ARTICLE IN PRESS

Neurocomputing 72 (2009) 3462–3468

Contents lists available at ScienceDirect

Neurocomputing

0925-23

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/neucom

Fusion ANFIS models based on multi-stock volatility causality forTAIEX forecasting

Ching-Hsue Cheng �, Liang-Ying Wei, You-Shyang Chen

Department of Information Management, National Yunlin University of Science and Technology, 123, Section 3, University Road, Touliu, Yunlin 640, Taiwan, ROC

a r t i c l e i n f o

Available online 11 June 2009

Keywords:

Multi-stock

TAIEX forecasting

ANFIS

Volatility causality

12/$ - see front matter & 2009 Elsevier B.V. A

016/j.neucom.2008.09.027

esponding author. Tel.: +886 5 5342601x5312

ail address: chcheng@yuntech.edu.tw (C.-H. C

a b s t r a c t

Stock market investors value accurate forecasting of future stock price from trading systems because of

the potential for large profits. Thus, investors use different forecasting models, such as the time-series

model, to assemble a superior investment portfolio. Unfortunately, there are three major drawbacks to

the time-series model: (1) most statistical methods rely on some assumptions about the variables; (2)

most conventional time-series models use only one variable in forecasting; and (3) the rules mined

from artificial neural networks are not easily understandable. To address these shortcomings, this study

proposes a new model based on multi-stock volatility causality, a fusion adaptive-network-based fuzzy

inference system (ANFIS) procedure, for forecasting stock price problems in Taiwan. Furthermore, to

illustrate the proposed model, three practical, collected stock index datasets from the USA and Taiwan

stock markets are used in the empirical experiment. The experimental results indicate that the proposed

model is superior to the listing methods in terms of root mean squared error, and further evaluation

reveals that the profits comparison results for the proposed model produce higher profits than the

listing models.

& 2009 Elsevier B.V. All rights reserved.

1. Introduction

Taiwan is an island country that lacks a natural energy source,and thus the degree of dependence on the global economic systemis very high. In particular, the foreign sector (net export) is over60% of Taiwan’s gross domestic product. Hence, Taiwan is a vitalmember of the international economic society. Under suchcircumstances, the impact of world economic fluctuations onTaiwan is significant, especially those originating in the USA.Additionally, Dickinson [10] demonstrated that internationalstock markets influence the movements of other global stockindexes.

Stock market investing is an exciting and challenging monetaryactivity. Market climates can dramatically change in a second, andgain–loss can be realized with a momentary decision; thus,accurate information while planning investments is crucial forinvestors. Volatility causality in a multi-stock market is animportant indicator when forecasting another stock market; inparticular, the predictive value of the US stock market’s volatilityhas been demonstrated. Thus, we utilize volatility causality in amulti-stock market in this study.

ll rights reserved.

; fax: +886 5 5312077.

heng).

For a long time, conventional time-series models have beenapplied to forecasting application problems of the real world, suchas Engle’s [11] autoregressive conditional heteroscedasticity(ARCH) model, Bollerslev’s [2] Generalized ARCH (GARCH) modelto refine the ARCH model, Box and Jenkins’ [3] autoregressivemoving average (ARMA) model and the autoregressive integratedmoving average model. However, traditional time-series requiresmore historical data along with some assumptions like normalitypostulates [16].

Furthermore, fuzzy time-series models have been used forstock-price forecasting. Song and Chissom [22] first proposed theoriginal model of the fuzzy time-series, and the following researchfocused on the two major processes of the fuzzy time-seriesmodel: (1) fuzzification and (2) establishment of fuzzy relation-ships and forecasting. In the fuzzification process, the length ofintervals for the universe of discourse can affect forecastingaccuracy, and Huarng [12] proposed distribution-based andaverage-based length to approach this issue. In addition, Chen[5] proposed a new method that used genetic algorithms to tunethe length of linguistics intervals. In the process of establishingfuzzy relationships and forecasting, Yu [25] argued that recurrentfuzzy relationships should be considered in forecasting andrecommended that different weights be assigned to various fuzzyrelationships. Therefore, Yu [25] proposed a weighted fuzzy time-series method to forecast the Taiwan Stock Exchange Capitalization

ARTICLE IN PRESS

C.-H. Cheng et al. / Neurocomputing 72 (2009) 3462–3468 3463

Weighted Stock Index (TAIEX). TAIEX is an index that reflects theoverall market movement and is weighted by the number ofoutstanding shares. As of 1966, its base year value was set at 100.Further, Cheng et al. [7] proposed a methodology that incorpo-rates trend-weighting into the fuzzy time-series model. To takeadvantage of neural networks (nonlinear capabilities), Huarng andYu [13] chose a neural network to establish fuzzy relationships inthe fuzzy time-series model, which is also nonlinear, but theprocess of mining fuzzy logical relationships is not easily under-standable [6]. Moreover, the models mentioned above have beenlimited to one variable application [26]. Recently, Chen et al. [6]proposed a comprehensive fuzzy time-series, which factors inrecent periods of stock prices and fuzzy logical relationships intothe forecasting processes. Jilani and Burney [16] proposed asimple time-variant fuzzy time-series method to forecast TAIEXand enrollments at the University of Alabama. Cheng et al. [8]proposed a new fuzzy time-series method, which is based on aweighted-transitional matrix, and also proposed two new fore-casting methods: the expectation method and the grade-selectionmethod. Yu and Huarng [26] proposed a bivariate model, whichapplies neural networks to fuzzy time-series forecasting.

From the literature review, we can conclude the following: (1)fuzzy time-series, where time-series data are represented byfuzzy sets instead of crisp values, can model the qualitativeaspects of human knowledge and can be applicable to humanrecognition; (2) fuzzy time-series are appropriately applied tolinguistic values datasets to generate forecasting rules and can gethigher accuracy. On the contrary, traditional time-series methodsfail to forecast the problems with linguistic value; (3) to improveaccuracy, the expert rules validated by expert group should beutilized in the forecasting procedure; and (4) constructing theappropriate fuzzy logical relationships for forecasting is critical.

Based on the above information, there are three majordrawbacks to the time-series model: (1) most statistical methodsrely upon some assumptions about the variables used in theanalysis, so they have limited application to all datasets [16]; (2)most conventional time-series models use only one variable inforecasting. However, there is a lot of noise caused by changes inmarket conditions; therefore, financial analysts should considermany market variables in forecasting. For this reason, forecastingmodels should use more variables to improve forecasting accuracy[26]; and (3) artificial neural networks (ANN) is a black-boxmethod, and the rules mined from ANN are not easily under-standable [6].

To address these drawbacks, this study considers that thevolatility of American stock indexes can significantly affect thevolatility of TAIEX. Because this forecasting model uses therelation between the volatility of the American stock index andthe volatility of TAIEX, the analytical results can approximate thereal world. Furthermore, a fuzzy inference system employingfuzzy if-then rules can model the qualitative aspects of humanknowledge and can be applicable for investors.

Based on the concept above, this study proposes a newvolatility model to forecast the Taiwan stock index. Firstly, thisstudy calculates the volatility of the NASDAQ stock index and theDow Jones stock index. Then, it uses the fuzzy inference system toforecast the Taiwan stock index; it considers the multi-stock index(NASDAQ stock index (t) and Dow Jones stock index (t) and TAIEX(t)) to forecast the TAIEX (t+1). Secondly, this study optimizes thefuzzy inference system parameters by adaptive network, whichcan overcome the limitations of statistical methods (data needobey some mathematical distribution). Thus, we expect that thismodel is viable and useful for investors and will provide higheraccuracy in forecasting the TAIEX in the Taiwan stock exchangemarket or the Taiwan Weighted Average Index stock index inTaiwan’s Futures Market, resulting in huge profits.

This rest of the paper is organized as follows: Section 2describes related studies, Section 3 briefly presents the proposedmodel, Section 4 describes the experiments and comparisons, andSection 5 presents the findings and discussion. Finally, theconclusions of the study are in Section 6.

2. Related studies

This section reviews related studies of different forecastingmodels for the stock market, including the adaptive-network-based fuzzy inference system (ANFIS), fuzzy C-means clustering(FCM) and subtractive clustering (Subclust).

2.1. Different forecasting models for the stock market

Researchers have presented many different methods to dealwith forecasting stock price problems. For instance, a study byDickinson [10] shows that the stock price indexes in differentcountries influence each other. Huarng et al. [14] have used thevolatility of the NASDAQ (the largest USA electronic stock market)stock index and the Dow Jones (Dow Jones Industrial Average)stock index to forecast the Taiwan stock index.

Time-series models have been applied to handle economicforecasting, such as stock index forecasting, and various modelshave been proposed. Engle [11] proposed the ARCH model, whichhas been used by many financial analysts, and the GARCH model[2]. While Box and Jenkins [3] proposed the ARMA model, whichcombines a moving average process with a linear differenceequation. During the past years, many researchers have applieddata mining techniques to financial analysis. Huarng and Yu [13]applied the backpropagation neural network to establish fuzzyrelationships in fuzzy time-series for forecasting stock price.Kinoto et al. [19] developed a prediction system for the stockmarket by using neural network. Nikolopoulos and Fellrath [20]combined genetic algorithms and neural network to develop ahybrid expert system for investment advising. Kim and Han [17]proposed a genetic algorithms approach to feature discretizationand the determination of connection weights for ANN to predictthe stock price index. Roh [21] integrated neural network andtime-series models for forecasting the volatility of the stock priceindex. Thawornwong and Enke [24] proposed redeveloped neuralnetwork models for predicting the directions of future excessstock return. Kim [18] applied support vector machine to predictthe stock price index.

Summarily, this provides the following arguments: (1) in thefuzzy time-series process, the concept of fuzzy logic wasintroduced to cope with the ambiguity and uncertainly of mostof the real-world problems; (2) traditional, crisp time-seriesforecasting methods cannot deal with the historical data repre-sented by linguistic values, but a fuzzy time-series model canovercome the drawback of traditional forecasting methods; (3)the fuzzy time-series are appropriately applied to linguistic valuesdatasets to generate high accuracy forecasting rules; and (4) thefuzzy time-series models are easily understandable for theresearchers.

2.2. Fuzzy C-means clustering

Clustering has become popular as an efficient data analysis toolto understand and visualize data structures. The prevalentformulation of this task is to use c feature vectors vjðj ¼

1;2; . . . ; cÞ to represent the c clusters, such that a sample xt isclassified into the j-th cluster according to some measure ofsimilarity and its corresponding objective function. FCM, proposed

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C.-H. Cheng et al. / Neurocomputing 72 (2009) 3462–34683464

by Bezdek [1], is the most famous and basic fuzzy clusteringalgorithm. FCM attempts to find a fuzzy partition of the dataset byminimizing the following within group least-squares errorobjective function with respect to fuzzy memberships uit andcenter vi:

Jm0 ðX;U;VÞ ¼Xc

i¼1

Xn

t¼1

um0

it d2ðxt;viÞ, (1)

where m041 is the fuzziness index used to tune out the noise inthe data, n is the number of feature vectors xt , c42 is the numberof clusters in the set and dðxt;viÞ is the similarity measurebetween a datum and a center. Minimization of Jm0 under thefollowing constraints:

ð1Þ 0 � uit � 1; 8i; t;

ð2Þ 0oPnt¼1

uit � n; 8i;

ð3ÞPci¼1

uit ¼ 1; 8t;

(2)

yields an iterative minimization pseudo-algorithm well known asthe FCM algorithm. The components vij of each center vi and themembership degrees uit are updated according to the expressions

vij ¼

Pnt¼1um0

it xkjPni¼1um0

it

and uit ¼1

Pcj¼1

dðxt ;viÞ

dðxt;vjÞ

� �2=m0�1, (3)

where j is a variable on the feature space, i.e., j ¼ 1,2,y,m.

2.3. Subtractive clustering

Chiu [9] developed subtractive clustering, a type of fuzzyclustering, to estimate both the number and initial locations ofcluster centers. Consider a set T of N data points in a D-dimensional hyperspace, where each data pointWiði ¼ 1;2; . . . ;NÞ. Wi ¼ ðxi; yiÞ, where xi denotes the p inputvariables and yi denotes the output variable. The potential valuePi of data point is calculated by Eq. (4)

Pi ¼XN

j¼1

e�akWi�Wjk2

, (4)

where a ¼ 4=r2, r is the radius defining a Wi neighborhood andk � k denotes the Euclidean distance.

The data point with many neighboring data points is chosen asthe first cluster center. To generate the other cluster centers, the

x

y

A1

A2Π

B1

B2

Π

Layer 1 Layer 2 Laye

W1

W2

Fig. 1. The architecture

potential Pi is revised of each data point Wi by Eq. (5)

pi ¼ pi � p�1 expð�bjjWi �W�1jj

2Þ, (5)

where b is a positive constant defining the neighborhood whichwill have measurable reductions in potential. W�

1 is the firstcluster center and P�1 is its potential value.

From Eq. (5), the method selects the data point with thehighest remaining potential as the second cluster center. For thegeneral equation, we can rewrite Eq. (5) as Eq. (6).

pi ¼ pi � p�k expð�bkWi �W�kk

2Þ, (6)

where W�k ¼ ðx

�k; y�kÞ is the location of the k’th cluster center and P�k

is its potential value.At the end of the clustering process, the method obtains q

cluster centers and D corresponding spreads Si, i ¼ (1,y,D). Thenwe define their membership functions. The spread is calculatedaccording to b.

2.4. ANFIS: adaptive-network-based fuzzy inference system

Jang [15] proposed ANFIS, which is a fuzzy inference systemimplemented in the framework of adaptive networks. Forillustrating the system, we assume the fuzzy inference systemconsists of five layers of adaptive network with two inputs x and y

and one output z. The architecture of ANFIS is shown in Fig. 1.Then, we suppose that the system consists of two fuzzy if–then

rules based on Takagi and Sugeno’s type [23]:Rule 1. If x is A1 and y is B1, then f 1 ¼ p1xþ q1yþ r1.Rule 2. If x is A2 and y is B2, then f 2 ¼ p2xþ q2yþ r2.The node in the i-th position of the k-th layer is denoted as Ok;i,

and the node functions in the same layer are of the same functionfamily as described below:

Layer 1: This layer is the input layer, and every node i in thislayer is a square node with a node function (see Eq. (7)). O1;i is themembership function of Ai, and it specifies the degree to whichthe given x satisfies the quantifier Ai. Usually, we select the bell-shaped membership function as the input membership function(see Eq. (8)), with maximum equal to 1 and minimum equal to 0.

O1;i ¼ mAiðxÞ for i ¼ 1;2, (7)

mAiðxÞ ¼1

1þx� ci

ai

� �2" #bi

, (8)

where ai, bi, and ci are the parameters, b is a positive value and c

denotes the center of the curve.

w2 f2

w1 f1

w2

w1

N

N

r 3 Layer 4 Layer 5

f

X Y

X Y

∑

of ANFIS network.

ARTICLE IN PRESS

Collect datasets

Define and partition the universe of discourse (B1)

Calculate multivariate volatility (NASDAQ

and Dow Jones)

Generate fuzzy inference system

Train fuzzy inference system

Forecast the testing TAIEX (t+1) by the four types

of forecasting models and calculate RMSE

FCM Subtractive Clustering

Set output MF (B2)

Linear Constant

Select the best model based on minimal RMSE and compare the

forecasting performances of the different models

Constant Linear

Fig. 2. Flowchart of proposed procedure.

C.-H. Cheng et al. / Neurocomputing 72 (2009) 3462–3468 3465

Layer 2: Every node in this layer is a square node labeled Pwhich multiplies the incoming signals and sends the product outby Eq. (9).

O2;i ¼ wi ¼ mAiðxÞ � mBiðyÞ for i ¼ 1;2. (9)

Layer 3: Every node in this layer is a square node labeled N. Thei-th node calculates the ratio of the i-th rule’s firing strength to thesum of all rules’ firing strengths by Eq. (10). Output of this layercan be called normalized firing strengths.

O3;i ¼ w̄i ¼wi

w1 þw2for i ¼ 1;2. (10)

Layer 4: Every node i in this layer is a square node with a nodefunction (see Eq. (11)). Parameters in this layer will be referred toas consequent parameters.

O4;i ¼ w̄if i ¼ w̄iðpi þ qi þ riÞ, (11)

where pi, qi and ri are the parameters.Layer 5: The single node in this layer is a circle node labeled

Pthat computes the overall output as the summation of allincoming signals (see Eq. (12))

O5;i ¼X

i

w̄if i ¼

Pi¼1wifPi¼1wi

¼ overall output. (12)

3. Proposed model

As stated in Section 1, there are three major drawbacks to thetime-series model: (1) statistical methods rely upon someassumptions [16]; (2) time-series models use only one variable[26]; and (3) ANN is a black-box method, and the rules minedfrom ANN are not easily understandable [6]. Nevertheless, theforecasting rules are useful for investors buying and selling stocks.To reconcile these drawbacks, this study considers that thevolatility of American stock indexes can significantly affectthe volatility of TAIEX. Because this forecasting model uses therelation between the volatility of the American stock index andthe volatility of TAIEX, the analytical results can approximate thereal world. Furthermore, a fuzzy inference system employingfuzzy if-then rules can model the qualitative aspects of humanknowledge and can be applicable for investors.

Based on the concept above, this study proposes a newvolatility model to forecast the Taiwan stock index. Firstly, thisstudy calculates the volatility of the NASDAQ stock index and theDow Jones stock index by Eqs. (1) and (2). Then, using the fuzzyinference system to forecast the Taiwan stock index, it considersmulti-stock indexes (NASDAQ stock index (t) and Dow Jones stockindex (t) and TAIEX (t)) to forecast TAIEX (t+1). Secondly, this studyoptimizes the fuzzy inference system parameters by adaptivenetwork, which can overcome the limitations of statisticalmethods (data need obey some mathematical distribution). Theoverall flowchart of the proposed model is shown in Fig. 2.

This section uses some numerical data as an example, and thecore concept of the proposed algorithm is shown step by step.

Step 1: Collect datasetsIn this section, we choose TAIEX data from 1997 to 2003 (7

sub-datasets) to illustrate the proposed model (such as year-2000sub-datasets, which contains 271 transaction days). Training dataare from January to October, and the remaining data (fromNovember and December) are used for testing.

Step 2: Calculate multivariate volatility (NASDAQ stock indexand Dow Jones stock index)

In this section, we define two variables, namely (1) theNASDAQ (N) and (2) the Dow Jones (D), and calculate the volatilityof the two variables by Eqs. (13) and (14). Table 1 lists thedifferences in the variables NASDAQ and Dow Jones. In Table 1,

some data under the NASDAQ and Dow Jones are empty becausethere were no transactions on those days. For this reason, thisstudy fills in the last volatility as the differences.

diffðNðtÞÞ ¼ NðtÞ � Nðt � 1Þ. (13)

diffðDðtÞÞ ¼ DðtÞ � Dðt � 1Þ. (14)

Step 3: Define and partition the universe of discourse for inputvariables (see B1 block of Fig. 2)

Firstly, we define each universe of discourse for three variables(TAIEX (t), diff(N(t)), diff(D(t))) according to the minimum andmaximum value in each variable. Secondly, we partition theuniverse of discourse into three linguistic intervals by using FCMclustering [1] (triangular membership function) and subtractiveclustering [9] (Gaussian membership function), respectively.

Step 4: Set the type of membership function for outputvariables (see B2 block of Fig. 2)

There are two types of membership functions for outputvariables as follows:

(1)

Lineal type: a typical rule in a Sugeno fuzzy model has theform as follows: If x (TAIEX (t)) ¼ Ai, y (diff(N(t))) ¼ Bj and z

(diff(D(t))) ¼ Ck, then output is fl ¼ pl x+ql y+rl z+sl, where x

ARTICLE IN PRESS

Table 1Differences in variables.

Date NASDAQ diff(N(t)) Dow Jones diff(D(t))

2000/1/3 4131.15 11357.51

2000/1/4 3901.69 �229.46 10 997.93 �359.58

2000/1/5 3877.54 �24.15 11122.65 124.72

2000/1/6 3727.13 �150.41 11 253.26 130.61

2000/1/7 3882.62 155.49 11522.56 269.3

2000/1/8 155.49 269.3

2000/1/9 155.49 269.3

2000/1/10 4049.67 167.05 11572.2 49.64

2000/1/11 3921.19 �128.48 11511.08 �61.12

2000/1/12 3850.02 �71.17 11551.1 40.02

2000/1/13 3957.21 107.19 11582.43 31.33

2000/1/14 4064.27 107.06 11722.98 140.55

2000/1/15 107.06 140.55

2000/1/16 107.06 140.55

2000/1/17 107.06 140.55

2000/1/18 4130.81 66.54 11560.72 �162.26

2000/1/19 4151.29 20.48 11489.36 �71.36

2000/1/20 4189.51 38.22 11351.3 �138.06

2000/1/21 4235.4 45.89 11 251.71 �99.59

2000/1/22 45.89 �99.59

2000/1/23 45.89 �99.59

2000/1/24 4096.08 �139.32 11008.17 �243.54

2000/1/25 4167.41 71.33 11029.89 21.72

2000/1/26 4069.91 �97.5 11032.99 3.1

2000/1/27 4039.56 �30.35 11028.02 �4.97

2000/1/28 3887.07 �152.49 10 738.87 �289.15

2000/1/29 �152.49 �289.15

2000/1/30 �152.49 �289.15

2000/1/31 3940.35 53.28 10 940.53 201.66

C.-H. Cheng et al. / Neurocomputing 72 (2009) 3462–34683466

(TAIEX (t)), y (diff(N(t))), z (diff(D(t))) are linguistic variables, Ai,Bj and Ck are the linguistic labels (high, middle, low), fl denotesthe l-th output value, pl, ql, rl and sl are the parameters (i ¼ 1,2, 3; j ¼ 1, 2, 3; k ¼ 1, 2, 3; and l ¼ 1,2,y,27).

(2)

Constant type: a zero-order Sugeno model, the output level f l

is a constant (pi ¼ qi ¼ si ¼ 0).

Step 5: Generate fuzzy inference systemFrom steps 3 and 4, we can obtain four types of forecasting

models: (1) fuzzy C-means with linear type (FCM_L), (2) fuzzy C-means with constant type (FCM_C), (3) subtractive clustering withlinear type (Subclust_L) and (4) subtractive clustering withconstant type (Subclust_C). Then, we can generate the fourdifferent fuzzy inference systems according to the four types offorecasting models, respectively.

The detailed steps of generating fuzzy inference system aredescribed as follows: Firstly, from step 3, we can get the linguisticintervals as input membership functions, and the output member-ship functions are set by step 4. Secondly, we can generate fuzzyif–then rules, where the linguistic values (Ai,Bi and Ci) from inputmembership functions are used as the if–condition part and theoutput membership functions (f i) as the then part.

FCM case: the input membership partitioned by FCM cluster-ing, we generate 27 rules (3�3�3). The general rule is describedas follows: If x(TAIEX (t)) ¼ Ai, y (diff(N(t))) ¼ Bj and z

(diff(D(t))) ¼ Ck, then output is fl ¼ pl x+ql y+rl z+sl.Where x (TAIEX (t)), y (diff(N(t))) and z (diff(D(t))) are linguistic

variables, Ai, Bj and Ck are the linguistic labels (high, middle, low),fl denotes the l-th output value, pl, ql, rl and sl are the parameters(i ¼ 1, 2, 3; j ¼ 1, 2, 3; k ¼ 1, 2, 3, and l ¼ 1,2,y,27).

The output membership function is constant whenpl ¼ ql ¼ rl ¼ 0.

Subclust case: the input membership partitioned by subtrac-tive clustering, we generate three rules. The general rule isdescribed as follows: If x (TAIEX (t)) ¼ Ai, y (diff(N(t))) ¼ Bi and z

(diff(D(t))) ¼ Ci, then fi ¼ pi x+qi y+ri z+si. Where x (TAIEX (t)), y

(diff(N(t))), z (diff(D(t))) are linguistic variables, Ai, Bi, Ci are thelinguistic labels (high, middle, low), fi denotes the i-th outputvalue, pi, qi, ri, si are the parameters (i ¼ 1,2,3).

The output membership function is constant whenpi ¼ qi ¼ ri ¼ 0.

Step 6: Train fuzzy inference system parameters from trainingdatasets

In this section, we employ a combination of the least-squaresmethod and the backpropagation gradient-descent method fortraining four types of forecasting models and use fuzzy inferencesystem membership function parameters to emulate a giventraining dataset. This study sets epoch as 50 (the process isexecuted for the predetermined fixed number (50) of iterationsunless it terminates while the training error converges) for thetraining stopping criterion and then obtains the parameters forthe selected output membership function.

Step 7: Forecast the testing TAIEX (t+1) by the four types offorecasting models and calculate root mean squared error (RMSE)

Firstly, the fuzzy inference system parameters of the four typesof forecasting models are determined when the stopping criterionis reached from step 6, then the four training forecasting modelsare used to forecast T (t+1) for the target testing datasets,respectively. Secondly, the four RMSE values are calculated intesting datasets by Eq. (15).

RMSE ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPnt¼1

jactualðtÞ � forecastðtÞj2

n

vuuut. (15)

Where actualðtÞ denotes the real TAIEX value, forecastðtÞ denotesthe predicting TAIEX value, and n is the number of data.

Step 8: Select the best model based on minimal RMSE andcompare the forecasting performances of the different models

Based on minimal RMSE for the target testing datasets fromstep 7, the best forecasting model among the four models can beobtained. Then the minimal RMSE is taken as evaluation criterionto compare with different models.

4. Experiments and comparisons

This section provides an evaluation of the model’s accuracyand comparison to other models, as well as profit evaluations andcomparisons. To verify the proposed model, experimentation,using the TAIEX from 1997 to 2003 (7 sub-datasets), wasimplemented. The sub-datasets for the first 10-month period areused for training, and those from November to December areselected for testing.

4.1. Accuracy evaluations and comparisons

After the experiments, we generate 7 forecasting performancesfor the 7 testing sub-datasets. Then, this study compares theperformances of the proposed model with the conventional fuzzytime-series model, Chen’s [4] model. Furthermore, to examinewhether the proposed model surpasses the latest fuzzy time-series model, the performance of Yu’s [25] model is comparedwith the proposed model. The forecasting performances of Chen’smodel, Yu’s model and the proposed model are listed in Table 2.From Table 2, we can see that the proposed model outperformsthe performances of the listing model.

4.2. Profit evaluations and comparisons

For making simulation trades and showing the profits, we settwo trade rules by the Taiwan Futures Exchange (TAIFEX) [27], use

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Table 2The performance comparisons of different models (TAIEX).

Models Year

1997 1998 1999 2000 2001 2002 2003

Yu’s model [25] 165 164 145 191 167 75 66

Chen’s model [4] 154 134 120 176 148 101 74

Proposed model 130a 113a 103a 138a 118a 67a 50a

a The best performance among three models.

Table 3The profits comparisons of different models (TAIEX).

Year a Models

Yu’s model [25] Chen’s model [4] Proposed model

1997 0.02 �107 �127 935a

1998 0.03 �864 661 878a

1999 0.03 �756 �702 124a

2000 0.01 �200 �106a�154

2001 0.02 450a�340 91

2002 0.005 �96 �17 466a

2003 0.005 �190 �238 481a

Cumulated profits �1763 �869 2821a

a The best profits among three models.

C.-H. Cheng et al. / Neurocomputing 72 (2009) 3462–3468 3467

the two trade rules to calculate profits, and assume that theprofits unit is equal to one. Therefore, the profit formula is definedas Eq. (16).

Rule 1: sell rule

IFjforecastðtÞ � actualðtÞj

actualðtÞ� a And

forecastðt þ 1Þ � actualðtÞ40 Then sell

Rule 2: buy rule

IFjforecastðtÞ � actualðtÞj

actualðtÞ� a And

forecastðt þ 1Þ � actualðtÞo0 Then buy

where a denotes threshold parameter (0oa � 0:07, the thresholdparameter depends on daily fluctuation of TAIEX).

Definition of profit:

Profit ¼Xp

ts¼1

ðactualðt þ 1Þ � actualðtÞÞ

þXq

tb¼1

ðactualðtÞ � actualðt þ 1ÞÞ (16)

where p represents the total number of days for selling, q

represents the total number of days for buying, ts represents thet-th day for selling and tb represents the t-th day for buying.

The optimal threshold parameter a is obtained when theforecasting performance reaches best profits in the trainingdataset. From the optimal threshold parameter a and Eq. (16),the profits for different models are calculated, and the profitsresults are shown in Table 3. Based on Table 3, we can see that theproposed model has higher profits than the listing models in fivetesting periods (excluding 2000 and 2001).

5. Findings and discussion

Based on the verification and comparison results, the proposedmodel outperforms the listing methods. Also of interest is the

degree of influence of other countries’, particularly neighboringcountries’ (e.g., China, Japan and Hong Kong), stock indexes; thus,in the future, we will further verify the influence of the stockindexes (e.g., A-share index, Nikkei 225 and Hang Seng index) ofdifferent countries on the proposed model.

From the experimental results, there are two findings:(1) According to Table 2, it is evident that the proposed model

is superior to the listing methods in terms of RMSE. The mainreason is that the proposed model takes into account multi-stockvolatility causality with ANFIS learning for TAIEX forecasting.

(2) From the empirical results, the best performance in fourtypes of forecasting models is subtractive clustering with lineartype (Subclust_L). That is, in Subclust_L, the parameters of themodel are not equal to 0 (output function). This means that theAmerican stock indexes significantly affect the volatility of TAIEX.These results confirm the study of Dickinson [10].

6. Conclusions

A new model, based on multi-stock volatility causality joinedto the fusion ANFIS procedure, was proposed to forecast stockindex problems in Taiwan; furthermore, the proposed model wascompared with two different models, Chen’s model and Yu’smodel, to evaluate the results. This proposed model mainly usesinput variables of stock index (e.g., Dow Jones, NASDAQ andTAIEX) to forecast the TAIEX in the next trading day for investors.To illustrate the proposed model, three practical, collected stockindex datasets from the USA and the Taiwan stock market, theDow Jones, NASDAQ and TAIEX, were employed in this empiricalexperiment, all of which consist of datum from 1997 to 2003 (7years in total). Each stock index dataset is split into 7 sub-datasets, based on year. The first 10 months of the sub-dataset foreach year, January to October, were used for training data, and thelast 2 months, November to December, were used for testing data.From Table 2, the experimental results of three datasets indicatethat the proposed model outperforms the listing models in termsof RMSE. Moreover, the results of this study should be useful andviable for stock investors, decision makers and future research.Investors can utilize this forecasting model to uncover superiortarget investments in the stock market.

Generally, the proposed model is based on the following,which overcome the drawbacks mentioned in Section 1: (1) ANFIScombines the advantages of ANN and fuzzy logic system; thus, itdoes not pre-assume the data distribution; (2) the study usesmore variables, which addresses the issue of only one variable inforecasting accuracy and better approximates actual marketconditions and environments, to improve forecasting perfor-mance; and (3) the study generates the decision rules regardedas references of investment for stock investors.

For subsequent research, we can use datasets of othercountries, such as China, Japan and Hong Kong, to further validatethe proposed model. Moreover, there are two methods suitable forintegration into the proposed model that will improve theforecasting accuracy: (1) employ data discretization in thepreprocessing step, which granulates (partitions) attributes toenhance performance of the proposed model (can generate fewerrules by data discretization) and (2) validate the generated rulesby expert group to improve accuracy.

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CHING-HSUE CHENG received the Bachelor’s degree inMathematics from Chinese Military Academy in 1982,the Master’s degree in Applied Mathematics fromChung-Yuan Christian University in 1988, and thePh.D. degree in System Engineering and Managementfrom National Defense University in 1994. He is nowprofessor of Information Management Department inNational Yunlin University of Science and Technology.His research is mainly in the field of fuzzy logic, fuzzytime series, soft computing, reliability, and datamining. He has published more than 200 papers(include 97 significant journal papers).

LIANG-YING WEI received the Bachelor’s degree inEnvironmental Engineering and Science from FengChia University in 1998 and the Master’s degree inInformation Management from Huafan University in2005. He is now a Ph.D. student of InformationManagement Department in National Yunlin Univer-sity of Science and Technology and works in nationalgenotyping center at academia sinica for research. Hisresearch is mainly in the field of fuzzy time series, softcomputing, machine learning, data mining and bioin-formatics.

YOU-SHYANG CHEN received the Bachelor’s degreefrom National Taiwan University of Sciences andTechnology in 1988, and the Master’s degree fromNational Yunlin University of Science and Technologyin 2006. He is now a Ph.D. student in National YunlinUniversity of Science and Technology and majors ininformation management. His research interests in-clude financial analysis and bankruptcy prediction.