace data can

be found in Table A.8.

5.2.1.3 Discussion of Training Results

The step-size for learning is set to a very small number (0.02) since the training

algorithm performs online learning. In other words, the parameters are updated

after each data pair is presented to the system.

138

The results are compared with the results taken from [14]. RMSE (root mean

square error) is used as the error criterion in order to compare the accuracy of

the system with other systems. The formula for RMSE is in formula (4.4).

RMSE =

√

∑Kk=1(Yk − Yk)

K(5.1)

In RMSE, K is the number of samples in the data set. Yk is the expected

output for the input given and Y k is the system’s response. RMSE is computed

after each epoch, i.e. after all data set is trained. The size of the data set is 292.

When a smaller set having data pairs is used, the RMSE value decreases to 0.583.

This is much smaller than the RMSE computed when the whole data set is used

which is 0.662 as seen in the Table 5.53. The reason is that the identification of

a function which holds for the whole data set is more difficult when the number

of data points increase.

Table 5.53: Comparison of RMSE values for different Neuro-Fuzzy Systems andANFIS unfolded in time

Model nfMod NFIDENT ANFIS System ANFIS unfoldedin time

RMSE 0.485 0.623 0.241 0.367 0.662Number of rules 26 21 49 26 16

In the results shown in Table 5.53, the error is computed for one-step-ahead

predictions. This means that the model is using the data pair < xt−4, yt−1 >

and producing the result yt rather than taking < xt−4, yt−4 > and producing

yt, as ANFIS unfolded in time does. This is not one-step ahead prediction but

four-step ahead prediction. Since other models are all multi-layer feed-forward

neural network models they take input values only in the layer 0. It is not

logical to take the values of variables at different time instances as input to the

neural network, when temporal data processing is concerned. Since the values

at different time instances are important for different variables, the neuro-fuzzy

model must perform learning and prediction concerning the time interval needed.

ANFIS unfolded in time is a neuro-fuzzy structure which performs the processing

139

for a predefined time interval. This is advantageous when time series prediction

and control applications are concerned.

44

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Out

put

Sample Number

Output of systemExpected output

Figure 5.16: Train results of ANFIS unfolded in time for Gas-Furnace data.

The training results for 296 points in time is displayed in Figure 5.16. It is

observed that there is no significant deviation of output produced by the system

from expected output. 292 data points are used as input since four-step ahead

prediction is performed. The output of the system Y and the expected output Y

can also be seen in Table 5.54.

5.2.1.4 Experiment using Fuzzy MAR result for Gas Furnace Data

Another experiment is performed by using Gas Furnace data in order to show

Fuzzy MAR yields compatible results for the data set given. In the experiment

ANFIS and ANFIS unfolded in time are tested by using the output of Fuzzy

MAR Algorithm for Gas Furnace data. In this experiment, Yt is denoted by x1,t

(The convention used in Fuzzy MAR algorithm). Fuzzy MAR Algorithm gives

x1,t−1, x1,t−4, x0,t−1 as the output for the variable x1,t. According to this result,

140

Table 5.54: Training output in tabular format (for 292 data points in column-wiseorder). Y is the output of ANFIS unfolded in time and Y is the expected output.

Y Y Y Y Y Y Y Y Y Y Y Y

56.57 53.40 47.89 48.30 53.43 52.40 53.32 53.90 54.31 55.60 56.93 56.4054.81 53.10 48.14 47.90 53.46 53.00 54.33 54.10 57.88 58.00 56.30 56.3053.26 52.70 47.39 47.20 52.79 53.40 52.30 54.00 59.35 59.50 57.85 56.4051.54 52.40 47.04 47.20 53.46 53.60 52.48 53.60 59.90 60.00 56.53 56.4052.88 52.20 47.32 48.10 54.07 53.70 53.14 53.20 61.61 60.40 54.95 56.0051.95 52.00 50.76 49.40 52.92 53.80 54.06 53.00 61.60 60.50 54.33 55.2052.88 52.00 50.24 50.60 53.85 53.80 53.19 52.80 59.15 60.20 54.08 54.0051.99 52.40 51.59 51.50 53.12 53.80 51.33 52.30 59.60 59.70 53.59 53.0053.23 53.00 51.48 51.60 53.90 53.30 52.23 51.90 60.50 59.00 52.09 52.0053.52 54.00 50.84 51.20 52.77 53.00 51.32 51.60 57.41 57.60 51.20 51.6055.31 54.90 50.82 50.50 53.52 52.90 52.17 51.60 57.04 56.40 52.60 51.6055.26 56.00 49.61 50.10 53.55 53.40 51.05 51.40 54.74 55.20 50.22 51.1057.43 56.80 50.31 49.80 54.00 54.60 52.29 51.20 55.36 54.50 50.51 50.4056.72 56.80 49.54 49.60 56.71 56.40 50.64 50.70 54.14 54.10 50.13 50.0057.66 56.40 50.44 49.40 56.98 58.00 50.38 50.00 54.41 54.10 49.32 50.0055.89 55.70 49.53 49.30 60.09 59.40 49.71 49.40 53.52 54.40 52.50 52.0053.61 55.00 48.54 49.20 58.85 60.20 48.56 49.30 55.02 55.50 52.76 54.0053.71 54.30 49.84 49.30 60.76 60.00 50.81 49.70 57.26 56.20 54.22 55.1054.29 53.20 49.58 49.70 57.82 59.40 50.94 50.60 56.50 57.00 52.98 54.5052.69 52.30 51.19 50.30 58.06 58.40 50.91 51.80 57.99 57.30 53.62 52.8050.45 51.60 51.27 51.30 58.82 57.60 52.60 53.00 56.95 57.40 52.56 51.4050.67 51.20 52.89 52.80 56.63 56.90 54.56 54.00 57.51 57.00 52.32 50.8051.10 50.80 53.27 54.40 57.22 56.40 54.03 55.30 55.98 56.40 51.24 51.2050.14 50.50 55.42 56.00 56.10 56.00 55.78 55.90 56.93 55.90 52.51 52.0050.94 50.00 57.00 56.90 55.16 55.70 55.45 55.90 55.68 55.50 52.29 52.8049.39 49.20 56.14 57.50 55.46 55.30 54.68 54.60 54.60 55.30 54.10 53.8047.11 48.40 57.32 57.30 53.81 55.00 53.62 53.50 55.97 55.20 53.66 54.5047.53 47.90 55.84 56.60 54.59 54.40 52.30 52.40 54.79 55.40 55.17 54.9048.47 47.60 56.17 56.00 52.65 53.70 53.00 52.10 56.41 56.00 53.37 54.9047.76 47.50 53.99 55.40 52.87 52.80 52.13 52.30 55.56 56.50 53.33 54.8047.13 47.50 54.99 55.40 50.82 51.60 53.97 53.00 57.49 57.10 54.34 54.4048.33 47.60 56.58 56.40 50.26 50.60 54.14 53.80 56.23 57.30 53.49 53.7048.07 48.10 57.45 57.20 50.28 49.40 53.95 54.60 56.01 56.80 54.50 53.3049.31 49.00 56.70 58.00 49.00 48.80 55.40 55.40 56.49 55.60 52.90 52.8049.13 50.00 57.90 58.40 48.92 48.50 55.43 55.90 54.36 55.00 51.48 52.6050.76 51.10 59.12 58.40 48.20 48.70 56.87 55.90 54.77 54.10 51.28 52.6051.93 51.80 57.36 58.10 50.04 49.20 54.83 55.20 54.95 54.30 52.18 53.0051.70 51.90 57.91 57.70 50.07 49.80 54.02 54.40 54.97 55.30 55.66 54.3052.59 51.70 55.94 57.00 49.29 50.40 54.87 53.70 57.73 56.40 55.93 56.0050.51 51.20 56.60 56.00 50.34 50.70 53.86 53.60 57.15 57.20 56.20 57.0050.30 50.00 55.01 54.70 51.42 50.90 52.92 53.60 58.15 57.80 57.32 58.0048.05 48.30 52.03 53.20 50.45 50.70 53.54 53.20 57.10 58.30 58.31 58.6046.43 47.00 52.27 52.10 51.20 50.50 51.69 52.50 58.09 58.60 59.00 58.5047.30 45.80 50.98 51.60 50.06 50.40 51.76 52.00 59.76 58.80 57.26 58.3046.35 45.60 51.45 51.00 51.16 50.20 52.89 51.40 58.73 58.80 57.99 57.8046.18 46.00 49.91 50.50 50.43 50.40 52.14 51.00 57.71 58.60 57.72 57.3047.87 46.90 51.05 50.40 50.12 51.20 51.15 50.90 58.15 58.00 56.54 57.0047.25 47.80 50.27 51.00 52.13 52.30 51.27 52.40 58.17 57.4048.71 48.20 51.24 51.80 53.66 53.20 54.07 53.50 56.21 57.00

141

Table 5.55: Training and Recognition Results for Gas Furnace Data Experimentby Fuzzy MAR

ANFIS ANFIS Unfolded in timeTrain. Recog. Train. Recog.

RMSE 0.109253 1.406189 0.380676 0.918988Number of rules 27 25

ANFIS unfolded in time must be constructed for four lags, and the input vari-

ables must be x0 and x1. ANFIS unfolded in time is trained by using 25 rules and

x0,t−4 and x1,t−4 as input. The output is obtained for four-ahead time instance,

so the output is x1,t. Obtained model is the same for benchmark problem and

output model of Fuzzy MAR. ANFIS is trained by using 27 rules. It takes x1,t−1,

x1,t−4 and x0,t−1 as input and gives x1,t as output. The data set contains samples

for 292 time instances. 150 samples are used in training phase and all of the data

set is used in recognition phase. The experimental results are shown in Table

5.55.

Experimental results show that

1. The training error of the model obtained by Fuzzy MAR Algorithm is

less than the Benchmark model in Section 5.2.1.2 ANFIS. ANFIS yields

0.109253 training error for Fuzzy MAR output and 0.241 for the Bench-

mark problem. ANFIS Unfolded in time yields 0.380676 for Fuzzy MAR

output.

2. The training error obtained by using ANFIS is smaller than the one ob-

tained by ANFIS unfolded in time. On the other hand, the recognition

rate obtained by using our model is smaller than ANFIS. Our model yields

0.918988 recognition error whereas ANFIS yields 1.406189 recognition error.

This experiment shows that ANFIS Unfolded in time gives better recogni-

tion results than ANFIS.

Figure 5.17 shows the recognition output by using ANFIS Unfolded in time.

In this figure, expected output and the change in error are also shown. The error

142

varies around zero. Moreover, the obtained output changes in time similarly to

the expected output.

-10

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0 50 100 150 200 250 300

Out

put

Sample Number

Expected OutputOutput

Error

Figure 5.17: ANFIS unfolded in time output for gas furnace data

Another experiment is performed by using ANFIS only. Both the benchmark

model and the output of Fuzzy MAR are tested by using ANFIS. 292 samples

(whole data set) and 64 rules are used in training. In other words, the number of

rules and the data set are the same for both of the experiments. ANFIS produces

0.223727 for benchmark problem and 0.165645 for the output of Fuzzy MAR.

5.2.2 Comparison of MAR and Fuzzy MAR Algorithm

MAR Algorithm is tested by using variables x2 and x3 of Agriculture data as in

[1]. The algorithm yields linear functions

x2,t = f(x2,t−1) (5.2)

and

x3,t = f(x3,t−1, x1,t−1, x1,t−2). (5.3)

143

Fuzzy MAR Algorithm gives

x2,t = f(x2,t−1, x2,t−2, x3,t−2, x1,t−1) (5.4)

and

x3,t = f(x0,t−1, x0,t−2, x1,t−1, x1,t−2, x3,t−1, x3,t−2). (5.5)

The functions obtained by the two methods are using in training phase of both

ANFIS and ANFIS Unfolded in time. Training data set contains samples for first

90 time instances. Recognition is performed on the whole data set. Data set size

varies according to the lagged variables.

ANFIS directly gets the input data, whereas our model gets the data cor-

responding to a specific time instance. The results shown in Table 5.56 show

that

1. Both ANFIS and ANFIS unfolded in time yields less training and recog-

nition errors for Fuzzy MAR Algorithm than MAR Algorithm. In other

words, Fuzzy MAR produces slightly better models than non-fuzzy MAR

Algorithm for both x2 and x3.

2. ANFIS yields better training errors, but worse recognition errors than AN-

FIS Unfolded in time. Our model uses on-line learning and it produces a

cumulative error for each of the samples.

Table 5.56: Training and Recognition Results for Agriculture Data

ANFIS ANFIS Unfolded in timeTrain. Recog. Train. Recog.

MAR-X2 3.632772 860.033991 6.023887 10.661MAR-X3 9.122629 65033.375375 1061.65014 1912.19Fuzzy-X2 1.461472 177.453338 5.934202 8.91102Fuzzy-X3 5.529144 5241.348344 860.494170 1049.02

144

5.2.3 Real Data Experiments

In this section, the results obtained by using real data sets are presented. All

the thirteen data sets are used in the real data experiments. Variable x1 in

Gas Furnace data is used in another experiment and the results can be seen in

Section 5.2.1.4 also. Variables x2 and x3 in Agriculture data are also used in

another experiment in order to compare MAR and Fuzzy MAR algorithm. The

results can be seen in Section 5.2.2. The rest of the data sets are tested by using

ANFIS and ANFIS Unfolded in time. First half of the observations are used in

training phase. All the data set is used in recognition phase. The training and

recognition results are presented in Table 5.57. The figures between 5.18 and 5.30

show the expected output, obtained output and error (RMSE) for all data sets.

ANFIS unfolded in time performs better prediction for variable x0 than x1 in

the AAA-CP bonds data set as seen in Figure 5.18. In Table 5.57, the recognition

rate for x0 is slightly better than the recognition rate for x1 which are 0.650826

and 1.146950 respectively.

For the Agriculture data, the recognition results are very different as shown in

Table 5.57. x0 and x2 are recognized better than x1 and x2. This is also validated

in Figure 5.19.

As seen in Figure 5.20, the recognition phase yields similar patterns for the

variables x0, x1 and x2 in Flour Prices data set. All the variables are recognized

with recognition errors close to training errors as seen in Table 5.57.

For the Forestry data set, it can be said that the recognition errors for x1 and

x3 are superior to the errors for x0 and x2 given as in Table 5.57. The output

figure for x3 shows the best recognition result in Figure 5.21.

Variable x0 of Gas Furnace is also used in real data experiments. Figure 5.22

shows that the expected and obtained x0 are very close to each other. This can

also be seen in Table 5.57. The recognition error is 0.201765.

For the Grain Prices data set, the recognition errors changes between 0.075

and 0.17 which are very close results given as Table 5.57. In Figure 5.23, it can

be seen that the output for x0 simulates successfully the behavior of the original

x0 data. The other variables also yields error very close to zero.

145

Table 5.57: RMSE for Experiments with ANFIS and ANFIS Unfolded in timeusing Real Data

ANFIS ANFIS Unfolded in time

Series Var. Train. Recog. Train. Recog.

x0 0.119902 0.652238 0.129453 0.650826AAA Bonds Interest Rates

x1 0.173633 4.317652 0.409995 1.146950

x0 0.406124 38.933538 1.491038 1.56818

x1 174.645974 24059.645974 1622.169764 2181.88Agriculture

x2 1.461472 177.453338 5.934202 8.91102

x3 5.529144 5241.348344 860.494170 1049.02

x0 2.960263 28.606507 7.628765 7.875690

Flour Price Indices x1 2.689367 71.649687 8.334503 8.080240

x2 9.378680 9.485944 10.297852 8.69711

x0 30.690742 978622.038395 53.744128 537.123

x1 0.016055 307.751859 0.056393 0.441184Forest

x2 2.345635 7352.330969 18.841375 68.5216

x3 0.604859 0.848928 1.049248 0.895668

x0 0.213040 0.514342 0.211783 0.201765Gas Furnace

x1 0.109253 1.406189 0.380676 0.918988

x0 0.107664 0.41797 0.389521 0.179064

x1 1.544528 2.923548 0.048258 0.086044Grain Price Indices

x2 0.041943 0.307117 0.094808 0.139154

x3 0.050431 0.092057 0.068364 0.0756004

x0 0.000747 243.336994 6.911277 12.946600Housing Starts and Sold

x1 2.667528 584.922654 5.700290 11.561000

x0 0.156515 1.104897 0.222275 0.745151

Interest Rates x1 0.143651 8.526677 0.2177 0.374334

x2 0.153290 0.665197 0.255971 0.503007

x0 2.538655 9.461350 14.156823 6.830420Investment and Inventories

x1 1.737847 1980.411870 4.490392 6.38886

x0 0.000006 1.262477 0.268479 0.295973Mink-Muskrat Furs

x1 0.241326 0.414645 0.284571 0.492810

x0 0.000026 2.430449 0.592158 0.963655

Power Station x1 0.000175 19.510901 0.825107 0.871221

x2 0.000118 5.237194 0.486922 0.947947

x0 1.065657 1.851736 1.339888 1.783490Production and Billing

x1 1.285181 18.492608 4.48082 7.087390

x0 1.361111 260.3560 0.000028 29.644100Umemployment and GDP

x1 0.211697 9.657877 1.027380 9.2844100

146

The Housing data set has approximately same recognition errors for both x0

and x1 as seen in Table 5.57 which are 12.9466 and 11.561. Also as seen in Figure

5.24, both for x0 and x1, the output of ANFIS unfolded in time follows the same

pattern with the original output but misses the local peak data points.

The Interest Rates data set variables x0, x1 and x2 yields promising recognition

errors in Table 5.57. Figure 5.25 validates this situation. The outputs obtained

are very close to the expected output.

The recognition errors for x0 and x1 in Investment and Inventories data set

are very close to each other in Table 5.57. Out-of-sample recognition error is

slightly higher than the error for training sample for both variables given as in

Figure 5.26.

The recognition errors are slightly worse for x1 than the error for x0 in Mink-

Muskrat data set in Table 5.57. In Figure 5.27, the outputs are close to the

expected results for both of the variables.

In Table 5.57, for the Power Station data, x0 and x2 has approximately the

same recognition errors (0.963655 and 0.947947 respectively), whereas x1 yields

better recognition error (0.871221). Besides to that x1 yields the best figure

among other variables in Figure 5.28.

For Production and Billing data set, the recognition error for x0 is small

compared to x1 in Table 5.57. In Figure 5.29, the output x0 yields a linear figure

missing most of the peak values, whereas x1 adapts itself to the fluctuations in

expected output values for x1.

x0 in Unemployment and GDP data gives worse recognition error than x1 as

in Table 5.57. For the out-of-sample values, the recognition gets worse for x1 as

seen in Figure 5.30. On the other hand, x0 simulates the expected results better.

The experimental results show that

1. ANFIS yields smaller training errors than ANFIS Unfolded in time. This

is an expected result, since during the training phase, our model uses online

learning. The error obtained for a sample is a cumulative error containing

the error in the whole time interval.

2. Our model gives better recognition results than ANFIS. This is also not

147

surprising, since the neuro-fuzzy model is performing t-ahead prediction,

the recognition results are promising.

148

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Figure 5.18: Output for AAA-CP Bonds data (a) x0 (b) x1

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Figure 5.19: Output for Agriculture data (a) x0 (b) x1

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Figure 5.19: Output for Agriculture data(cont’d) (c) x2 (d) x3

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Figure 5.20: Output for Flour Price data (a) x0 (b) x1

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Figure 5.20: Output for Flour Price data(cont’d) (c) x2

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Figure 5.21: Output for Forestry data (a) x0 (b) x1

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Figure 5.21: Output for Forestry data(cont’d) (c) x2 (d) x3

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Figure 5.22: Output for Gas Furnace data variable x0

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Figure 5.23: Output for Grain Price data (a) x0 (b) x1

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Figure 5.23: Output for Grain Price data(cont’d) (c) x2 (d) x3

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Figure 5.24: Output for Housing data (a) x0 (b) x1

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(b)

Figure 5.25: Output for Interest Rate data (a) x0 (b) x1

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(c)

Figure 5.25: Output for Interest Rate data(cont’d) (c) x2

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Figure 5.26: Output for Investment and Inventories data (a) x0 (b) x1

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(b)

Figure 5.27: Output for Mink and Muskrat Furs data (a) x0 (b) x1

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(b)

Figure 5.28: Output for Power Station data (a) x0 (b) x1

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Figure 5.28: Output for Power Station data(cont’d) (c) x2

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(b)

Figure 5.29: Output for Production and Billing data (a) x0 (b) x1

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Figure 5.30: Output for Unemployment and GDP data (a) x0 (b) x1

167

CHAPTER 6

CONCLUSION AND FUTURE WORK

Time series identification and prediction are studied in the scope of this thesis.

An approach combining time series analysis with rule-based systems is proposed

for forecasting problems. A time series analysis approach which is based on fuzzy

linear regression is designed in order to construct rules for defining variables in a

multivariate time series data. Moreover, a neuro-fuzzy system is constructed for

predicting the time series data by means of a temporal learning algorithm and a

temporal neuro-fuzzy model.

The resulting approach is a hybrid approach which combines time series anal-

ysis with neuro-fuzzy systems. Fuzzy multivariate time series analysis is used

to extract fuzzy temporal rules defining the time series data in order to predict

the future values. The resulting algorithm is named Fuzzy MAR. These rules are

used to determine the time intervals to be used in the neuro-fuzzy system called


FLR is applied to multivariate time series analysis in order to find a linear

function in a multivariate time series data. The algorithm is based on finding

fuzzy linear functions and comparing them by using BIC. The function containing

the optimum number of variables and lags is found as a result of execution of the

algorithm. The functions provide necessary information for any fuzzy rule-based

system. In this thesis, a neuro-fuzzy system is designed which is convenient for

online learning of temporal data. The output of Fuzzy MAR is processed by

168


Since the recurrent neural network structure seems to be convenient for time

series analysis, unfolding-in-time approach is useful to represent a recurrent neural

network as a feed-forward neural network. Unfolding-in-time approach is applied

to a neuro-fuzzy system in order to predict multivariate time series data in this

thesis. The neuro-fuzzy system which is basically a black box of feed-forward

neural networks is duplicated for T time intervals in this approach. The number

of time intervals is provided to the neuro-fuzzy system as an argument. As an

alternative, tests can be iteratively performed to find the best number of time

intervals for the time series data given.

Fuzzy MAR algorithm is developed to find the number of time intervals. Fuzzy

MAR algorithm produces the best number of previous instances and number of

variables by means of a predetermined information criterion. This algorithm is

based on Fuzzy Linear Regression Analysis. In Fuzzy Linear Regression, the

difference between the actual and the expected values is treated as the vagueness

of the system. The aim of the fuzzy linear regression study is to find the fuzzy

parameters which minimizes the cost function and finds the best parameters.

There is a system interface module which combines two components of the

main neuro-fuzzy system. The system interface is an important component which

forms a bridge between different modules. It takes the output of the Fuzzy MAR

Algorithm and converts it to a format acceptable to the ANFIS unfolded in time.

ANFIS unfolded in time contains neural network structures duplicated for the

time intervals obtained as a result of Fuzzy MAR Algorithm and transmitted by

the system interface. ANFIS unfolded in time accepts another input externally,

which is the multivariate time series data to be trained. The input is trained by

the neuro-fuzzy black box and forecasting is performed continuously.

6.1 Concluding Remarks

The proposed model covers time series data analysis by combining neuro-fuzzy

concepts. In order to show the applicability of the model both of the components

are tested by using many multivariate time series data. Fuzzy MAR Algorithm

169

aims to find the best fuzzy equation for a variable, and the equation is used in

the neuro-fuzzy system ANFIS Unfolded in time for forecasting.

Since the resulting model can be used for small time intervals T, it can be

applied to areas including short-term prediction, such as

• Financial or meteorological data forecasting can be an application area,

since the model is convenient for forecasting problems.

• Sequence detection given by Rumelhart [37] is a possible application of

unfolding-in-time concept.

• Image processing, specifically motion detection can be a different appli-

cation area for ANFIS unfolded in time because of the usage of temporal

expert system rules.

6.2 Future Directions

As the future study, the model can be enhanced in order to improve the accuracy

and robustness. The followings are some tasks that can be achieved in the future

study.

• The equation obtained at the end of execution of Fuzzy MAR may be further

processed in order to extract the surplus variables. Since the algorithm

is applied to the raw time series data, sometimes excess information is

obtained for a variable.

• Different error measures can be tried in order to find the best-fit model for

unfolding-in-time approach.

• Learning algorithms other than back-propagation may be used for temporal

learning.

• Partial data forecasting can be performed by using the neuro-fuzzy system.

An approach by Pal and Mitra [34] is introduced in Chapter 2. Time series

data may have incomplete data. Neuro-fuzzy approach to process partial

or incomplete data may be integrated to the system interface module.

170

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174

APPENDIX A

TIME SERIES DATA

Table A.1: Syntectical Data produced by xt = 0.67xt−1 + 0.47xt−2

Time X1-X50 Time X51-X100 Time X101-X150 Time X151-X200

1 0.500000000000 51 57.266612917789 101 6148.146686750887 151 660065.3636713338782 0.600000000000 52 62.880842460277 102 6750.890676688923 152 724775.9913108010083 0.637000000000 53 69.045472519746 103 7412.725696154495 153 795830.6351037636634 0.708790000000 54 75.814462544560 104 8139.444834467306 154 873851.2414355981165 0.774279300000 55 83.247061989136 105 8937.409116285708 155 959520.7302606196616 0.851898431000 56 91.408328928664 106 9813.603180111058 156 1053588.9727493464027 0.934683219770 57 100.369699517099 107 10775.696415328692 157 1156879.3549645533798 1.026630019816 58 110.209613272929 108 11832.110092922421 158 1270295.9850184435499 1.127143226569 59 121.014199665899 109 12992.091077462508 159 1394831.606795697240

10 1.237702071114 60 132.878032014429 110 14265.792765573418 160 1531576.28951178561011 1.359017704134 61 145.904955292640 111 15664.363959341570 161 1681726.96916687418712 1.492261835193 62 160.208995092850 112 17200.046452578357 162 1846597.92541234497913 1.638553750523 63 175.915355699750 113 18886.282184118038 163 2027632.28553470200914 1.799194075391 64 193.161516012472 114 20737.830896070915 164 2226414.65625205263515 1.975580293258 65 212.098432907239 115 22770.899326902992 165 2444684.99389018537516 2.169260011916 66 232.891862573712 116 25003.283070178335 166 2684353.83434488903717 2.381926945815 67 255.723811390789 117 27454.522340663891 167 2947519.01613946305618 2.615443259297 68 280.794129041473 118 30146.073011228626 168 3236484.04295553825819 2.871852648262 69 308.322257811458 119 33101.494417635207 169 3553778.24636575812520 3.153399606205 70 338.549153383169 120 36346.655575093042 170 3902178.92525416100421 3.462548480840 71 371.739393938109 121 39909.961611600884 171 4284735.65571219380922 3.802005297079 72 408.183496028623 122 43822.602400066324 172 4704796.98419662565023 4.174741335038 73 448.200457490088 123 48118.825565496853 173 5166039.73759646993124 4.584019184103 74 492.140549651812 124 52836.236256914061 174 5672501.20676204934725 5.033421280817 75 540.388383287055 125 58016.126307915940 175 6228614.48520091362326 5.526881274676 76 593.366275138679 126 63703.835667053288 176 6839247.27226277533927 6.068718456017 77 651.537944487831 127 69949.149261646191 177 7509744.48046048916928 6.663675564629 78 715.412572122026 128 76806.732768817994 178 8245975.01987203210629 7.316960302629 79 785.549257231038 129 84336.611108081765 179 9054383.16913069225830 8.034290918137 80 862.561911242147 130 92604.693843759247 180 9942044.98265741951831 8.821946257387 81 947.124631430826 131 101683.352096117131 181 10916730.22787189669932 9.686820723974 82 1039.977601342463 132 111652.052010965330 182 11986970.39452315866933 10.636484626035 83 1141.933569671939 133 122598.050332521831 183 13162133.37143030762734 11.679250439711 84 1253.884964311156 134 134617.158167943329 184 14452505.44428419135535 12.824245568843 85 1376.811703834286 135 147814.579628807289 185 15869381.33224265277436 14.081492237789 86 1511.789774795215 136 162305.832690234238 186 17425163.05141614750037 15.461995216674 87 1660.000649914909 137 178217.760327996366 187 19133468.47060286626238 16.977838146933 88 1822.741629596740 138 195689.640784167656 188 21009250.50946950912539 18.642289310282 89 2001.437197289823 139 214874.406679550622 189 23068928.02252791822040 20.469917766947 90 2197.651488094649 140 235939.983643857704 190 25330529.51454437524141 22.476720879687 91 2413.101979749631 141 259070.760180773446 191 27813850.94533285498642 24.680264339855 92 2649.674525836738 142 284469.201633731311 192 30540629.00520886853343 27.099835921156 93 2909.439862792941 143 312357.622379563516 193 33534731.37779638543744 29.756614306907 94 3194.671735214538 144 342980.131762161269 194 36822365.65557174384645 32.673854468571 95 3507.866798106423 145 376604.770799042890 195 40432308.73679737001746 35.877091218189 96 3851.766470282136 146 413525.858363574545 196 44396158.71177295595447 39.394362716415 97 4229.380930199050 147 454066.567379145126 197 48748611.44318264722848 43.256455892546 98 4644.015464265967 148 498581.753574907256 198 53527764.26146566122849 47.497175924721 99 5099.299398251751 149 547461.061563386116 199 58775449.43347784131850 52.153642139060 100 5599.217865033678 150 601132.335427675163 200 64537600.323319017887

175

Table A.2: Syntectical Data with Uniform Random Noise

Time X0-X49 Time X50-X99 Time X100-X149 Time X150-X199

0 0.500000000000 50 59.053504597779 100 6182.625988781500 150 685016.3387166829781 0.600000000000 51 61.868006636520 101 6981.920567941279 151 728976.1538851738442 0.626498624025 52 71.508018219946 102 7498.560755214894 152 796504.4144489567263 0.711820150927 53 76.988596612863 103 8395.377603181158 153 885604.7971196193254 0.803544100815 54 83.337102563641 104 9088.845823180218 154 984721.5552095117755 0.839407019255 55 91.758647164220 105 10120.178053208219 155 1093479.1101413071166 0.920626540495 56 101.320421216112 106 10970.878027172137 156 1160419.1885258655537 1.067570710928 57 111.297202773563 107 11759.250908586931 157 1318798.6183827556678 1.151979928548 58 125.541731357713 108 12962.735468500754 158 1408543.2829401269569 1.213325841048 59 134.526912215200 109 14514.619971773634 159 1565978.768943440635

10 1.399839483734 60 150.781322591974 110 15410.740128917792 160 1748103.70503105968211 1.508580586436 61 159.161258550673 111 16955.752814175470 161 1847771.61321087065212 1.635367147803 62 181.497045936689 112 19036.888020339502 162 2042783.86169778578913 1.854634470208 63 196.199769985326 113 21518.988834515909 163 2304291.88371169287714 1.968284826005 64 214.860267297981 114 23443.071269066244 164 2500962.28664805227915 2.255864028691 65 236.460917875923 115 25853.230022887441 165 2641206.79325872799416 2.457732487415 66 254.658163960774 116 27251.612968149468 166 2969425.78437052806817 2.612546396424 67 291.441766902149 117 31263.688567146557 167 3355084.09240595949818 2.856675931529 68 309.519012851873 118 34077.481245226161 168 3548424.57292992994219 3.211874819838 69 350.674383559545 119 37706.697475480869 169 3877704.59644405497220 3.569279009704 70 366.108685955140 120 39193.218651800620 170 4319055.00350032467421 3.855651059849 71 405.221439065847 121 43708.002462441640 171 4660114.63356059230922 4.276120391471 72 450.815902581423 122 48871.862545686228 172 5082518.62570625823023 4.571132730451 73 503.322969487533 123 52861.008687478170 173 5588386.82964650169024 5.074528681202 74 540.204849935177 124 59003.245068426237 174 6293930.50482050981425 5.714713759887 75 609.191125507313 125 65590.075577727970 175 7055405.40996747836526 6.000668494013 76 662.547704201242 126 69383.832818912400 176 7598752.47499739378727 6.754173426700 77 720.063265073372 127 77726.044098463462 177 8447700.37031914666328 7.192109376263 78 816.444199128776 128 84286.205372067532 178 9344011.91091013886029 8.062838045621 79 888.409972625435 129 90871.804225523316 179 10296045.91542307473730 8.787072771165 80 973.697556219250 130 104815.733127727450 180 11221582.28671600297131 9.680237886140 81 1071.730325276762 131 114444.515314577657 181 12344043.63154543936332 10.873758562171 82 1180.598745635320 132 126951.430185852572 182 13180581.00690791942233 11.674126585777 83 1283.629600827744 133 134938.175506607455 183 14290230.59509716555534 13.126840536746 84 1385.098820726074 134 150246.709481542726 184 15920778.14674775302435 14.364582805839 85 1504.141627758675 135 164327.110347972950 185 17210233.07701188698436 15.537732032386 86 1639.341005942036 136 177081.525127817062 186 18877712.32223168760537 17.462666813112 87 1887.433029760783 137 194626.469926275196 187 21561763.39969836175438 19.014807990789 88 2051.091594561224 138 220524.692820748838 188 22742449.43236260488639 20.202444432163 89 2164.615650233885 139 238621.431518274389 189 25747555.98936603218340 23.125439794474 90 2432.166056965779 140 267363.429652638908 190 28057519.52919278293841 24.927564623919 91 2697.980487220107 141 293033.040707894485 191 30092203.40137823298642 27.584080927856 92 2904.865792815849 142 317718.265575727448 192 34778835.98120837658643 29.367309936772 93 3193.488574715840 143 342974.518676676962 193 38156559.20885486900844 32.701110325481 94 3486.679244608566 144 387064.171137345664 194 39635624.13192006945645 37.103230461107 95 3972.901932720978 145 416288.533303717326 195 45289376.20349620282646 38.875946569074 96 4377.217215297515 146 451331.746406622871 196 47781424.79164095222947 42.938931023570 97 4689.493416515005 147 488800.645693974569 197 53941433.01296208798948 47.167949533482 98 5249.061017487089 148 549516.294566310127 198 59760759.36663043499049 52.234777750266 99 5592.700930818339 149 591190.642984349863 199 64450683.342476345599

176

Table A.3: Quarterly AAA Corporate Bonds and Commercial Interest Rates

AAA Commercial AAA Commercial

Year Quarter Bond Rate Paper Rate Year Month Bond Rate Paper Rate

1953 1 3.070 2.327 1962 1 4.410 3.2432 3.323 2.623 2 4.297 3.2033 3.270 2.747 3 4.337 3.3334 3.133 2.373 4 4.257 3.263

1954 1 2.957 2.037 1963 1 4.197 3.3102 2.877 1.633 2 4.220 3.3173 2.883 1.363 3 4.287 3.6974 2.887 1.310 4 4.333 3.907

1955 1 2.980 1.613 1964 1 4.370 3.9502 3.033 1.967 2 4.407 3.9333 3.100 2.327 3 4.410 3.9104 3.117 2.833 4 4.430 4.063

1956 1 3.097 3.000 1965 1 4.420 4.3002 3.263 3.263 2 4.443 4.3803 3.423 3.350 3 4.497 4.3804 3.677 3.630 4 4.613 4.470

1957 1 3.700 3.630 1966 1 4.813 4.9702 3.773 3.683 2 5.003 5.4273 4.070 3.953 3 5.320 5.7904 3.997 3.993 4 5.383 6.000

1958 1 3.607 2.917 1967 1 5.120 5.4502 3.580 1.817 2 5.263 4.7173 3.870 2.130 3 5.617 4.9734 4.093 3.213 4 6.027 5.303

1959 1 4.130 3.303 1968 1 6.127 5.5802 4.353 3.603 2 6.253 6.0803 4.473 4.193 3 6.077 5.9634 4.570 4.760 4 6.243 5.963

1960 1 4.553 4.678 1969 1 6.700 6.6572 4.453 4.073 2 6.887 7.5403 4.313 3.373 3 7.063 8.4874 4.320 3.270 4 7.467 8.620

1961 1 4.270 3.013 1970 1 7.893 8.5532 4.283 2.860 2 8.140 8.1673 4.437 2.897 3 8.220 7.8374 4.410 3.057 4 7.907 7.293

177

Table A.4: Monthly Agriculture Data E: First difference of the logarithm ofExchange rate, P: Price, SA: Logarithm of levels of sales, SH: Logarithm ofshipments

Year E SA P SH

1978:02 74.9710 3330.5000 158.0548 3477.50001978:03 73.6915 4673.6001 161.8651 3531.10011978:04 72.1436 4233.6001 173.4470 3854.30001978:05 73.6363 5134.2998 168.2341 5701.70021978:06 72.5750 3708.5000 158.0340 5052.29981978:07 70.0887 5487.2998 142.5968 5550.50001978:08 68.0279 2735.8000 135.8864 3982.80001978:09 67.9020 2762.8999 135.8470 4097.39991978:10 66.4023 4899.1001 137.0749 4062.50001978:11 68.2956 5689.2002 141.4187 3630.20001978:12 68.9532 3599.2000 139.7100 3551.50001979:01 70.8264 4075.6001 147.2295 3745.20001972:02 71.6424 4875.1001 146.1185 3130.30001979:03 72.2849 2768.8999 147.4998 3779.89991979:04 73.3875 8038.7998 146.9257 5637.20021979:05 73.8351 7214.7002 148.8361 4370.70021979:06 74.3947 9728.0000 157.9763 5500.00001979:07 73.6612 11197.4004 168.5065 7013.89991979:08 74.2412 2791.6001 153.3172 5320.20021979:09 74.3801 5573.1001 148.9070 4541.79981979:10 75.2657 4900.5000 151.0461 5317.89991979:11 77.8164 5554.2998 146.3190 5204.20021979:12 77.1804 4477.6001 144.9154 6290.89991980:01 77.1146 6170.2998 131.6884 4416.20021980:02 77.8007 3916.6001 134.5485 4578.89991980:03 79.3713 3916.6001 130.4765 6877.20021980:04 79.7922 2781.3999 125.9952 5198.89991980:05 76.4102 4753.8999 127.5110 4112.10011980:06 74.8877 6771.2002 129.1551 5259.39991980:07 74.7106 6520.1001 146.9212 4496.70021980:08 75.5064 10375.9004 157.9019 6412.20021980:09 73.7472 7935.5000 157.5575 4603.60011980:10 73.3667 8073.1001 155.6899 6386.00001980:11 74.8438 5984.3999 160.0905 5895.00001980:12 74.9620 3824.8000 160.7453 5245.20021981:01 73.7306 3613.3000 162.9302 6361.39991981:02 75.6072 3015.3999 162.2236 5301.79981981:03 76.0088 1441.0000 148.5428 5333.29981981:04 76.9838 3568.7000 149.0356 4871.39991981:05 79.1158 3703.0000 145.3769 5574.29981981:06 80.3512 1973.4000 140.6853 3789.10011981:07 82.5629 3370.2000 141.9630 4077.50001981:08 83.8029 4375.1001 134.4076 3130.00001981:09 82.2222 6127.2002 123.5228 4143.50001981:10 82.2338 4026.6001 117.8251 5347.39991981:11 81.0846 3687.5000 113.1628 4531.60011981:12 80.6541 2401.8999 111.1705 4099.79981982:01 81.4489 4483.8999 108.9824 4732.50001982:02 84.3711 5615.2002 115.1846 3900.60011982:03 87.9434 4702.1001 116.6017 4550.00001982:04 88.5295 4962.1001 120.5543 5692.20021982:05 86.9241 3101.5000 119.9183 5253.70021982:06 91.2189 2686.1001 116.9227 4051.70001982:07 92.9265 5032.2002 112.5357 3523.20001982:08 95.9211 4222.1001 105.1904 2707.60011982:09 96.9391 6372.1001 102.3565 2863.00001982:10 96.3476 2314.2000 93.5086 4557.00001982:11 95.6618 4339.2002 105.1904 4384.20021982:12 91.8009 4868.2998 106.5480 4588.39991983:01 90.7740 5220.3999 108.8314 4269.20021983:02 91.4701 4315.0000 117.5161 4427.29981983:03 91.6144 4463.1001 123.9069 3830.50001983:04 91.9752 4403.3999 133.3176 4405.79981983:05 91.6901 3923.3000 133.5697 3677.00001983:06 93.4442 3966.1001 134.4745 3559.39991983:07 94.8545 5834.3999 139.5170 3604.00001983:08 96.1456 4648.7002 153.5271 3062.89991983:09 95.6553 5069.7998 148.2086 4003.50001983:10 93.8139 3139.6001 145.9925 4558.70021983:11 94.8224 4497.1001 145.7499 5010.79981983:12 95.3536 2635.2000 141.2319 5215.89991984:01 95.4012 2457.0000 139.9417 3942.39991984:02 94.5507 3493.2000 133.5152 3848.50001984:03 93.1276 4894.7002 143.2249 4927.39991984:04 93.1344 2595.8000 144.2228 3908.50001984:05 94.6948 2683.3999 141.0589 3500.20001984:06 95.0941 2993.7000 141.5730 3507.39991984:07 98.0450 9208.2998 137.2361 2421.50001984:08 98.7482 4667.7998 134.8748 3866.39991984:09 99.9553 3096.3999 130.5609 2339.20001984:10 100.4023 3567.3999 118.9555 3919.10011984:11 99.3523 3128.2000 115.7184 7126.00001984:12 100.8191 2189.7000 113.0435 4940.79981985:01 101.9379 3102.8999 117.0213 4747.60011985:02 103.3904 4010.5000 116.1665 3945.39991985:03 103.4036 5448.3999 118.3317 5090.70021985:04 100.7625 1819.4000 118.1026 3787.00001985:05 101.5515 2739.1001 114.0097 3696.7000

178

Table A.4: Monthly Agriculture Data (cont’d)

Year E SA P SH

1985:06 101.4990 2304.1001 113.2623 2166.89991985:07 99.2322 3066.5000 113.3721 1969.10001985:08 100.0082 2659.3999 103.2132 2470.80001985:09 100.6550 4841.5000 100.8815 2025.40001985:10 95.8412 3409.6001 97.1817 2908.39991985:11 94.1735 4749.0000 105.4159 6241.79981985:12 93.6765 3158.3999 107.1429 3499.80001986:01 91.7611 2634.5000 104.6512 5092.70021986:02 87.5968 1214.3000 103.2448 3082.60011986:03 85.7529 873.4000 100.6979 2096.50001986:04 84.6784 1594.2000 102.4096 1237.50001986:05 83.0553 2008.5000 106.0000 1426.40001986:06 83.9655 1733.7000 106.1061 1306.20001986:07 81.9692 1402.7000 85.5131 907.30001986:08 80.7844 3308.0000 74.5217 1458.80001986:09 80.2596 2724.2000 67.4044 2183.30001986:10 80.1017 2742.8999 67.2016 3587.30001986:11 81.8333 2479.0000 74.1483 2948.30001986:12 82.7806 2770.8000 74.2227 2328.50001987:01 80.1827 2333.8999 69.6517 3019.89991987:01 79.7820 5212.7002 68.3168 2526.89991987:01 79.2150 5929.7002 72.1344 3630.30001987:01 76.5630 3410.3000 74.5829 4380.00001987:01 75.3233 3964.3000 79.9220 5060.50001987:01 75.9641 2313.7000 79.6116 2863.70001987:01 77.3026 3645.3999 74.3961 3949.70001987:01 76.5162 1829.2000 69.3642 2724.39991987:01 74.9921 3045.7000 71.3597 3051.89991987:01 74.6192 5668.2002 76.8492 4093.80001987:01 71.9004 2868.1001 79.6545 3344.70001987:01 69.7991 2538.3000 80.6142 3176.89991988:01 69.3949 6061.7002 83.1740 3805.00001988:02 69.7908 3908.8999 83.9695 3255.50001988:03 68.9160 4595.3999 86.7493 3447.10011988:04 67.8466 2938.5000 85.0662 5182.89991988:05 68.0343 2665.8000 84.5070 3916.10011988:06 69.0880 6328.5000 110.0746 3307.50001988:07 71.0494 4239.3999 120.4819 3810.20001988:08 71.2644 2464.3000 110.1852 3301.70001988:09 71.3566 5501.2998 112.8585 4327.79981988:10 69.5652 2977.1001 111.8299 4321.70021988:11 67.1983 3791.1001 105.2632 4113.50001988:12 66.9505 6568.2998 108.2569 4191.29981989:01 68.1120 3006.6001 107.6923 4721.10011989:02 68.3782 2646.1001 106.4982 4212.10011989:03 70.3833 6668.5000 106.7265 5279.20021989:04 69.8679 4633.6001 103.2947 4562.20021989:05 71.8275 3050.6001 105.1237 4562.00001989:06 73.4650 4009.1001 101.8601 6491.89991989:07 71.9767 3377.7000 95.7447 3016.00001989:08 73.0268 2952.1001 91.0714 2719.10011989:09 74.2547 7446.1001 91.6370 3339.00001989:10 72.7193 8540.5996 94.8582 3934.30001989:11 73.3739 6546.5000 97.6043 7061.20021989:12 72.6713 3899.3999 97.3451 6881.20021990:01 72.6099 4712.3999 92.2541 5958.39991990:02 72.5494 4081.8999 93.5315 4498.29981990:03 74.2819 5316.6001 96.3222 5487.10011990:04 74.6865 5511.7002 104.2945 4706.70021990:05 73.2494 5235.2998 106.4572 4519.70021990:06 73.8561 2980.1001 104.9869 5749.00001990:07 74.5365 3937.1001 100.4367 3104.50001990:08 73.6267 2599.3999 94.4206 3828.39991990:09 71.8571 1941.9000 86.1487 2622.80001990:10 69.6079 2675.3999 82.7815 2393.70001990:11 69.1751 4403.2998 84.0966 5289.79981990:12 70.0241 2245.5000 87.6158 3001.10011991:01 69.9899 7502.6001 89.9160 3641.30001991:02 68.8145 3074.8999 92.1502 4468.29981991:03 72.6329 3153.8000 94.6644 5100.50001991:04 73.3007 2603.8999 95.6897 3277.30001991:05 73.4972 2647.3000 90.9871 3093.00001991:06 74.6456 3548.3000 90.2062 2520.89991991:07 74.1126 4637.3999 91.3006 3723.50001991:08 73.6676 3029.8999 95.5250 3958.20001991:09 72.2758 4131.7002 93.8846 3142.70001991:10 71.3388 2506.6001 94.5017 3284.20001991:11 70.1196 3336.6001 92.7835 3613.30001991:12 69.4182 1869.9000 92.3210 2702.00001992:01 68.2921 3279.6001 95.0142 3107.80001992:02 69.0452 3216.0000 98.7591 2825.50001992:03 71.1068 4378.3999 100.7086 3145.20001992:04 70.5836 2504.0000 94.4423 3252.10011992:05 67.7603 4118.2002 94.0530 3422.89991992:06 67.6915 2948.1001 93.7490 3212.89991992:07 67.3393 6108.8999 87.1503 3975.10011992:08 67.3503 2867.5000 83.2847 3222.00001992:09 66.7731 3671.6001 83.4066 3355.30001992:10 67.5400 6018.1001 80.6692 4178.29981992:11 68.7410 2537.8000 81.5430 4473.70021992:12 68.7429 3641.8000 81.0122 3625.5000

179

Table A.5: Monthly US Interest Rate Series for the Federal Funds Rate, 90-DayTreasury Bill Rate, and the One-Year Treasury Bill Rate

Federal

Year Month Funds 90-Day One-Year

1960 1 3.99 4.35 4.952 3.97 3.96 4.453 3.84 3.31 3.684 3.92 3.23 3.835 3.85 3.29 4.016 3.32 2.46 3.17 3.23 2.3 3.038 2.98 2.3 2.829 2.6 2.48 2.86

10 2.47 2.3 2.9211 2.44 2.37 2.8712 1.98 2.25 2.64

1961 1 1.85 2.24 2.632 2.14 2.42 2.753 2.02 2.39 2.764 1.5 2.29 2.745 1.98 2.29 2.726 1.73 2.33 2.87 1.56 2.24 2.798 2 2.39 2.919 1.88 2.28 2.88

10 2.26 2.3 2.911 2.62 2.48 2.912 2.33 2.6 2.97

1962 1 2.14 2.72 3.192 2.37 2.73 3.213 2.7 2.72 2.984 2.69 2.73 2.95 2.29 2.68 2.916 2.68 2.73 2.897 2.71 2.92 3.178 2.93 2.82 3.19 2.9 2.78 2.99

10 2.9 2.74 2.911 2.94 2.83 2.9412 2.93 2.87 2.94

1963 1 2.91 2.91 32 3 2.92 33 2.98 2.89 2.974 2.9 2.9 3.035 3 2.92 3.066 2.99 2.99 3.117 3.02 3.18 3.48 3.49 3.32 3.59 3.48 3.38 3.57

10 3.5 3.45 3.6111 3.48 3.52 3.6712 3.38 3.52 3.69

1964 1 3.48 3.52 3.682 3.48 3.53 3.713 3.43 3.54 3.784 3.47 3.47 3.755 3.5 3.48 3.716 3.5 3.48 3.77 3.42 3.48 3.648 3.5 3.5 3.679 3.45 3.53 3.73

10 3.36 3.57 3.7911 3.52 3.64 3.8612 3.85 3.84 3.96

1965 1 3.9 3.81 3.912 3.98 3.91 43 4.04 3.93 4.024 4.09 3.93 45 4.1 3.89 3.966 4.04 3.8 3.897 4.09 3.84 3.898 4.12 3.84 3.969 4.01 3.92 4.09

10 4.08 4.03 4.1611 4.1 4.09 4.2312 4.32 4.38 4.56

1966 1 4.42 4.59 4.692 4.6 4.65 4.813 4.65 4.59 4.814 4.67 4.62 4.765 4.9 4.64 4.856 5.17 4.5 4.787 5.3 4.8 4.948 5.53 4.96 5.349 5.4 5.37 5.8

10 5.53 5.35 5.5211 5.77 5.32 5.4912 5.4 4.96 5

180

Table A.5: Monthly US Interest Rate Series for the Federal Funds Rate, 90-DayTreasury Bill Rate, and the One-Year Treasury Bill Rate (cont’d)

Federal


1967 1 4.94 4.72 4.612 5 4.56 4.573 4.53 4.26 4.184 4.05 3.84 3.95 3.94 3.6 3.886 3.98 3.54 4.167 3.79 4.21 4.98 3.89 4.27 5.059 4 4.42 5.1

10 3.88 4.56 5.2211 4.12 4.73 5.3912 4.51 4.97 5.59

1968 1 4.60 5 5.32 4.72 4.98 5.243 5.05 5.17 5.424 5.76 5.38 5.475 6.12 5.66 5.846 6.07 5.52 5.687 6.02 5.31 5.398 6.03 5.09 5.159 5.78 5.19 5.18

10 5.92 5.35 5.3311 5.81 5.45 5.5112 6.02 5.96 5.97

1969 1 6.3 6.14 6.072 6.64 6.12 6.213 6.79 6.02 6.24 7.41 6.11 6.045 8.67 6.04 6.116 8.9 6.44 6.887 8.61 7 7.198 9.19 6.98 7.289 9.15 7.09 7.35

10 9 7 7.2111 8.85 7.24 7.3612 8.97 7.82 7.62

1970 1 8.98 7.87 7.512 8.98 7.13 7.053 7.76 6.63 6.54 8.1 6.51 6.535 7.94 6.84 7.126 7.6 6.68 7.077 7.21 6.45 6.628 6.61 6.41 6.559 6.29 6.13 6.39

10 6.2 5.91 6.2311 5.6 5.28 5.3912 4.9 4.87 4.84

1971 1 4.14 4.44 4.42 3.72 3.7 3.843 3.71 3.38 3.64 4.15 3.86 4.095 4.63 4.14 4.646 4.91 4.75 5.337 5.31 5.4 5.748 5.57 4.94 5.529 5.55 4.69 5.19

10 5.2 4.46 4.7511 4.91 4.22 4.4912 4.14 4.01 4.4

1972 1 3.5 3.38 3.822 3.29 3.2 4.063 3.83 3.73 4.434 4.17 3.71 4.655 4.27 3.69 4.466 4.46 3.91 4.717 4.55 3.98 4.98 4.80 4.02 4.99 4.87 4.66 5.44

10 5.04 4.74 5.3911 5.06 4.78 5.212 5.33 5.07 5.28

1973 1 5.94 5.41 5.582 6.58 5.6 5.933 7.09 6.09 6.534 7.12 6.26 6.515 7.84 6.36 6.636 8.49 7.19 7.057 10.4 8.01 7.978 10.5 8.67 8.329 10.78 8.29 8.07

10 10.01 7.22 7.1711 10.03 7.83 7.412 9.95 7.45 7.01

181

Table A.5: Monthly US Interest Rate Series for the Federal Funds Rate, 90-DayTreasury Bill Rate, and the One-Year Treasury Bill Rate (cont’d)

Federal


1974 1 9.65 7.77 7.012 8.97 7.12 6.513 9.35 7.96 7.344 10.51 8.33 8.085 11.31 8.23 8.216 11.93 7.9 8.167 12.92 7.55 8.048 12.01 8.96 8.889 11.34 8.06 8.52

10 10.06 7.46 7.5911 9.45 7.47 7.2912 8.53 7.15 6.79

1975 1 7.13 6.26 6.272 6.24 5.5 5.563 5.54 5.49 5.74 5.49 5.61 6.45 5.22 5.23 5.916 5.55 5.34 5.867 6.1 6.13 6.648 6.14 6.44 7.169 6.24 6.42 7.2

10 5.82 5.96 6.4811 5.22 5.48 6.4812 5.2 5.44 6.16

1976 1 4.87 4.87 5.442 4.77 4.88 5.533 4.84 5 5.824 4.82 4.86 5.545 5.29 5.2 5.986 5.48 5.41 6.127 5.31 5.23 5.828 5.29 5.14 5.649 5.25 5.08 5.5

10 5.03 4.92 5.1911 4.95 4.75 512 4.65 4.35 4.64

1977 1 4.61 4.62 52 4.68 4.67 5.163 4.69 4.6 5.194 4.73 4.54 5.15 5.35 4.96 5.436 5.39 5.02 5.417 5.42 5.19 5.578 5.9 5.49 5.979 6.14 5.81 6.13

10 6.47 6.16 6.5211 6.51 6.1 6.5212 6.56 6.07 6.52

1978 1 6.7 6.44 6.82 6.78 6.45 6.863 6.79 6.29 6.824 6.89 6.29 6.965 7.36 6.41 7.286 7.6 6.73 7.537 7.81 7.01 7.798 8.04 7.08 7.739 8.45 7.85 8.01

10 8.96 7.99 8.4511 9.76 8.64 9.212 10.03 9.08 9.44

1979 1 10.07 9.35 9.542 10.06 9.32 9.393 10.09 9.48 9.384 10.01 9.46 9.285 10.24 9.61 9.276 10.29 9.06 8.817 10.47 9.24 8.878 10.94 9.52 9.169 11.43 10.26 9.89

10 13.77 11.7 11.2311 13.18 11.79 11.2212 13.78 12.04 10.92

182

Table A.6: Monthly Flour Price Indices for Three US Cities

Year Month Buffalo Minnesota Kansas

1972 8 107.1 106.5 110.99 113.5 112.4 114.6

10 112.7 111.8 115.511 114.7 113.3 117.012 123.4 124.5 135.0

1973 1 123.6 124.3 132.82 116.3 116.5 122.63 118.5 118.6 123.84 119.8 119.6 128.95 120.3 119.4 126.76 127.4 128.6 139.37 125.1 126.3 135.78 127.6 126.8 135.69 129.0 125.7 146.0

10 124.6 120.8 140.711 134.1 127.9 147.012 146.5 147.6 163.9

1974 1 171.2 169.8 194.32 178.6 177.6 200.83 172.2 172.5 193.44 171.5 170.1 190.35 163.6 171.3 188.06 185.6 189.9 196.17 198.8 206.9 215.08 195.7 197.4 201.69 190.3 195.0 203.4

10 207.9 214.2 222.111 212.8 219.2 228.712 199.9 205.6 216.1

1975 1 185.3 193.4 200.22 183.0 185.1 189.63 173.5 174.0 173.34 172.2 173.2 169.75 165.3 164.5 161.06 159.9 158.9 151.77 170.3 169.7 167.18 172.2 174.4 174.49 184.5 186.2 189.7

10 185.0 184.7 187.411 177.7 176.4 178.412 169.1 167.6 165.8

1976 1 174.7 170.9 164.92 169.4 168.3 171.83 177.8 176.4 175.44 170.1 168.6 165.95 167.1 164.6 157.36 171.4 170.1 161.47 172.3 169.4 159.28 152.6 149.6 142.89 144.1 139.5 138.5

10 143.5 137.3 134.211 135.6 129.7 126.112 135.4 128.4 124.2

1977 1 134.5 126.9 122.72 136.1 128.8 123.53 135.6 126.5 118.34 122.8 116.6 112.35 119.0 113.4 105.76 108.5 102.8 97.77 113.3 107.7 105.88 114.8 109.4 106.99 120.9 114.9 110.0

10 123.7 117.5 114.311 127.8 120.0 118.812 125.4 117.6 117.2

1978 1 131.5 124.0 126.12 127.7 119.7 120.53 131.2 125.0 125.64 145.2 141.1 132.05 141.9 137.0 134.66 139.3 132.3 130.37 141.1 134.8 137.08 135.9 129.7 136.69 136.5 128.7 137.0

10 137.2 129.9 138.411 143.8 137.2 142.912 138.7 132.8 140.4

1979 1 133.9 127.5 136.02 137.7 131.2 140.13 143.8 137.1 148.24 140.8 135.5 146.45 153.4 147.1 158.56 157.5 151.6 163.57 179.5 173.6 187.18 177.5 171.6 181.79 178.0 170.8 181.5

10 176.8 172.4 181.911 179.8 174.9 190.912 174.2 168.1 186.9

183

Table A.6: Monthly Flour Price Indices for Three US Cities (cont’d)

Year Month Buffalo Minnesota Kansas

1980 1 171.1 164.7 180.12 175.9 170.0 184.83 172.2 164.9 174.84 164.7 157.9 169.05 175.7 169.2 178.46 177.4 168.6 175.37 187.5 179.8 178.28 190.7 179.0 182.09 190.4 179.2 188.6

10 192.4 181.4 190.811 192.9 181.8 192.2

184

Table A.7: Monthly forestry data. Q: Lumber Production, P: Lumber price, H:Housing Starts, I: Disposable Income

Year Q P H I

1969:03 635.0000 1.3693 131.9000 4.30111969:04 664.0000 1.3796 159.0000 4.04251969:05 631.0000 1.3662 155.5000 4.40561969:06 617.0000 1.2997 147.3000 4.46781969:07 609.0000 1.2123 125.2000 4.75141969:08 575.0000 1.1681 124.9000 5.17651969:09 563.0000 1.1285 129.3000 5.12291969:10 636.0000 1.1253 123.4000 5.18941969:11 545.0000 1.1247 94.6000 5.06371969:12 523.0000 1.1185 84.1000 4.75481970:01 579.0000 1.1205 66.4000 4.51781970:02 556.0000 1.0954 74.3000 1.49321970:03 592.0000 1.0817 114.7000 1.28341970:04 631.0000 1.0815 128.4000 5.23101970:05 628.0000 1.0788 125.0000 4.82071970:06 589.0000 1.0732 135.2000 4.48511970:07 613.0000 1.0647 140.8000 4.93531970:08 594.0000 1.0813 128.7000 5.45531970:09 616.0000 1.0916 130.9000 5.29651970:10 682.0000 1.1051 140.9000 5.08891970:11 608.0000 1.0943 126.9000 5.06201970:12 607.0000 1.0836 121.4000 5.21021971:01 597.0000 1.1019 110.6000 5.56031971:02 658.0000 1.1379 102.2000 4.94691971:03 710.0000 1.1772 167.9000 4.96831971:04 750.0000 1.1873 201.1000 4.84961971:05 694.0000 1.2021 198.5000 4.77951971:06 731.0000 1.2225 193.8000 5.70941971:07 718.0000 1.2663 194.3000 4.86681971:08 721.0000 1.2883 204.5000 4.75061971:09 715.0000 1.2977 173.8000 5.10971971:10 756.0000 1.2950 179.7000 5.14621971:11 694.0000 1.2950 173.7000 5.31851971:12 688.0000 1.2876 152.1000 5.24611972:01 691.0000 1.2835 149.1000 4.67531972:02 730.0000 1.2985 152.2000 4.39291972:03 782.0000 1.3240 203.9000 4.47451972:04 770.0000 1.3435 211.6000 4.60811972:05 776.0000 1.3418 225.8000 4.65571972:06 803.0000 1.3350 223.1000 3.93201972:07 744.0000 1.3300 206.5000 4.42001972:08 802.0000 1.3367 228.6000 4.82041972:09 770.0000 1.3433 203.0000 4.89301972:10 815.0000 1.3491 216.5000 6.00501972:11 710.0000 1.3573 185.7000 6.52111972:12 697.0000 1.3309 150.5000 5.78101973:01 659.0000 1.3197 146.6000 5.08411973:02 640.0000 1.3561 138.0000 4.28771973:03 731.0000 1.3825 200.0000 3.19821973:04 643.0000 1.4564 205.0000 2.78901973:05 705.0000 1.4674 234.0000 2.06521973:06 649.0000 1.4989 202.6000 1.14731973:07 628.0000 1.4944 202.6000 1.78841973:08 689.0000 1.4611 197.2000 9.28211973:09 644.0000 1.5246 148.4000 0.26771973:10 684.0000 1.5313 147.1000 1.25491973:11 618.0000 1.5161 133.3000 1.41291973:12 557.0000 1.4831 90.4000 0.29751974:01 599.0000 1.4102 84.5000 8.33471974:02 573.0000 1.3640 109.4000 6.68201974:03 670.0000 1.3636 124.8000 5.53951974:04 681.0000 1.3784 159.8000 5.17651974:05 671.0000 1.3185 149.0000 4.49611974:06 619.0000 1.2635 147.6000 4.32311974:07 589.0000 1.2000 126.6000 2.94811974:08 567.0000 1.1091 111.1000 1.38101974:09 505.0000 1.0877 98.3000 1.37391974:10 577.0000 1.0510 96.7000 0.87351974:11 443.0000 1.0314 75.1000 0.22651974:12 350.0000 1.0000 55.1000 0.25481975:01 466.0000 0.9826 56.1000 9.45821975:02 460.0000 1.0087 54.7000 9.87411975:03 549.0000 1.0299 80.2000 0.10721975:04 615.0000 1.0800 97.9000 0.51301975:05 606.0000 1.1244 116.1000 2.54751975:06 586.0000 1.0966 110.3000 0.99661975:07 615.0000 1.0664 119.3000 9.89951975:08 609.0000 1.0525 117.3000 0.02031975:09 599.0000 1.0455 111.9000 9.97811975:10 700.0000 1.0217 123.6000 0.01501975:11 584.0000 1.0303 96.9000 0.12941975:12 538.0000 1.0553 76.1000 0.0000

185

Table A.7: Monthly forestry data (cont’d)

Year Q P H I

1976:01 636.0000 1.1169 72.5000 0.30051976:02 583.0000 1.1686 89.9000 0.51591976:03 623.0000 1.2767 118.4000 0.56001976:04 634.0000 1.2475 137.2000 0.17161976:05 598.0000 1.2434 147.9000 0.08221976:06 664.0000 1.2304 154.2000 9.76311976:07 710.0000 1.2338 136.6000 9.67691976:08 743.0000 1.2687 145.9000 9.81761976:09 716.0000 1.2864 151.8000 9.56801976:10 761.0000 1.2843 148.4000 9.46851976:11 656.0000 1.2710 127.1000 9.77101976:12 663.0000 1.2896 107.4000 9.82081977:01 651.0000 1.3010 81.3000 9.34081977:02 702.0000 1.2945 112.5000 8.46931977:03 787.0000 1.3167 173.6000 8.62251977:04 778.0000 1.3190 182.2000 8.27741977:05 729.0000 1.3129 201.3000 8.15181977:06 728.0000 1.3277 197.6000 8.44771977:07 708.0000 1.4132 189.8000 8.83721977:08 759.0000 1.5138 194.0000 9.14621977:09 742.0000 1.5988 177.7000 9.05511977:10 764.0000 1.5427 193.1000 8.82471977:11 629.0000 1.5076 154.8000 8.88151977:12 621.0000 1.5181 129.2000 8.72051978:01 608.0000 1.5284 88.6000 8.40271978:02 622.0000 1.5319 101.3000 8.26671978:03 728.0000 1.5389 172.1000 1.31861978:04 730.0000 1.5435 197.5000 1.02901978:05 735.0000 1.5468 211.0000 0.74531978:06 728.0000 1.5357 216.0000 0.47291978:07 669.0000 1.5085 192.2000 0.40911978:08 733.0000 1.5185 190.9000 7.70171978:09 688.0000 1.5211 180.5000 7.47891978:10 737.0000 1.5042 192.1000 7.26881978:11 663.0000 1.4979 158.6000 7.16091978:12 646.0000 1.4952 119.5000 6.96701979:01 654.0000 1.4553 88.2000 6.45391979:02 642.0000 1.4312 84.5000 6.01871979:03 742.0000 1.4565 152.9000 5.74541979:04 665.0000 1.4512 161.0000 5.15341979:05 669.0000 1.4245 189.1000 4.67871979:06 673.0000 1.4141 191.8000 4.51151979:07 670.0000 1.4167 164.2000 4.32451979:08 726.0000 1.4661 170.3000 4.13821979:09 638.0000 1.4858 163.7000 3.52661979:10 743.0000 1.4677 169.0000 3.14491979:11 594.0000 1.4443 118.7000 3.04481979:12 522.0000 1.3765 91.6000 2.77821980:01 633.0000 1.3216 73.1000 2.49411980:02 599.0000 1.2969 79.9000 1.62371980:03 640.0000 1.2674 85.1000 1.14291980:04 425.0000 1.1196 96.2000 0.79381980:05 493.0000 1.1099 91.7000 0.41561980:06 553.0000 1.1330 116.4000 0.29091980:07 566.0000 1.1484 120.1000 0.01331980:08 614.0000 1.1628 129.9000 9.70711980:09 575.0000 1.1254 138.3000 9.85821980:10 626.0000 1.0647 152.7000 9.77371980:11 519.0000 1.0773 112.9000 9.76291980:12 515.0000 1.0949 95.9000 9.78681981:01 569.0000 1.0725 84.5000 9.25321981:02 492.0000 1.0635 71.9000 8.93651981:03 552.0000 1.0629 107.8000 8.71651981:04 604.0000 1.1010 123.0000 8.35001981:05 546.0000 1.1170 109.9000 8.20451981:06 559.0000 1.0995 105.8000 8.21731981:07 557.0000 1.0495 99.9000 8.37371981:08 512.0000 1.0303 86.3000 5.44141981:09 485.0000 0.9889 84.1000 8.48681981:10 488.0000 0.9676 87.2000 8.50151981:11 364.0000 0.9453 64.6000 8.44941981:12 415.0000 0.9899 59.1000 8.29761982:01 366.0000 0.9749 47.2000 7.96491982:02 419.0000 0.9579 51.3000 8.00001982:03 487.0000 0.9618 78.2000 8.13861982:04 515.0000 1.0020 84.1000 8.47191982:05 490.0000 0.9920 98.8000 8.34771982:06 556.0000 1.0440 91.1000 6.09701982:07 547.0000 1.0627 106.8000 8.22511982:08 582.0000 1.0060 96.0000 8.20341982:09 643.0000 1.0060 106.4000 8.24701982:10 563.0000 0.9750 110.5000 8.19161982:11 513.0000 0.9940 108.9000 8.24031982:12 505.0000 1.0209 82.9000 8.2647

186


Year Q P H I

1983:01 521.0000 1.0579 91.3000 8.39621983:02 515.0000 1.0955 96.3000 8.25271983:03 561.0000 1.1125 134.6000 8.34961983:04 550.0000 1.1195 135.8000 8.71711983:05 584.0000 1.1300 174.9000 6.46131983:06 594.0000 1.1594 173.2000 8.42381983:07 570.0000 1.1392 161.6000 8.69501983:08 598.0000 1.1110 176.8000 8.40571983:09 551.0000 1.0569 154.9000 8.53731983:10 586.0000 1.0734 159.3000 8.75541983:11 536.0000 1.0833 136.0000 9.19391983:12 471.0000 1.1222 108.3000 9.15351984:01 577.0000 1.1351 109.1000 9.17201984:02 610.0000 1.1444 130.0000 9.44481984:03 661.0000 1.1319 137.5000 9.43981984:04 626.0000 1.1250 172.7000 9.44231984:05 615.0000 1.0768 180.7000 9.39391984:06 636.0000 1.0663 184.0000 9.57501984:07 605.0000 1.0701 162.1000 9.61321984:08 657.0000 1.0732 147.4000 9.88151984:09 617.0000 1.0435 148.5000 0.17991984:10 893.0000 1.0329 152.3000 0.01451984:11 824.0000 1.0203 126.2000 0.02601984:12 679.0000 1.0300 98.9000 0.32081985:01 769.0000 1.0261 105.4000 0.40141985:02 783.0000 0.9961 95.4000 0.24881985:03 890.0000 1.0029 145.0000 0.09311985:04 992.0000 0.9903 175.8000 0.79281985:05 1039.0000 1.1024 170.2000 1.21061985:06 894.0000 1.1752 163.2000 0.56821985:07 920.0000 1.0882 160.7000 0.63571985:08 920.0000 1.0117 160.7000 0.72251985:09 895.0000 0.9863 147.7000 0.88251985:10 944.0000 0.9631 173.0000 0.81341985:11 783.0000 0.9458 124.1000 0.83951985:12 761.0000 0.9431 120.5000 0.87361986:01 881.0000 0.9787 115.6000 1.04941986:02 825.0000 0.9931 107.2000 1.69621986:03 968.0000 1.0608 151.0000 2.44961986:04 878.0000 1.1396 188.2000 3.09941986:05 949.0000 1.1090 186.6000 2.82001986:06 955.0000 1.0721 183.6000 2.69071986:07 798.0000 1.0644 172.0000 2.90441986:08 919.0000 1.0695 163.8000 2.95071986:09 676.0000 1.0392 154.0000 2.96281986:10 1014.0000 1.0100 154.8000 2.73021986:11 846.0000 1.0130 115.6000 2.66231986:12 833.0000 1.0181 113.0000 2.85661987:01 966.0000 1.0299 105.1000 2.64971987:02 912.0000 1.0574 102.8000 2.71391987:03 1008.0000 1.1018 141.2000 2.61261987:04 1092.0000 1.1070 159.3000 1.13151987:05 988.0000 1.0867 158.0000 1.95421987:06 1071.0000 1.1272 162.9000 1.77771987:07 1086.0000 1.1874 152.4000 1.71301987:08 1007.0000 1.1628 143.6000 1.66961987:09 1004.0000 1.1649 152.0000 1.70201987:10 1102.0000 1.0922 139.1000 2.05381987:11 929.0000 1.0893 118.8000 1.81191987:12 994.0000 1.1123 85.4000 2.20441988:01 938.0000 1.1291 78.2000 2.11851988:02 966.0000 1.1412 90.2000 2.29011988:03 1124.0000 1.1287 128.8000 2.38231988:04 1085.0000 1.1210 153.2000 1.82611988:05 1081.0000 1.0864 140.2000 1.82541988:06 1127.0000 1.0746 150.2000 1.73791988:07 1033.0000 1.0649 137.0000 1.56161988:08 1042.0000 0.9843 136.8000 1.56671988:09 1056.0000 0.9399 131.1000 1.52451988:10 1089.0000 0.9510 135.1000 1.90301988:11 1028.0000 1.0018 113.0000 1.66021988:12 971.0000 1.0046 94.2000 1.63951989:01 1067.0000 0.9955 100.1000 1.21451989:02 945.0000 0.9901 85.8000 1.41341989:03 1047.0000 0.9677 117.8000 1.28971989:04 1064.0000 0.9448 129.4000 0.67761989:05 1123.0000 0.9231 131.7000 0.47611989:06 1153.0000 0.9637 143.2000 0.60411989:07 988.0000 1.0027 134.7000 0.66581989:08 1049.0000 0.9571 122.4000 0.87951989:09 1038.0000 0.9386 109.3000 0.68241989:10 1054.0000 0.9929 130.1000 0.90781989:11 990.0000 0.9388 96.6000 0.91841989:12 904.0000 0.9372 75.0000 0.9000

187


Year Q P H I

1990:01 1065.0000 0.9304 99.2000 0.57701990:02 1006.0000 0.9615 86.9000 0.83221990:03 1080.0000 0.9842 108.5000 0.96581990:04 1121.0000 1.0237 119.0000 1.01141990:05 1091.0000 1.0428 121.1000 1.60991990:06 1142.0000 1.0149 117.8000 0.86091990:07 1239.0000 1.0157 111.2000 0.84981990:08 1114.0000 0.9785 102.8000 0.19231990:09 990.0000 0.9333 93.1000 9.73651990:10 1147.0000 0.8609 94.2000 8.87671990:11 947.0000 0.8718 81.4000 9.23981990:12 867.0000 0.8660 57.4000 9.87951991:01 950.0000 0.8731 52.5000 0.33451991:02 924.0000 0.8558 59.1000 9.95481991:03 1024.0000 0.8916 73.8000 0.35541991:04 1102.0000 0.9483 99.7000 0.38621991:05 1071.0000 0.9511 97.7000 0.32451991:06 1123.0000 1.0524 103.4000 0.41151991:07 1021.0000 1.0724 103.5000 0.36181991:08 1049.0000 0.9690 94.7000 0.30901991:09 1039.0000 0.9483 86.6000 0.36861991:10 1169.0000 0.9390 101.8000 0.39001991:11 992.0000 0.9536 75.6000 0.26631991:12 903.0000 0.9810 65.6000 0.79901992:01 1250.0000 1.0182 71.6000 0.79071992:02 1083.0000 1.1078 78.8000 0.89221992:03 1177.0000 1.1533 111.6000 0.89661992:04 1180.0000 1.1668 107.6000 0.95791992:05 1067.0000 1.1451 115.2000 0.79521992:06 1170.0000 1.0712 117.8000 0.61611992:07 1228.0000 1.0475 106.2000 0.64801992:08 1115.0000 1.0867 109.9000 0.82751992:09 1143.0000 1.1551 106.0000 0.78641992:10 1215.0000 1.0999 111.8000 0.96361992:11 1108.0000 1.1154 84.5000 1.04331992:12 1076.0000 1.2024 78.6000 0.85971993:01 1115.0000 1.3144 70.5000 0.95511993:02 1169.0000 1.3978 74.6000 0.85471993:03 1151.0000 1.5535 95.5000 0.92251993:04 1125.0000 1.5432 117.8000 1.03351993:05 1088.0000 1.4386 120.9000 0.95991993:06 1244.0000 1.3088 128.5000 0.92641993:07 1264.0000 1.2743 115.3000 0.91021993:08 1237.0000 1.3142 121.8000 1.35801993:09 1260.0000 1.3437 118.5000 1.32601993:10 1322.0000 1.4349 123.2000 1.32581993:11 1092.0000 1.5282 102.3000 1.52231993:12 1179.0000 1.5938 98.7000 1.8209

188

Table A.8: Gas furnace data (columnwise order). X: Gas Flow Rate, Y: Carbondioxide concentration

X Y X Y X Y X Y X Y X Y

-0.109 53.8 1.608 46.9 -0.288 51.0 -0.049 53.2 -2.473 55.6 0.185 56.30.000 53.6 1.905 47.8 -0.153 51.8 0.060 53.9 -2.330 58.0 0.662 56.40.178 53.5 2.023 48.2 -0.109 52.4 0.161 54.1 -2.053 59.5 0.709 56.40.339 53.5 1.815 48.3 -0.187 53.0 0.301 54.0 -1.739 60.0 0.605 56.00.373 53.4 0.535 47.9 -0.255 53.4 0.517 53.6 -1.261 60.4 0.501 55.20.441 53.1 0.122 47.2 -0.229 53.6 0.566 53.2 -0.569 60.5 0.603 54.00.461 52.7 0.009 47.2 -0.007 53.7 0.560 53.0 -0.137 60.2 0.943 53.00.348 52.4 0.164 48.1 0.254 53.8 0.573 52.8 -0.024 59.7 1.223 52.00.127 52.2 0.671 49.4 0.330 53.8 0.592 52.3 -0.050 59.0 1.249 51.6

-0.180 52.0 1.019 50.6 0.102 53.8 0.671 51.9 -0.135 57.6 0.824 51.6-0.588 52.0 1.146 51.5 -0.423 53.3 0.933 51.6 -0.276 56.4 0.102 51.1-1.055 52.4 1.155 51.6 -1.139 53.0 1.337 51.6 -0.534 55.2 0.025 50.4-1.421 53.0 1.112 51.2 -2.275 52.9 1.460 51.4 -0.871 54.5 0.382 50.0-1.520 54.0 1.121 50.5 -2.594 53.4 1.353 51.2 -1.243 54.1 0.922 50.0-1.302 54.9 1.223 50.1 -2.716 54.6 0.772 50.7 -1.439 54.1 1.032 52.0-0.814 56.0 1.257 49.8 -2.510 56.4 0.218 50.0 -1.422 54.4 0.866 54.0-0.475 56.8 1.157 49.6 -1.790 58.0 -0.237 49.4 -1.175 55.5 0.527 55.1-0.193 56.8 0.913 49.4 -1.346 59.4 -0.714 49.3 -0.813 56.2 0.093 54.50.088 56.4 0.620 49.3 -1.081 60.2 -1.099 49.7 -0.634 57.0 -0.458 52.80.435 55.7 0.255 49.2 -0.910 60.0 -1.269 50.6 -0.582 57.3 -0.748 51.40.771 55.0 -0.280 49.3 -0.876 59.4 -1.175 51.8 -0.625 57.4 -0.947 50.80.866 54.3 -1.080 49.7 -0.885 58.4 -0.676 53.0 -0.713 57.0 -1.029 51.20.875 53.2 -1.551 50.3 -0.800 57.6 0.033 54.0 -0.848 56.4 -0.928 52.00.891 52.3 -1.799 51.3 -0.544 56.9 0.556 55.3 -1.039 55.9 -0.645 52.80.987 51.6 -1.825 52.8 -0.416 56.4 0.643 55.9 -1.346 55.5 -0.424 53.81.263 51.2 -1.456 54.4 -0.271 56.0 0.484 55.9 -1.628 55.3 -0.276 54.51.775 50.8 -0.944 56.0 0.000 55.7 0.109 54.6 -1.619 55.2 -0.158 54.91.976 50.5 -0.570 56.9 0.403 55.3 -0.310 53.5 -1.149 55.4 -0.033 54.91.934 50.0 -0.431 57.5 0.841 55.0 -0.697 52.4 -0.488 56.0 0.102 54.81.866 49.2 -0.577 57.3 1.285 54.4 -1.047 52.1 -0.160 56.5 0.251 54.41.832 48.4 -0.960 56.6 1.607 53.7 -1.218 52.3 -0.007 57.1 0.280 53.71.767 47.9 -1.616 56.0 1.746 52.8 -1.183 53.0 -0.092 57.3 0.000 53.31.608 47.6 -1.875 55.4 1.683 51.6 -0.873 53.8 -0.620 56.8 -0.493 52.81.265 47.5 -1.891 55.4 1.485 50.6 -0.336 54.6 -1.086 55.6 -0.759 52.60.790 47.5 -1.746 56.4 0.993 49.4 0.063 55.4 -1.525 55.0 -0.824 52.60.360 47.6 -1.474 57.2 0.648 48.8 0.084 55.9 -1.858 54.1 -0.740 53.00.115 48.1 -1.201 58.0 0.577 48.5 0.000 55.9 -2.029 54.3 -0.528 54.30.088 49.0 -0.927 58.4 0.577 48.7 0.001 55.2 -2.024 55.3 -0.204 56.00.331 50.0 -0.524 58.4 0.632 49.2 0.209 54.4 -1.961 56.4 0.034 57.00.645 51.1 0.040 58.1 0.747 49.8 0.556 53.7 -1.952 57.2 0.204 58.00.960 51.8 0.788 57.7 0.900 50.4 0.782 53.6 -1.794 57.8 0.253 58.61.409 51.9 0.943 57.0 0.993 50.7 0.858 53.6 -1.302 58.3 0.195 58.52.670 51.7 0.930 56.0 0.968 50.9 0.918 53.2 -1.030 58.6 0.131 58.32.834 51.2 1.006 54.7 0.790 50.7 0.862 52.5 -0.918 58.8 0.017 57.82.812 50.0 1.137 53.2 0.399 50.5 0.416 52.0 -0.798 58.8 -0.182 57.32.483 48.3 1.198 52.1 -0.161 50.4 -0.336 51.4 -0.867 58.6 -0.262 57.01.929 47.0 1.054 51.6 -0.553 50.2 -0.959 51.0 -1.047 58.01.485 45.8 0.595 51.0 -0.603 50.4 -1.813 50.9 -1.123 57.41.214 45.6 -0.080 50.5 -0.424 51.2 -2.378 52.4 -0.876 57.01.239 46.0 -0.314 50.4 -0.194 52.3 -2.499 53.5 -0.395 56.4

189

Table A.9: Monthly US Grain Price Data

Wheat

Year Month Flour Corn Wheat Rye

1961 1 5.58 1.12 1.62 1.0752 5.46 1.15 1.58 1.0643 5.50 1.18 1.56 1.1584 5.53 1.09 1.39 1.0735 5.63 1.15 1.28 1.1256 5.65 1.13 1.28 1.0707 5.70 1.18 1.36 1.2988 5.68 1.15 1.46 1.1689 5.75 1.11 1.45 1.205

10 5.66 1.11 1.47 1.29811 5.70 1.14 1.51 1.32012 5.75 1.12 1.53 1.295

1962 1 5.75 1.09 1.52 1.2982 5.78 1.11 1.52 1.2933 5.72 1.11 1.54 1.2534 5.80 1.12 1.59 1.2285 5.95 1.16 1.65 1.2306 6.09 1.15 1.63 1.2787 6.25 1.12 1.62 1.1258 6.37 1.14 1.57 1.1459 6.45 1.13 1.59 1.12

10 6.44 1.12 1.55 1.14811 6.46 1.10 1.59 1.16412 6.39 1.17 1.58 1.21

1963 1 6.03 1.21 1.6 1.2582 6.05 1.21 1.63 1.2283 6.02 1.2 1.64 1.1954 6.15 1.21 1.68 1.2335 6.09 1.23 1.63 1.1966 6.05 1.30 1.38 1.227 5.70 1.34 1.3 1.2058 5.28 1.34 1.3 1.1859 5.76 1.36 1.47 1.355

10 5.98 1.22 1.64 1.43311 6.41 1.18 1.69 1.4212 6.83 1.23 1.69 1.423

1964 1 6.45 1.26 1.78 1.4632 6.9 1.23 1.75 1.3763 6.64 1.24 1.48 1.3034 6.68 1.26 1.66 1.2805 6.7 1.29 1.54 1.2636 6.53 1.26 1.43 1.2657 7.09 1.23 1.46 1.2458 6.7 1.25 1.44 1.219 6.73 130 1.52 1.280

10 6.84 1.25 1.5 1.23611 6.9 1.19 1.56 1.23312 6.9 1.28 1.59 1.185

1965 1 6.68 1.29 1.57 1.1842 6.7 1.31 1.58 1.163 6.7 1.34 1.56 1.194 6.7 1.35 1.56 1.175 6.7 1.37 1.46 1.1736 6.9 1.36 1.45 1.17 7.07 1.33 1.43 1.0018 7.1 1.29 1.53 1.0889 7.1 1.31 1.54 1.111

10 7.2 1.27 1.53 1.10511 7.28 1.17 1.66 1.12512 7.21 1.24 1.74 1.180

1966 1 7.22 1.32 1.71 1.2432 7.22 1.32 1.75 1.2253 7.22 1.26 1.67 1.1554 7.19 1.3 1.65 1.2685 7.2 1.32 1.67 1.1536 7.63 1.31 1.77 1.137 8.07 1.42 1.90 1.288 8.09 1.45 1.89 1.2489 7.92 1.46 1.92 1.243

10 7.72 1.4 1.69 1.15811 7.73 1.35 1.75 1.1612 7.62 1.44 1.83 1.218

1967 1 7.35 1.44 1.80 1.2032 7.48 1.39 1.68 1.1633 7.55 1.42 1.81 1.2084 7.27 1.38 1.8 1.1885 7.43 1.39 1.67 1.186 7.34 1.38 1.57 1.167 7.4 1.33 1.48 1.1918 7.29 1.21 1.42 1.0999 7.27 1.2 1.45 1.173

10 7.24 1.2 1.49 1.1411 7.18 1.08 1.44 1.11812 7.21 1.18 1.5 1.148

190

Table A.9: Monthly US Grain Price Data (cont’d)

Wheat

Year Month Flour Corn Wheat Rye

1968 1 7.18 1.15 1.52 1.1542 7.04 1.17 1.56 1.193 6.94 1.14 1.51 1.1794 6.76 1.16 1.46 1.1455 6.78 1.21 1.47 1.136 6.77 1.16 1.26 1.0737 6.73 1.15 1.29 1.1258 6.72 1.08 1.19 1.0469 6.85 1.08 1.18 1.119

10 6.89 1.1 1.28 1.14511 6.9 1.17 1.36 1.16912 6.85 1.15 1.34 1.168

1969 1 6.85 1.19 1.43 1.2052 6.83 1.17 1.40 1.2013 6.76 1.17 1.35 1.2154 6.75 1.22 1.36 1.2155 6.75 1.31 1.4 1.246 6.75 1.32 1.33 1.2057 6.92 1.3 1.3 1.1858 6.86 1.29 1.28 1.0359 6.78 1.27 1.33 1.055

10 6.89 1.27 1.33 1.10511 6.95 1.16 1.42 1.11512 7.14 1.18 1.48 1.135

1970 1 7.14 1.26 1.5 1.1352 7.17 1.26 1.52 1.1353 7.19 1.25 1.54 1.1454 7.19 1.29 1.54 1.1655 7.19 1.32 1.55 1.1656 7.2 1.36 1.4 1.2157 7.16 1.39 1.43 1.0858 7.21 1.37 1.45 1.0659 7.2 1.45 1.68 1.085

10 7.35 1.44 1.67 1.08511 7.43 1.42 1.74 1.14512 7.5 1.33 1.63 1.165

1971 1 7.45 1.39 1.72 1.1552 7.45 1.4 1.74 1.1653 7.45 1.36 1.68 1.1354 7.51 1.33 1.66 1.1455 7.58 1.31 1.66 1.1856 7.55 1.38 1.63 1.1657 7.52 1.3 1.44 1.0358 7.48 1.29 1.31 0.8759 7.48 1.17 1.33 0.945

10 7.48 1.12 1.39 0.96511 7.48 1.09 1.49 0.89512 7.48 1.26 1.6 0.98

1972 1 7.28 1.23 1.58 1.0302 7.15 1.21 1.52 1.0503 7.15 1.23 1.58 1.0304 7.15 1.25 1.68 1.0405 7.07 1.28 1.67 1.0606 7.07 1.28 1.36 0.9907 7.07 1.3 1.47 0.988 7.66 1.29 1.6 0.969 8.39 1.39 1.86 0.99

10 8.57 1.31 1.92 1.010

191

Table A.10: Monthly Housing Starts and Housing Sold Data

Housing Housing Housing Housing

Year Month Starts Sold Year Month Starts Sold

1965 1 52.1 38 1970 1 33.4 342 47.2 44 2 41.4 293 82.2 53 3 61.9 364 100.9 49 4 73.8 425 98.4 54 5 74.8 436 97.4 57 6 83.0 447 96.5 51 7 75.5 448 88.8 58 8 77.3 489 80.9 48 9 76.0 45

10 85.8 44 10 79.4 4411 72.4 42 11 67.4 4012 61.2 37 12 69.0 37

1966 1 46.6 42 1971 1 54.9 452 50.4 43 2 58.3 493 83.2 53 3 91.6 624 94.3 49 4 116.0 625 84.7 49 5 115.6 586 79.8 40 6 116.9 597 69.1 40 7 107.7 648 69.4 36 8 111.7 629 59.4 29 9 102.1 50

10 53.5 31 10 102.9 5211 50.2 26 11 92.9 5012 38.0 23 12 80.4 44

1967 1 40.2 29 1972 1 76.2 512 40.3 32 2 76.3 543 66.6 41 3 111.4 604 79.8 44 4 119.8 655 87.3 49 5 135.2 646 87.6 47 6 131.9 637 82.3 46 7 119.1 638 83.7 47 8 131.3 729 78.2 43 9 120.5 61

10 81.7 45 10 117.0 6511 69.1 34 11 97.4 5112 47.0 31 12 73.2 47

1968 1 45.2 35 1973 1 77.1 542 55.4 43 2 73.6 583 79.3 46 3 105.1 664 98.0 46 4 120.5 635 86.8 43 5 131.6 646 81.4 41 6 114.8 607 86.4 44 7 114.7 538 82.5 47 8 106.8 529 80.1 41 9 84.5 44

10 85.6 40 10 86.0 4011 64.8 32 11 70.5 3612 53.8 32 12 46.8 28

1969 1 51.3 34 1974 1 43.3 362 47.9 40 2 57.6 423 71.9 43 3 77.3 534 85.0 42 4 102.3 535 91.3 43 5 96.4 556 82.7 44 6 99.6 487 73.5 39 7 90.9 478 69.5 40 8 79.8 439 71.5 33 9 73.4 39

10 68.0 32 10 69.5 3311 55.1 31 11 57.9 3012 42.8 28 12 41.0 23

192

Table A.11: Quarterly Fixed Investment and Changes in Business Inventories

Changes in Changes in

Year Quarter Invest Inventories Year Quarter Invest Inventories

1947 1 69.6 0.1 1960 1 104.9 13.52 67.6 -0.9 2 101.8 4.93 69.5 -2.9 3 98.8 3.04 74.7 2.7 4 98.6 -3.9

1948 1 77.1 4.1 1961 1 97.7 -3.82 77.4 5.6 2 99.2 1.93 76.6 6.9 3 101.3 6.64 76.1 5.3 4 104.6 6.7

1949 1 71.8 -0.3 1962 1 106.1 10.62 68.9 -7.1 2 109.9 9.23 68.5 -2.5 3 111.1 8.04 70.6 -7.7 4 110.1 4.7

1950 1 75.4 4.4 1963 1 110.7 7.62 82.3 7.7 2 116.0 7.03 88.2 8.0 3 118.5 9.34 86.9 22.1 4 122.0 7.1

1951 1 83.4 13.4 1964 1 124.0 6.12 80.3 19.9 2 124.0 8.03 79.4 14.6 3 124.9 7.34 78.6 7.0 4 126.4 7.9

1952 1 79.3 7.3 1965 1 133.4 13.42 80.3 -2.7 2 137.9 10.63 75.3 5.4 3 140.1 12.44 80.6 7.2 4 143.8 8.8

1953 1 83.9 3.9 1966 1 147.5 13.52 84.2 5.1 2 146.2 17.83 84.4 1.9 3 145.0 15.14 83.8 -5.0 4 139.7 20.5

1954 1 82.8 -3.4 1967 1 136.4 14.62 84.1 -4.1 2 139.6 7.53 87.0 -2.7 3 141.1 12.24 88.5 1.5 4 145.5 13.8

1955 1 92.1 5.9 1968 1 148.9 6.32 96.1 8.0 2 148.9 11.83 98.3 7.8 3 150.7 9.24 98.8 9.2 4 155.0 7.6

1956 1 96.6 7.5 1969 1 159.1 9.82 97.4 5.5 2 158.4 12.23 97.6 4.9 3 158.1 13.44 96.6 5.4 4 154.3 6.8

1957 1 96.2 2.5 1970 1 151.8 2.92 95.3 2.9 2 150.0 4.83 96.4 3.7 3 150.4 6.34 94.9 -3.0 4 149.5 3.3

1958 1 90.0 -6.8 1971 1 154.3 7.92 87.2 -6.2 2 158.4 10.03 88.0 0.3 3 162.1 5.04 93.0 5.3 4 166.0 3.7

1959 1 98.3 5.02 101.6 13.03 102.6 -0.44 101.4 8.2

193

Table A.12: Natural Logarithms of the Annual Sales of Mink Furs and MuskratFurs

Year Log Mink Log Musk Year Log Mink Log Musk

1850 10.2962 12.0752 1881 10.4957 13.62801851 9.9594 12.1791 1882 10.7277 13.84441852 10.1210 12.5863 1883 10.7687 13.88241853 10.1327 13.1102 1884 10.8646 13.89531854 10.6543 13.1466 1885 11.4167 13.61341855 10.8364 12.7531 1886 11.2451 12.75721856 11.0281 12.4638 1887 11.0714 12.84831857 11.0341 12.6191 1888 11.3269 12.75091858 11.2415 12.6556 1889 10.6152 12.31771859 11.0551 12.4461 1890 10.4800 12.68281860 10.7084 12.0855 1891 10.2914 13.26171861 10.3448 12.2357 1892 10.6517 13.60001862 10.8088 12.7230 1893 10.9711 13.74791863 10.6911 12.7857 1894 10.8359 13.38271864 11.0305 13.1417 1895 10.8452 13.42221865 11.0077 12.9441 1896 11.1595 13.60871866 10.8475 12.6786 1897 11.2433 13.22081867 10.9759 12.9292 1898 11.1620 13.25151868 11.2061 13.3344 1899 10.6416 13.46101869 11.2164 12.9096 1900 10.7359 13.55121870 10.2295 12.3556 1901 10.7751 13.74101871 10.3730 13.0036 1902 10.9616 14.31641872 10.5781 13.4657 1903 11.1057 14.21311873 10.7086 13.5514 1904 10.9091 13.73691874 11.0092 13.4180 1905 10.9330 13.87021875 11.1882 13.1689 1906 11.0030 13.45181876 11.2799 13.2765 1907 10.5756 12.91771877 11.2780 12.9880 1908 9.9774 12.51511878 11.3415 13.0940 1909 9.7902 12.61881879 11.0444 13.1218 1910 9.9891 13.52671880 10.4652 13.0775 1911 10.4045 13.7784

194

Table A.13: Power Station Data from a 50 Megawatt Turbo-Alternator of In-Phase Current Deviations, Out-of-Phase Current Deviations and Frequency De-viations of Voltage Generated

In- Out-of- In- Out-of-

Period Phase Phase Freq Period Phase Phase Freq

1 9.2 -7.5 47.5 51 3.7 14.0 52.42 9.5 -9.2 47.5 52 3.0 14.9 52.03 9.9 -10.4 47.4 53 2.7 14.9 52.04 10.7 -10.1 47.5 54 3.0 14.7 52.05 11.3 -8.3 47.5 55 3.3 15.4 52.06 11.6 -6.4 47.7 56 3.3 16.4 52.17 11.5 -5.5 47.8 57 3.4 16.0 52.08 10.8 -4.6 47.6 58 3.6 14.6 51.59 9.7 -3.1 48.1 59 3.8 14.1 51.5

10 8.9 -1.4 48.1 60 4.1 14.1 51.411 8.5 -0.2 48.4 61 4.6 13.9 51.012 7.6 1.2 48.8 62 5.0 12.6 51.113 6.5 3.2 49.4 63 5.0 10.4 51.014 5.9 5.4 49.6 64 4.7 8.9 51.015 5.9 6.8 49.9 65 4.4 9.9 50.916 5.8 7.2 50.5 66 4.7 12.1 51.117 5.8 7.7 51.1 67 5.2 13.1 50.818 6.0 8.2 51.1 68 5.2 12.5 51.019 6.0 8.7 51.0 69 5.0 11.3 51.020 5.5 9.1 51.0 70 5.0 10.4 51.521 5.4 9.8 50.5 71 4.9 9.8 51.522 6.0 11.0 50.0 72 4.8 9.3 51.523 6.2 11.7 50.1 73 4.7 8.6 51.024 5.4 10.9 50.0 74 4.6 7.9 51.525 5.0 10.0 50.0 75 4.3 8.0 51.026 5.8 10.6 50.1 76 4.2 9.1 51.027 6.5 11.7 50.4 77 4.6 11.0 51.028 6.5 11.8 50.9 78 5.2 12.6 51.129 6.9 9.6 50.9 79 5.8 12.6 51.130 7.4 6.4 50.6 80 6.1 11.0 51.031 7.6 6.0 50.6 81 6.6 9.2 51.132 7.2 8.0 50.4 82 7.2 7.9 51.133 6.9 8.4 50.1 83 7.2 6.7 50.834 7.1 6.8 49.9 84 6.8 5.3 50.535 8.1 6.3 49.7 85 7.0 4.4 50.136 9.6 7.4 49.9 86 7.7 4.3 49.037 10.9 8.1 50.0 87 7.5 3.9 49.038 11.0 6.7 50.1 88 6.4 3.2 49.539 10.5 3.8 49.7 89 5.6 2.9 49.440 10.4 1.7 48.8 90 5.0 3.4 49.041 10.5 1.7 48.1 91 4.0 3.8 49.642 10.3 2.9 48.1 92 2.8 3.7 49.743 10.0 4.1 47.9 93 2.4 3.3 49.944 9.2 5.2 47.9 94 2.6 2.6 50.845 8.1 6.7 48.5 95 2.1 2.5 51.646 7.3 7.7 48.8 96 1.6 3.1 52.147 6.0 8.0 49.0 97 1.8 4.0 51.948 4.8 8.6 50.0 98 1.9 4.7 51.949 4.5 10.1 51.1 99 1.8 4.5 52.050 4.4 12.2 51.5 100 1.9 3.6 51.6

195

Table A.14: Weekly Production Schedule and Billing Figures

Period Prod. Billing Period Prod. Billing Period Prod. Billing

1 50.900 103.000 35 50.533 103.838 68 48.454 104.4102 49.112 103.916 36 50.628 110.873 69 49.372 111.1243 48.791 100.115 37 51.531 110.924 70 46.517 104.9564 50.114 101.577 38 48.488 108.998 71 47.724 95.5255 52.127 100.812 39 46.747 105.282 72 49.661 92.1636 50.706 96.455 40 49.301 105.015 73 48.638 83.4647 51.100 98.459 41 51.849 94.262 74 52.285 82.0838 50.164 101.856 42 49.635 85.900 75 51.087 89.4179 49.998 98.898 43 50.549 86.393 76 48.851 100.043

10 51.269 99.189 44 48.722 97.544 77 47.761 111.95011 48.894 99.455 45 49.824 101.861 78 45.251 115.78612 52.673 98.775 46 51.045 106.499 79 49.121 109.12813 53.406 103.087 47 50.943 104.903 80 48.933 102.14714 51.192 100.569 48 50.249 104.643 81 47.942 88.77015 50.114 105.557 49 50.538 103.684 82 48.715 90.12616 49.968 109.592 50 50.569 103.667 83 48.746 92.81217 53.321 105.883 51 50.671 103.087 84 49.058 92.24618 51.683 102.970 52 51.360 96.766 85 51.504 95.29819 49.843 94.108 53 49.646 95.956 86 48.549 100.04420 51.464 97.793 54 49.415 95.719 87 49.037 102.80621 51.671 99.117 55 48.940 96.644 88 49.361 112.43722 50.456 101.337 56 50.746 93.025 89 50.291 106.93423 50.395 101.926 57 48.912 92.576 90 47.061 106.28324 52.591 103.925 58 49.358 95.205 91 47.768 102.77625 51.916 103.674 59 47.812 100.016 92 47.585 100.18126 49.967 100.041 60 49.362 101.049 93 48.311 92.54127 51.100 101.488 61 50.630 104.115 94 48.049 86.05628 50.214 97.475 62 52.009 101.299 95 45.933 85.62329 47.217 88.786 63 48.907 98.391 96 49.883 89.01630 48.172 88.079 64 49.114 99.020 97 46.565 96.53731 50.618 90.521 65 52.602 103.847 98 44.731 95.56132 51.055 92.665 66 53.808 99.947 99 47.420 106.05733 52.299 92.890 67 49.900 97.395 100 49.694 103.59034 51.282 98.968

196

Table A.15: Quarterly Unemployment and GDP in UK (UN: unemployment,GDP: gross domestic product)

Year Quarter UN GDP Year Quarter UN GDP Year Quarter UN GDP

1955 1 225 81.37 1960 1 363 92.30 1965 1 306 108.072 208 82.60 2 342 92.13 2 304 107.643 201 82.30 3 325 93.17 3 321 108.874 199 83.00 4 312 93.50 4 305 109.75

1956 1 207 82.87 1961 1 291 94.77 1966 1 279 110.202 215 83.60 2 293 95.37 2 282 110.203 240 83.33 3 304 95.03 3 318 110.904 245 83.53 4 330 95.23 4 414 110.40

1957 1 295 84.27 1962 1 357 95.07 1967 1 463 111.002 293 85.50 2 401 96.40 2 506 112.103 279 84.33 3 447 96.97 3 538 112.504 287 84.30 4 483 96.50 4 536 113.00

1958 1 331 85.07 1963 1 535 96.16 1968 1 544 114.302 396 83.60 2 520 99.79 2 541 115.103 432 84.37 3 489 101.14 3 547 116.404 462 84.50 4 456 102.95 4 532 117.80

1959 1 454 85.20 1964 1 386 103.96 1969 1 532 116.802 446 87.07 2 368 105.28 2 519 117.803 426 88.40 3 358 105.81 3 547 119.004 402 90.03 4 330 107.14 4 544 119.60

197

VITA

Nuran Arzu Sisman Yılmaz was born in Luleburgaz on March 21, 1970. She

received her B.S. degree in Computer Engineering from Middle East Technical

University in July 1992. She received her M.S. degree from the same department

in September 1995. She worked at ISBANK between 1992 and 1994. Then,

she worked at the Computer Engineering Department of Middle East Technical

University as a teaching assistant between 1994-1999. Since 1999 she has been

working at Central Bank of Turkey.

198

a temporal neuro-fuzzy approach for time series …

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