UNIVERSITY OF CALGARY

Comparison of Ordinary Least Square Regression, Spatial Autoregression, and

Geographically Weighted Regression for Modeling Forest Structural Attributes Using a

Geographical Information System (GIS)/Remote Sensing (RS) Approach

By

Prasanna Man Shrestha

A DOCUMENT

SUBMITTED TO THE FACULTY OF GRADUATE STUDIES

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE

DEGREE OF MASTER OF GEOGRAPHIC INFORMATION SYSTEMS

DEPARTMENT OF GEOGRAPHY

CALGARY, ALBERTA

SEPTEMBER, 2006

© Prasanna Man Shrestha 2006

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Approval Page

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Abstract

The performances of three modeling techniques: (i) ordinary least square (OLS)

regression, (ii) spatial autoregression (SAR) and (iii) geographically weighted regression

(GWR) were compared for the task of predicting a key forest structural parameter –

crown closure – across a study area in west-central Alberta using a series of spectral and

topographic variables. The OLS model performed moderately well (log-

likelihood=47.6954, AIC=-87.391, R2=47.95%), but exhibited clear signs of spatial

autocorrelation (Moran’s I =0.2851) and spatial non-stationarity (Breusch-Pagan’s

statistic =26.17). The SAR model produced a statistically-significant improvement in

model fit (log-likelihood=81.94, AIC=-153.89, R2=59.81%) and a substantial reduction

of the spatial autocorrelation problem (Moran’s I=0.0252), however, the model still

displayed significant issues with spatial non-stationarity (Breusch-Pagan’s statistic

=40.98). The best results were obtained with the GWR model (log-likelihood=138.656,

AIC=-201.286, R2=72.65%) with an almost complete removal of the effects of spatial

autocorrelation (Moran’s I = -0.0046) and spatial non-stationarity. The results suggest

that GWR provides an effective vehicle for modeling forest structural variables across

diverse natural landscapes, as well as an intriguing new tool for detecting and mapping

the spatial variability of model elements. However, the interpretation of spatially-varying

model results requires the use of contextual and other underlying spatial information. In

addition, the task of inverting the GWR for locations beyond the training points is

problematic.

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Acknowledgements

First of all, I wish to express my sincere thanks to all of those who directly or indirectly

contributed to the completion of this study. I am very greatly indebted to my main

supervisor, Dr. Gregory J. McDermid, Department of Geography, University of Calgary,

for persistent and invaluable supervision throughout the work, starting with proposal

writing and ending with final language ‘polishing’.

I extend sincere thanks to all the staffs of Master in GIS Program, especially Dr.

Nigel Waters and Dr. Shelly Alexander for their support and co-operation that I received.

My special thanks are also due to Bart Hulshof, Paulina Medori, David Nuell and Lynn

Moorman for their continuous administrative and technical support throughout the study

period.

I sincerely express my heartfelt gratitude to my “Idol Guru” Professor Shivcharn

S. Dhillion, University of Oslo, who was my supervisor for former Master Degree in

Natural Resource Management, Agricultural University of Norway. I reach up to this

level because of him.

This project would not have been possible if Dr. McDermid did not allow me to

access to data from The Foothills Model Forest Grizzly Bear Research Program. So

thanks especially to all the individuals who contributed to this data set and to the many

government and industrial partners who support this research program.

Thanks for being supportive and understanding in difficult times to my all

colleagues at department especially Charles Omoregie. My sincere thanks to Asha

Omoregie, my in-law’s family (Binod K.C., Nisha Dhaubhadel and Ashish K.C.) for their

inspiration and support I received in various ways for the completion of this project, and

during my stay in Calgary.

Finally, but not least, I am extremely grateful to my family at Kathmandu, Nepal

for their continuous inspiration and moral support. Last but not least, my words of

appreciation go to my wife, Rashmi Dhaubhadel for her invaluable moral support in

materializing this thesis, and my ever-charming daughter, the “Greatest Sharon”, without

whom I can not imagine this degree.

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Dedication

To my beloved parents

Dr. Jeevan Man Shrestha and Mrs. Shanta Laxmi Shrestha,

for your support, by being true to all the ideals and values

that you tried to teach me

Thank you forever for standing by me

I love and appreciate you forever.

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Table of Contents

Approval Page ................................................................................................................... ii Abstract............................................................................................................................. iii Acknowledgements .......................................................................................................... iii Dedication .......................................................................................................................... v Table of Contents ............................................................................................................. vi List of Tables .................................................................................................................. viii List of Figures................................................................................................................... ix CHAPTER 1: INTRODUCTION................................................................................... 1

1.1 Rational of Study ................................................................................................ 1 1.2 Study Objectives ................................................................................................. 7

CHAPTER 2: LITERATURE REVIEW....................................................................... 8

2.1 Remote Sensing of Forest Structure ................................................................... 9 2.1.1 Estimating Forest Structural Attributes .................................................... 10 2.1.2 Thematic Mapper Spectral Bands and Forest Structural Attributes ......... 13 2.1.3 Spectral Vegetation Index and Forest Structural Attributes ..................... 15 2.1.4 Digital Elevation Models and Forest Structural Attributes....................... 19

2.2 Review of Selected Forest Structural Attributes............................................... 22 2.2.1 Crown Closure .......................................................................................... 22

2.3 Review of Regression Techniques.................................................................... 28 2.3.1 Overview of Models ................................................................................. 29

2.3.1.1 Ordinary Least Square Regression........................................................ 29 2.3.1.2 Spatial Autoregression .......................................................................... 34 2.3.1.3 Geographically Weighted Regression................................................... 37

2.3.2 Theoretical Background............................................................................ 46 2.3.2.1 Ordinary Least Square Regression........................................................ 46 2.3.2.2 Spatial Autoregression .......................................................................... 53 2.3.2.3 Geographically Weighted Regression................................................... 56

CHAPTER 3: STUDY AREA....................................................................................... 65

3.1 Description of Study Area ................................................................................ 65 CHAPTER 4: METHODS ............................................................................................ 69

4.1 Data Acquisition ............................................................................................... 69 4.1.1 Ground Crown Closure Data .................................................................... 69 4.1.2 Landsat Imagery........................................................................................ 70 4.1.3 Digital Elevation Model (DEM) ............................................................... 74

4.2 Data Analysis .................................................................................................... 75 4.2.1 Descriptive Analysis ................................................................................. 75 4.2.2 Arcsine Transformation ............................................................................ 75 4.2.3 Correlation Analysis ................................................................................. 76 4.2.4 Selection of Predictor Variables ............................................................... 76

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4.2.5 Model Building ......................................................................................... 77 4.2.5.1 OLS model ............................................................................................ 77 4.2.5.2 SAR model............................................................................................ 78 4.2.5.3 GWR model .......................................................................................... 79

4.2.6 Comparison of Models.............................................................................. 80 4.2.6.1 Analysis of model diagnostics .............................................................. 80 4.2.6.2 Analysis of model residuals .................................................................. 80 4.2.6.3 Analysis of Moran’s I ........................................................................... 81

4.2.7 Mapping of Parameter Estimates and Model Statistics ............................ 81 4.2.8 Mapping of Observed, Predicted and Residual Values ............................ 82

CHAPTER 5: RESULTS AND DISCUSSION ........................................................... 83

5.1 Descriptive Statistics of Model Variables......................................................... 83 5.2 Correlation Analysis Among Model Variables................................................. 84 5.3 Selection of Predictor Variables ....................................................................... 84 5.4 Models Fitting................................................................................................... 86

5.4.1 OLS Model................................................................................................ 86 5.4.2 SAR Model ............................................................................................... 90 5.4.3 GWR Model.............................................................................................. 91

5.5 Comparison of OLS, SAR and GWR Models .................................................. 94 5.5.1 Analysis of Model Diagnostics................................................................. 94 5.5.2 Analysis of Model Residuals .................................................................... 97 5.5.3 Analysis of Moran’s I ............................................................................. 101

5.6 Spatial Variation of Estimated Parameters across the Study Area ................. 104 5.7 Mapping Observed, Predicted and Residual Values Across the Study Area.. 111

CHAPTER 6: CONCLUSIONS ................................................................................... 119 REFERENCES.............................................................................................................. 122 APPENDIX A: Output of OLS model ........................................................................ 141 APPENDIX B: Output of SAR model......................................................................... 149 APPENDIX C: Output of GWR model ...................................................................... 163 APPENDIX D: List of Acronyms…………………………………………………….193

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List of Tables

Table 3.1: A summary of the natural regions and subregions found in the study area, including dominant vegetation and characteristic geology and landforms. Modified after Achuff, 1992. .................................................................................................................... 68 Table 4.1: Coefficients for calculating the tasseled cap components from 8-bit, at-satellite reflectance imagery. ............................................................................................ 74 Table 5.1: Description of model variables. ...................................................................... 83 Table 5.2: Descriptive statistics of model variables......................................................... 83 Table 5.3: Correlation analysis of model variables.......................................................... 84 Table 5.4: Statistical results for selecting the independent variables............................... 85 Table 5.5: Statistical results of OLS model for predicting ArcCC. ................................. 87 Table 5.6: Statistical results of SAR model for predicting ArcCC. ................................. 91 Table 5.7: Comparative analysis of parameter estimates fitted OLS and GWR models. 92 Table 5.8: Monte Carlo test for spatial variability. .......................................................... 93 Table 5.9: Goodness of fit (F-test) for improvement of GWR over OLS model. ............ 93 Table 5.10: Evaluation of OLS, SAR and GWR based on model diagnostics. ............... 95

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List of Figures

Figure 3.1: Location of the study area in west-central Alberta, Canada.......................... 66 Figure 3.2: Natural regions and subregions within the study area................................... 67 Figure 4.1: Plot layout used to characterize vegetation and ground cover across a 30-metre Landsat pixel (McDermid, 2005)............................................................................ 70 Figure 4.2: Landsat TM5 covering the study area and sample plots. .............................. 72 Figure 5.1: Plots of OLS fitted model- a) Histogram of residual b) Normal PP plot of residual c) Detrended Normal QQ Plot of residual.......................................................... 89 Figure 5.2: Spatial correlogram for the variables ArcCC, Wt, Gr and EL. ..................... 90 Figure 5.3: Scatter diagram between observed ArcCC and fitted ArcCC by OLS, SAR and GWR .......................................................................................................................... 96 Figure 5.4: Residual plot of the observed ArcCC and predicted ArcCC fitted by OLS, SAR and GWR.................................................................................................................. 99 Figure 5.5: Comparison of models across crown classes: a) Model residuals and b) Absolute residuals........................................................................................................... 100 Figure 5.6: Moran’s I on residuals for three different models. ...................................... 102 Figure 5.7: Spatial correlogram of residuals for three different models. ....................... 103 Figure 5.8: Estimated value and corresponding interpolated surface fitted by GWR model: a) Intercept, b) t-value for intercept, c) Elevation, d) t-value for elevation, e) Greenness, f) t-value for greenness, g) Wetness and h) t-value for wetness .................. 107 Figure 1.9: Local R2 value and corresponding interpolated surface…………………..109 Figure 1.10: Variation of GWR estimates of intercept, parameters, residual and local R2 across three different types of forest……………………………………………….110 Figure 1.11: Maps of observed, predicted and residual crown closure using OLS Model……………………………………………………………………………...113 Figure 1.12: Maps of observed, predicted and residual crown closure using SAR model……………………………………………………………………………...114 Figure 1.13: Maps of observed, predicted and residual crown closure using GWR model…………………………………………………………………………….115

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CHAPTER 1: INTRODUCTION

1.1 Rational of Study

Forests comprise a central role in both Canadian and international economies, while also

supporting a variety of critical ecosystem processes such as wildlife habitat, biodiversity,

and the global climate system (Gedney and Valdes, 2000; Castro et al., 2003; Lu et al.,

2004; Couteron et al., 2005). Intelligent management and understanding of these

resources requires detailed information on forest structure and its attributes at both local

and regional scales (Aplin, 2005).

The term forest structure is vague, and includes attributes such as age, diameter,

height, density, basal area, species composition, biomass, crown closure, and leaf area

index (Spies, 1998; Stone and Porter, 1998; Lu et al., 2004; Maltamo et al., 2005).

These attributes are interrelated with each other, and play an important role in

understanding and estimating of many ecological processes, including forest health

(Clark et al., 2004; Leckie et al., 2004), net primary production (Melillo et al., 1993;

Gower et al., 1999; Kotchenova et al., 2004), light interception (Sellers et al., 1992;

Nagler et al., 2004), water flux (Mohamed et al., 2004; Leuning, 2005), carbon flux

(Potter, 1999; Potter et. al, 2003; Tuyl et al., 2005), fire dynamics (Andersen et al., 2005;

Hall et al., 2005), habitat dynamics (DeGraaf et al., 1998; Hyde, 2005), forest disturbance

(Chittibabu and Parthasarathy, 2000; Bhuyan et al., 2002), and forest succession (Hall et

al., 1991; Fiorella and Ripple, 1993). It is well established that accurate mapping and

monitoring of forest structural parameters provides an important foundation for

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sustainable forest management strategies (Franklin, 2001; Neeff et al, 2003; Mäkelä and

Pekkarinen, 2004).

Field methods, air photo interpretation, and digital remote sensing have all been

used to estimate forest structural attributes (Martin et al., 1998). The specific application

purpose, desired accuracy, resources availability and site accessibility are some key

factors in determining the methods of choice (Martin et al., 1998; Aplin, 2005). However,

conventional practices (field measurement and manual photo interpretation) are time- and

labour-intensive processes that are often impractical over large and inaccessible areas

(Woodcock et al., 1994; McRoberts et al., 2002; Pu et al., 2003; Lu et al., 2004; Mäkelä

and Pekkarinen, 2004; Sivanpillai et al., 2006). As an alternative, digital remote sensing

strategies have shown great promise (e.g. Sader et al., 1990; Lucas et al., 1993;

Steininger, 2000; Gerylo et al., 2002; Franklin et al., 2003), particularly with recent

technological improvements (Dymond et al., 2002; McRoberts et al., 2002; Lu et al.,

2004). In the past two decades, various techniques have been developed and used for the

estimation of the forest structural attributes from airborne optical sensors (Franklin and

McDermid, 1993; King, 2000), satellite optical sensors (Turner et al., 1999; Eklundh et

al., 2001; Franco-Lopez et al., 2001; Lu et al., 2004), RADAR (Harrell et al., 1995;

Fransson and Israelsson, 1999; Castel et al., 2002; Santos et al., 2003), hyperspectral

(Underwood et al., 2003; Schmid et al., 2004) and LiDAR (Næsset, 1997; Drake et al.,

2003; Lovell et al., 2003; Clark et al., 2004). However among others, Landsat data is

still widely used for practical purposes, including wide availability, easy data processing,

low cost of acquisition (e.g., Cohen and Goward, 2004; Lu et al., 2004; Aplin, 2005;

Boyd and Danson, 2005).

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The physical principles underlying most of our strategies for extracting vegetation

information from remote sensing involve the characteristic patterns in which plant

materials absorb, transmit, and reflect electromagnetic radiation at different wavelengths;

many of which can be captured by digital sensor systems (Boyd and Danson, 2005;

Jensen, 2005). To date, most ecological remote sensing applications are directed towards

developing models that relate spectral response patterns extracted from remote sensing

imagery to in-situ measurements acquired in the field (Cohen et al., 1995; Kimes et al.,

1996; Turner et al., 1999; Eklundh et al., 2001; Franco-Lopez et al., 2001). Many

scholars have attempted to establish statistical relationships between spectral response

and in-situ measurements using original Landsat bands (Gemmell, 1995; Jakubauskas

and Price, 1997; Gerylo et al., 2002; Franklin et al., 2003), extracted spectral indices

(Sader et al., 1990; Frank and Aase, 1994; Schmidt and Karnieli, 2001), and transformed

bands (Cohen and Spies, 1992; Cohen et al., 1995; Collins and Woodcock, 1996;

Dymond et al., 2002). Moreover, several studies (e.g., Anderson et al., 1993; Lu et al.,

2004; Healey et al., 2006; Sivanpillai et al., 2006) have examined comparative analysis of

the relationships using these multiple information sources. However conclusions derived

are mixed and varied, depending upon stand and site characteristics (Lu et al., 2004).

Such variation may be a result of external factors such as orientation/inclination of the

viewing angle, sun elevation, wind speed, clouds, terrain and moisture condition, as well

as internal factors such as phenology, physical temperature, area coverage, species

composition, vigor and geometry (Treitz and Howarth, 1999; Wigneron et al., 2003).

Moreover, improvements on the estimation of forest structural attributes have been

reported when remote sensing data is enhanced with additional topographic and climatic

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variables (e.g., Davis and Goetz, 1990; Moisen and Edwards, 1999; Fahsi et al., 2000;

Hörsch, 2003).

When the forest structural attribute to be modeled is a continuous variable

measured at the ratio level, regression techniques involving ordinary least square (OLS)

methods are far and away the most commonly encountered strategy in the literature (e.g.,

Franklin and McDermid, 1993; Cohen et al., 2001; Berterretche et al., 2005; McDermid,

2005). Using OLS techniques, previous studies have demonstrated the relationship

between spectral response and in-situ measurements of forest structural attributes (e.g.,

Fassnacht et al., 1997; Salvador and Pons, 1998; Geyrlo et al., 2002). However, OLS

techniques are subject to assumptions of independent observations and constant variance

across space (Montgomery et al., 2001) that may limit their suitability (e.g., Foody, 2003;

Zhang and Shi, 2004; Zhang et al., 2005). Several scholars (e.g., Miller and Franklin,

2002; Foody, 2004; Hi et al., 2006) have argued that these assumptions are commonly

violated in ecological settings, leading to biased parameter estimates, misleading

significance tests and suboptimal prediction results (Fox et al., 2001; Foody, 2003; Miller

and Franklin, 2002). Under these conditions, there is a strong need to consider the role of

alternative regression techniques.

It is well known that autocorrelation - the lack of independence among successive

observations - is present in ecological variables across both spatial and temporal domains

(Legendre, 1993; Legendre and Legendre, 1998). Consequently, a variety of statistical

modeling techniques, both spatial and non-spatial, have been explored (e.g., Austin and

Meyers, 1996; Lehmann, 1998; Guisan et al., 2002; Moisen and Frescino, 2002), and

improved results have been reported (e.g., Frescino et al., 2001; Cohen et al., 2003).

5

Spatial models present attractive strategies that have the potential to account for

the spatial structuring present in natural ecosystems (Legendre and Legendre, 1998;

Lichstein et al., 2002). Spatial autoregression (SAR) is one brand of spatial statistical

models which takes into account spatial autocorrelation by incorporating spatial linkages

or proximity of the observation (i.e., spatial weight matrix). SAR model has been

successfully used to analyze spatial datasets in ecology exhibiting strong spatial

autocorrelation, which yielding better classification and prediction accuracy (Legendre

and Legendre, 1998; Lichstein et al., 2002; Dark, 2004)

While SAR techniques provide methods for capturing spatial autocorrelation, they

are not able to contend with non-stationarity: the variability of model parameters across

space (e.g., Zhang and Shi, 2004, Zhang et al., 2004). For spatial modeling, both spatial

autocorrelation and spatial non-stationarity are equally important, and needs to take into

account (Brunsdon et al., 1996; Fotheringham et al., 1998; 2002). Spatial non-stationarity

is an important inherent component of natural environment due to its complex and

dynamic biological processes as well as broad-scale physical processes (Legendre and

Legendre, 1998; Frescino et al., 2001; Fernandes and Leblanc, 2005), and previous

research has shown that models designed to account for these patterns have improved

prediction capabilities (e.g., Foody, 2003; Zhang and Gove, 2005; Zhang et al., 2005).

Failure to take into account of non-stationarity in the modeling process may lead to

biased parameter estimates, misleading significance tests and suboptimal prediction

(Zhang and Shi, 2004). Recently, a new regression technique called geographically

weighted regression (GWR) has been developed which explicitly incorporates the issue

of spatial location, and is thus able to take into account spatial autocorrelation and non-

6

stationarity (Brunsdon et al., 1996; Fotheringham et al., 1998; 2003). GWR enables the

estimation of model parameters and statistics at each sample location, thereby allowing

the description of patterns and relationships between variables in a manner that was

unobservable with OLS and SAR (Fotheringham et al., 1998; 2003). In addition, GWR is

compatible with Geographical Information System (GIS) tools that facilitate the mapping

of estimated parameters and model statistics across space, thereby allowing the in-depth

investigation of spatial variability (Platt, 2004; Wang et al, 2005). With these advantages,

the use of GWR has been increasing across many disciplines (e.g., Nakaya, 2001; Huang

and Leung, 2002; Hanham and Spiker, 2004; Benson et al., 2005; Malczewski and Poetz,

2005) including ecological studies (e.g., Foody, 2004; Zhang et al., 2005; Shi et al.,

2006). However, the topic has received very little attention in the remote sensing

literature (but see Foody, 2003; 2004; Wang et al., 2005). In particular, there is a paucity

of studies that provide empirical comparisons of OLS, SAR and GWR techniques in a

remote sensing/forest structure context, as has been done in other disciplines (Hanham

and Spiker, 2004; Yu and Wu, 2005).

The purpose of this study is to examine the capabilities of OLS, SAR, and GWR

for modeling crown closure in the foothills of west-central Alberta. The work is designed

to analyze the impacts of autocorrelation and non-stationarity on forest structure models

derived from spectral and topographic variables, and to provide a comparative analysis of

each modeling technique. In addition, it is hoped that this effort provides an effective a

case study that helps to illuminate the roles of SAR and GWR in the disciplines of remote

sensing and ecology.

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1.2 Study Objectives

The overall objective is to model the relationship between in-situ measurement of crown

closure and spectral components extracted from Landsat-5 Thematic Mapper data and

topographic components generated from a digital elevation model (DEM). Specific

objectives include:

• Applying OLS, SAR and GWR to model relationships between crown closure and

spectral/topographic variables

• Evaluating the performance of OLS, SAR and GWR models

• Exploring local variation across the study area using GIS

To fulfill the above objectives, the following research questions are posed:

• What are the important variables that can be used as predictors for modeling

crown closure across the study area?

• Does the remote sensing data provide enough information to establish a

statistically-valid relationship with field measurements?

• Does the inclusion of topographic variables extracted from a DEM improve the

model fit?

• Are there any statistical measures to compare OLS, SAR and GWR?

• How does GWR perform in relation to OLS and SAR?

• Can we visualize local variation of the relationship across space?

• Does the local variation have any local management implications?

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CHAPTER 2: LITERATURE REVIEW

The purpose of this chapter is twofold: first, to provide a brief background of remote

sensing of forest and its structural attributes with emphasis on the crown closure attribute,

and second, to provide an overview and theoretical background on the three regression

techniques evaluated in this research: OLS, SAR, and GWR. In first section, the review

begins with a brief discussion regarding concept of forest structure and its importance.

Next, different techniques involved in the estimation of forest structural attributes as well

as different remote sensing data currently being used are identified and described.

Following this, a brief discussion is undertaken regarding the principle of remote sensing

and the manner in which spectral bands are related to forest structural attributes. Next,

the use of vegetation indices and transformed bands are highlighted in view of ecological

perspectives, and the influence of endogenous and exogenous factors is described.

Finally, the role of topographic variables as well as DEM in the ecological modeling is

highlighted. In the second section, the review is mainly concerned with crown closure

attribute in regards to Landsat (wherever possible) with emphasis on its importance from

the ecological perspectives, different techniques of estimation, relationship with spectral

bands, vegetation indices and DEM, and factors affecting the relationship. In the third

section, fundamental concepts and theoretical background of the three specific regression

techniques: OLS, SAR and GWR are reviewed. Moreover, comparative analyses of these

regressions in ecological applications are reviewed.

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2.1 Remote Sensing of Forest Structure

Several definitions of the term forest structure presently exist in the scientific literature

(Spies, 1998; Stone and Porter, 1998; Maltamo et al., 2005). Spurr and Barnes (1980)

defined it as the above-ground organization of plant materials, as the result of light, water

and nutrients in a particular location. Kimmins (1996 in Stone and Porter, 1998) defined

forest structure as the vertical and horizontal organization of plants - a simple geometric

description of the forest stand- whereas Oliver and Larson (1990) defined it as the

physical and temporal distribution of trees in a stand - implying the aggregation of

individual plant measures such as distribution, density, patterns, size and age. Spies

(1998) summarized forest structure into four main components: tree size/age distribution,

vertical foliage distributions, horizontal canopy distribution, and dead wood. Stone and

Porter (1998) characterized the forest resource in terms of inventory parameters, which

provide detailed data on the location and extent, and biophysical parameters, which

provide data on the productivity, structure and amount. McElhinny et al. (2005)

categorized the forest structure according to its measures: abundance, relative abundance,

richness, size variation, and spatial variation.

Regardless of the specific definition adopted, it is well established by now that

these forest structural attributes are related with many ecosystem processes (Wulder,

1998), and knowledge concerning them provides key information about forest health and

condition (Clark et al., 2004; Leckie et al., 2004), net primary production (Melillo et al.,

1993; Gower et al., 1999; Kotchenova et al., 2004), light interception (Sellers et al., 1992;

Nagler et al., 2004), water flux (Mohamed et al., 2004; Leuning, 2005), carbon flux

(Potter, 1999; Potter et al., 2003; Tuyl et al., 2005), fire behavior and effects (Andersen

10

et al., 2005; Hall et al., 2005), wildlife habitat analysis (DeGraaf et al., 1998; Hyde,

2005), forest disturbance (Chittibabu and Parthasarathy, 2000; Bhuyan et al., 2002),

forest succession (Hall et al., 1991; Fiorella and Ripple, 1993), and many other important

processes. As a result, the mapping and monitoring of forest structural attributes provide

key component of sustainable forest management strategies (Franklin, 2001; Neeff et al.,

2003; Mäkelä and Pekkarinen, 2004).

2.1.1 Estimating Forest Structural Attributes

Several techniques for the estimation of forest structural attributes presently exist in the

literature, which can be grouped mainly into three categories: (i) field measurement, (ii)

photographic interpretation, and (iii) digital remote sensing. The choice of estimation

methods depend upon the availability of resources, the accessibility of area sampled, size

of the study area, and accuracy required (Martin et al., 1998). Although, field

measurement is generally considered to be the most accurate method (Aplin, 2005), it is

an expensive and time-consuming process, and often impractical over large and

inaccessible areas (Lu et al., 2004; Mäkelä and Pekkarinen, 2004). Manual interpretation

of air photos provides a means of optimizing the use of expensive field data and

expanding analysis into inaccessible areas, but its accuracy is dependent on the

experience of photo interpreters, and consistency issues are often experienced when

multiple personnel contribute to projects over large study areas (Pu et al., 2003). As an

alternative, digital remote sensing is gaining increasing favour over the last two decades,

following developments in computer technology, sensor design, and processing methods

(Sader et al., 1990; Lucas et al., 1993; Steininger, 2000; Franco-Lopez et al., 2001;

Santos et al., 2002; Boyd and Danson, 2005). Current sensor systems are capable of

11

providing high-quality information over large areas at ever-increasing spectral and spatial

resolutions, delivered to end users in digital format at a reasonable cost (Leckie et al.,

1995; Wynne et al., 2000; Franklin et al., 2003; Lu et al., 2004). Franklin (2001) divided

the use of remote sensing in forestry applications into three board categories: forest cover

type mapping, forest structure and change analysis, and forest inventory assessment.

The general strategy used to extract forest information from remote sensing

imagery is to establish relationships between the structural attributes representing the

forest condition and spectral information contained within the image. When estimating

forest structural attributes over large areas, it is common practice to develop statistical

models by integrating data collected through field measurement with coincident spectral

information derived from digital remote sensing. Field measurements are considered to

be the most reliable source for calibrating and validating the spectral information

contained in remotely sensed data (Dougill and Trodd, 1999; Lambin, 1999). A variety of

studies (e.g., Franklin, 1986; Lathrop and Pierce, 1991; Cohen et al., 1995; Kimes et al.,

1996; Turner et al., 1999; Eklundh et al., 2001; Franco-Lopez et al., 2001; Lu et al.,

2004) have developed empirical relationships in this manner. While satellite optical

imagery is most commonly used in these studies, other remote sensing data alternatives

are being increasingly explored (Aplin, 2005), such as airborne optical sensors (Franklin

and McDermid, 1993; King, 2000), RADAR (Harrell et al., 1995; Fransson and

Israelsson, 1999; Castel et al., 2002; Santos et al., 2003), hyperspectral (Underwood et

al., 2003; Schmid et al., 2004) and LiDAR (Næsset, 1997; Drake et al., 2003; Lovell et

al., 2003; Clark et al., 2004).

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The choice of remote sensing data in a particular study relies on various

practicalities, including interested area of coverage, extent of information needed,

availability of funds, processing capabilities, and time constraints (Aplin, 2005; Boyd and

Danson, 2005). Several review papers regarding the application of remote sensing on

forestry/ecological studies (e.g., Iverson et al., 1989; Wulder, 1998; Cohen and Goward,

2004; Aplin, 2005; Boyd and Danson, 2005) conclude that coarse-spatial-resolution

remotely sensed data remains the most popular choice, particularly for studies involving

large areas. Although fine-spatial-resolution imagery is capable of producing accurate

information over small areas, the extrapolation of these results is difficult because of

spatial variability over space (Boyd and Danson, 2005). In addition, these imagery are

highly expensive, have higher storage requirements, low spatial coverage, and complex

processing requirements in comparison (Hyppa et al., 2000; Lee et al., 2004). Salvador

and Pons (1998) reported that although airborne data have advantages over satellite data

in terms of higher spatial, spectral, and radiometric resolution, and possibility of selecting

image acquisition time, satellite imagery have many practical advantages including ease

of geometrical correction, low acquisition costs, wider spatial coverage, and the ability

for multitemporal analysis. Several studies (e.g., Rignot et al., 1997; Hyyppa et al., 2000;

Lefsky et al., 2001; Thenkabail et al., 2003; Lee et al., 2004) have performed comparative

analyses of the performance of different satellite systems in different forestry

applications, and have concluded that Landsat data provides comparable and even

stronger predictions of certain forest structural attributes. Moreover, other scholars (e.g.,

Hudak et al., 2002; Chen et al., 2004; Lucas et al., 2006) have reported improved

discrimination results when Landsat data are combined with ancillary variables. These

13

studies serve to highlight the practical advantages of Landsat and its role in estimating

and mapping of forest structural attributes.

2.1.2 Thematic Mapper Spectral Bands and Forest Structural Attributes

The application of remote sensing is based on the principal that different materials

absorb, transmit, and reflect incident electromagnetic radiation in different amounts, and

that these patterns can be captured by remote sensing devices (Jensen, 2005). The

spectral reflectance of vegetation is directly related to the absorption pigments in plants

such as chlorophyll, carotenoid, anthocyaninm, lutein, and physical structure of the

surface and the cell in the leaves (Zwiggelaar, 1998). Because of these, there is large

absorption of energy in visible-red portion of the spectrum and large reflectivity of

energy in the near-infrared region (Zwiggelaar, 1998; Byod et al., 2002). In addition to

these, spectral reflectance varies with external factors such as orientation/inclination of

the viewing angle, sun elevation, wind speed, clouds, terrain, and moisture condition, as

well as internal factors such as phenology, physical temperature, area coverage, species

composition, vigor, and geometry (Guyot et al., 1989; Treitz and Howarth, 1999;

Wigneron et al., 2003). All of the factors combined determine how vegetation and its

attributes respond to incident energy, which in turn provides the basis for discriminating

the forest structural attributes (Boyd and Danson, 2005).

For the estimation of forest structural attributes, it is important to have a good

understanding of how optical spectral bands behaves, and to select appropriate

wavelengths/wavebands that would provide the appropriate information (Zwiggelaar,

1998; Lu et al., 2004). Several scholars earlier have attempted to relate TM bands with

different structural attributes, however conclusions are mixed. For example, Jakubauskas

14

and Price (1997) on their study of the lodgepole pine forests of Yellowstone National

Park observed that the physical structure of the forest canopy (height, basal area and

biomass) were best predicted using a combination of visible and middle-infrared TM

bands, whereas other over storey factors (density, size diversity, mean diameter and

number of over story species) and under storey factors (number of seedlings, number of

under story species, total cover by forbs, grasses, and shrubs, and non-living cover) were

poorly explained by spectral and transformed spectral data. Gemmell (1995) on his study

in a mountainous mixed-conifer species site in southeast British Columbia found that TM

Bands 4 and 5 were strongly inversely related to stand timber volume. He concluded that

because timber volume was strongly related to the bulk reflectance characteristics of the

stand including stand height, age, species composition, slope, and aspect, it could be also

possible to estimate other stand parameters from the remote sensing data. Gerylo et al.

(2002) observed that near-infrared spectral response was the most highly correlated with

stand height for three tree species in the Northwest Territories: jack pine, white spruce

and trembling aspen. They explained this as a result of the larger dynamic range of

spectral response values of near-infrared band (50 for jack pine, 52 for white spruce and

68 for trembling aspen) as compared to very low values of visible bands (below than 15)

in all of these stands. Similarly, Lu et al. (2004) observed TM Band 5 was strongly

correlated with various forest stand parameters (R=0.57-0.92). They also observed that

the complex forest stand structures and associated canopy shadows weakened

relationships between TM spectral bands and forest stand parameters. Steininger (2000)

found the highest correlations of middle-infrared reflectance with stand basal area

(R2=0.71) and biomass (R2=0.70). Similarly, Thenkabail et al. (2003) observed the best

15

relationships of ETM+ mid-infrared bands (5 or 7) with basal area (R2 = 0.52) and stem

density (R2=0.54). However, Sivanpillai et al. (2006) observed an inverse relationship

between ETM + band 4 and stand age whereas a direct relationship with band 7. They

also observed that a linear combination of these two bands- 4 and 7 was the best predictor

(R2 =0.68) of the stand age as compared to other bands. They found that the spectral

reflectance in bands 4 and 1 increased with stand density whereas reflectance in band 7

decreased. The regression model with these three bands: 1, 4, and 7 was able to predict

stand density (adjusted R2 = 0.60) as compared to other combinations of bands. Franklin

et al. (2003) observed spectral variability for stand age and stand height within individual

forest species groups. They found a strong high correlation of TM band 4 with stand

height for both species-Jack pine (R=-0.67) and White spruce (R=-0.45). For stand age,

TM band 2 was the most correlated for Jack pine (R=-0.55) whereas least correlated

(R=0.01) for White spruce. From above review, we can conclude that there is no single

band which has consistent relationship with particular attributes. All of these

relationships may rely on variations of stand structure, site characteristics, and other

external factors as mentioned by Lu et al. (2004).

2.1.3 Spectral Vegetation Index and Forest Structural Attributes

An analysis of literature to date in ecological studies reveals that the use of spectral

vegetation indices (VIs) plays dominant role in the remote sensing of forest structural

attributes. Vegetation indices are simple mathematical expressions and quantitative

measures (Byod et al., 2002), usually derived from combinations of several spectral

bands, whose values are added, divided, or multiplied in order to yield a single value.

The uses of VIs exploit the spectral variation of the reflectance from the vegetation

16

canopies, and thus provide more information about the forest structural parameters as

compared to original bands (Kalácska et al., 2005; Rautinent et al., 2005). In addition,

VIs may serve to enhance the vegetation information by minimizing external factors such

as soil, atmospheric, and solar irradiance effects (Myneni et al., 1995; Byod et al., 2002).

As mentioned previously, vegetated surfaces have distinctive spectral properties

in the visible and near-infrared wavelengths with corresponding large absorption and

large reflectivity of incident energy (Franklin, 2001). In general, the greater the green

vegetation biomass, the larger the contrast between the red and near-infrared regions will

be observed (Kalácska et al., 2005). It is this principal that explains the basis of many

VIs on the ratio between red and near-infrared spectral bands (Byod et al., 2002; Lu et al.,

2004; Kalácska et al., 2005; Wang et al., 2005a).

The simplest and most widely-used vegetation indices in ecological applications

are the Simple Ratio (SR=ρred /ρnir) and the Normalized Difference Vegetation Index

(NDVI=ρnir -ρred /ρnir + ρred) (Schmidt and Karnieli, 2001; Lu et al., 2004; Jenson, 2005).

These indices have been used for the estimation of structural attributes in many studies;

however the conclusions derived from these studies are varied, revealing complex

dependencies on the specific attributes of interest and the characteristics of the study area

(Lu et al., 2004). For example, Frank and Aase (1994) observed that both SR and NDVI

were significantly related to dry matter accumulation on moderately and heavily grazed

pastures. They concluded that these vegetation indices can be a useful tool for predicting

the forage production. Similarly, Freitas et al. (2005) on their study in Atlantic

Rainforest using Landsat ETM+ imagery observed that NDVI was a good indicator of

green biomass in deciduous and dry forests. Justice et al. (1985) on their study of

17

phenology of global vegetation observed that NDVI serves as a valuable tool for

vegetation mapping and monitoring at regional and global scales. However, Anderson et

al. (1993) observed that no significant relationships exist between biomass and SR and

NDVI, and concluded that these indices could explain only 1.8%-16.2% of total variation

in green biomass. Similarly, Sader et al. (1990) on their study in Luquillo Mountains

observed that the NDVI was a good predictor of low biomass in an even aged plantation

forest, but could not forecast total biomass in an uneven aged mixed broadleaf forest.

They also concluded that the NDVI was influenced significantly by local topographic

effects, which need to be considered during analysis. Hall et al. (1995) observed in

Superior National Forest near Minnesota that NDVI did not provide reliable estimates of

stand biophysical characteristics, since it was highly correlated to the fraction of sunlit

canopy, which was itself relatively insensitive to both biomass and LAI. Similarly, Lu et

al. (2004) observed NDVI and SR were only weakly correlated with the forest stand

parameters such as aboveground biomass, basal area, stand diameter and stand height.

The variation and inconsistency of SR and NDVI to predict forest structural

attributes directly has led to the development of correction factors based on mid-infrared

wavelength (Nemani et al., 1993; Brown et al., 2000), and the application of new indices

such as the corrected normalized difference vegetated index, the soil adjusted vegetated

index, the modified soil adjusted vegetated index, the global environmental monitoring

index, and the perpendicular vegetation index (Fassnacht et al., 1997; McDonald et al.

1998; Paruelo Steininger, 2000; Schmidt and Karnieli, 2001; Cohen and Goward, 2004).

Furthermore, several scholars (e.g., Cohen et al., 1995; Byod et al., 2002; Lu et al., 2004)

have argued that indices derived only from only two or three bands are ignoring

18

information from other parts of the spectrum, and thereby limiting their predictive

capacity. As an alternative, other types of indices such as those derived from the

Tasseled Cap Transformation (TCT) (Crist and Cicone, 1984; Crist, 1985), which fully

utilizes all bands of information, has gained increasing acceptance (Lillesand et al., 2000;

Dymond et al., 2002; Huang et al., 2002). By using the six non-thermal bands of Landsat

TM or ETM+, the TCT generates three orthogonal indices usually known as brightness,

greenness, and wetness, which are linear combinations of the original bands, and have the

added benefit of being easier to display and interpret (Franklin, 2001; Jensen, 2005).

Generally, brightness is responsive to changes in features that reflect strongly to all

bands, and thus represents difference in soil properties; greenness is constructed as a

contrast between visible bands and the two infrared bands (TM 4 and 5), and thus

represents vegetation characteristics; while wetness is essentially a ratio between the mid-

infrared bands (TM 5 and 7) and remaining bands, and thus responds to differences in

soil and plant moisture (Crist and Cicone, 1984; Franklin, 2001; Lu et al., 2004). By

using these three components, vegetation classification and discrimination can be

improved (Dymond et al., 2002). For example, Cohen and Spies (1992) on their study of

Douglas-Fir/Western Hemlock forest stands found that Tasseled Cap wetness was

substantially more valuable than the brightness/greenness, and most highly correlated to

all sixteen observed stand attributes. The authors cited the fact that wetness was less

sensitive to topographically induced illumination angle due to the ratio effect inherent to

its construction. Similarly, Cohen et al. (1995) observed that because wetness is largely

unaffected by topographic variation, it is more useful for discriminating classes of closed-

canopy forest than brightness or greenness. Collins and Woodcock (1996) found that

19

Tasseled Cap wetness was a good indicator of conifer mortality. They observed that

wetness was able to capture the mid-infrared changes, and thus provided the most

consistent single indicator of forest change. Using a linear combination of NDVI, SR and

the brightness component of TCT, Sivanpillai et al. (2006) were able to predict 78% of

the variability in stand age – a better result than that obtained from original ETM+ bands

– whereas alternative vegetation indices, such as NDVI, SR, and the brightness and

wetness components of TCT were only able to achieve 60.1% variability; a result that

was in fact lower than that achieved with the original ETM+ bands. Lu et al. (2004)

found that the tasseled cap transformation provided overall higher correlations as than

other indices such as (SR/NDVI) in all three stands and sites. The brightness component

of the Tasseled Cap had the highest correlation (R2=0.55-0.89) with measured forest

stand parameters, but complex forest stand structures and associated canopy shadows

weakened relationships. Healey et al. (2006) found that harvesting intensity was highly

correlated with the TCT wetness index, but weakly correlated with TCT greenness and

brightness. From the above review, it is clear that the conclusion drawn about the

relationship about stand attributes and transformed bands are complex. As a result, the

appropriateness of vegetation indices/transformed bands may depend upon a variety of

endogenous (such as canopy geometry, terrain factors, species composition) and

exogenous factors (sun elevation angle, zenith view angle, atmospheric conditions) that

affected spectral response as reported by Treitz and Howarth (1999).

2.1.4 Digital Elevation Models and Forest Structural Attributes

Optical remote sensing data often suffer from issues related to spectral, spatial, and

temporal resolution, which may limit the ability of these data for discriminating specific

20

plant characteristics (Fashi et al., 2000; Keane et al., 2002). For example, the topographic

effect can produce errors in vegetation discrimination especially in high relief areas due

shadow and shading problems (Franklin, 1990; Gu and Gillespi, 1998; Fashi et al., 2000).

Moreover, several studies (e.g., Franklin, 1995, 1998; Ohmann and Spies, 1998;

Ostendorf and Reynolds, 1998) have indicated that plant ecology and its functionalities

are often related to topographic/environmental variables. For these reasons, there is an

increasing trend towards models that use remotely-sensed data in conjunction with other

topographic/environmental variables in ecological applications (Sagers and Lyon, 1997;

He et al., 1998; Fashi et al., 2000). However, the availability of accurate

topographic/environmental data is often a crucial problem (Dirnböck et al., 2002; Keane

et al., 2002). The growing availability of digital elevation models (DEMs) with improved

resolution and reliability has made it possible to extract and use topographic variables for

modeling purposes (Fashi et al., 2000; Keane et al., 2002). When used alongside remotely

sensing data, DEMs have been shown to enhance the information content of explanatory

data sets (Florinsky, 1998), thus providing improved abilities to accurately map and

model vegetation attributes (Fashi et al., 2000). Davis and Goetz (1990) in their study of

oak forests near Lompoc, California found that vegetation patterns were highly associated

with topographic variables such as elevation, slope, aspect, clear-sky solar radiation and

drainage area derived from a DEM. Similarly, Moisen and Edwards (1999) in their study

of Northern Utah’s Mountain Ecoregion concluded that the inclusion of auxiliary

variables (topography or geographic position) derived from DEMs substantially

improved existing maps of both discrete and continuous forest response variables. By

stratifying the image into shadowed and non-shadowed areas with solar incidence angle,

21

and then performing K-means clustering with elevation derived from a DEM, Wulder et

al. (2004) observed increased overall accuracy of land cover classification in a high-relief

forested area of British Columbia, Canada. Dymond and Johnson (2002) observed that

the vegetation spatial patterns of subalpine forest species were largely controlled by

variation in temperature, water and solar radiation resources. Employing standard land

cover classification techniques using Landsat TM imagery, they reported an overall

accuracy just 68.3%. However when they added biophysical gradients derived from a

DEM, the accuracy of classification increased to 83.2%. Fahsi et al. (2000) examined the

contribution of DEMs in improving land cover classification using Landsat-TM data over

a rugged area in the Atlas Mountains, Morocco, and observed that the variation of

coefficient (brightness value variability) was greatly reduced (varied from 9% for TM1 to

45% for TM7) for topographically-corrected imagery as compared to the uncorrected

image. Lower reductions in shorter wavelengths were attributed to their reduced

topographic sensitivity as compared to longer wavelengths. The authors observed an

improvement of overall accuracy by 28%, Kappa coefficient by 43% and individual class

accuracies by 63.6% for areas in rugged terrain. They reported that integration of DEM

with the TM data reduced the shadow effect by decreasing the brightness values of

surfaces facing towards the sun and increasing those of surfaces facing away from the

sun, which brought the corrected values of a surface of any slope and aspect to the value

of a horizontal surface, thus helps in vegetation discrimination. After his study to analysis

complex interaction between spatial distribution of site factors and vegetation cover

types, Hörsch (2003) reported that DEM was found to be valuable in vegetation analyses

as it was highly correlated with temperature, moisture, geomorphological processes and

22

disturbance factors. He concluded that it is important to use DEM for discriminating

habitat cover in a landscape characterized by high topodiversity and a patchwork of

microclimatic habitats. Similarly, Keane et al. (2002) reported that integration of remote

sensing, simulation modeling, and topographic variables provided an efficient and

successful approach for developing maps for broad-scale assessments and ecosystem

management. They concluded that topographic variables enhance geographic

distributions of vegetation communities and highlight spatially important processes on

the landscape, and when coupled with remote sensing data, could produce accurate

ecological maps for land management planning. Each of the above examples highlights

the importance of DEMs for creating and enhancing ecological modeling.

2.2 Review of Selected Forest Structural Attributes

As mentioned above, different remote sensing techniques have been used in estimating

various attributes of forest structure. A detailed review of each of these attributes and

remote sensing techniques is beyond the scope of this project. This document is chiefly

concerned with crown closure attribute in regards to Landsat remote sensing (wherever

possible). In this review, emphasis is given on its’ importance from the ecological

perspectives, different estimation techniques, their relationship with spectral bands,

vegetation indices and DEM, and factors affecting the relationship.

2.2.1 Crown Closure

Crown structure play a vital roles in regulating the spatial and temporal variability of

stand dynamics and microclimate, and influences many ecosystem processes such as light

interception, photosynthesis, carbon influx, primary production, nutrient and hydrological

23

cycles, fire dynamics, and biodiversity (Chen et al., 1999; Herwitz et al., 2000; Prescott,

2002; Ishii et al., 2004; Nadkarni and Sumera, 2004; Nilson and Kuusk, 2004; Falkowski

et al., 2005). Synonymous terms such as canopy closure, crown cover, or canopy cover

are common in the literature, though slight differences in definitions among these terms

may exist (Rautiainen et al., 2005). Most commonly, crown closure is defined as the

percentage of forest canopy projected to a horizontal plane per unit ground area (Xu et

al., 2003).

Conventional field sampling generally involves establishing a set of quadrants

over an area based on sampling stratification. Within each quadrant, crown closure is

determined by measuring the crown sectional area of all trees in a stand and dividing by

the quadrant area, which is generally expressed as a percentage. Field measurement is

difficult and time-consuming especially for large area of coverage (Xu et al., 2003).

Crown closure is often estimated from vertical aerial photographs, though photo

interpretation relies on the experience of photo interpretations, requires intensive human

involvement are not particularly cost effective over large areas (Pu et al., 2003). Also,

visual interpretation of the photographs is subjective and time consuming, and requires

cumbersome filing systems for storage (Woodcock et al., 1994; McRoberts et al., 2002;

Sivanpillai et al., 2006). As an alternative, digital estimation of crown closure offers

increased accuracy and reasonable costs, even over large areas and inaccessible terrain

(Dymond et al., 2002; McRoberts et al., 2002), and has generated substantial activity in

the literature (e.g. Gerylo et al., 2002; Franklin et al., 2003; Pu et al., 2003; Xu et al,

2003). Most of these earlier studies are directed towards building relationships between

24

crown closure measured at the field and spectral response extracted from the image

covering that area. However, the conclusions drawn about the relationship are mixed.

Butera (1985 in Iverson et al., 1989) conducted a study to identify the relationship

between the crown closure and the simulated TM spectral bands in Colorado, USA in

which he observed strong relationship between crown closure and all TM bands, with the

highest correlation for TM band 5(R=0.80). He was able to generate a canopy closure

map using TM band 5 with observed accuracies of 71%, 74%, and 54% for canopy

closures of 0-25%, 25-75% and 75- 100% respectively. Falkowski et al. (2005) observed

that green and red band had strong negative correlations with crown closure (R2=0.71 and

R2=0.75 respectively), while the NIR band had a weak negative correlation with crown

closure (R2=0.08). Similarly, Pu et al. (2003) observed higher individual band

correlations with crown cover in visible and short wave infrared regions. The reason for

this result was due to spectral absorption features - pigments in visible and water and

other biochemical’s in infrared region. This finding is concurrent with the finding of Xu

et al. (2003), where they observed the highest statistically significant correlation

(R2=0.69) of crown closure with TM 3 band as compared to other infrared or mid-

infrared bands. When they developed regression model with all six bands, they found a

considerable amount of improvement in variability explanation over any individual band,

though the coefficient of determination (R2=0.803) was similar to the model (R2=0.802)

developed with only three bands: NIR, red and blue. In their study conducted in the

Canadian Northwest Territories, Franklin et al. (2003) found species-specific variability

in the relationship between spectral response and crown closure. They observed high

positive correlation of crown closure with TM band 4 (R=0.81) and band 5 (R=0.55) for

25

Jack Pine yet documented high negative correlation with band 5 (R=-0.57) and band 7

(R=-0.45). Band 4 displayed the least correlation with crown closure 4 (R=-0.02).

Similarly, Gerylo et al. (2002) observed three unique species spectral variability trends

between crown closure and Landsat TM spectral response during their study of forest

stand near Fort Simpson, Northwest Territories. Jack Pine crown closure was positively

related to TM spectral response values, White Spruce crown closure was negatively

related to TM spectral response values, and Trembling Aspen crown closure was

unrelated to any TM spectral response values. For Jack Pine, higher correlation

coefficients were observed for band 4 (R=0.76) and band 2 (R=0.50) whereas for White

Spruce, higher correlation coefficients were found for bands 5 (R=-0.55) and 7(R=-0.45).

From above review, it is difficult to generalize which TM bands work better for a

particular situation.

The use of spectral vegetation indices (VIs) for estimation of crown closure is

common in forestry/ecological studies. The most commonly used vegetation indices are

NDVI and the tasseled cap components (Qi et al., 1995; Dymond et al., 2002). As with

most VIs, these indices are simple to compute, and improves crown closure

discrimination as compared to single bands (Lillesand and Kiefer, 1994). For example,

Salvador and Pons (1998) in their crown closure modeling using twenty five independent

variables-Landsat TM spectral bands and spectral ratios in a Mediterranean study area

found that models incorporating all six spectral bands produced a weak significant

correlation (R2=0.34), whereas the addition of spectral ratios improved correlation results

drastically (R2=0.61). Similarly, Chen and Cihlar (1996) found that NDVI from Landsat

TM data was highly correlated to field measurements of the crown closure in boreal

26

conifer forests of Canada. Confirming these results, Pereira et al. (1995 in Carreiras et

al., 2006) observed the best correlation of NDVI with percent canopy cover (R2=0.65).

This is concurrent with the finding of Xu et al. (2003) who observed that NDVI can

improve the estimation of crown closure (R2=0.70) over individual bands. However,

several scholars (e.g., Cohen et al., 1995; Byod et al., 2002; Lu et al., 2004) have argued

that since NDVI is derived only from two bands (red and near infrared), the index does

not fully exploit the total information content in all bands, and thus may limit the

predictive power of the relationship. As an alternative, the Tasseled Cap Transformation

(TCT) (Crist and Cicone, 1984; Crist, 1985), which utilizes information from all TM

bands, has been increasingly used in ecological mapping and applications (Dymond et al.,

2002; Huang et al., 2002). For example, Hansen et al. (2000) in their study in

Revelstoke, British Columbia found a strong positive relationship between crown closure

and the Tasseled Cap component wetness (adjusted R2 =0.73). The positive relation was

explained by the sensitivity of wetness to the amount of moisture contained within the

tree foliage, as first reported by Cohen and Spies (1992).

Several studies (e.g., Florinsky, 1998; Kirkman, 2001; Keane et al., 2002;

Falkowski et al., 2005) have demonstrated the role of environmental/topographic

variables in ecological/forestry modeling. Many models observed in the ecological /

forestry applications involve the use of slope, aspect, and elevation as topographic

explanatory variables. One of the reasons to incorporate topographic attributes in the

modeling process is that these attributes are easily measured and static, and do not require

repeated sampling (Franklin, 1995; Keane et al., 2002). With improved resolution,

reliability and easy availability, the digital elevation model (DEM) has made it possible

27

to extract and use topographic variables such slope, elevation, aspect for the vegetation

modeling including crown closure (Fashi et al., 2000; Keane et al., 2002). In summary, it

is clear that satellite remote sensing has been used as a valuable, complementary

supplement to field work in many ecological applications, with the benefit of providing

large-area coverage at reasonable time and cost over otherwise inaccessible terrain. With

steady advancements in spectral, spatial, temporal and radiometric resolution; processing

capabilities; and new algorithms, digital remote sensing can be expected to play larger

role in future ecological applications, including estimation of forest structural attributes.

In order to improve the effectiveness use of remote sensing technology, it is

important to develop methods that integrate all available components, including field

measurements, spectral variables and indices, and topographic/ environmental attributes

from DEMs. In addition, attempts should be directed towards minimizing endogenous

and exogenous factors that may undermine the model relationship. One of the chief

problems associated with remote sensing studies in ecological/forestry applications is that

observed spectral response patterns are subject to a wide variety of factors (Lu et al.,

2004, Ingram et al., 2005) that may limit our ability to generalize these patterns across

large spans of time and space (Woodcock et al., 2001, Lu et al., 2004). Several scholars

have documented this issue, including Foody et al. (2003), who investigated the

transferability of predictive relations for estimating tropical forest biomass from Landsat

TM data between sites in Brazil, Malaysia and Thailand. The authors observed strong

relationships between predicted biomass and measured biomass for individual sites, but

reported problems in transferring specific relationships from one site to the next,

documenting relationships that differed markedly in both strength and direction.

28

Similarly, Lu et al. (2004) explored forest stand parameters and TM spectral relationships

at three different characteristics study areas in the eastern Amazon basin. They observed

that forest structure and associated canopy shadow has significant affect on the

relationship between the forest stand parameters and TM spectral response relationships,

and concluded that different stand structures affected the relationships between forest

stand parameters and TM spectral response. Additional research is needed to fill this

critical knowledge gap. We require more effort directed towards exploring the

relationship between spectral response and forest structural attributes across varying sites,

and incorporating improved techniques and parameters that can be replicated across time

and space (Lu et al., 2004; Ingram et al., 2005).

2.3 Review of Regression Techniques

Regression is the most widely used statistical technique for examining and modeling the

relationships between variables (Montgomery et al., 2001), and has an immense

application in almost every domain, including ecological and remote sensing studies

(Foody et al., 2003). For example, considerable effort has been spent developing

relationships through regression analysis between the forest structural attributes collected

from the field measurements and spectral response extracted from coincident remote

sensing imagery (Foody et al., 2003; Lu et al., 2004). Once developed, these models can

be inverted to make predictions of forest structural attributes at other sites using spectral

information as input variables (Foody et al., 2003; Zhang and Shi, 2004). Franklin (2001)

pointed out seven steps of the regression modeling process: (i) establishment of field

measurement sites in a forest area, (ii) collection of data for forest structural attributes on

those sites, (iii) acquisition of remote sensing imagery covering those sites, (iv) location

29

of field measurement sites on the image, (v) extraction of spectral information from the

image for those sites, (vi) construction of model that relates to field measurement and

spectral information, and (vii) utilization of the model to predict the attributes for all

forest pixels using spectral information. Various regression approaches have been used to

establish the relationship, and the review of each of these techniques is beyond the scope

of this project. This section is mainly focused on review of three specific regression

techniques: ordinary least squares (OLS) regression, spatial autoregression (SAR) and

geographically weighted regression (GWR) in relation to mainly ecological applications

wherever possible. In this review, fundamental concepts and theoretical background of

the techniques are presented. Moreover, comparative analyses of these regressions for

ecological applications are reviewed.

2.3.1 Overview of Models

2.3.1.1 Ordinary Least Square Regression

Ordinary least square (OLS) regression is far and away the most widely-used regression

technique for estimating forestry structural parameters in ecological applications (Zhang

and Shi, 2004; Berterretche et al., 2005). Through OLS regression, many studies (e.g.,

Peterson et al., 1987; Fassnacht et al., 1997; Tunner et al., 1999; Geyrlo et al., 2002) have

establish relationships between field measurement as response variables and spectral

response as predictor variables. For example, Peterson et al. (1987) observed a strong

positive empirical relationship between field-based measurement of LAI and SR

(R2=0.83 with OLS and R2=0.91 with log-linear OLS). Similarly, Fassnacht et al. (1997)

reported a good relationship between field based measurement of LAI and spectral

response from Landsat (R2=0.60, p<0.00l), and observed that the inclusion of multiple

30

variables improved the ability of spectral data to account for the variability observed in

ground measurements. Using NDVI and SR extracted from Landsat data, Tunner et al.

(1999) observed a good relationship with LAI (R2=0.53-0.74) through multivariate OLS.

Using OLS, Gerylo et al. (2002) developed empirical relationships between crown

closure and spectral response variables extracted from Landsat image for Jack pine

(R2=0.55) and for White spruce (R2=0.44). Salvador and Pons (1998) used multiple OLS

regression to model crown closure (highest R2=0.61) and LAI (R2=0.68). Similarly,

Cohen et al. (2001) mapped forest vegetation attributes as continuous variables using

Tasseled Cap Transformations (TCT) extracted from the Landsat TM, and was able to

model four attributes through OLS: percent green vegetation (R2=0.74), percent conifer

cover (R2=0.69), conifer crown diameter (R2=0.51) and stand age (R2=0.56). They

concluded that regional forest attributes maps can be produced from Landsat data through

OLS at acceptable error, reasonable time and cost. In spite of these wider successes, OLS

regressions have been criticized across many disciplines including ecological studies.

OLS regression relies on determining the dependent variable (Y) by producing

unbiased minimum sum of error square in Y in regards to the independent variable (X).

For unbiased observations, the prediction should be equal to the expected value of the

dependent variable for a particular set of data (Fernandes and Leblanc, 2005). OLS is

based on a set of assumptions: normality, homogeneity and independence of residuals

(Montgomery et al., 2001). Violation of these assumptions leads to inefficient and biased

estimators, thus resulting in inaccurate estimation of model parameters by OLS (Fox et

al., 2001; Montgomery et al., 2001). Several scholars (e.g., Cohen et al., 2003; Foody,

2003) have pointed out the assumption of no error in measurements of vegetation

31

reflectance and/or biophysical parameters of interest by OLS regression are almost

impossible to achieve in reality. With violation of the fundamental assumption, however,

it could be argued that is not appropriate to use OLS regression in remote sensing for

forestry applications (Foody, 2003). While calibrating remote sensing models, Fernandes

and Leblanc (2005) criticized OLS for the two problems of errors in both regressors and

response variables - errors due to systematic biases in measurement or theory, and

random errors in observations due to noise in measurement methods or collocation

problems. Moreover, Frescino et al. (2001) pointed out that vegetation communities do

not display normal distributions along environmental gradients, and thus many statistical

models which hold this assumption, such as OLS, may misrepresent true vegetation

patterns. These authors advocate the use of alternative models with more realistic a-priori

assumptions that are capable of handling the non-linear response of vegetation along

environmental gradients.

Such criticisms have led to development of many alternative regression

techniques, including reduced major axis regression, linear mixed model, general additive

model, classification and regression trees, multivariate adaptive regression splines,

artificial neural networks (e.g., Austin and Meyers, 1996; Lehmann, 1998; Moisen and

Edwards, 1999; Frescino et al., 2001; Guisan et al., 2002; Moisen and Frescino, 2002;

Cohen et al., 2003). Several scholars (e.g., Frescino et al., 2001; Miller and Franklin,

2002; Cohen et al., 2003) have conducted comparative analyses of different regression

techniques based on the model performance, accuracy and errors. For example, using

vegetation indices extracted from Landsat ETM+, Cohen et al. (2003) analyzed three

different regression models to predict LAI for an agro-ecosystem, and tree canopy cover

32

for evergreen boreal forest: OLS (Y on X) regression, inverse (X on Y) OLS regression

and an orthogonal reduced major axis (RMA) regression. They observed that RMA was

able to preserve the variance of the observations in the prediction better than OLS and

inverse OLS regression, which respectively produced lower and inflated variance in the

prediction. Using non-linear generalized additive models, Frescino et al. (2001) analyzed

predictability of the forest attributes integrating forest resource inventory field data with

satellite and digital environmental/topographic data. The authors were able to predict the

presence of forest and lodgepole pine with accuracies of 88% and 80%, respectively, and

generated models of basal area, shrub cover and snag density with average accuracies of

62% with a 15% deviation from the field observations. They concluded that improvement

of model prediction can be possible when satellite data used in conjunction with digital

topographic/environmental data. Similarly, using generalized linear models (GLM) and

classification trees (CT), Miller and Franklin (2002) modeled the distribution of four

vegetation alliances from environmental/topographic data, and observed higher

classification accuracy with CT models for training data, but, less predictive capacity for

test data as compared to those of with GLM. They also observed spatial dependency in

vegetation distribution, and thus were able to improve both models with the addition of a

kriged dependence term. However, while comparing the ability of three regression

methods (OLS, seemingly unrelated regression (SUR), and partial least squares (PLS))

for estimating the biophysical properties of forest stands, Næsset et al. (2005) reported

only minor discrepancies among the three approaches in terms of bias and precision.

They concluded that in spite of violating assumptions when both response and regressor

variables are correlated, OLS can be considered comparatively the best technique on the

33

basis of its simplicity, ease of interpretation and the availability of powerful tools for

selecting efficient variables. Similarly, Schwarz and Zimmermann (2005) compared two

regression techniques: generalized linear models (GLM-an extensive of OLS) and

regression tree model (RTM) for mapping tree cover continuous fields, and observed that

GLM were comparatively better than the tree model. They concluded that GLMs were

easy to calibrate, and were capable of taking account of non-linear relationships present

in the data. For the prediction of forest characteristics, Moisen and Frescino (2002)

compared five modeling techniques: linear models (OLS), generalized additive model

(GAM), classification and regression tree (CART), multivariate adaptive regression

spline (MARS) and artificial neural network (ANN). When applied to real data, they

observed that there was no significant difference among these techniques, and concluded

that MARS and GAM performed marginally better for the prediction of forest

characteristics. However, in some instances, they observed better performance in OLS as

compared to other more complex models. They suspected that these results may have

been caused by noise in the dataset or lack of nonlinear relationship between the response

and predictor variables. For estimating of LAI using simple ratio indices, Fernandes and

Leblanc (2005) compared regression techniques: three parametric (ordinary least square

(OLS), geometric mean (GM), modified least squares (MLS)) and one non-parametric

(Theil-Sen (TS)). They concluded that OLS and GM were sub-optimal estimators of

linear relationships and should not be applied without strong justification, whereas MLS

could be used as back-up solution only if data are known to be free from outlier and a

priori knowledge on measurement errors are available. They recommended TS approach

in remote sensing application as potential alternative to commonly used OLS techniques

34

on the following grounds: simple to compute and analytical estimates of confidence

intervals, robust to outliers, testable residual assumptions and require limited information

about measurement errors. However in their study, they did not consider spatial pattern

associated with predictions and residuals over space.

From the above discussion, it is evident that several scholars have developed

different techniques to overcome problem of OLS assumptions, and achieved some

successes over OLS depending upon characteristics of sites and biophysical variables.

However, all of these regression techniques are non-spatial in nature. From an ecological

point of view, the potential ability of spatial models to capture the spatial structuring

present in natural systems is particularly attractive (Legendre, 1993; Legendre and

Legendre, 1998).

2.3.1.2 Spatial Autoregression

Several scholars (e.g., Zhang and Shi, 2004; Zhang et al., 2005; Shi et al., 2006) have

pointed out the flaws of OLS assumptions in terms of spatial autocorrelation (or spatial

dependency) in ecological studies. In this regards, Miller and Franklin (2002) cited two

important laws: Tobler’s law of geography, everything is related to everything, but near

things are more related than distant things and ecology theory, elements of an ecosystem

close to one another are most likely to be similar as result of same generating process.

Both of these concepts imply that when attributes are measured across space, nearby

variables are more related than those of apart. This phenomenon explains the issue of

spatial autocorrelation – the lack of independence amongst closely-spaced observations –

observed in all ecological variables in natural systems (Legendre, 1993; Legendre and

Legendre, 1998). Zhang and Shi (2004) illustrated the issue of spatial autocorrelation in

35

a forest ecosystem, which may be positive (trees are surrounded by similar-sized trees) as

a result of micro-site conditions, or negative (larger trees are surrounded by smaller trees

and vice versa) as a result of inter-tree competition. The presence of spatial

autocorrelation in a dataset is troublesome for OLS regression, which assumes

independence of observations, and may lead to detrimental consequences such as

inefficient parameter estimators, biased hypothesis test results, and inaccurate predictions

(Legendre and Legendre, 1998; Carroll and Pearson, 2000; Fortin and Payette, 2000; Fox

et al., 2001; Jetz and Rahbek, 2002; Miller and Franklin, 2003; Zhang et al., 2004). This

issue has prompted several authors (e.g., Legendre, 1993; Legendre and Legendre, 1998;

Fox et al., 2001; Foody, 2003; Dark, 2004; Jetz et al., 2005) to cast skepticism on the

reliability of ecological models based on OLS, and promote the use of alternatives

methods. In addressing this issue, Legendre (1993) indicated that spatial autocorrelation

might initiate a paradigm shift in geographical ecology that favors analytical techniques

that offer better understanding of spatial patterns as well as avoiding the major pitfalls of

OLS regression. Spatial autoregression (SAR) is one type of commonly-used spatial

models which take into account the effects of spatial autocorrelation by assuming it as an

intrinsic part of the ecological process (Legendre and Legendre, 1998; Anselin, 2005).

Dark (2004) compared ordinary least squares (OLS) regression and spatial

autoregressive (SAR) models to assess the relationship between alien plant species

distribution and a series of explanatory variables: native plant species richness, road

density, population density, elevation, area of sample unit, and precipitation. She found

that the OLS model for invasive alien plants included two significant effects; native plant

species richness and elevation whereas the SAR model for invasive alien plants included

36

three significant effects; elevation, road density, and native plant species richness. These

results illustrated the importance of testing for spatial autocorrelation in ecological

studies and point out the values of using SAR models over OLS when conditions are

appropriate. Similarly, Lichstein et al. (2002) examined breeding habitat relationships for

three common Neotropical migrant songbirds in the southern Appalachian Mountains of

North Carolina and Tennessee, USA and found that residuals from OLS models were

autocorrelated, indicating that the assumption of independent errors was violated,

whereas residuals from SAR models showed little spatial pattern, indicating better fit.

Moreover, the authors observed that the abundance of all three songbird species was

correlated with both local- and landscape-scale habitat variables when they did not

account for spatial effects. However, the magnitude of habitat effects tended to decrease

and relative importance of different habitat variables increased when they incorporated

broad-scale spatial trend (trend surface analysis) and fine-scale autocorrelation

(autoregressive spatial covariance matrix) in the modeling process. Likewise, Wang et al.

(2005) examined the relationships between net primary production of Chinese forest

ecosystems and environmental variables, and observed that all the model variables used

had significant autocorrelation, which violates assumptions of OLS. As a result, they

found significant improvement of model fit by adding spatial lag term in the modeling

process (to AIC = 5001, R2=0.60 from AIC=5036 and R2=0.58).

Several scholars (e.g., Fox et al., 2001; Zhang and Shi, 2004, Zhang et al., 2004)

have pointed out that while significant efforts have been expended dealing with issues of

spatial autocorrelation in ecological modeling, the challenge of spatial non-stationarity

had largely been neglected. Despite this, it could be argued that both spatial

37

autocorrelation and spatial non-stationarity are equally important issues, and

sophisticated environmental modeling efforts must deal with both. Spatial non-

stationarity indicates the lack of stability over space of the relationships under study

(Legendre and Legendre, 1998; Zhang and Shi, 2004). The existence of non-stationarity

may lead to violations in the OLS assumption of constant variance, and lead to biased

parameter estimates, misleading significance tests and suboptimal prediction (Zhang and

Shi, 2004). Spatial non-stationarity may result from imperfect datasets with missing

information (Fotheringham, 1997) or systematic biological, environmental and physical

processes in the natural ecosystem (Legendre and Legendre, 1998; Zhang et al., 2005). In

the latter scenarios, it is theoretically possible to use spatial location as proxy variables

for revealing spatial non-stationarity (Fotheringham et al., 2002; Zhang and Shi, 2004).

This premise has been the foundation for a new regression technique called

geographically weighted regression (GWR), whose popularity has been increasing across

many disciplines because of its ability to capture both spatial dependency and spatial

non-stationarity in the modeling process.

2.3.1.3 Geographically Weighted Regression

Wheeler and Tiefelsdorf (2005) pointed out the development of GWR originated from

local regression and smoothing techniques (Brunsdon et al., 1996; Fotheringham et al.,

1998), which were further improved with continuous development of statistical measures

such as maximum likelihood estimation of the kernel bandwidths (Páez et al., 2002a),

spatial autocorrelation among the residuals (Páez et al., 2002b), generalized linear model

specifications (Fotheringham et al., 2002), Bayesian GWR (LeSage, 2004) and test

statistics for spatial non-stationarity and heterogeneity of the local model parameters

38

(Leung et al., 2000). In fact, GWR is an extended version of traditional OLS regression,

which takes into account spatial heterogeneity by incorporating the spatial location of

data (Hanham and Spiker, 2004; Shi et al., 2006). Because of the spatial stationary

assumption, parameters derived from OLS and SAR models are usually applied globally

across the space over which measurements are taken. Due to spatial non-stationarity in

the relationship, however, the parameters derived from the global (OLS and SAR) models

may not represent true situation, and do not account consideration of local variation over

space (Foody, 2003). So for local variation over space, it is appropriate to perform local

analysis (i.e., GWR) that attempts to capture true local reality rather than global analysis

(i.e., OLS and SAR), which assumes spatial stationarity (Fotheringham et al., 2002;

Foody 2003).

Hanham and Spiker (2004) pointed out that GWR is based on three interrelated

principles. Firstly, spatial data are not often stationary. Secondly, the spatial structure of

data greatly influences the estimation of relationships between variables under study.

Thirdly, the relationships between variables may be localized and varied across space.

GWR attempts to capture local spatial variation across space by calibrating multiple

regressions at each sampled points (Zhang et al., 2004). In GWR, the fitting of regression

at each sample points is based on a spatial proximity approach used as a weighting

function for an observation, which means observations near the location at which

estimation are to be made have more influencing weight on the analysis than those

observations further away (Fothringham et al., 2002; Foody, 2003). It is possible to

specify kernel bandwidth as a threshold that controls distance beyond which neighboring

observations have no long effect on the local parameter estimation (Wang et al., 2005).

39

For each sampled point across space, GWR produces a set of local equations and

parameter estimates as well as goodness-of-fit measures (R2, residuals). This stands in

sharp contrast to OLS or SAR models, which produce global equations and goodness-of-

fit measures for the entire study area. Many statistical tests can be performed within the

GWR framework, such including F tests for determining whether there is significant

improvement in GWR estimates as compared to global estimate, Monte Carlo

permutation tests or Leung’s test for determining whether model parameters are

significantly non-stationary (for details see Fothringham et al., 2002). Most importantly,

GWR is compatible with GIS, which helps in visualizing and mapping of estimates,

diagnostics and residuals, and thus allows in-depth investigating local variation in the

relationships across space (Fotheringham and Brunsdon, 1999; Fothringham et al., 2002).

An analysis of the literature on GWR shows that the technique has gaining

popularity across many disciplines, especially those concerning house price modeling

(e.g., Brunsdon et al., 1999; Kestens et al., 2005; Malczewski and Poetz, 2005; Xiaolu

and Yasushi, 2005), regional/urban planning modeling (e.g., Nakaya, 2001; Huang and

Leung, 2002; Platt, 2004; Yu, 2005) and demographical modeling (e.g., Hanham and

Spiker, 2004; Benson et al., 2005; Farrow et al., 2005; Kam et al., 2005). However, only

a few studies to date have applied GWR to ecological applications, including forest stand

analysis (e.g., Zhang and Gove, 2004; Zhang et al., 2004; Zhang et al., 2005; Shi et al.,

2006) and wild species analysis (e.g., Foody, 2004; Shi et al., 2006). For example, Zhang

et al. (2004) conducted modeling of the spatial variation in tree diameter-height

relationships using OLS and GWR, and observed a significant improvement (p-

value=0.0036) of GWR over OLS. The overall R2 (0.94) of the GWR model was higher

40

than those of OLS (0.85) while the overall absolute residual (0.074) was 43% smaller

than that of OLS (0.129). They explained the outcome as a result of spatial variation of

local growth condition and competition that were taken into account by the GWR model.

Similarly, Zhang and Shi (2004) examined the relationship between tree growth and

diameter for 48 sampled plots using GWR, and observed a significant improvement over

OLS (p-value<0.05 for 45 sampled plots). Further, after detailed analysis in three selected

plots they reported that the standard errors of the means of the residuals were almost half

in the case of GWR as compared to OLS, and that GWR residuals were normally

distributed in two plots as compared to OLS residuals. They concluded that use of GWR

over OLS not only better predicted individual tree growth but also provided useful

information about the growth variation caused by neighboring competitors and

surrounding environmental factors, which could then be visualized with the aid of GIS

tools and had wider implications in ecological studies. Zhang and Gove (2005) performed

a spatial assessment of model errors using four regression techniques (ordinary least

square (OLS), linear mixing model (LMM), general additive model (GAM) and

geographically weighted regression (GWR) for estimating of tree crown using tree

diameter, and observed that there was significant improvement for all three models over

OLS (pvalue<0.001) and that GWR had the least error sum of square (SSE=4799.40) and

Akaike Information Criterion (AIC=4523.4). Moreover, they observed clusters of

positive or negative residuals with OLS and GAM, indicating underestimated or

overestimated response variables as compared to more desirable spatial distribution of

errors with LMM and GWR. They explained the better results in case of GWR and LMM

through their able to take account of local spatial autocorrelations while estimating model

41

coefficients. Similarly, Zhang et al. (2005) analyzed spatial distribution and heterogeneity

in the model residuals using six difference regression techniques (ordinary least square

(OLS), linear mixed model (LMM), generalized additive model (GAM), multi-layer

perceptron (MLP) neural network, radial basis function (RBF) neural network, and

geographically weighted regression (GWR) taking tree crown area as the response

variable and tree diameter as the regressor, and reported three important things. Firstly,

all five alternative regression techniques had better prediction that the OLS model,

though the spatial patterns of model residuals for four regression techniques (LMM,

GAM, MLP, RBF) were similar to those of OLS. Secondly, LMM, GAM, MLP, RBF

and OLM models each exhibited residual clusters of similar values, which is an

indication of either underestimation or overestimation of the response variable. Thirdly,

GWR produced the most accurate predictions as well as the most desirable spatial

distribution for model residuals.

So far as a literature review has revealed, only three studies (Foody, 2003; 2004;

Wang et al., 2005) has reported the use of GWR with remote sensing data for ecological

studies. Foody (2003) analyzed the relationship between NDVI and rainfall using OLS

and GWR for North Africa and the Middle East over an 8-year period, and observed

spatially non-stationary relationships for each year. He reported significant relationships

for each year with GWR (minimum R2=0.9633) as compared to OLS (minimum

R2=0.6683). Most importantly, however, using GWR, they were able to highlight areas of

local variation observed in both of the model parameters. He concluded that location

variation is important in remote sensing to investigate the relationship between variables,

and that GWR could make important contribution to remote sensing communities in

42

coming years. Similarly, Wang et al. (2005) examined the relationship between net

primary production and other environmental variables (altitude, temperature,

precipitation) and NDVI in Chinese forest ecosystems using three regression techniques:

ordinary least squares (OLS) regression, spatial autoregression (SAR) and geographically

weighted regression (GWR). They observed that GWR provided significantly better

predictions (AIC=4891 and R2=0.66) of NPP as compared to SLM (AIC=5001 and

R2=0.60) and OLS (AIC=5036 and R2=0.58). Moreover, they observed that there was

significant spatial autocorrelation in OLS model residuals and positive autocorrelation at

small spatial scale in SLM residuals where as no autocorrelation existed across spatial

scale with GWR model residuals, indicating the ability of GWR to account for spatial

variation in the relationship of NPP with other response variables. Foody (2004) explored

spatial non-stationarity in the relationship between species richness and a set of

environmental determinants (temperature, precipitation and NDVI) in sub-Saharan Africa

using GWR and OLS. He observed that GWR maintained strong relationship between

explanatory variables and species richness (R2=0.79-0.96 depending of selected band

width and minimum AIC difference =735.98 with OLS) as compared to OLS (R2=0.59).

Moreover, they observed spatially non-stationary and scale dependent (p<0.0001)

relationships between variables. They concluded that since spatial non-stationarity in the

relationship had an important role in ecological studies, the ability of GWR to explore

spatial non-stationary could provide important local management implications.

Outside of the ecological arena, Hanham and Spiker (2004) analyzed urban

sprawl detection using Landsat imagery and GWR. They found an improvement (p<0.01)

of GWR band 4 model (R2=0.54) over global model (R2=0.37), and (p <0.10) of GWR

43

band 5 model (R2=0.63) over global model (R2=0.59). They concluded that since spatial

non-stationarity of data from remote sensing data can influence the change detection of

urban sprawl, it is desirable to modify an image change detection technique by

incorporating the spatial structure of land cover data, in which the role of GWR is

apparent. Similarly, Yu and Wu (2005) conducted a study to understand population

segregation issues utilizing Landsat ETM+ and two regression techniques: OLS and

GWR. They found that remote sensing information can provide important input for

understanding of population segregation, however, the explanatory power of OLS for

explaining the information remains fairly poor (R2= 0.4). This was substantially increased

when they applied GRW (more than 90% of the block groups exhibit a local R2 higher

than OLS and almost 60% of them have a local R2 greater than 0.8). They concluded that

GWR could identify local patterns and specific unusual locations in a microscopic way

(Fotheringham and Brunsdon, 1999), which could aid on a more appropriate

understanding of the relationships between remotely sensed information and population

segregation, and thus deliver local insight for management implications.

From the above discussion, it is clear that spatial non- stationarity is an important

yet often overlooked component of spatial modeling in many disciplines including

ecological studies, and that GWR can act as an important explanatory tool to studying

this regard. Development of an integrated approach (local spatial modeling tools like

GWR combined with other analytical and visualization tools like GIS/Remote sensing)

are capable of producing better models, thus helping in our understanding of complex and

dynamics spatial processes (Zhang and Griffith, 2000; Zhang and Shi, 2004). However,

no studies have yet been conducted to establish relationships between field-based

44

measurement of forest structural attributes (i.e., crown closure which is focus of this

project) and digital variables from Landsat imagery and DEMs.

In addition to the attractive qualities of GWR noted above, it is also important to

examine the limitations and criticisms of the technique. For example, Zhang and Shi

(2004) pointed out two limitations of GWR. Firstly, the estimated local model parameters

are based on spatial weighted proximity of observations, and thus for some points over

space, the effective number of observations used to derive parameter estimates may be

very low. Secondly, outliers have tremendous impacts on estimating localized model

parameters. Yu and Wu (2005) indicated that although the GWR model is capable of

identifying important spatial non-stationarity in the relationships, the interpretation of

such non-stationarity requires more contextual information and underlying information.

Platt (2004) reported that the interpretation of spatial variation coefficients as produced

by GWR is not straight forward and often regionally specific. This change may or may

not be important for a specific area of interest. As a result, he mentioned that local

analysis would aid as supplementary to other planning data rather than as a basis for

policy. Wang et al. (2005) pointed out that parameter estimation in GWR highly relies on

the weighting function of the bandwidth of the kernel used. The selection of the

weighting function and bandwidth should be handled properly to get desired local

variation. As the bandwidth increases, the parameter estimates will approach same as

those from a global model. Foody (2003) reported three limitations of GWR. Firstly,

regression models and diagnostic statistics from GWR can be interpreted in a similar

manner as those of traditional OLS regression, but local R2 should not be interpreted as

confidence as global R2. Secondly, GWR is not suitable for extrapolating a relationship

45

beyond the region at which model has been developed. Thirdly, there is susceptibility

about selection of suitable kernel and bandwidth for a particular application. Wheeler

and Tiefelsdorf (2005) put forward concerns about GWR being established as a standard

tool in exploratory spatial data analysis. In their study, they observed that the effects of

multicollinearity were substantially stronger in the GWR model than in the global

regression model. They claimed that the correlation of local regression coefficients

potentially invalidates any interpretation of individual GWR parameter estimates and can

facilitate misleading conclusions if the underlying situations are not properly diagnosed.

Moreover, with GWR being in the early developmental phase, the authors harboured

suspicions regarding the effects of spatially autocorrelated errors on the model

parameters, the effects of different bandwidths, alternative specifications of the spatial

weights function, interaction with exogenous variables, the error term and the potential

spatial structure effects. As a result, they called for further investigation before drawing

any definitive conclusions. Following the work of Foody (2004), Jetz et al. (2005)

heavily criticized the statistical and conceptual validation of the comparison, and

concluded three points on GWR. Firstly, general inferences as offered by global OLS

models can not be obtained by local GWR models. Secondly, GWR models do not fully

address spatial autocorrelation and must be interpreted with caution. Thirdly, GWR is a

useful supplement but not alternative to global modeling. However the authors did

acknowledge the two advantages of GWR: (i) identification of missing variables/

interaction terms by mapping of local variation in parameter estimates, and (ii)

determination in change in strength of relationship with spatial resolution of the analysis.

46

In conclusion, progress in the advancement of statistical modeling techniques

(and indeed, science in general) is based on a continuous process of hypothesis testing,

constructive criticism, and refinement. A modeling process can be considered good

enough if its positive aspects outweigh negative aspects. However, one should always

strive for improvement by exploring new techniques and methodology.

2.3.2 Theoretical Background

2.3.2.1 Ordinary Least Square Regression

Before beginning a modeling project, it is important to review some basic statistical

descriptors: mean, minimum, maximum, standard deviation, standard error of the mean,

kurtosis and skewness.

Mean is one measure of central tendency in a distribution. It can be computed by

adding all observations, then dividing by the number of observations.

Standard error of mean is a measure of how much the value of the mean may vary

from sample to sample taken from the same distribution. It equals to the standard

deviation of the population divided by square root of sample size.

Standard deviation measures the amount of variability in the distribution of a

variable. The more that the individual data points differ from each other, the larger the

standard deviation will be. Conversely, if there is a great deal of similarity between

points, the standard deviation will be quite small. The standard deviation describes the

deviations of observations about the mean.

Skewness measures the degree and direction of asymmetry. A symmetric

distribution such as a normal distribution has a skewness of 0. A distribution that is

47

skewed to the left (when mean is less than median) has a negative skewness whereas

skewed to the right (when mean is more than median) has positive skewness.

Kurtosis is a measure the peakedness of distribution. A normal distribution has a

kurtosis of 3. Heavy-tailed distributions (peaked, leptokurtic) will have kurtosis values

greater than 3, while light-tailed distributions (flat, platykurtic) will have kurtosis values

less than 3 (Rogerson, 2004).

The general equation of OLS model can be written as

ij

p

jijoi Xy εββ ++= ∑

=1

Where

=oβ Intercept coefficient

=jβ Slope coefficient for the jth independent variable Xj

=iε Random error term with ),0( 2 IN σ

=I n × n identity matrix

In matrix notation, the model can be represented by

εβ += XY

Under the assumption of independent assumption and constant variance across

space, the least square estimate of β is

yXXX TT 1)( −∧

=β

Where

T = Transpose matrix

48

=−1)( XX T Inverse matrix

Analysis of variance (ANOVA) can be performed to check whether the regression

model is statistically significant or not (F-value and corresponding p-value).

T-test (p-value) for each parameter estimates can be performed to see whether

they are statistically significant or not.

∧

=xxss

T/

β

Where

xx

xy

ss

=∧

β

))((__yyxxsxy −−= ∑

2

_

)(∑ −= xxsxx

Coefficient of determination can be calculated to see how well the model is

successful at explaining variability as

yyxx

xy

SSS

R2

2 =

Where

2

_)(∑ −= yys yy

Multicollinearity refers to higher linear inter-correlation among model variables

Multicollinearity expose the redundancy of model variables and the need to remove

49

variables from the analysis. The high multicollinearity leads to large variances and

covariance, large confidence intervals, and insignificant significance coefficients, leading

to statistical measures unreliable and unpredictable. In order to check for

multicollinearity, a tolerance and variance inflation factor (VIF) can be calculated. The

tolerance for each variable can be estimated by 1 - R-squared for the regression of that

variable on all the other independents, ignoring the dependent. When tolerance is close to

0, there is high multicollinearity of that variable with other independents and the b and

beta coefficients will be unstable. VIF can be estimated by the reciprocal of tolerance.

When VIF is high, there is high multicollinearity and instability of the b and beta

coefficients (Garson, 2006). Typically a VIF value greater than 10 is of concern

(Montgomery, 2001).

The histogram, normal PP plots and detrended normal QQ plot of residual can be

checked to see normality assumption. The histogram plot provides a univariate (one-

variable) description, which displays the frequency distribution for the residuals of data.

The PP plot plots a variable's cumulative proportions against the cumulative proportions

of the test distribution. The straighter the line formed by the PP plot, the more the

variable's distribution conforms to the normal distribution. The detrended normal plots

emphasize the deviations from the theoretical straight line. If the scores are normally

distributed then some of data points will fall above the horizontal line, others will fall

below it, and the pattern of scores above and below the horizontal line seem to be

random. A skewed distribution typically shows a "J" or an inverted "U" distribution.

50

The Shairo-Wilk test can be performed to see whether normality assumption is

statistically valid or not. The Shapiro-Wilk test calculates a W statistic using the

following equation (NIST/SEMATECH, 2003)

2

1

_2

1)(

)(∑

∑

=

=

−

⎟⎠

⎞⎜⎝

⎛

=n

i i

n

iii

xx

xaW

Where

=)(ix Ordered sample values (x(1) is the smallest)

ai = constants generated from the means, variances and covariances of the order

statistics of a sample of size n from a normal distribution

The Shapiro-Wilk test tests the null hypothesis that a sample x1, ..., xn came from

a normally distributed population. The test rejects the null hypothesis if W is too small.

The Jarque-Bera test can also be performed to see whether normality assumption is

statistically valid or not. The Jarque-Bera test calculates test statistics JB is based on the

sample kurtosis and skewness using the following equation (WIKIPEDIA, 2006)

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

46

22 KSnJB

Where

S = Skewness

K = Kurtosis

n = Number of observations

51

The statistic has an asymptotic chi-squared distribution with two degrees of freedom and

can be used to test the null hypothesis that the data are from a normal distribution.

Samples from a normal distribution have an expected skewness of 0 and an expected

kurtosis of 3. Any deviation from this increases the JB statistic, indicating the problem in

normality assumption.

The Breusch-Pagan test on random coefficients and White test on specification

robust can be performed to check the presence of spatial heteroscedasticity (i.e. spatial

non-stationarity). The Breusch-Pagan statistic tests whether the estimated variance of the

residuals from a regression depend on the values of the independent variables or not. It

uses an auxiliary regression approach (US, 2006) where the residuals from the regression

are used to run a second regression which allows us to conduct a test. Let’s assume a

regression based on

iii XY εβα ++=

Where

i = 1,……,N

E( iε ) = 0

Then, auxiliary regression is given by

iii vXZ ++= δφ2

Where

NuSi

/22 ∑∧

=

222 / SuZii ∑

∧

=

The testable hypotheses are expressed as follows:

52

Ho: δ = 0 (homoscedastic error variance)

Ha: δ ≠ 0 (heteroscedastic error variance)

If the null is upheld, the empirical error variances are only a function of the constant term

whereas the alternative hypothesis is allowing the error variances to be a function of the

X variable.

The White test also uses an auxiliary regression approach. It starts with the null

hypothesis of homoscedastic errors and tests for departures from it (US, 2006). The

residuals are computed from the original model and an empirical measure for the error

variances is constructed by squaring them. Let us consider the regression model

ii WXY εββα +++= 21

The auxiliary regression becomes

iiiiii vWXWXWXu +×+++++=∧

52423212 δδδδδφ

The testable hypotheses are expressed as follows:

Ho: δ = 0 (homoscedastic error variance)

Ha: δ ≠ 0 (heteroscedastic error variance)

The White test is implicitly based on a comparison of the sample variance of the least

squares estimators under homoscedasticity and under heteroscedasticity. If the null

hypothesis cannot be rejected, then the differences in the variances should be attributable

to sampling fluctuations.

53

To see spatial autoregression as alternative specification, Lagrange Multiplier-Lag

statistics and Robust Lagrange Multiplier-Lag statistics were calculated and statistically

tested. The Lagrange Multiplier lag test ( LagLM ) has a 2χ distribution with one degree of

freedom (Anselin, 1992) and can be expressed as

{ })'(/)'()/'(

22

22

WWWtrsMWXWXbsWyeLM Lag ++

=β

Where

=tr Trace matrix operator

')'( 1 XXXXIM −−= M

=y Vector containing the dependent variable

=e Vector of OLS residuals

=W Spatial weights matrix

Nees /'2 = (ML estimate for residual variance)

=b Vector of OLS estimates

2.3.2.2 Spatial Autoregression

When there exist spatially autocorrelation in the dataset, we need to incorporate an

autocorrelation in the modeling process, which can be done in two ways: as an additional

regressor in the form of a spatially lagged dependent variable, and in the error structure

(Anselin, 1999). Spatial autoregression can be represented as

iXWyy εβρ ++=

Where

54

y = n by 1 vector of observations on the dependent variable,

W = n by n spatial weights matrix that formalizes

ρ = spatial autoregressive parameter

X = n by k matrix of observations on the exogenous variables with an associated

k by 1 regression coefficient vector β

ε = a vector or random error terms

The log-likelihood for the spatial autoregression can be estimated by

)()')(2/1(ln 2 βρβρσ XWyyXWyyL −−−−−=

The least square estimator and variance for spatial autoregression (see for details

Anselin, 1988) can be estimated by

)(')'( 1 WyyXXXML λβ −= −∧

Neeee LoLoML /)()'( ρρσ −−=∧

Where

oo Xye∧

−= β

LL Xye∧

−= β

Generally, the spatial linkages or proximity of the observations are measured by

defining a spatial weight matrix, which represents the strength of the potential interaction

between locations (Bao, 1998) and an essential part of the computation of spatial

autocorrelation tests and the specification of spatial regression models (Anselin, 2005).

55

However, the determination of the proper specification for the elements of a spatial

weight matrix is considered to be one of the most difficult and controversial

methodological issues in spatial data analysis (Bao, 1998). There are many ways to

assign spatial weights, the choice of which relies on the type of spatial application and on

the research question. The most commons are row standardization, length of common

boundary and distance functions (Anselin, 2005). In construction of distance-based

spatial weights, the definition of neighbor indicates the distance between points or

between polygon centroids (Anselin, 2005).

Moran’s I is one the most popular measures of spatial autocorrelation (Bao,

1998), which indicates similarity between observations for a given variables as a function

of spatial distance, and is given as

∑∑

∑∑ −−= n

i

n

jij

n

i

n

jjiij

WS

xxxxWI

)(

))((

2

__

Where

∑ −=n

ii xx

nS 2

_2 )(1

=ijx Observed value at location i,

x = Average of the {xi} over the n locations

=ijW Spatial weight matrix

The Moran’s I is positive when the observed value of variables within a certain

distance tend to be similar, negative when they tend to be dissimilar, and approximately

56

zero when the observed values are arranged randomly and independently over space

(Bao, 1998). The expected value and variance of the Moran’s I for samples of size n can

be estimated according to the assumed pattern of spatial data distribution (Bao, 1998).

Under the normal test for the null hypothesis of no spatial autocorrelation, the expected

value of Moran’s I approximates near to zero, whereas positive and negative values

indicate the presence of positive and negative autocorrelation, respectively.

For the spatial autoregression model, a Likelihood Ratio test on the spatial

autoregressive coefficient can be carried out which corresponds to twice the difference

between the log likelihood in this model and the log- likelihood in the standard regression

model with the same independent variables with l equaling zero. The Likelihood Ratio

test is 2χ distributed with one degree of freedom.

2.3.2.3 Geographically Weighted Regression

Let us consider ),( ii vu be a set of location co-ordinates of ith point over space. By

utilizing location coordinates, GWR extends global OLS regression framework as

∑=

++=p

jiiijijiioi vuXvuy

1

),(),( εββ

Where,

=),( iik vuβ Continuous function of the location ),( ii vu at point i

=iε Random error term with ),0( 2 IN σ

This allows a continuous surface of parameter values taken at certain locations. In

GWR, the weighting of an observation is not constant but varies with i. That means data

57

from the observations close to i are weighted more than those of observations farther

away. In matrix notation, the GWR notation can be written as

εβ +⊗= XY

Where

⊗ = Logical multiplication operator in which each element of matrix β is

multiplied by the corresponding element of matrix X.

The matrix β consists of n sets of local parameters, which is given as

),(),(),(

),(),(),(),(,(),(

10

22221220

11)111110

nnknnnn

k

k

vuvuvu

vuvuvuvuvuvu

βββ

ββββββ

β

L

LLLL

L

L

=

The GWR estimator of β i is given by

yWXXWX iT

iT

i1)( −

∧

=β

Where

=iW Weighting matrix, given by

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=

in

i

i

i

w

ww

W

L

MOMM

L

K

00

0000

2

1

58

If Wi = I (identity matrix), the GWR resembles same as OLS (Zhang et al., 2004). Thus,

the weighting matrix can be computed for each location i on the basis that closer a point

is to location i, more weight is given for estimation of the parameters for location i. Thus,

this generates a set of estimates of spatially varying parameters without specifying a

function form for the spatial variation (Zhang and Shi, 2004). For calculating the

weighting matrix, it is important to specify kernel types and bandwidth, which largely

impact the estimated parameters.

Two types of kernels are available-fixed kernel, which uses a constant bandwidth

at each regression point over space, and adaptive kernel, which uses variable bandwidth.

In fixed kernel, the weight of kth data point at the ith regression point is given by

2)/( hikdewik−=

Where

=ikd Euclidean distance between data point k and regression point i

=h Bandwidth

If k coincides with i ( )0=ikd , the weighting of a data point is unity, and the weighting of

other data points will decrease according to Gaussian curve as dij increases.

In adoptive kernel, where the regression points are closely spaced, smaller

bandwidths are applied than those of when the regression points are widely spaced. The

weight of kth data point at the ith regression point is given by

22

1⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−=

i

ikik h

dw when ,iik hd ≤

= 0 when iik hd >

59

At the regression point )0( =ikd , the weighting of a data point is unity, and becomes zero

when the distance equals to bandwidth )( hdik = . The bandwidth is selected such that

there are the same number of points with non-zero points at each regression point i across

space.

In GWR, the bandwidth acts as smoothing function. There are three methods of

bandwidth selection. In the first method, a predefined bandwidth can be selected based on

existing knowledge. Although it makes computation easy, its performance solely relies

on user’s knowledge and ability to judge best bandwidth (Shi et al., 2006). When

bandwidth is too high, an over-smoothed model will result, producing parameters with

too little variation across space. Alternatively, if the bandwidth is too low, an under-

smoothed model will be produced; generating greater variation in model parameters over

space than what is realistic. In case of fixed kernel, the bandwidth needs to be specified in

terms of the same units as the location variables, whereas in case of an adaptive kernel,

the bandwidth is specified as the number of data points in the local sample used to

estimate the parameters. With a very large data set, bandwidth selection can be made

using the desired percentage of the sample points to save time or otherwise can used all

data.

The second method of bandwidth selection in GWR requires no prior knowledge,

and estimates bandwidth through a cross-validation technique which minimizes the

squared error as

[ ]21 )(∑ ≠−=

ii hyyz

60

Where

=≠ )(1 hy Fitted value of yi with observation for location i omitted from

the estimation process

This technique is only possible when the regression point locations are the same as the

data point locations.

The third method for bandwidth selection in GWR employs a technique which

minimizes Akaike Information Criterion (AIC) for fitting the best regression model as

⎭⎬⎫

⎩⎨⎧

−−+

++=∧

)(2)()2(log)(log2Strn

StrnnnnAIC eec πσ

Where

=n Sample size

∧

σ = Estimated standard deviation of the error term

)(Str = Tract of the hat matrix

=c Subscript for corrected AIC estimate

As a general rule, the lower the cAIC , the better is the reflection of the model to

reality. Therefore, the smaller the cAIC , the better is the model. In general, a complex

model (GWR) is preferred over a simpler one (OLS) only if there is significant

improvement (Yu and Wu, 2005). As a result, it is important to compare between global

OLS and local GWR models; a process which can be performed in two ways: a local non-

stationarity test or an F-test (Shi et al., 2006).

61

The aim of a non-stationarity test is to examine whether the local variations as

generated by GWR are significantly different over space. If there are no significant

differences in local variations, then there is no advantage of using GWR over OLS. There

are three ways of performing the test. Firstly, it can be done by comparing the range of

the local parameter estimates with a confidence interval around the OLS estimate of the

equivalent parameter (Wang et al., 2005). The parameter is considered to be non-

stationary if the interquartile range of GWR estimates is greater than ± 1 standard

deviation of the same global parameter. A second non-stationarity test can be performed

through a Monto Carlo test, which uses a pseudo-random number generator to relocate

the observations across the space. The observed value of test statistics is compared with

n-1 simulated one, which determines the significance of local variation.

The third non-stationarity test for GWR is Leung’s parameter variation test, which

involves a complex theoretical derivation (see for details Leung et al., 2000 and

Fotheringham et al., 2002). The F-test for checking improvement of GWR over OLS

involves an analysis of variance (ANOVA). This method is based on the assumption that

the distribution of residual sum of squares of the GWR model divided by the effective

number of parameters can be approximated by χ2 distribution with effective degrees of

freedom equal to number of parameters (Wang et al., 2005). Thus

ww

o

RSSoRSS

Fνν

//

=

Where

oRSS = Residual sum of squares for OLS

62

wRSS = Residual sum of squares for GWR

oν = Degree of freedoms for OLS

wν = Degree of freedoms for GWR

GWR provides four important diagnostics statistics at each point, which helps to

understand information locally. Firstly, it provides a set of the standardized residuals.

Residual is the difference between the calculated value of Y and the actual observed

value of Y for a given value of the explanatory variable. Thus, the residuals tell us how

well the model fits the data. However, residual values depend on the scale and units used,

which often need to be standardized. Standardized residuals values greater than ±3 are

unusual observations and thus should be examined carefully. It is calculated as residuals

divided by standard error of the residuals.

s

ii MS

er

Re

= i = 1,2,……,n

The second diagnostic statistic provided by GWR is a set of local variations of the

R2 statistics. These values tell us how a local model can replicate the data observed in the

vicinity of the regression points. These values should be interpreted very carefully

because the model calibrated at one location may not be suitable to replicate the data at

other locations. The local R2 statistics are calculated as

w

wwi TSS

RSSTSSr

−=2

Where

63

=wTSS Geographically weighted total sum of squares, given by

_

2)( yyWTSS jj

ijw −= ∑

=wRSS Geographically weighted residual sum of squares, given by

2)(∧

−= ∑ jjj

ijw yywRSS

ijw = Weight of data point j at regression point i

The third set of diagnostic statistics produced by GWR is leverage values, which

measure the influence of an observation on the model calibration. Leverage values are a

standardized measure of the distance of the ith observation from the centroid of the X-

space. A high leverage value indicates that the particular observation is distant from the

centre of the X observations. High-leverage points have the potential to dominate a

regression analysis but are not necessarily influential. However, observations with large

leverage values and large residual values are likely to be influential. Generally if the

leverage values exceed 2p/n (p= no of variables and n= number of observation), then it is

considered as cut off points for leverage (Montgomery et al., 2001). The leverage of an

observation at xi of the X matrix can be written as

Ti

Tiii xXXxh 1)( −=

The third diagnostic statistic produced by GWR is Cook’s Distance, which is another

indication of the influence of an observation. An observation that is influential will have

a large value of Cook’s Distance. Usually, attention should be paid to those observations

64

whose values are larger than 1. An observation with a large Cook’s Distance is influential

because it has a large studentized residual, a large leverage or both. The Cook’s Distance

can be calculated as

⎟⎟⎠

⎞⎜⎜⎝

⎛−

=ii

iiii h

hp

rD

1

2

Where

=ir Studentized residual

=iih Leverage

=p Number of variables

65

CHAPTER 3: STUDY AREA

3.1 Description of Study Area

The study area for this project is situated along the front range of the Rocky Mountains in

the west-central part of Alberta, Canada (Figure 3.1). This region is one of the western

Canada’s most physiographically and biologically diverse landscapes and the study area

is adjacent to a number of provincially and federally protected reserves including Banff

and Jasper National Parks. Outside of these protected areas, the region is subject to

intensive resource extraction for oil, gas, forestry, mining and agriculture; activities

which provide the foundation for the Alberta economy.

The study area contains two of the province’s six natural regions: Rocky

Mountains and Foothills (Achuff, 1992). The Rocky Mountains natural region covers up

27% of the study area whereas the Foothills natural region covers the remaining 73%.

The natural regions are subdivided into subregions for finer-scale landscape descriptions

and patterns. On the basis of vegetation patterns, soil and climates the Rocky Mountains

natural region is divided into three subregions –namely Alpine, Subalpine and Montane

(Achuff, 1992). Similarly, the Foothills natural region is divided into two subregions-

namely Upper Foothills and Lower Foothills. Table 3.1 provides a summary of the

common vegetation, geology and landform patterns occurring in each of these natural

subregions, which also gives a general overview of the study area.

66

Figure 3.1: Location of the study area in west-central Alberta, Canada.

67

Figure 3.2: Natural regions and subregions within the study area.

68

Table 3.1: A summary of the natural regions and subregions found in the study area, including dominant vegetation and characteristic geology and landforms. (Modified after Achuff, 1992)

69

CHAPTER 4: METHODS

4.1 Data Acquisition

This research revolves around the development and analysis of statistical models

generated between crown closure measurements acquired in the field (response variable)

and spectral/topographic information extracted from spatially-coincident Landsat

imagery and a DEM (predictor variables). Each of these three main data sources is

described in the following subsections. The data sets used in this project were provided

by the Foothills Model Forest Grizzly Bear Research Program, and are described more

fully in McDermid (2005).

4.1.1 Ground Crown Closure Data

Measurements of a variety of structural forest stand attributes, including crown closure,

were collected through a total of four field campaigns conducted from 2000 to 2003. The

field sample plots were generated with stratified random sample across the study area.

The ground protocol involved observations and measurements of vegetation composition

(species, percent cover) and structure (height, crown closure and volume) using standard

vegetation sampling and timber cruise methods establishing a 30-meter plot at each

sampling location equivalent to a Landsat TM pixel in size (Figure 4.1). A prism sweep

was conducted from the centre of each plot using a basal area factor 2 or 4 prism. When

trees were present, they were labeled, identified and measured for diameter at breath

height (DBH). From these, three representative trees of each species in the sweep were

chosen for more detailed measurements (height, height to live crown and coring with an

increment bore). Five 5-metre radius subplots located in the centre and four corners of

70

each field plot were used as measurement stations to acquire crown closure estimates

using a spherical densitometer.

Figure 4.1: Plot layout used to characterize vegetation and ground cover across a 30-metre Landsat pixel (McDermid, 2005).

4.1.2 Landsat Imagery

Muiltispectral TM imagery from Landsat 5 was acquired for this study to extract spectral

information for modeling crown closure. The study area is covered by World Reference

System (WRS) scene path 44 row 23 acquired on July 10, 2003 (Figure 4.2). For this

study, the Landsat image was available in fully rectified form. However, a series of pre-

processing were applied by McDermid (2005) to the image, a summary of which are

71

described here. A relative calibration procedure was used to perform atmospheric

normalization of the Landsat image following the work of Hall et al. (1991b), who

proposed that a “slave” image could be radiometrically normalized to a “master” through

a series of empirical transformation models developed for each spectral band using

pseudo-invariant features (PIFs) in each scene. The models are expressed in the form of

iOriginaliiiNormalized DNbaDN )()( +=

Where

=iNormalizedDN )( DN value of a pixel in the normalized image

iOriginalDN )( = DN value of the same pixel in the original image

ib = Coefficient that accounts for differences in solar irradiance,

downwelling sky radiance and atmospheric transmission

ia = Differences in sensor calibration and path radiance

The WRS path 45, row 23 image dated September 3, 2003 (which was collected

as part of The Foothills Model Forest Grizzly Bear Research Program) was chosen as the

master image because of its high radiometric quality acquired through an exceptionally

clear atmosphere. Calculating TOA reflectance values first required converting the digital

numbers (DNs) back to the original 32-bit radiance values measured by the sensor with

the following equation:

λλλλ BiasDNGainL +×=

Where

72

λ = TM band number

L = at-satellite radiance

Gain = band-specific gain obtained from the header file

Bias = band-specific bias obtained from the header file

Figure 4.2: Landsat TM5 covering the study area and sample plots.

Once the physical radiance values were obtained, at-satellite reflectance was calculated as

)sin(

2

θπ

ρλ

λ

×××

=ESUN

dL

Where

73

λ = TM band number

L = at-satellite radiance

ρ = TOA reflectance

ESUN = mean solar exo-atmospheric irradiance

θ = sun elevation angle obtained from the header file

After the master image was prepared, the image of this study area was radiometrically

corrected by using the linear transformation procedure. No topographic corrections were

conducted.

Finally, a tasseled cap transformation (Crist and Ciccone, 1984) was performed to

the rectified image for generating the standard orthogonal components brightness,

greenness and wetness (Table 4.1). In addition, geometric pre-processing of the image

was performed in order to achieve precise integration with other geographic data in a GIS

environment. Orthorectification was done using the satellite orbital math model (Toutin,

1995) in Geomatica OrthoEngine. Orthorectified Landsat 7 imagery downloaded from

the Geogratis website maintained on the Internet by Natural Resources Canada was used

as reference for image-to-image ground point collection, with geometric models

generated to within 0.5 pixels using 50 ground points. The imagery was re-sampled using

bilinear interpolation to 30-metre pixels in UTM Zone 11, NAD83 Datum based on

GRS80 ellipsoid. For verification, the geometric quality of the resulting images was

inspected visually using roads, cut lines and other linear features in a GIS environment.

74

Table 4.1: Coefficients for calculating the tasseled cap components from 8-bit, at-satellite reflectance

imagery.

4.1.3 Digital Elevation Model (DEM)

The DEM for this study was created through interpolating of the National Topographic

Database 1:50,000 digital map contours, spot heights, and water body polygons. A

smoothing algorithm following Hutchinson (1989) was used to eliminate ‘stepping’ and

‘pit’ artefacts commonly associated with similar medium-quality elevation models. The

model was created at 30-metre resolution, and distributed by 1:250,000 map sheet. Wall-

to-wall DEM coverage of the study area was conducted by mosaicking 1:250,000 map

sheets using ArcView.

Further morphometric processing was performed to derive topographic variables

of slope and angle of incidence for use in subsequent modeling activities. Slope was

extracted over a 3x3 neighborhood surrounding each pixel in the image as the slope of

the plane formed by the vector connecting the left and right neighbors and the vector

connecting the upper and lower neighbors of the pixel, and ranges between 0 and 90

degrees. Angle of incidence was calculated using a light source at 152° azimuth and 57°

elevation – typical for summer day at the latitude of the study area during which the

imagery in this study was acquired. Together, the measures of elevation, slope, and

75

incidence provided a basic description of topography for each pixel in the elevation

model.

4.2 Data Analysis

4.2.1 Descriptive Analysis

Descriptive analysis of model variables was performed using SPSS statistical software,

which provided the general overview of the dataset. Sample size, mean, minimum,

maximum, standard deviation, standard error of the mean, kurtosis and skewness were

checked to understand the descriptive qualities of model variables.

4.2.2 Arcsine Transformation

Crown closure data, like most proportional attributes, typically exhibits I -shaped

variance pattern (McDermid, 2005). Thus, it is common practice to perform the arcsine

transformation (arcsine of the square root of the percentage expressed as a decimal) in

ecological studies (e.g., Brockley, 2005; Kenefic et al., 2005; Lehtonen, 2005) before the

proportional data are analyzed using standard parametric statistical tests. So, proportional

crown closure (CC %) data were first transformed (ArcCC) using the arcsine

transformation to approximate a normal distribution using the equation

py 1sin −=

Where

y = the response variable in radians

p = the proportion (e.g. cover) in percent

76

4.2.3 Correlation Analysis

Correlation analysis measures the strength of the linear association between variables

(Rogerson, 2004). Using SPSS, correlation analysis was performed between the response

variable and predictor variables to see which one predictor variable was linearly related

to the response variable, and among the predictor variables to reveal problems of

multicollinearity. Strong multicollinearity results in large variances and covariances for

the least-squares estimators of the regression coefficients (Montgomery et al., 2001). In

general, only effective number of predictor variables should be retained in the modeling

process; those which have higher Pearson correlation coefficients with the response

variable (Wang et al., 2005).

4.2.4 Selection of Predictor Variables

Akaike’s Information Criterion (AIC) is one of the most commonly-used methods for

selecting best subset of predictor variables furthering a multiple regression analysis

(Insightful Corporation, 2005; Malczewski and Poetz, 2005). Using S-Plus statistical

software, AIC value was calculated on each individual and possible combination of

predictor variables for both Tasseled Cap and topographic components. To check the

combined effect, the best subset of topographical and Tasseled Cap component was

integrated together. According to the criterion, the combination of independent variables,

which give the lowest AIC is best for the further model analysis (Burnham and Anderson,

1998). AIC was calculated as

AIC = -2loglike + 2K

Where

77

loglike = the goodness of fit or loglikelihood for a specific model

K = the number of parameters estimated for the specific model

4.2.5 Model Building

The best subset of predictor variables was used to calibrate ordinary least square (OLS)

regression, spatial autoregression (SAR) and geographically weighted regression (GWR),

which were later evaluated to see their performance in modeling crown closure.

4.2.5.1 OLS model

Using the best subset of the predictor variables, the OLS model was constructed in SPSS.

Analysis of variance (ANOVA) was conducted to check whether regression was

statistically significant or not (F-value and corresponding p-value). Coefficient of

parameter estimates both in magnitude and direction were checked to see how much

influence each parameter had on the model. T-tests (p-value) were performed for each

parameter estimates to see whether they were statistically significant or not. Coefficients

of determination were examined to reveal how well the model explained variability. The

tolerance and variance inflation factor (VIF) for each variable was calculated to reveal

potential multicollinearity problems.

Throughout the modeling process, it is important to review the assumptions of

each analysis procedure and inspect the data for violations (Montgomery, 2001). Plots

can aid in the validation of the assumptions of normality, linearity, and equality of

variances (SPSS, 2006). The histogram, normal PP plots and detrended normal QQ plot

of residuals were examined to test the normality assumption. The Shairo-Wilk test and

78

Jarque-Bera test were performed to see whether normality assumption was statistically

valid or not.

Using Geoda statistical software, the Breusch-Pagan test on random coefficients

and the White test on specification robust were conducted to check the presence of spatial

heteroscedasticity (i.e. spatial non-stationarity). To check spatial autocorrelation, spatial

correlograms were calculated on each model variables using S-Plus, which is a measure

of standardized spatial covariance as a function of distance, ranging between -1 and 1.

Spatially weight is an essential part of the computation of spatial autocorrelation tests and

the specification of spatial regression model (Anselin, 2005). A distance-based spatial

weight was calculated in Geoda where the definition of neighbor is based on the distance

between points (see for details Anselin, 2005). Using spatial weight, Moran’s I statistic

was calculated and tested to see whether it was statistically significant or not. To examine

the spatial lag model as an alternative specification, Lagrange Multiplier-Lag statistics

and Robust Lagrange Multiplier-Lag statistics were calculated and statistically tested.

4.2.5.2 SAR model

Using the best subset of the predictor variables and spatial weight, the SAR model was

calibrated in Geoda. The spatial autocorrelation coefficient was tested statistically to

check whether the addition of a spatial lag term had any influence or not. In addition, the

coefficient of parameter estimates and corresponding significance levels were examined

and compared to the OLS model to check whether the addition of a spatial weight matrix

resulted in a meaningful change in the value of the R2.

The Likelihood Ratio Test (LRT) is a statistical test of the goodness-of-fit

between two models. A relatively more complex model is compared to a simpler model

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to see if it fits a particular dataset significantly better. The LRT was performed to see

whether there was any improvement or not by comparing the null model (OLS regression

specification) to the alternative spatial autoregression model (Anselin, 2005). The

Breusch-Pagan test on random coefficients of the SAR model was conducted to check

whether the added spatial weight matrix could account for the problem of spatial non-

stationarity.

4.2.5.3 GWR model

Using the best subset of the predictor variables, the GWR model was calibrated GWR 3.0

statistical software using an adaptive kernel and AIC minimization bandwidth

(Fotheringham et al., 2002).

A non-stationarity test was performed to determine if parameter estimates in

GWR model were significantly different across the study area by comparing OLS

parameter estimates with corresponding GWR estimates (Fortheringham et al., 2002). If

the inter-quartile (i.e., 25%-75% quartiles) range of the local estimates was greater than

that of ± 1 standard deviation of the equivalent global parameter (which is simply 2 x

S.E. of each global estimate), then the parameter under investigation might be non-

stationary. Even though variation over space occurs, it is important to know whether it is

statistically significant or not. If the estimates are not significantly different, the GWR

model is essentially the same as the OLS model (Shi et al., 2006). For this, a Monte Carlo

test was performed in which a pseudo-random number generator was used to relocate the

observations across space.

Given the greater flexibility of the GWR coefficients over space, GWR can be

expected to provide a better model fit than OLS in terms of residual sum of square.

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However, it is important to test whether the GWR model offers a significantly

improvement over the OLS model (Shi et al., 2006). An approximate F-test (Goodness of

fit) was conducted to see whether GWR was statistically better than OLS or not

(Fortheringham et al., 2002).

4.2.6 Comparison of Models

4.2.6.1 Analysis of model diagnostics

One of the ways to compare performance of the models is to evaluate model diagnostics

(Fortheringham et al., 2002; Anselin, 2005). Thus, the values of R2, log-likelihood and

AIC of the three candidate crown closure models were compared. In general, the higher

the value of R2 and log-likelihood, and lower the value of AIC, the better would be model

fit and performance. The scatter plots between the observed arcsine-transformed crown

closure (ArcCC) and fitted ArcCC of all three models were analyzed to see how the

model performance visually. In general, the model which gives the prediction closest to

and around the regression line is better one.

4.2.6.2 Analysis of model residuals

Another way to determine the model performance is to compare model residual errors

(e.g., Moisen and Frescino, 2002; Zhang and Gove, 2005; Zhang et al., 2005). Residual

plots of all three models were inspected for evidence of non-stationarity. Similarly,

residuals and absolute residuals for all three models were calculated across ten crown

closure classes at interval of 10%, and compared graphically. The individual residuals

were calculated as the difference of observed and predicted crown closure value whereas

the absolute residual was calculated by taking the absolute value of individual residuals.

Lower residual and absolute residual values indicate better prediction for that class

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(Zhang et al., 2005). Also, the values of residual sum of square and residual standard

error for all three models were compared, wherein low value indicated better model

performance.

4.2.6.3 Analysis of Moran’s I

Moran’s I not only detect spatial autocorrelation, but has also great power in detecting

mis-specfications in the model (Anselin, 2005; Malczewski and Potez, 2005). For all

three model’s residual, Moran’s I values were calculated and compared. The Moran’s I

around zero value is preferred (Anselin, 2005). To check for spatial autocorrelation of

residuals, spatial correlograms at various lag distances were calculated for each model

residual and inspected visually. The distribution around the zero value is preferred (Wang

et al., 2005).

4.2.7 Mapping of Parameter Estimates and Model Statistics

As compared to a single set of constant values across space as in the case of OLS and

SAR, GWR produce set of parameter estimates and model statistics at each point, which

can be visualized in a GIS to investigate spatial variation (Brunsdon et al., 1996;

Fotheringham and Brunsdon, 1999; Fortheringham et al., 2002). The output of GWR

(text format) was converted into compatible format using ArcGIS software. These values

were symbolized in standard mapping layout format so as to highlight variation across

study area. As the values were available only at the point level, an interpolated surface

across study area was created using inverse distance weighting (IDW) technique in

ArcGIS’s Geostatistical Wizard. The spatial variation across three forest types:

broadleaved forests, coniferous forest and mixed forest were calculated in SPSS and

described.

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4.2.8 Mapping of Observed, Predicted and Residual Values

Maps of observed, predicted and residual crown closure values fitted by OLS, SAR and

GWR models were created in ArcGIS. The point values were extrapolated across

Theissen polygons created in Geoda and described. The area within the polygon may or

may not be a representation of the actual value for that polygon, but the polygons provide

a convenient basis for comparison.

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CHAPTER 5: RESULTS AND DISCUSSION

5.1 Descriptive Statistics of Model Variables

The descriptive information and statistics of model variables are shown in Table 5.1 and

Table 5.2 respectively, which provides general overview of dataset. There was great

variation in the mean values of the various parameters as a result of differences in

measurement units. The standard deviation reveals variability in the dispersion of the

variables. The variables ArcCC, In and Wt had negative skewness whereas El, Sl, Br and

Gr had positive skewness. Only ArcCC had platykurtic distribution whereas all other

variables had leptokurtic distribution, especially larger in case of In and Sl, which were

more than the general threshold value 3 (Rogerson, 2004).

Table 5.1: Description of model variables

Variables Description CC% Proportional Crown closure measured in the field plots ArcCC Arcsine transformation of proportional crown closure data to approximate normal

distribution (Response Variable) El Elevation extracted from DEM (Predicting Variable) In Angle of incidence extracted from DEM (Predicting Variable) Sl Slope extracted from DEC (Predicting Variable) Br Brightness component of Tasseled Cap derived from Landsat TM (Predicting Variable) Gr Greenness component of Tasseled Cap derived from Landsat TM (Predicting Variable) Wt Wetness component of Tasseled Cap derived from Landsat TM (Predicting Variable)

Table 5.2: Descriptive statistics of model variables.

Parameters CC% ArcCC El In Sl Br Gr Wt Min 0.020 0.142 900 19 0 20 107 150 Mean 0.665 0.976 1277.439 53.791 5.255 43.962 133.944 173.376Max 0.980 1.429 2122 77 34.684 72 177 186 Std Dev 0.245 0.284 264.736 5.047 6.227 8.512 11.850 6.158 SE Mean 0.014 0.017 15.626 0.298 0.368 0.502 0.670 0.363 Skewness -0.719 -0.589 1.128 -0.770 2.045 0.522 0.708 -1.074 Kurtosis -0.470 -0.344 1.007 10.400 4.804 0.388 0.313 1.362

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5.2 Correlation Analysis Among Model Variables

Table 5.2 shows correlation among the model variables. The predictor variables El, In

and Br were negatively correlated with the response variable ArcCC whereas other

variables Sl, Gr and Wt were positively correlated. However, only three predictor

variables El, Gr and Wt had significant correlation with ArcCC, which means that only

these variables would be able to contribute enough in predicting ArcCC. Moreover, the

predictor variables were strongly correlated with each other, which suggest if they were

all included in the model, there might be the problems of multicollinearity. Strong

multicollinearity results in large variances and covariances for the least-square estimators

of the regression coefficients, leading to unrealistic prediction (Montgomery et al., 2001).

In order to avoid such issues, it is important to determine the best subset of predicted

variables to be included into the model building process (Malczewski and Poetz, 2005;

Wang et al., 2005).

Table 5.3: Correlation analysis of model variables. Variables ArcCC El In Sl Br Gr WtArcCC 1 El -0.141* 1 In -0.060 0.148* 1 Sl 0.047 0.559** 0.039 1 Br -0.025 -0.156** 0.258** 0.009 1 Gr 0.338** -0.128* 0.162** 0.076 0.847** 1 Wt 0.447** 0.029 -0.295** 0.023 -0.761** -0.341** 1

* Correlation is significant at 0.05 level (2-tailed) ** Correlation is significant at 0.01 level (2-tailed)

5.3 Selection of Predictor Variables

Table 5.3 shows the statistical results of Akaike Information Criterion (AIC) for selecting

the independent variables to be included into multiple regression analysis. The

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combination of variables which gives the minimum AIC value is assumed to be the best

one (Insightful Corporation, 2005). Among the Tasseled Cap components, the

combination of Gr and Wt was observed to be the best subset, in which both were

statistically significant and had the lowest AIC (-899.47). Similarly, among the

topographical components, the combination of El and Sl was observed to be the best

subset, in which both were statistically significant and had the lowest AIC (-728.97). The

best Tasseled Cap and topographic variables were combined together, which gave

combination of Gr,

Table 5.4: Statistical results for selecting the independent variables. Combination of Variables AIC Tasseled Cap Component Br* -718.64 Gr -753.19 Wt -782.38 Br, Gr -890.99 Br, Wt -881.08 Gr, Wt -899.47 Br*, Gr, Wt -899.37 Topographical Component El -724.20 Sl* -715.47 In* -719.48 El, Sl -728.97 El, In* -722.65 Sl*, In* -718.18 El, Sl, In* -727.26 Combined Gr+Wt+El+Sl* -901.24 Gr+Wt+El -901.82

* indicates the variables which are not statistically significant.

Wt and El as the best subset of the variables, in which all three variables were statistically

significant and had lowest AIC (-901.82). This result was expected as only these three

explanatory variables were significantly correlated with the response variable ArcCC in

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the preliminary analysis (Table 5.2). While this result indicates that the addition of the

topographical variable El improves the model fit, the reduction of AIC (from -899.47 to -

901.82) and residual sum of square (from 0.2076 to 0.2064) is relatively small. Several

earlier studies (e.g., Franklin, 1995, 1998; Ohmann and Spies, 1998; Ostendorf and

Reynolds, 1998) have revealed the topographic influence of plant ecology and its

functionalities and demonstrated the value of topographic variables in ecological

modeling (Sagers and Lyon, 1997; He et al., 1998; Fashi et al., 2000). For comparative

analysis, the best subset of variables (Gr, Wt and El) were calibrated with three types of

models, namely ordinary least square (OLS) regression, spatial autoregression (SAR) and

geographically weighted regression (GWR).

5.4 Models Fitting

5.4.1 OLS Model

Table 5.4 shows statistical results of fitted OLS model. The F statistics (F=86.874; p-

value=0.0000) showed that the overall regression was highly significant. However, the

model explained only 47.94% of the variance, which indicated that the model was mis-

specified, and other variables, not taken into consideration, explained the bulk of the

variance (Foody, 2003; Malczewski and Potez, 2005). The values of tolerance and

variance inflation factor (VIF) indicated that there was no problem of multicollinearity in

the model specification, and stability of model coefficient (Table 5.4). The variable Wt

had the largest influence in predicting the ArcCC because of its largest coefficient value.

This was expected, as Wt was highly correlated with ArcCC as compared to Br and El

(Table 5.3). This result is concurrent with the findings of other studies in ecological

modeling (e.g., Cohen and Spies, 1992; Cohen et al., 1995; Collins and Woodlock, 1996;

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Hansen et al., 2000) where they observed that wetness was the most important spectral

component of Tasseled Cap, and showed high correlation with structural stand attributes

due to its sensitivity to the amount of moisture associated within tree foliage and general

insensitivity to topographically-induced illumination differences. The regression

coefficients of Wt and El were statistically significant (p-value=0.0000), and had positive

contribution on the response variable i.e. the increase in these variables leads to the

increase in the response variable. Similarly, the coefficient of El was significant at 5% of

significance level (p-value=0.0386) and had negative contribution on the response

variable i.e. the increase in the variable leads to decrease in the response variable.

Table 5.5: Statistical results of OLS model for predicting ArcCC. Predictor variables

Coefficient value

Standard error

t-value Significance Pr (> | t |)

Tolerance Variance Inflation Factor (VIF)

Intercept 0.5731 0.4459 -12.851 0.0000 El -0.0009 0.0005 -2.0781 0.0386 0.9830 1.0172 Gr 0.0130 0.0011 11.801 0.0000 0.8701 1.1501 Wt 0.0290 0.0021 13.909 0.0000 0.8834 1.1325

Note: Response variable = ArcCC, R2=0.4794, R2 adjusted=0.4739, F=86.874 (p-value=0.0000)

During the model building process, it is important to validate model adequacy by

checking model assumptions (Montgomery et al., 2001). The histogram, normal PP plots

and detrended normal QQ plot of residual (Figure 5.1) indicated that there was no

problem regarding the normality assumption. These were further supported by statistical

test of normality, Shairo-Wilk (W=0.992, p-value=0.111) and Jarque-Bera (JB=1.277, p-

value=0.528), which did not reject the normality of the residual distribution. However,

the Breusch-Pagan test on random coefficients (BP=26.1738, p-value=0.0000) and the

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White test on specification robust (W=33.9586, p-value=0.0000) confirmed the presence

of spatial heteroscedasticity (i.e. spatial non-stationarity). Moreover, the spatial

correlogram (Figure 5.2) clearly indicated spatial autocorrelation for all the model

variables. It is evident that for all the variables at short distances there was positive

autocorrelation whereas negative autocorrelation at large distances. The Wt variable was

positively autocorrelated up to lag distances of 27000 m whereas the variable El was

positively autocorrelated approximately up to lag distance 70000 m. Similarly, other two

variables Gr and ArcCC were positively autocorrelated approximately up to lag distance

58000 m. Furthermore, the Moran’s I score of 0.2851 was highly significant (p-

value=0.0000) indicating strong spatial autocorrelation in the model. Similarly, both

Lagrange Multiplier-Lag (LM=153.232) and Robust Lagrange Multiplier-Lag

(RL=13.721) were highly significant (p-value=0.0000 and p-value=0.0002 respectively),

which are both clear indications of spatial autocorrelation in the dataset. Such spatial

autocorrelation and non-stationarity are violations of the core assumptions of OLS, and

can be expected to contribute to biased estimates in the modeling process, misleading

significance tests, and suboptimal prediction (Fox et al., 2001; Zhang et al., 2004).

However, it is important to note that such violations are common when OLS regression is

applied to spatial data sets, and are largely ignored in the literature (Zhang et al., 2004,

2005). As previously noted, spatial variation is a common phenomenon that results from

systematic environmental heterogeneity and complex vegetation inter-competition in a

forest ecosystem (Foody et al., 2003; Zhang and Shi, 2004; Zhang et al., 2004), and

should not be ignored. In such conditions, the use of OLS modeling procedures are not

appropriate and alternatives need to be evaluated (Foody, 2003).

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5.4.2 SAR Model

Table 5.5 shows the statistical results of the fitted SAR model with maximum likelihood

approach while controlling for spatial autocorrelation. In comparing the SAR parameters

to the OLS model (Table 5.4), a spatial weight matrix (W_ArcCC) appears as additional

indicator. The spatial autoregressive coefficient had positive effect on the model

(Rho=0.5598), and was highly significant (t-value=8.5573, p-value=0.0000); a clear

indication of the average influence of neighboring observations. There was a notable

difference in the significance level of the variable El, whose p-value changed from

0.0386 to 0.4302. Moreover, the magnitude of all the estimated coefficients was

decreased in absolute terms, with the difference made up by the coefficient of the added

spatial weight matrix. The R2 (0.5981) in SAR model increased as compared to the OLS

model (0.4794; Table 5.3). The Likelihood Ratio Test, which compares OLS as null

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model with the alternative SAR model confirmed the strong significance of the spatial

autoregressive coefficient (LR=68.497, p-value=0.0000), and suggested an improvement

of model fit with the addition of the spatial weight matrix. However, the Breusch-Pagan

test for non-stationarity in the error terms (BP=40.978, p-value=0.0000) confirmed

existence of non-stationarity, suggesting that even though the SAR model exhibited

significant improvements over OLS due its ability to account for spatial autocorrelation,

issues of non-stationarity still exist.

Table 5.6: Statistical results of SAR model for predicting ArcCC. Predictor variables

Coefficient value

Standard error

t-value Significance Pr (> | t |)

W_ArcCC 0.5598 0.0653 8.5573 0.0000 Intercept -0.5319 0.3954 -13.4528 0.0000 El -0.0003 0.0004 -0.7888 0.4302 Gr 0.0102 0.0010 9.8253 0.0000 Wt 0.0255 0.0019 13.2679 0.0000

Note: Response variable = ArcCC, R2=0.5981

5.4.3 GWR Model

In contrast to a single set of constant values over the study area generated by the OLS and

SAR models, GWR produced set of parameter estimates and model statistics at each

sampled point (see for details Appendix C). Table 5.6 provides a summary of the

parameter estimates fitted by GRW, including the median, upper quartiles, lower

quartiles, range, minimum and maximum. For a general understanding of non-stationarity

relationship across space, Fotheringham et al. (2002) suggested comparing the range of

the GWR local parameter estimates with confidence intervals (CI) around OLS global

estimate of the equivalent parameter. If the inter-quartile (i.e., 25%-75% quartiles) range

of the local estimates is greater than that of ± 1 standard deviation of the equivalent

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global parameter (which is simply 2 x S.E. of each global estimate), then the parameter

under investigation might be non-stationary. It is clearly evident that all of the

interquartile values were greater than their corresponding 2 x S.E. of the global estimates

(Table 5.6), indicating the presence of non-stationarity of relationships over the study

area (Fotheringham et al., 2002; Zhang et al., 2004). Even though spatial variation of

relationships exists, it is important to test whether the variation is significant or not (Shi

et al., 2006). If the relationships do not vary significantly, there is no advantage to using

the GWR model over the simplified OLS model (Shi et al., 2006). The Monte Carlo test

on the parameter estimates suggested that all the parameter estimates were highly

significant at 0.1% of significance level (Table 5.7). This confirms the existence of

significant local non-stationary relationships over study area (Fotheringham et al., 2002).

If we apply a global model which assumes stationarity under such conditions, such as

OLS or SAR, there is certainly miss-specification of the true situation, and under-

prediction of the model (Fotheringham et al., 2002; Zhang et al., 2004; Wang et al.,

2005).

Table 5.7: Comparative analysis of parameter estimates fitted OLS and GWR models.

Estimated Parameters Model Statistics Intercept Elevation Greenness Wetness GWR Model Minimum -9.57921 -0.00106 -0.00058 0.00505 25% Quartile -6.71280 -0.00030 0.00931 0.01706 Median -5.89840 -0.00008 0.01335 0.02739 75% Quartile -2.97710 0.00004 0.01458 0.03564 Interquartile range 3.73566 0.00033 0.00528 0.01858 Maximum -0.56349 0.00123 0.01784 0.04886 OLS Model Estimate -5.73080 -0.00010 0.01303 0.02933 Standard error (SE) 0.44594 0.00005 0.00110 0.00211 2 * Standard error (2SE) 0.89188 0.00010 0.00220 0.00422

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Given the ability of GWR coefficients to vary over space, GWR can often

produce better residual sum of squares, which is an indication of better model fit

(Brunsdon et al., 1996; Fotheringham et al., 2002; Zhang and Shi, 2004). As a result, it is

important to test statistically whether or not GWR can achieve significant improvement

over OLS model results (Zhang and Shi, 2004; Shi et al., 2006). In general, the simpler

model (OLS) is preferred over the complex one (GWR) if there is no significant

improvement (Yu and Wu, 2005). An approximate F-test in Table 5.8 clearly indicated

that GWR model significantly improved model fitting over the OLS model (F=6.6119, p-

value<0.0001). This provided further evidence that linear relationships between the

response and predictor variables were not constant across the study area (Fotheringham et

al., 2002; Zhang et al., 2005; Shi et al., 2006).

Table 5.8: Monte Carlo test for spatial variability. Parameters p-value Intercept 0.0000* Wt 0.0000* Gr 0.0000* El 0.0000*

* Significant at 0.1% Table 5.9: Goodness of fit (F-test) for improvement of GWR over OLS model. Source Sum of

square Degree of freedom

Mean Square

F statistics p-value

OLS Residuals 12.1 4 GWR Improvement 5.7 34.01 0.1682 GWR Residuals 6.3 248.99 0.0254 6.6119 <0.0001

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5.5 Comparison of OLS, SAR and GWR Models

5.5.1 Analysis of Model Diagnostics

The central question in the modeling process is which one model performs the best? One

way to access the model performance is to compare model diagnostics (Anselin, 2005).

Table 5.9 shows comparative analysis of the three models based on the model

diagnostics. It is evident that when spatial weight matrix is added, R2 value increased

from 0.4795 for OLS model to 0.5981 for SAR model. However, comparing R2 alone is

not appropriate as R2 in the case of spatial model only represents pseudo-R2, which is not

directly comparable with the coefficient of variation from the OLS model (Anselin,

2005). So, other measures of fit (Log-likelihood and AIC) are needed to thoroughly

evaluate model performance. The added spatial weight matrix of the SAR model

noticeably increased the Log-likelihood value (from 47.6954 for OLS to 81.944 for SAR)

and decreased the AIC value (from -87.391 to -153.888) relative to OLS model. As a rule

of thumb, an absolute differential AIC value of 3 or more is clear evidence of improved

model performance (Fotheringham et al., 2001). Based on these comparisons, we can say

that there was an improvement of fit over OLS model with adding the spatial weight

matrix. However, both of these models (OLS and SAR) assumed stationarity

relationships across study area in contrast to non-stationarity in the case of the GWR

model. Once again, it was evident (Table 5.6 and Table 5.7) that the model parameters

varied across the study area. So it is interesting to observe the influence of spatial non-

stationarity on the model fit by incorporating the spatial locations of data as in the case of

GWR model. It is clear from the Table 5.9 that GWR had the lowest AIC (-201.286), and

the highest R2 (0.7265) and log-likelihood (138.656); all three of which are indicators of

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better model performance. Thus, we can conclude that by accounting for spatial non-

stationarity in the modeling process, GWR appears capable producing better models of

crown closure than OLS and SAR, which assume spatial stationarity. This result is

confirmed visually from the scatter plots (Figure 5.3) of observed ArcCC versus fitted

ArcCC, which reveals a tighter cluster of predicted points close to and around the

regression line for the GWR model as compared to OLS and SAR (Wang et al., 2005).

Table 5.10: Evaluation of OLS, SAR and GWR based on model diagnostics. Parameters OLS SAR GWR Coefficient of Determination (R2) 0.4795 0.5981 0.7265 Log-likelihood 47.6954 81.944 138.656 Akaike Information Criterion (AIC) -87.391 -153.888 -201.286

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Figure 5.3: Scatter diagram between observed ArcCC and fitted ArcCC by OLS, SAR and GWR

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5.5.2 Analysis of Model Residuals

Another way to assess the model performance is to compare model residual errors (e.g.,

Zhang and Gove, 2005; Zhang et al., 2005). Residual plot of the OLS model (Figure 5.4)

had an almost triangular in shape, which was an indication of inequity of variances, or

non-stationarity (Shi et al., 2006). Even after adding the spatial weight matrix, the shape

of residual plot of SAR model (Figure 5.4) was close to triangular, indicating the

continued presence of non-stationarity. However, the shape of the residual plot of the

GWR model was much improved, displaying a more regular shape (Figure 5.4). This

result provides evidence that GWR has successfully reduced non-stationarity by taking

spatial location into account in the modeling process.

Figure 5.5a shows model residuals for OLS, SAR and GWR across ten crown

closures classes. In general, all the models produced negative residuals (over-estimation)

at lower crown closure classes (< 70-80%) and positive residuals (under-estimation) for

higher crown closure classes (>70-80%). However, it is evident that in almost crown

closure classes (except for 40-50% and 70-80%) GWR model produced less residual

errors (residual sum of square=6.33 and residual standard error=0.1595) as compared to

those of the SAR (=9.31 and =0.1801) and OLS (=12.05 and=0.2064) models. Figure

5.5b shows the absolute model residuals for OLS, SAR and GWR across ten crown

closure classes, which also depicts substantially lower residuals for the GWR model for

almost all classes (except 60-70% crown class) as compared to those for the OLS and

SAR models. The overall average of absolute model residual (0.131) for the GWR model

was 24% smaller than those for the SAR model (0.172) and 32% smaller than those of

the OLS model (0.194). Again, this is an indication of better fit for the GWR model.

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Reporting results similar to these, Zhang et al. (2005) found the least residual and

absolute residuals in GWR models as compared to other models while predicting tree

crown by diameter.

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>90

Ave

rage

Crown Closure Class (CC%)

Abs

olut

e R

esid

uals

OLSSARGWR

b

Figure 5.5: Comparison of models across crown classes: a) Model residuals and b) Absolute residuals

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5.5.3 Analysis of Moran’s I

Moran’ I is one method for detecting the problem of misspecification in the relationships

described by the model (Fortheringham et al., 2002; Anselin, 2005; Malczewski and

Potez, 2005). From the Figure 5.6, it is evident that the residuals for OLS model exhibit

significant positive autocorrelation (Moran’s I=0.2851, p=0.0001). By adding the spatial

weight matrix, the Moran’s I was sharply reduced to 0.0252 in case of SAR model.

Similarly, there was a substantial reduction of the Moran’s I value (-0.0046) on residual

for the GWR model. This indicated that not only GWR was able to improve model

performance and take into account of the problem of spatial non-stationarity; it also

alleviated the problem of spatially autocorrelated error terms (Fotheringham et al., 2002;

Malczewski and Potez, 2005; Wang et al., 2005). Figure 5.7 shows spatial correlograms

of residual for OLS, SAR and GWR models at various lag distances. The spatial

correlogram of residuals for the OLS model showed positive autocorrelation at short

distances and negative autocorrelation at larger distances. There was not much difference

between the spatial correlogram of residuals for SAR and GWR models at large lag

distances, but at a short lag distances, high autocorrelation on residuals was observed for

the SAR model as compared to the GWR model. The high positive autocorrelation at

short distance might lead to distortion of statistical tests and unreliability of model

interpretation (Legendre and Legendre, 1998). On the basis of Moran’s I comparison, we

can conclude that by incorporating the spatial lag term in the modeling process, SAR

model performed better than OLS model, but not quite as well as the GWR model. This

finding was consistent with Wang et al. (2005), who they compared same three types of

models for predicting net primary production.

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Figure 5.6: Moran’s I on residuals for three different models.

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Figure 5.7: Spatial correlogram of residuals for three different models.

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5.6 Spatial Variation of Estimated Parameters across the Study

Area

One of advantage of GWR is that the set of model parameters and statistics produced by

the technique can be visualized in a Geographical Information System (GIS), providing a

powerful tool for investigating local variation across the study area (Brunsdon et al.,

1996; Fotheringham and Brunsdon, 1999; Fortheringham et al., 2002). Figure 5.8 shows

the spatial distribution of estimated values and corresponding t-values, along with the

interpolated (inverse weighted distance) surface fitted by the GWR model. Figure 5.8a

shows spatial variation in the intercept, which has a median of -5.898 and range from -

9.579 to -0.563 for the GWR model in contrast to constant intercept values (-5.731 for

OLS and -5.318 for SAR) for the OLS and SAR models across study area. Higher

coefficient values (positive or negative), indicate greater influence of the particular

predictor variable on that plot, whereas lower values point to reduced influence for the

particular variable. An inspection of the map reveals higher negative values towards the

NE and NW regions and lower negative values towards the SW. Similarly, Figure 5.8b

shows spatial variation in t-values of the intercept. Earlier results indicated that the

relationship between ArcCC and the intercept was highly significant, with t-values of -

12.851 for OLS, (Table 5.4) and -13.453 for the SAR model (Table 5.5). However, a

visual inspection of Figure 8.5b reveals that the relationship was not significant

throughout study area (shown by red star and corresponding yellow color interpolated

surface). Similarly, earlier statistics from the global model suggested a negative

relationship between elevation and ArcCC (coefficients =-0.00009 for OLS and -0.00003

for SAR model). This result could be interpreted to mean that an increase in elevation

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corresponded would be accompanied by a decrease of crown closure throughout the study

area. However, the GWR model revealed that the relationship between crown closure and

elevation was more complex, with a median estimated value of 0.00008 and range from -

0.00106 to 0.00123. A positive relationship existed between ArcCC and elevation (El)

especially towards NE (Figure 5.8c, shown by red circle and corresponding interpolated

surface). Figure 5.8d shows spatial variation in t-values of estimated elevation, which

depicted a mixed relationship (significant relationship with blue star and non-significant

relationship with red star) as compared to the constant significant relationship suggested

by the OLS model (t-value=0.039) and non-significant relationship for the SAR model (t-

value=0.430) throughout the study area. Likewise, the estimated value of greenness (Gr)

ranged from -0.0006 to 0.0178 with median of 0.0133 for the GWR model, as compared

to constant value (0.0130 for OLS and 0.0102 for SAR) for the other two models. Also,

GWR revealed that the t-value of estimated greenness was not significant throughout the

study area as indicated by global model (t-value=11.801 for OLS model and 9.825 for

SAR model), especially in the SW portion of the study area (Figure 5.8f shown by red

star and corresponding yellow colored interpolated surface). Similarly, Figure 5.8g shows

that the estimated value for wetness (Wt) ranged from 0.005 to 0.049 with a median of

0.027 instead of the constant values (0.029 for OLS model and 0.026 for SAR model)

suggested by the global models. In general, the SW portion of study area had the lowest

values of estimated wetness as compared to other portions. This is likely why the SW

portion of study area also had the lowest t-values; some of which were insignificant

(Figure 5.8h as indicated by red star and corresponding yellow colored interpolated

surface). Predictably, these intricacies were lost on the global models, which estimated

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wetness as highly significant across the entire study area with a constant t-value (13.909

for OLS model and 13.268 for SAR model). The results noted by the GWR model are a

reflection of the variability of the relationship between the response and predictor

variables across different parts of the study area. Figure 5.9 shows the local variation in

coefficient of determination (R2), which varies from 0.3050 to 0.8659 in the GWR model

as compared to the constant values (0.4795 for OLS and 0.5981 for SAR) generated by

the global models. In fact, about 210 plots (roughly 73% of the total plots) have R2 higher

than the OLS model, while 109 plots (about 40%) have R2 values higher than the SAR

model. From a spatial standpoint, the SW and central portions of study area had lower

value of R2 as compared to the other regions. This made sense, since as these areas also

displayed lower values of estimated Wt and Gr (Figure 5.8e and 5.8g): the two most

highly-contributing variables in the prediction of crown closure. It is important to note,

however, that the local R2 should be not interpreted with as much confidence as global R2

values (Fotheringham et al., 2002).

107

Figure 5.8: Estimated value and corresponding interpolated surface fitted by GWR model: a)

Intercept, b) t-value for intercept, c) Elevation, d) t-value for elevation, e) Greenness, f) t-value for greenness, g) Wetness and h) t-value for wetness

108

Figure 5.8 (continued): Estimated value and corresponding interpolated surface fitted by GWR

model: a) Intercept, b) t-value for intercept, c) Elevation, d) t-value for elevation, e) Greenness, f) t-value for greenness, g) Wetness and h) t-value for wetness

109

Figure 5.9: Local R2 value and corresponding interpolated surface

Figure 5.10 shows the mean and range (each bar stretches two standard deviations on

either side of the mean) values for the intercept, estimated parameters and local R2 fitted

by GWR model across three forest types: broadleaf, coniferous and mixed. From this

figure, we can observe that spatial variability occurs within and across all three forest

types.

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46118123N =

Forest Types

MixedConiferousBroadleaf

Mea

n +-

1 S

D (I

nter

cept

)

0

-2

-4

-6

-8

-1046118123N =

Forest Types

MixedConiferousBroadleaf

Mea

n +-

1 S

D (E

leva

tion)

.0006

.0004

.0002

-.0000

-.0002

-.0004

-.0006

46118123N =

Forest Types

MixedConiferousBroadleaf

Mea

n +-

1 S

D (G

reen

ness

) .018

.016

.014

.012

.010

.008

.006

.004

.002

46118123N =

Forest Types

MixedConiferousBroadleaf

Mea

n +-

1 S

D (W

etne

ss)

.05

.04

.03

.02

.01

0.00

46118123N =

Forest Types

MixedConiferousBroadleaf

Mea

n +-

1 S

D (R

esid

ual)

.3

.2

.1

-.0

-.1

-.2

-.3

46118123N =

Forest Types

MixedConiferousBroadleaf

Mea

n +-

1 S

D (L

ocal

R2)

.8

.7

.6

.5

.4

.3

Figure 5.10: Variation of GWR estimates of intercept, parameters, residual and local R2 across three

different types of forest

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5.7 Mapping Observed, Predicted and Residual Values Across

the Study Area

The maps of observed, predicted and residual crown closure values fitted by OLS, SAR

and GWR model are shown in Figure 5.11, 5.12 and 5.13 respectively. It is important to

mention here that the crown closure values displayed in the figures were extrapolated

using a Theissen polygon. The area within the polygon may or may not be a

representation of the actual crown closure for that polygon. However, the polygons

provide convenient units for performing visual comparisons. From the observed crown

closure map (Figure 5.11), it is evident that there is a large variation of crown closure in

the plots measured across study area (from 0.02 to 0.98). In general, greater crown

closure values are concentrated in the central part of the study area, whereas lower crown

closure values are located on south western portion of the study area. The range of crown

closure values predicted by the GWR model range from 0.04 to 0.98, as compared to 0.04

to 0.96 for the SAR model and 0.01 to 0.95 for the OLS model. These observations are

similar is an indication that all three regression methods have the ability to predict crown

closure over a wide range. However, the coefficients of determination for the GWR

model (R2 =72.65%) suggested that this method captured more of the observed variability

than that of the SAR model (R2= 59.81%) or the OLS model (47.95%) (Table 3). This

higher predictability of the GWR model is confirmed by comparing the observed and

predicted crown closure maps (Figure 5.13), which match quite closely as compared to

the observed and predicted maps generated by SAR (Figure 5.12) and OLS (Figure 5.11).

The unexplained portion (27.35 %) of the GWR model appears as residual, varying from

-0.3925 to 0.4111, which was quite low as compared to those of the SAR model (from -

112

0.5055 to 0.6147) and the OLS model (from -0.5464 to 0.6777). It is evident from the

residual maps (lower parts of Figures 5.11 through 5.13) that some plots had negative

values whereas others had positive values; the residuals being an indication of

overestimation and underestimation by the model, respectively. In general the area which

had the highest observed crown closure range also had the highest positive residuals,

which suggest that the model underestimates these values. In contrast, the areas that

displayed the lower crown closure values, tended to have large negative residuals,

revealing the tendency of the model overestimate these values. The higher unexplained

variance of OLS model (52.05%) and SAR model (40.19%) may be attributed to the

assumption of constant relationships over space. The distribution of residuals appeared

more heterogeneous in the GWR model as compared to SAR and OLS. When mapping of

the residuals with non-spatial techniques (i.e. OLS), they tend naturally to form ‘clumps’,

where neighboring residuals tend to be more similar than distant ones, a pattern which

affects both the significance and (less appreciated) the values of model parameters

(Legendre, 1993; Diniz-Filho et al., 2003).

113

Figure 5.11: Maps of observed, predicted and residual crown closure using OLS model

114

Figure 5.12: Maps of observed, predicted and residual crown closure using SAR model

115

Figure 5.13: Maps of observed, predicted and residual crown closure using GWR model

116

From the above discussion, it is clear that spatial variation is exhibited in the model,

which might be due to imperfections in the dataset itself and missing variables in the

modeling process (Fotheringham et al., 1997; Foody, 2003; Shi et al, 2006). While these

issues could disappear if the dataset was perfect and all the important variables were

included in the modeling process (Zhang and Shi, 2004; Shi et al., 2006), such a scenario

is almost impossible in a practical sense. For example, the variability of spectral

reflectance patterns are influenced by a variety of exogenous factors, such as orientation

and inclination of the viewing angle, sun elevation, wind speed, clouds, terrain and

moisture condition, as well as endogenous factors such as phenology, temperature,

foliage coverage, species composition, vigor and geometry (Treitz and Howarth, 1999;

Wigneron et al., 2003); all of which are extremely difficult to control. Similarly,

unintentional and non-measurable personal and instrumental errors are always expected

while collecting and preprocessing model variables (Jensen, 2005). Moreover, in many

circumstances models need to be built without complete information of the forest stands

due to various practical constraints (Zhang and Shi, 2004). In such scenarios, GWR can

serve as better model by incorporating the factor of spatial location, a readily-available

attribute, as a proxy for missing variables and imperfect datasets (Fotheringham et al.,

2002; Foody, 2004; Zhang and Shi, 2004). We can conclude from the crown closure

modeling conducted here that GWR was able to account for both spatial autocorrelation

and spatial non-stationary, thereby providing a better foundation for prediction and

explanatory ability than corresponding OLS and SAR models. This finding is concurrent

with the results of other ecological studies (e.g., Foody, 2003, 2004; Zhang et al., 2004,

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2005; Shi et al., 2006) that reported significant variation in ecological parameters, and

better prediction of GWR model over alternative regression procedures.

Crown closure is interrelated with many forest stand attributes including age,

diameter, height, density, basal area, species composition, biomass, and leaf area index

(Spies, 1998; Stone and Porter, 1998), and an important contributor to various ecological,

hydrological and climate models (Herwitz et al., 2000; Nadkarni and Sumera, 2004;

Nilson and Kuusk, 2004; Falkowski et al., 2005). By predicting accurate and reliable

estimation of crown closure, it can be expected that GWR is capable of providing better

insights to stand attribute relationships and ecosystem processes (Shi et al., 2006).

Moreover, with the help of GWR, it is possible to estimate and map site-specific

individual parameters and models, which can provide better reflection of local variation

across space, and can be used to develop and implement local site specific management

activities (Atkinson et al., 2003; Foody, 2004; Zhang and Shi, 2004). Mapping local

variation in parameter estimates may also facilitate the identification of potential causes

of missing variables or interaction terms and low estimation efficiencies (Jetz et al., 2005,

Shi et al., 2006), which could provide positive implications towards future ecological

data collection and research activities (Zhang and Shi, 2004). Realizing these advantages,

the use of GWR has been increasing across many disciplines (e.g., Nakaya, 2001; Huang

and Leung, 2002; Hanham and Spiker, 2004; Benson et al., 2005; Malczewski and Poetz,

2005) including ecological studies (e.g., Foody, 2004; Zhang and Gove, 2004; Zhang et

al., 2004; Zhang et al., 2005; Shi et al., 2006). However, the technique has received only

limited exposure in remote sensing (Foody, 2003; 2004; Wang et al., 2005) and

ecological disciplines (Hanham and Spiker, 2004; Yu and Wu, 2005). However, with the

118

growing concern regarding the role of spatial variation in ecological/remote sensing

applications, GWR has the potential to play a leading role in this important new research

area (Foody, 2003; 2004; Hanham and Spiker, 2004).

Despite the promising results obtained in this and other studies, it is important to

recognize that the GWR technique is not free from limitations. The full utilization of

modeling process relies on its ability to generalize across varied large areas over time and

space, which is difficult to achieve in the case of GWR (Fotheringham et al., 2002; Zhang

and Gove, 2005). GWR is not suitable for extrapolating relationship beyond the region

and over the time at which model has been developed because of the potential for

complex spatial-size interrelationships (Wang et al., 2005; Zhang and Gove, 2005).

Another concern revolves around the sensitivity of local parameter estimates to the

proper selection of suitable kernel and bandwidth values (Foody, 2003; Wang et al.,

2005). Finally, the interpretation of spatially-varying model coefficients is more complex

in the case of GWR than it is for other global models (Fotheringham et al., 2002; Foody,

2003), and requires the use of contextual and other underlying spatial information (Platt,

2004; Yu and Wu, 2005).

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CHAPTER 6: CONCLUSIONS

This study explored the use of ordinary least squares (OLS) regression, spatial

autoregression (SAR) and geographically-weighted regression (GWR) for modeling

crown closure in the foothills of west-central Alberta using spectral and topographic

variables in a geographical information system/remote sensing approach. It was observed

that spectral response patterns extracted from Landsat imagery, as captured in the

Tasseled Cap coefficients of brightness, greenness, and wetness, could provide enough

information to model crown closure at statistically-significant levels. Model fit was

increased when spectral response variables were combined with topographic information

extracted from a DEM, but the improvement was not statistically significant. An analysis

of the predictor variables revealed significant correlation, an indication of

multicollinearity. The combination of elevation, greenness and wetness appeared as a

best subset of the predictor variables with lowest AIC values and residual sums of square.

Using the best subset of predictor variables, a comparative analysis was performed of the

three candidate regression models; OLS, SAR, and GWR. The work was designed to

analyze the impacts of autocorrelation and non-stationarity on forest structure models

derived from spectral and topographic variables.

Although the OLS model fitted the crown closure data moderately well, evidence

suggested that it violated two of the core assumptions of OLS: independence and constant

variance. Spatial autocorrelation and spatial non-stationarity were clearly exhibited in the

model variables. With addition of spatial weight matrix, the SAR model was able to

achieve a statistically significant improvement of model fitting over OLS, with an almost

complete removal of the spatial autocorrelation problem. However, the SAR model

120

could not account for the spatial non-stationarity problem. Due to the presence of non-

stationarity over the study area, global models such as OLS and SAR could be expected

to generate misleading significance tests and suboptimal prediction results. By taking

spatial location into consideration, GWR was not only provided improved explanatory

and exploratory abilities, but was also able to detect the spatial variability amongst

relationships across the study area. The local spatial variations could be mapped easily

using visualization tools within a GIS, which might have positive implications towards

understanding ecological process and relationships, developing local management

strategies and identifying missing variables for future modeling efforts.

Recently, there has been growing concerns regarding the effects of spatial

variation on the modeling process in many disciplines, including ecological/remote

sensing applications. Spatial variation is a common phenomenon that may result from

imperfections in the dataset, missing relevant information in the modeling process and

prevailing systematic and environmental heterogeneity. Ignoring spatial variations under

these conditions could result in biased estimation, misleading significant tests and sub-

optimal predictions. In such scenarios, GWR provides an effective alternative option by

making use of spatial location as proxy variables. However, patterns of both spectral

response and stand characteristics can be expected to vary across time and space. As a

result, the full utilization of the modeling process relies on its ability to generalize across

large areas over time and space, an objective which is difficult to achieve with GWR.

However, it is expected that the development of an integrated approach to analysis (i.e.

local spatial modeling tools like GWR compatible with other analytical and visualization

tools like GIS/Remote sensing), in conjunction with further refinement in the GWR

121

methodology, will lead the technique towards playing a leading role in the understanding

of complex and dynamic spatial processes and relationships.

122

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APPENDIX A: OUTPUT OF OLS MODEL

SUMMARY OF OUTPUT: ORDINARY LEAST SQUARES ESTIMATION Data set : CT Dependent Variable : CC_ARC Number of Observations: 287 Mean dependent var : 0.975564 Number of Variables : 4 S.D. dependent var : 0.284029 Degrees of Freedom : 283 R-squared : 0.479461 F-statistic : 86.8892 Adjusted R-squared : 0.473943 Prob(F-statistic) :7.06454e-040 Sum squared residual: 12.052 Log likelihood : 47.6954 Sigma-square : 0.0425867 Akaike info criterion : -87.3909 S.E. of regression : 0.206365 Schwarz criterion : -72.753 Sigma-square ML : 0.0419931 S.E of regression ML: 0.204922 ----------------------------------------------------------------------- Variable Coefficient Std.Error t-Statistic Probability ----------------------------------------------------------------------- CONSTANT -5.730766 0.4459399 -12.85098 0.0000000 EL -9.659491e-005 4.648285e-005 -2.078076 0.0386035 GR 0.01303071 0.001104239 11.80062 0.0000000 WT 0.02932544 0.002108421 13.90872 0.0000000 ----------------------------------------------------------------------- REGRESSION DIAGNOSTICS MULTICOLLINEARITY CONDITION NUMBER 93.65152 TEST ON NORMALITY OF ERRORS

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TEST DF VALUE PROB Jarque-Bera 2 1.276614 0.5281859 DIAGNOSTICS FOR HETEROSCEDASTICITY RANDOM COEFFICIENTS TEST DF VALUE PROB Breusch-Pagan test 3 26.17384 0.0000088 Koenker-Bassett test 3 22.52657 0.0000507 SPECIFICATION ROBUST TEST TEST DF VALUE PROB White 9 33.95861 0.0000908 DIAGNOSTICS FOR SPATIAL DEPENDENCE FOR WEIGHT MATRIX : spatial.GWT (row-standardized weights) TEST MI/DF VALUE PROB Moran's I (error) 0.285063 17.0474245 0.0000000 Lagrange Multiplier (lag) 1 153.2329602 0.0000000 Robust LM (lag) 1 13.7209856 0.0002121 Lagrange Multiplier (error) 1 236.6217166 0.0000000 Robust LM (error) 1 97.1097420 0.0000000 Lagrange Multiplier (SARMA) 2 250.3427022 0.0000000 OBS CC_ARC PREDICTED RESIDUAL 1 0.90700 1.05443 -0.14743 2 1.04700 1.05671 -0.00971 3 1.13300 1.08845 0.04455 4 1.28400 1.27510 0.00890 5 1.15900 0.96383 0.19517 6 0.46400 0.09693 0.36707 7 0.86600 0.93006 -0.06406 8 0.91700 1.02908 -0.11208 9 0.60100 0.65455 -0.05355 10 0.99100 1.03391 -0.04291 11 1.02400 1.06107 -0.03707 12 1.23300 1.17665 0.05635 13 1.13300 1.23526 -0.10226

143

14 1.28400 0.99280 0.29120 15 1.13300 1.17035 -0.03735 16 1.02400 1.28213 -0.25813 17 0.14200 0.43139 -0.28939 18 0.24700 0.47144 -0.22444 19 0.32200 0.42412 -0.10212 20 0.52400 0.90571 -0.38171 21 0.54600 0.85291 -0.30691 22 0.56900 0.61725 -0.04825 23 0.58000 0.73214 -0.15214 24 0.59100 0.82176 -0.23076 25 0.59100 0.98654 -0.39554 26 0.60100 0.92621 -0.32521 27 0.62300 0.98524 -0.36224 28 0.63300 0.76439 -0.13139 29 0.65400 1.04752 -0.39352 30 0.66400 0.38458 0.27942 31 0.68500 1.09442 -0.40942 32 0.68500 0.87698 -0.19198 33 0.69500 1.15492 -0.45992 34 0.73500 1.11330 -0.37830 35 0.74500 0.85922 -0.11422 36 0.76500 0.85944 -0.09444 37 0.76500 1.02932 -0.26432 38 0.76500 0.80361 -0.03861 39 0.80500 0.78206 0.02294 40 0.83500 0.94393 -0.10893 41 0.84600 1.07399 -0.22799 42 0.85600 1.04134 -0.18534 43 0.86600 1.04345 -0.17745 44 0.86600 0.97430 -0.10830 45 0.86600 0.93751 -0.07151 46 0.86600 1.05053 -0.18453 47 0.86600 1.10728 -0.24128 48 0.88600 0.88219 0.00381 49 0.88600 0.98015 -0.09415 50 0.89600 0.85057 0.04543 51 0.89600 1.22720 -0.33120 52 0.89600 1.05623 -0.16023 53 0.90700 1.09215 -0.18515 54 0.90700 0.89003 0.01697 55 0.91700 1.03155 -0.11455 56 0.91700 1.09858 -0.18158 57 0.91700 1.07854 -0.16154 58 0.92700 0.96315 -0.03615 59 0.92700 1.03457 -0.10757 60 0.92700 1.11420 -0.18720

144

61 0.92700 1.05292 -0.12592 62 0.92700 1.24134 -0.31434 63 0.93800 0.95591 -0.01791 64 0.94800 1.01324 -0.06524 65 0.94800 1.14802 -0.20002 66 0.94800 0.76867 0.17933 67 0.95900 0.92494 0.03406 68 0.95900 1.11055 -0.15155 69 0.97000 0.84189 0.12811 70 0.97000 0.98013 -0.01013 71 0.98000 1.02627 -0.04627 72 0.99100 0.80041 0.19059 73 0.99100 0.94827 0.04273 74 0.99100 0.91346 0.07754 75 0.99100 1.28558 -0.29458 76 1.00200 1.00873 -0.00673 77 1.00200 0.87817 0.12383 78 1.00200 1.05044 -0.04844 79 1.01300 0.95466 0.05834 80 1.01300 0.97001 0.04299 81 1.01300 1.13400 -0.12100 82 1.01300 1.00587 0.00713 83 1.01300 0.95254 0.06046 84 1.01300 1.16702 -0.15402 85 1.02400 0.92330 0.10070 86 1.02400 0.93388 0.09012 87 1.03600 1.20166 -0.16566 88 1.03600 0.96540 0.07060 89 1.03600 1.03294 0.00306 90 1.03600 1.23764 -0.20164 91 1.03600 1.14638 -0.11038 92 1.03600 1.10563 -0.06963 93 1.04700 0.76507 0.28193 94 1.04700 1.02263 0.02437 95 1.04700 0.84993 0.19707 96 1.05900 1.11621 -0.05721 97 1.05900 0.97719 0.08181 98 1.05900 1.22745 -0.16845 99 1.05900 1.13777 -0.07877 100 1.07100 0.75810 0.31290 101 1.07100 0.85239 0.21861 102 1.08300 0.89244 0.19056 103 1.08300 0.94981 0.13319 104 1.08300 1.05693 0.02607 105 1.08300 1.12616 -0.04316 106 1.09500 0.94172 0.15328 107 1.09500 1.09333 0.00167

145

108 1.09500 1.17881 -0.08381 109 1.10700 1.04929 0.05771 110 1.10700 1.02027 0.08673 111 1.10700 1.12488 -0.01788 112 1.10700 0.97874 0.12826 113 1.10700 0.93216 0.17484 114 1.10700 1.08181 0.02519 115 1.12000 1.15101 -0.03101 116 1.12000 1.04734 0.07266 117 1.12000 1.15591 -0.03591 118 1.12000 1.12336 -0.00336 119 1.13300 1.01952 0.11348 120 1.13300 1.02229 0.11071 121 1.13300 0.94743 0.18557 122 1.13300 1.00847 0.12453 123 1.13300 1.11356 0.01944 124 1.13300 1.11093 0.02207 125 1.13300 0.99160 0.14140 126 1.13300 1.31234 -0.17934 127 1.14600 1.02993 0.11607 128 1.14600 1.09067 0.05533 129 1.15900 0.83300 0.32600 130 1.15900 1.15648 0.00252 131 1.15900 1.15722 0.00178 132 1.17300 1.19057 -0.01757 133 1.17300 1.14412 0.02888 134 1.17300 1.04221 0.13079 135 1.17300 1.05072 0.12228 136 1.17300 1.16460 0.00840 137 1.17300 0.94050 0.23250 138 1.17300 1.09665 0.07635 139 1.18700 1.03135 0.15565 140 1.18700 1.04378 0.14322 141 1.18700 1.06374 0.12326 142 1.20200 0.98080 0.22120 143 1.20200 0.45405 0.74795 144 1.20200 1.03566 0.16634 145 1.20200 1.07381 0.12819 146 1.20200 1.20259 -0.00059 147 1.20200 1.00277 0.19923 148 1.20200 1.05942 0.14258 149 1.20200 0.90496 0.29704 150 1.20200 0.98214 0.21986 151 1.20200 1.00406 0.19794 152 1.21700 1.02415 0.19285 153 1.21700 0.99286 0.22414 154 1.21700 0.96815 0.24885

146

155 1.21700 1.08862 0.12838 156 1.21700 1.12404 0.09296 157 1.21700 1.19427 0.02273 158 1.21700 1.20085 0.01615 159 1.23300 1.07982 0.15318 160 1.23300 1.07698 0.15602 161 1.23300 1.00194 0.23106 162 1.23300 1.11309 0.11991 163 1.23300 0.72023 0.51277 164 1.23300 1.18988 0.04312 165 1.23300 1.19240 0.04060 166 1.23300 0.78869 0.44431 167 1.24900 1.03586 0.21314 168 1.24900 1.10173 0.14727 169 1.24900 1.08455 0.16445 170 1.24900 1.09962 0.14938 171 1.24900 1.10821 0.14079 172 1.24900 1.26004 -0.01104 173 1.24900 1.09827 0.15073 174 1.26600 1.13977 0.12623 175 1.26600 0.95607 0.30993 176 1.26600 1.11626 0.14974 177 1.26600 1.25325 0.01275 178 1.26600 1.12091 0.14509 179 1.26600 1.14196 0.12404 180 1.28400 0.95446 0.32954 181 1.28400 1.24105 0.04295 182 1.28400 1.02861 0.25539 183 1.28400 1.08921 0.19479 184 1.28400 1.29542 -0.01142 185 1.28400 1.16029 0.12371 186 1.28400 1.17374 0.11026 187 1.28400 1.19701 0.08699 188 1.30300 0.93233 0.37067 189 1.30300 1.19588 0.10712 190 1.30300 1.09768 0.20532 191 1.30300 0.99694 0.30606 192 1.30300 1.25780 0.04520 193 1.30300 1.31499 -0.01199 194 1.32300 1.22788 0.09512 195 1.32300 1.04426 0.27874 196 1.32300 1.17896 0.14404 197 1.32300 1.14405 0.17895 198 1.34500 1.33615 0.00885 199 1.34500 1.24014 0.10486 200 1.34500 1.08428 0.26072 201 1.34500 1.25982 0.08518

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202 1.34500 1.18669 0.15831 203 1.34500 1.19282 0.15218 204 1.34500 1.16494 0.18006 205 1.36900 1.17169 0.19731 206 1.36900 1.29271 0.07629 207 1.36900 1.10005 0.26895 208 1.36900 1.08548 0.28352 209 1.36900 1.18151 0.18749 210 1.39700 1.07886 0.31814 211 1.39700 1.13630 0.26070 212 1.39700 0.83701 0.55999 213 1.39700 1.18225 0.21475 214 1.39700 1.13923 0.25777 215 1.39700 1.10627 0.29073 216 1.39700 1.22472 0.17228 217 1.42900 0.90835 0.52065 218 1.42900 1.05175 0.37725 219 1.42900 0.94296 0.48604 220 0.28700 0.47578 -0.18878 221 0.28700 0.47354 -0.18654 222 0.30500 0.43727 -0.13227 223 0.32200 0.60067 -0.27867 224 0.32200 0.75357 -0.43157 225 0.36900 0.96578 -0.59678 226 0.36900 0.74746 -0.37846 227 0.41200 0.82880 -0.41680 228 0.42500 0.56232 -0.13732 229 0.42500 0.56232 -0.13732 230 0.45100 0.83465 -0.38365 231 0.47600 0.87598 -0.39998 232 0.48800 0.90775 -0.41975 233 0.53500 0.92911 -0.39411 234 0.56900 0.79246 -0.22346 235 0.58000 0.71700 -0.13700 236 0.59100 0.68642 -0.09542 237 0.60100 0.65835 -0.05735 238 0.60100 0.75955 -0.15855 239 0.64400 0.65370 -0.00970 240 0.67400 0.92509 -0.25109 241 0.80500 0.99442 -0.18942 242 0.80500 0.90304 -0.09804 243 0.84600 0.89909 -0.05309 244 0.85600 1.05655 -0.20055 245 0.94800 0.87731 0.07069 246 0.69500 0.79293 -0.09793 247 0.71500 0.92919 -0.21419 248 0.79500 0.94125 -0.14625

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249 0.75500 0.87105 -0.11605 250 0.78500 0.78809 -0.00309 251 0.82500 0.78258 0.04242 252 0.63300 0.67055 -0.03755 253 0.20100 0.69233 -0.49133 254 0.73500 0.85793 -0.12293 255 0.75500 0.98014 -0.22514 256 0.65400 0.82007 -0.16607 257 0.78500 0.69479 0.09021 258 0.86600 1.04616 -0.18016 259 0.72500 0.65141 0.07359 260 0.72500 0.50810 0.21690 261 0.87600 1.23974 -0.36374 262 0.84600 1.19160 -0.34560 263 0.69500 0.65353 0.04147 264 0.67400 0.62534 0.04866 265 0.78500 0.98172 -0.19672 266 0.78500 0.95140 -0.16640 267 0.66400 0.82731 -0.16331 268 0.79500 0.85038 -0.05538 269 0.79500 0.77618 0.01882 270 0.74500 0.97509 -0.23009 271 0.80500 0.73553 0.06947 272 0.83500 0.70542 0.12958 273 0.78500 0.75374 0.03126 274 0.47600 0.81371 -0.33771 275 0.78500 0.64627 0.13873 276 0.79500 0.72558 0.06942 277 0.73500 0.75824 -0.02324 278 0.69500 0.61119 0.08381 279 0.77500 0.96892 -0.19392 280 0.69500 0.64837 0.04663 281 0.69500 0.65245 0.04255 282 0.55800 0.98591 -0.42791 283 0.64400 0.92957 -0.28557 284 0.65400 0.91558 -0.26158 285 0.50000 0.77657 -0.27657 286 0.65400 0.84557 -0.19157 287 0.69500 0.78082 -0.08582 ========================= END OF REPORT ==============================

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APPENDIX B: OUTPUT OF SAR MODEL

SPATIAL_REGRESSION SUMMARY OF OUTPUT: SPATIAL LAG MODEL - MAXIMUM LIKELIHOOD ESTIMATION Data set : CT Spatial Weight : spatial.GWT Dependent Variable : CC_ARC Number of Observations: 287 Mean dependent var : 0.975564 Number of Variables : 5 S.D. dependent var : 0.284029 Degrees of Freedom : 282 Lag coeff. (Rho) : 0.559833 R-squared : 0.598101 Log likelihood : 81.9438 Sq. Correlation : - Akaike info criterion : -153.888 Sigma-square : 0.0324222 Schwarz criterion : -135.59 S.E of regression : 0.180062 ----------------------------------------------------------------------- Variable Coefficient Std.Error z-value Probability ----------------------------------------------------------------------- W_CC_ARC 0.5598326 0.06526942 8.577257 0.0000000 CONSTANT -5.318876 0.3953735 -13.45279 0.0000000 EL -3.324423e-005 4.214313e-005 -0.7888411 0.4302048 GR 0.01020822 0.001038968 9.825348 0.0000000 WT 0.02550158 0.001922052 13.26789 0.0000000 -----------------------------------------------------------------------

150

REGRESSION DIAGNOSTICS DIAGNOSTICS FOR HETEROSCEDASTICITY RANDOM COEFFICIENTS TEST DF VALUE PROB Breusch-Pagan test 3 40.97797 0.0000000 DIAGNOSTICS FOR SPATIAL DEPENDENCE SPATIAL LAG DEPENDENCE FOR WEIGHT MATRIX : spatial.GWT TEST DF VALUE PROB Likelihood Ratio Test 1 68.49673 0.0000000 OBS CC_ARC PREDICTED RESIDUAL PRED ERROR 1 0.907 1.14755 -0.22175 -0.24055 2 1.047 1.14295 -0.06802 -0.09595 3 1.133 1.17416 -0.02237 -0.04116 4 1.284 1.33765 -0.01629 -0.05365 5 1.159 0.97742 0.11281 0.18158 6 0.464 0.18932 0.22181 0.27468 7 0.866 0.91721 -0.03824 -0.05121 8 0.917 1.11552 -0.18621 -0.19852 9 0.601 0.73098 -0.18984 -0.12998 10 0.991 1.02331 -0.03126 -0.03231 11 1.024 1.14649 -0.08295 -0.12249 12 1.233 1.23312 0.00059 -0.00012 13 1.133 1.31353 -0.14156 -0.18053

151

14 1.284 1.00718 0.24642 0.27682 15 1.133 1.23196 -0.08995 -0.09896 16 1.024 1.30674 -0.24957 -0.28274 17 0.142 0.32891 -0.25637 -0.18691 18 0.247 0.46678 -0.18931 -0.21978 19 0.322 0.35274 0.00127 -0.03074 20 0.524 1.02459 -0.47785 -0.50059 21 0.546 0.98409 -0.41447 -0.43809 22 0.569 0.69503 -0.17018 -0.12603 23 0.58 0.69503 -0.18585 -0.11503 24 0.591 0.76730 -0.25270 -0.17630 25 0.591 1.07142 -0.39447 -0.48042 26 0.601 0.91958 -0.19850 -0.31858 27 0.623 0.92100 -0.36330 -0.29800 28 0.633 0.74856 -0.16296 -0.11556 29 0.654 0.94010 -0.26027 -0.28610 30 0.664 0.40978 0.20321 0.25422 31 0.685 0.95025 -0.23796 -0.26525 32 0.685 0.87809 -0.15271 -0.19309 33 0.695 1.08305 -0.25326 -0.38805 34 0.735 1.12479 -0.35199 -0.38979 35 0.745 0.84324 -0.15688 -0.09824 36 0.765 0.84165 -0.13575 -0.07665

152

37 0.765 0.96621 -0.05300 -0.20121 38 0.765 0.84165 -0.01997 -0.07665 39 0.805 0.79513 -0.06491 0.00987 40 0.835 0.93675 -0.13881 -0.10175 41 0.846 1.02347 -0.03111 -0.17747 42 0.856 1.06713 -0.09823 -0.21113 43 0.866 1.06053 -0.24567 -0.19453 44 0.866 0.95300 -0.07866 -0.08700 45 0.866 0.91821 -0.02696 -0.05221 46 0.866 1.08768 -0.20566 -0.22168 47 0.866 1.11148 -0.15962 -0.24548 48 0.886 0.87315 -0.05350 0.01285 49 0.886 0.89093 -0.07501 -0.00493 50 0.896 0.82584 0.00792 0.07016 51 0.896 1.22192 -0.32336 -0.32592 52 0.896 1.04685 -0.10691 -0.15085 53 0.907 1.17476 -0.25520 -0.26776 54 0.907 0.91003 -0.01827 -0.00303 55 0.917 1.07584 -0.18608 -0.15884 56 0.917 1.06137 0.02859 -0.14437 57 0.917 1.09939 -0.08770 -0.18239 58 0.927 0.99550 -0.14514 -0.06850 59 0.927 1.15938 -0.20381 -0.23238

153

60 0.927 1.10325 -0.18642 -0.17625 61 0.927 0.95336 0.01765 -0.02636 62 0.927 1.13091 -0.05359 -0.20391 63 0.938 0.93389 0.18247 0.00411 64 0.948 1.09902 -0.15378 -0.15102 65 0.948 1.10942 0.01339 -0.16142 66 0.948 0.71311 0.41666 0.23489 67 0.959 1.07392 -0.09254 -0.11492 68 0.959 1.10907 0.01647 -0.15007 69 0.97 0.82573 0.08467 0.14427 70 0.97 0.99110 -0.09131 -0.02110 71 0.98 1.06097 -0.14219 -0.08097 72 0.991 0.83049 0.12464 0.16051 73 0.991 1.06880 -0.03966 -0.07780 74 0.991 0.92445 -0.01386 0.06655 75 0.991 1.16657 -0.11549 -0.17557 76 1.002 0.99547 -0.07329 0.00653 77 1.002 0.84293 0.09282 0.15907 78 1.002 0.80854 0.17723 0.19346 79 1.013 0.97210 -0.03414 0.04090 80 1.013 1.11458 -0.07999 -0.10158 81 1.013 1.21277 -0.16802 -0.19977 82 1.013 1.05820 -0.06824 -0.04520

154

83 1.013 0.98592 -0.00644 0.02708 84 1.013 1.20475 -0.20054 -0.19175 85 1.024 0.95083 0.04458 0.07317 86 1.024 0.94175 0.00951 0.08225 87 1.036 1.27491 -0.20219 -0.23891 88 1.036 1.02393 -0.01874 0.01207 89 1.036 1.06045 -0.04709 -0.02445 90 1.036 1.20701 -0.19059 -0.17101 91 1.036 1.21091 -0.17083 -0.17491 92 1.036 1.06601 -0.07641 -0.03001 93 1.047 0.76658 0.22909 0.28042 94 1.047 1.01649 -0.04108 0.03051 95 1.047 0.79814 0.18436 0.24886 96 1.059 1.07843 -0.01648 -0.01943 97 1.059 0.97490 0.00832 0.08410 98 1.059 1.18423 -0.13393 -0.12523 99 1.059 1.09948 -0.07065 -0.04048 100 1.071 0.81521 0.19358 0.25579 101 1.071 0.83218 0.18530 0.23882 102 1.083 0.91779 0.08464 0.16521 103 1.083 0.93082 0.09498 0.15218 104 1.083 1.03005 -0.01945 0.05295 105 1.083 1.09043 0.10474 -0.00743

155

106 1.095 0.93735 0.08255 0.15765 107 1.095 1.11987 0.04267 -0.02487 108 1.095 0.91328 0.28710 0.18172 109 1.107 1.14946 -0.02912 -0.04246 110 1.107 1.02464 0.02315 0.08236 111 1.107 1.11130 -0.07383 -0.00430 112 1.107 1.11277 0.03410 -0.00577 113 1.107 0.92822 0.10645 0.17878 114 1.107 1.12039 0.05993 -0.01339 115 1.12 1.17840 -0.06897 -0.05840 116 1.12 1.07075 -0.02472 0.04925 117 1.12 1.10407 -0.01023 0.01593 118 1.12 1.12977 0.05469 -0.00977 119 1.133 1.13385 0.03884 -0.00085 120 1.133 1.04110 0.07153 0.09190 121 1.133 0.98940 0.07643 0.14360 122 1.133 1.02897 0.06665 0.10403 123 1.133 1.11363 -0.05703 0.01937 124 1.133 1.12883 0.07375 0.00417 125 1.133 0.84829 0.34216 0.28471 126 1.133 1.31783 -0.13024 -0.18483 127 1.146 1.14610 0.03911 -0.00010 128 1.146 1.12478 0.09199 0.02122

156

129 1.159 0.85548 0.23977 0.30352 130 1.159 1.16192 -0.03797 -0.00292 131 1.159 1.26817 -0.07329 -0.10917 132 1.173 1.28170 -0.07605 -0.10870 133 1.173 1.17616 -0.03499 -0.00316 134 1.173 1.06773 0.09180 0.10527 135 1.173 1.05150 0.04362 0.12150 136 1.173 1.12061 0.07149 0.05239 137 1.173 0.87026 0.25957 0.30274 138 1.173 1.12172 0.12521 0.05128 139 1.187 1.17207 0.05259 0.01493 140 1.187 1.17272 0.05249 0.01428 141 1.187 1.08981 0.03740 0.09719 142 1.202 0.96508 0.17203 0.23692 143 1.202 0.53048 0.63190 0.67152 144 1.202 1.05324 0.11387 0.14876 145 1.202 1.10715 0.07086 0.09485 146 1.202 1.28406 -0.04518 -0.08206 147 1.202 1.09524 0.11290 0.10676 148 1.202 1.04933 0.07395 0.15267 149 1.202 0.97891 0.21148 0.22309 150 1.202 0.99343 0.13815 0.20857 151 1.202 0.98745 0.21510 0.21455

157

152 1.217 1.01912 0.16635 0.19788 153 1.217 1.02084 0.12268 0.19616 154 1.217 1.00489 0.16192 0.21211 155 1.217 1.11302 0.05461 0.10398 156 1.217 1.14153 0.01493 0.07547 157 1.217 1.13516 0.01431 0.08184 158 1.217 1.28168 -0.02624 -0.06468 159 1.233 1.05374 0.10873 0.17926 160 1.233 1.13968 0.11458 0.09332 161 1.233 0.98372 0.20315 0.24928 162 1.233 1.15709 0.07419 0.07591 163 1.233 0.77711 0.43473 0.45589 164 1.233 1.27110 -0.00132 -0.03810 165 1.233 1.14837 0.08419 0.08463 166 1.233 0.78445 0.39038 0.44855 167 1.249 1.07670 0.13613 0.17230 168 1.249 1.09863 0.11643 0.15037 169 1.249 1.08267 0.13760 0.16633 170 1.249 1.14516 0.08210 0.10384 171 1.249 1.13758 0.07877 0.11142 172 1.249 1.20811 0.02420 0.04089 173 1.249 1.05399 0.19257 0.19501 174 1.266 1.23640 0.06563 0.02960

158

175 1.266 0.94935 0.25311 0.31665 176 1.266 1.13644 0.09791 0.12956 177 1.266 1.19306 0.06187 0.07294 178 1.266 1.21100 0.09279 0.05500 179 1.266 1.25733 0.04593 0.00867 180 1.284 0.95256 0.26269 0.33144 181 1.284 1.32148 -0.00255 -0.03748 182 1.284 1.04594 0.20684 0.23806 183 1.284 1.13195 0.12564 0.15205 184 1.284 1.32744 -0.00897 -0.04344 185 1.284 1.21834 0.10837 0.06566 186 1.284 1.15836 0.09079 0.12564 187 1.284 1.27519 0.04607 0.00881 188 1.303 0.93369 0.29363 0.36931 189 1.303 1.28655 0.05379 0.01645 190 1.303 1.07398 0.22010 0.22902 191 1.303 0.99777 0.23609 0.30523 192 1.303 1.24115 0.03255 0.06185 193 1.303 1.32681 -0.01466 -0.02381 194 1.323 1.30500 0.04800 0.01800 195 1.323 1.06144 0.20919 0.26156 196 1.323 1.14236 0.14788 0.18064 197 1.323 1.22396 0.12077 0.09904

159

198 1.345 1.32101 0.00320 0.02399 199 1.345 1.17951 0.14806 0.16549 200 1.345 1.07906 0.20678 0.26594 201 1.345 1.27890 0.11287 0.06610 202 1.345 1.15716 0.12138 0.18784 203 1.345 1.20690 0.13731 0.13810 204 1.345 1.13933 0.17750 0.20567 205 1.369 1.25534 0.14630 0.11366 206 1.369 1.36478 0.03402 0.00422 207 1.369 1.12043 0.22859 0.24857 208 1.369 1.09850 0.25023 0.27050 209 1.369 1.27162 0.13775 0.09738 210 1.397 1.18408 0.25256 0.21292 211 1.397 1.24978 0.18748 0.14722 212 1.397 0.86351 0.52783 0.53349 213 1.397 1.17225 0.18374 0.22475 214 1.397 1.15648 0.21436 0.24052 215 1.397 1.12946 0.29794 0.26754 216 1.397 1.30985 0.12532 0.08715 217 1.429 0.93565 0.42612 0.49335 218 1.429 1.05737 0.34357 0.37163 219 1.429 0.96482 0.44912 0.46418 220 0.287 0.50174 -0.06094 -0.21474

160

221 0.287 0.55842 -0.22818 -0.27142 222 0.305 0.57606 -0.26878 -0.27106 223 0.322 0.73123 -0.41507 -0.40923 224 0.322 0.73791 -0.27548 -0.41591 225 0.369 0.92269 -0.42439 -0.55369 226 0.369 0.75432 -0.25043 -0.38532 227 0.412 0.84510 -0.26614 -0.43310 228 0.425 0.57164 -0.19149 -0.14664 229 0.425 0.57164 -0.19149 -0.14664 230 0.451 0.86404 -0.42746 -0.41304 231 0.476 0.85781 -0.24678 -0.38181 232 0.488 0.91277 -0.43800 -0.42477 233 0.535 1.00256 -0.46531 -0.46756 234 0.569 0.82171 -0.26593 -0.25271 235 0.58 0.74068 -0.16376 -0.16068 236 0.591 0.72442 -0.13747 -0.13342 237 0.601 0.69061 -0.09137 -0.08961 238 0.601 0.77580 -0.17815 -0.17480 239 0.644 0.76555 -0.11572 -0.12155 240 0.674 0.96980 -0.28691 -0.29580 241 0.805 0.99849 -0.21880 -0.19349 242 0.805 0.75613 0.08645 0.04887 243 0.846 1.01211 -0.15032 -0.16611

161

244 0.856 1.09119 -0.26629 -0.23519 245 0.948 0.87397 0.08802 0.07403 246 0.695 0.75357 0.00551 -0.05857 247 0.715 0.86184 -0.08584 -0.14684 248 0.795 0.87138 -0.01292 -0.07638 249 0.755 0.79572 0.02242 -0.04072 250 0.785 0.72687 0.11427 0.05813 251 0.825 0.72504 0.15748 0.09996 252 0.633 0.64567 0.05302 -0.01267 253 0.201 0.65019 -0.39972 -0.44919 254 0.735 0.76321 -0.01364 -0.02821 255 0.755 0.89077 -0.09349 -0.13577 256 0.654 0.67828 -0.05326 -0.02428 257 0.785 0.56862 0.18181 0.21638 258 0.866 0.87120 -0.04379 -0.00520 259 0.725 0.53279 0.15577 0.19221 260 0.725 0.42005 0.27133 0.30495 261 0.876 1.02220 -0.17116 -0.14620 262 0.846 0.97033 -0.14606 -0.12433 263 0.695 0.53351 0.12440 0.16149 264 0.674 0.51281 0.12409 0.16119 265 0.785 0.80396 -0.06485 -0.01896 266 0.785 0.79542 -0.04643 -0.01042

162

267 0.664 0.71320 -0.09420 -0.04920 268 0.795 0.72627 0.02480 0.06873 269 0.795 0.66546 0.08688 0.12954 270 0.745 0.85060 -0.14731 -0.10560 271 0.805 0.61191 0.15075 0.19309 272 0.835 0.59977 0.19476 0.23523 273 0.785 0.63135 0.11035 0.15365 274 0.476 0.68013 -0.24910 -0.20413 275 0.785 0.53426 0.21595 0.25074 276 0.795 0.59743 0.16125 0.19757 277 0.735 0.63128 0.06538 0.10372 278 0.695 0.49960 0.15942 0.19540 279 0.775 0.79966 -0.06576 -0.02466 280 0.695 0.53953 0.11825 0.15547 281 0.695 0.53702 0.12081 0.15798 282 0.558 0.83806 -0.33714 -0.28006 283 0.644 0.85403 -0.15367 -0.21003 284 0.654 0.84389 -0.13272 -0.18989 285 0.5 0.73235 -0.17528 -0.23235 286 0.654 0.79962 -0.08451 -0.14562 287 0.695 0.74640 0.01784 -0.05140 ========================= END OF REPORT ==============================

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APPENDIX C: OUTPUT OF GWR MODEL

************************************************* * Geographically Weighted Regression * * Release 3.0.1 * * Dated: 06-vii-2003 * * * * Martin Charlton, Chris Brunsdon * * Stewart Fotheringham * * (c) University of Newcastle upon Tyne * ************************************************* Program starts at: Tue Apr 25 11:15:31 2006 ** Program limits: ** Maximum number of variables..... 52 ** Maximum number of observations.. 80000 ** Maximum number of fit locations. 80000 GWR_CC ** Observed data file: F:\1_Project683\DataAnalysis\CT_excel.cs ** Prediction location file: Estimation at sample point locations ** Result output file: F:\1_Project683\DataAnalysis\GWR\GWR.e00 ** Variables in the data file... PL_ID X Y SC CC cc_per cc_arc El Gr Wt In Sl Br ** Dependent (y) variable..........cc_arc ** Easting (x-coord) variable.....X ** Northing (y-coord) variable.....Y ** No weight variable specified ** Independent variables in your model... El Gr Wt ** Kernel type: Adaptive ** Kernel shape: Bi-Square ** Bandwidth selection by AICc minimisation ** Use all regression points ** Calibration history requested ** Prediction report requested

164

** Output estimates to be written to .e00 file ** Monte Carlo significance tests for spatial variation ** Casewise diagnostics to be printed *** Analysis method *** *** Geographically weighted multiple regression ** Cartesian coordinates: Euclidean Distance *********************************************************** * * * GEOGRAPHICALLY WEIGHTED GAUSSIAN REGRESSION * * * *********************************************************** Number of data cases read: 287 Observation points read... Dependent mean= 0.975564778 Number of observations, nobs= 287 Number of predictors, nvar= 3 Observation Easting extent: 180340. Observation Northing extent: 194858. *Finding bandwidth... ... using all regression points This can take some time... *Calibration will be based on 287 cases *Adaptive kernel sample size limits: 14 287 *AICc minimisation begins... Bandwidth AICc 98.361639635000 -173.143087443589 150.500000000000 -157.143164948996 66.138360746561 -182.035034901394 46.223279124257 -170.427289899333 78.446558012695 -179.924932625568 58.531476480465 -180.818938470719 70.839673746599 -181.647798473498 63.232789514909 -181.383092880939 67.934102514947 -181.488335067533 65.028531296436 -181.866531636907 ** Convergence after 10 function calls ** Convergence: Local Sample Size= 66 ********************************************************** * GLOBAL REGRESSION PARAMETERS * **********************************************************

165

Diagnostic information... Residual sum of squares......... 12.052025 Effective number of parameters.. 4.000000 Sigma........................... 0.206365 Akaike Information Criterion.... -85.177375 Coefficient of Determination.... 0.479461 Adjusted r-square............... 0.472078 Parameter Estimate Std Err T --------- ------------ ------------ ------------ Intercept -5.730766329354 0.445939870470 -12.850984573364 El -0.000096594914 0.000046482851 -2.078076362610 Gr 0.013030707171 0.001104239252 11.800619125366 Wt 0.029325440663 0.002108421333 13.908719062805 ********************************************************** * GWR ESTIMATION * ********************************************************** Fitting Geographically Weighted Regression Model... Number of observations............ 287 Number of independent variables... 4 (Intercept is variable 1) Number of nearest neighbours...... 66 Number of locations to fit model.. 287 Diagnostic information... Residual sum of squares......... 6.332454 Effective number of parameters.. 38.012928 Sigma........................... 0.159477 Akaike Information Criterion.... -189.421918 Coefficient of Determination.... 0.726495 Adjusted r-square............... 0.684571 ********************************************************** * CASEWISE DIAGNOSTICS * ********************************************************** Obs Observed Predicted Residual Std Resid R-Square Influence Cook's D

166

----- -------------- -------------- -------------- ----------- ----------- ----------- ----------- 1 0.90700 1.06537 -0.15837 -0.425350 0.742266 0.067485 0.000344 2 1.04700 1.06083 -0.01383 -0.037224 0.752278 0.072072 0.000003 3 1.13300 1.13387 -0.00087 -0.002410 0.718810 0.127519 0.000000 4 1.28400 1.31379 -0.02979 -0.080815 0.816169 0.086068 0.000016 5 1.15900 1.06803 0.09097 0.246370 0.616244 0.082839 0.000144 6 0.46400 0.56409 -0.10009 -0.369852 0.699807 0.507324 0.003706 7 0.86600 0.87515 -0.00915 -0.025874 0.747920 0.158070 0.000003 8 0.91700 1.03946 -0.12246 -0.347240 0.815724 0.163429 0.000620 9 0.60100 0.74146 -0.14046 -0.537602 0.678655 0.540832 0.008955 10 0.99100 1.05780 -0.06680 -0.178091 0.719550 0.053622 0.000047 11 1.02400 0.97711 0.04689 0.139809 0.795930 0.243451 0.000165 12 1.23300 1.17753 0.05547 0.155497 0.669246 0.143931 0.000107 13 1.13300 1.27913 -0.14613 -0.407088 0.799144 0.133215 0.000670 14 1.28400 1.06236 0.22164 0.591738 0.696411 0.056251 0.000549 15 1.13300 1.21181 -0.07881 -0.219679 0.711278 0.134292 0.000197 16 1.02400 1.18673 -0.16273 -0.534378 0.802172 0.376234 0.004531 17 0.14200 0.28507 -0.14307 -0.458184 0.916665 0.344158 0.002898 18 0.24700 0.22773 0.01927 0.062986 0.903023 0.370091 0.000061 19 0.32200 0.19714 0.12486 0.408617 0.917809 0.371931 0.002601 20 0.52400 0.72974 -0.20574 -0.625571 0.840360 0.272378 0.003854 21 0.54600 0.66939 -0.12339 -0.385736 0.838489 0.311659 0.001772 22 0.56900 0.88360 -0.31460 -0.967493 0.734466 0.288743 0.009997

167

23 0.58000 0.79551 -0.21551 -0.584635 0.699991 0.085921 0.000845 24 0.59100 0.77310 -0.18210 -0.527244 0.697697 0.197577 0.001801 25 0.59100 0.81502 -0.22402 -0.672399 0.889956 0.253307 0.004035 26 0.60100 0.77169 -0.17069 -0.556150 0.846871 0.366352 0.004704 27 0.62300 0.93728 -0.31428 -0.858347 0.702641 0.098197 0.002110 28 0.63300 0.89179 -0.25879 -0.703915 0.704772 0.090783 0.001302 29 0.65400 0.91528 -0.26128 -0.747522 0.900234 0.178195 0.003187 30 0.66400 0.56202 0.10198 0.351967 0.703013 0.435289 0.002512 31 0.68500 0.91448 -0.22948 -0.702887 0.899760 0.282969 0.005129 32 0.68500 0.85882 -0.17382 -0.545945 0.770394 0.318136 0.003658 33 0.69500 0.98182 -0.28682 -0.770575 0.858196 0.068064 0.001141 34 0.73500 0.94237 -0.20737 -0.581541 0.887186 0.144651 0.001505 35 0.74500 0.94510 -0.20010 -0.544051 0.717750 0.090048 0.000771 36 0.76500 0.89160 -0.12660 -0.340896 0.721818 0.072211 0.000238 37 0.76500 0.91544 -0.15044 -0.420317 0.847596 0.138200 0.000745 38 0.76500 0.74672 0.01828 0.053524 0.772502 0.215205 0.000021 39 0.80500 0.93151 -0.12651 -0.345941 0.743966 0.100331 0.000351 40 0.83500 0.93333 -0.09833 -0.285738 0.727936 0.203426 0.000549 41 0.84600 0.89216 -0.04616 -0.123471 0.852661 0.059931 0.000026 42 0.85600 0.91827 -0.06227 -0.170625 0.891237 0.103944 0.000089 43 0.86600 1.07206 -0.20606 -0.561754 0.637748 0.094901 0.000870 44 0.86600 0.93103 -0.06503 -0.176486 0.746283 0.086811 0.000078 45 0.86600 0.93676 -0.07076 -0.189052 0.752956 0.057692 0.000058

168

46 0.86600 0.98464 -0.11864 -0.337835 0.799682 0.170383 0.000617 47 0.86600 0.81007 0.05593 0.172383 0.885400 0.291996 0.000322 48 0.88600 0.95119 -0.06519 -0.175483 0.720213 0.071670 0.000063 49 0.88600 0.88904 -0.00304 -0.008637 0.697170 0.167485 0.000000 50 0.89600 0.90170 -0.00570 -0.015265 0.696684 0.063288 0.000000 51 0.89600 1.29899 -0.40299 -1.106710 0.771892 0.108085 0.003905 52 0.89600 1.04202 -0.14602 -0.406728 0.852025 0.133006 0.000668 53 0.90700 1.10976 -0.20276 -0.541652 0.748923 0.057445 0.000470 54 0.90700 0.88332 0.02368 0.064140 0.732767 0.083106 0.000010 55 0.91700 1.10574 -0.18874 -0.506204 0.787276 0.064800 0.000467 56 0.91700 0.80066 0.11634 0.345258 0.858507 0.236171 0.000970 57 0.91700 0.94069 -0.02369 -0.064573 0.872953 0.094525 0.000011 58 0.92700 1.12082 -0.19382 -0.533227 0.602739 0.111249 0.000936 59 0.92700 1.12256 -0.19556 -0.547992 0.774622 0.143317 0.001322 60 0.92700 1.16689 -0.23989 -0.644896 0.699888 0.069203 0.000813 61 0.92700 0.99808 -0.07108 -0.202724 0.912829 0.172991 0.000226 62 0.92700 1.00425 -0.07725 -0.221309 0.845276 0.180373 0.000284 63 0.93800 0.78297 0.15503 0.422924 0.865118 0.096119 0.000500 64 0.94800 1.00752 -0.05952 -0.163172 0.740123 0.105119 0.000082 65 0.94800 0.94642 0.00158 0.004245 0.851868 0.070505 0.000000 66 0.94800 0.52206 0.42594 1.311143 0.850141 0.290109 0.018482 67 0.95900 1.02661 -0.06761 -0.219368 0.796195 0.361040 0.000715 68 0.95900 0.93941 0.01959 0.052789 0.867194 0.073906 0.000006

169

69 0.97000 0.95149 0.01851 0.049710 0.714977 0.066827 0.000005 70 0.97000 1.07453 -0.10453 -0.284887 0.615207 0.094406 0.000223 71 0.98000 1.07671 -0.09671 -0.279221 0.698618 0.193093 0.000491 72 0.99100 0.84485 0.14615 0.413035 0.677512 0.157750 0.000841 73 0.99100 0.98004 0.01096 0.030212 0.783187 0.115486 0.000003 74 0.99100 1.13877 -0.14777 -0.427587 0.569133 0.196556 0.001177 75 0.99100 1.19259 -0.20159 -0.639694 0.788642 0.331987 0.005350 76 1.00200 1.11043 -0.10843 -0.289664 0.756850 0.057465 0.000135 77 1.00200 0.94527 0.05673 0.151246 0.693538 0.053505 0.000034 78 1.00200 1.05390 -0.05190 -0.153014 0.915965 0.226214 0.000180 79 1.01300 1.05468 -0.04168 -0.117830 0.670291 0.158198 0.000069 80 1.01300 1.04927 -0.03627 -0.103162 0.869611 0.168381 0.000057 81 1.01300 1.13543 -0.12243 -0.330585 0.796319 0.077366 0.000241 82 1.01300 1.06169 -0.04869 -0.131459 0.733634 0.077026 0.000038 83 1.01300 1.06191 -0.04891 -0.130052 0.719861 0.048603 0.000023 84 1.01300 1.17453 -0.16153 -0.444870 0.847007 0.113116 0.000664 85 1.02400 0.92626 0.09774 0.265291 0.686970 0.086933 0.000176 86 1.02400 1.04622 -0.02222 -0.058913 0.743804 0.042677 0.000004 87 1.03600 1.21455 -0.17855 -0.483161 0.824290 0.081381 0.000544 88 1.03600 1.01578 0.02022 0.055383 0.768917 0.103502 0.000009 89 1.03600 1.02447 0.01153 0.032876 0.693770 0.173227 0.000006 90 1.03600 1.27482 -0.23882 -0.646029 0.724474 0.080718 0.000964 91 1.03600 1.16292 -0.12692 -0.346302 0.690051 0.096420 0.000337

170

92 1.03600 1.00115 0.03485 0.107884 0.734156 0.298142 0.000130 93 1.04700 0.94004 0.10696 0.293008 0.727266 0.103592 0.000261 94 1.04700 1.20466 -0.15766 -0.440144 0.610761 0.136870 0.000808 95 1.04700 0.87864 0.16836 0.471473 0.702046 0.142250 0.000970 96 1.05900 1.08687 -0.02787 -0.076848 0.738378 0.115006 0.000020 97 1.05900 1.08009 -0.02109 -0.056000 0.741438 0.046269 0.000004 98 1.05900 1.21319 -0.15419 -0.433106 0.729682 0.147449 0.000853 99 1.05900 1.14550 -0.08650 -0.276487 0.705849 0.341575 0.001043 100 1.07100 0.93912 0.13188 0.373046 0.695700 0.159335 0.000694 101 1.07100 0.90676 0.16424 0.442590 0.699482 0.073735 0.000410 102 1.08300 1.09668 -0.01368 -0.037359 0.622584 0.098044 0.000004 103 1.08300 1.00528 0.07772 0.213099 0.718587 0.105217 0.000140 104 1.08300 1.15756 -0.07456 -0.202916 0.734364 0.091814 0.000110 105 1.08300 1.06936 0.01364 0.037473 0.902761 0.108708 0.000005 106 1.09500 1.06292 0.03208 0.084838 0.740694 0.038161 0.000008 107 1.09500 1.02940 0.06560 0.180154 0.884850 0.107963 0.000103 108 1.09500 1.23993 -0.14493 -0.410825 0.932140 0.162861 0.000864 109 1.10700 1.13339 -0.02639 -0.073253 0.761400 0.127099 0.000021 110 1.10700 1.12478 -0.01778 -0.048049 0.709270 0.079183 0.000005 111 1.10700 1.23495 -0.12795 -0.360694 0.667907 0.153588 0.000621 112 1.10700 1.00756 0.09944 0.279707 0.826702 0.149741 0.000362 113 1.10700 1.04168 0.06532 0.173842 0.654973 0.050387 0.000042 114 1.10700 1.05908 0.04792 0.132211 0.905958 0.116297 0.000061

171

115 1.12000 1.26112 -0.14112 -0.386164 0.779490 0.101617 0.000444 116 1.12000 1.16429 -0.04429 -0.119375 0.625077 0.074146 0.000030 117 1.12000 1.16379 -0.04379 -0.121942 0.742970 0.132679 0.000060 118 1.12000 1.12004 -0.00004 -0.000113 0.832196 0.086575 0.000000 119 1.13300 1.01121 0.12179 0.331697 0.825286 0.093067 0.000297 120 1.13300 1.04317 0.08983 0.254809 0.709607 0.163922 0.000335 121 1.13300 1.03445 0.09855 0.271757 0.740810 0.115419 0.000253 122 1.13300 1.03867 0.09433 0.256965 0.669679 0.093447 0.000179 123 1.13300 1.24733 -0.11433 -0.314686 0.607117 0.112151 0.000329 124 1.13300 0.99844 0.13456 0.367290 0.886576 0.097132 0.000382 125 1.13300 1.00390 0.12910 0.375976 0.916335 0.206835 0.000970 126 1.13300 1.18465 -0.05165 -0.152140 0.871375 0.224592 0.000176 127 1.14600 1.04427 0.10173 0.272915 0.821223 0.065396 0.000137 128 1.14600 1.08513 0.06087 0.173260 0.886993 0.169803 0.000162 129 1.15900 1.03106 0.12794 0.350868 0.696746 0.105618 0.000382 130 1.15900 1.25494 -0.09594 -0.259582 0.803182 0.081046 0.000156 131 1.15900 1.29506 -0.13606 -0.368510 0.826708 0.082948 0.000323 132 1.17300 1.27482 -0.10182 -0.276391 0.817650 0.087065 0.000192 133 1.17300 1.25426 -0.08126 -0.220972 0.794681 0.090313 0.000128 134 1.17300 1.07776 0.09524 0.253732 0.710008 0.052262 0.000093 135 1.17300 1.16818 0.00482 0.012978 0.604437 0.073993 0.000000 136 1.17300 1.18186 -0.00886 -0.025438 0.743020 0.184558 0.000004 137 1.17300 1.04280 0.13020 0.463535 0.707476 0.469254 0.004998

172

138 1.17300 1.08789 0.08511 0.236491 0.907535 0.128765 0.000217 139 1.18700 1.14315 0.04385 0.121332 0.808601 0.121403 0.000054 140 1.18700 1.08793 0.09907 0.284848 0.782420 0.186349 0.000489 141 1.18700 1.18404 0.00296 0.007919 0.656312 0.061546 0.000000 142 1.20200 1.09559 0.10641 0.283011 0.785398 0.049101 0.000109 143 1.20200 0.80788 0.39412 1.159185 0.705669 0.222390 0.010109 144 1.20200 1.18098 0.02102 0.057150 0.763658 0.090363 0.000009 145 1.20200 1.15667 0.04533 0.120962 0.691641 0.055361 0.000023 146 1.20200 1.25124 -0.04924 -0.131562 0.830490 0.057682 0.000028 147 1.20200 1.09278 0.10922 0.294621 0.716137 0.075539 0.000187 148 1.20200 1.21610 -0.01410 -0.038826 0.597211 0.113414 0.000005 149 1.20200 0.90695 0.29505 0.803435 0.794853 0.092801 0.001737 150 1.20200 1.09916 0.10284 0.277661 0.685244 0.077179 0.000170 151 1.20200 0.98666 0.21534 0.574824 0.734141 0.055971 0.000515 152 1.21700 1.07659 0.14041 0.382452 0.699616 0.093353 0.000396 153 1.21700 1.13680 0.08020 0.218903 0.607670 0.097029 0.000135 154 1.21700 1.06283 0.15417 0.423449 0.678610 0.108323 0.000573 155 1.21700 1.17002 0.04698 0.127497 0.672040 0.086731 0.000041 156 1.21700 1.24386 -0.02686 -0.073170 0.650629 0.093544 0.000015 157 1.21700 1.36499 -0.14799 -0.434234 0.780001 0.218650 0.001388 158 1.21700 1.24972 -0.03272 -0.087117 0.822063 0.051015 0.000011 159 1.23300 1.09369 0.13931 0.393807 0.723918 0.158256 0.000767 160 1.23300 1.06265 0.17035 0.503242 0.835904 0.229194 0.001981

173

161 1.23300 1.10047 0.13253 0.365447 0.820207 0.115261 0.000458 162 1.23300 1.12424 0.10876 0.293850 0.794428 0.078541 0.000194 163 1.23300 0.90303 0.32997 1.030457 0.755998 0.310225 0.012563 164 1.23300 1.21114 0.02186 0.057926 0.816486 0.042409 0.000004 165 1.23300 1.22612 0.00688 0.019086 0.717408 0.127028 0.000001 166 1.23300 0.90161 0.33139 0.887738 0.705800 0.062598 0.001384 167 1.24900 1.10453 0.14447 0.391785 0.754614 0.085370 0.000377 168 1.24900 1.29280 -0.04380 -0.123975 0.809078 0.160303 0.000077 169 1.24900 1.27782 -0.02882 -0.081458 0.807012 0.158190 0.000033 170 1.24900 1.21936 0.02964 0.080936 0.825628 0.097543 0.000019 171 1.24900 1.19835 0.05065 0.136896 0.728583 0.079157 0.000042 172 1.24900 1.31450 -0.06550 -0.185880 0.718272 0.164759 0.000179 173 1.24900 1.06516 0.18384 0.515394 0.742437 0.144140 0.001177 174 1.26600 1.17184 0.09416 0.247720 0.817495 0.028115 0.000047 175 1.26600 1.05682 0.20918 0.558243 0.782810 0.055474 0.000481 176 1.26600 1.21340 0.05260 0.140724 0.683041 0.060032 0.000033 177 1.26600 1.29180 -0.02580 -0.071163 0.727876 0.116152 0.000018 178 1.26600 1.10139 0.16461 0.438266 0.821066 0.051016 0.000272 179 1.26600 1.27643 -0.01043 -0.028335 0.821992 0.088518 0.000002 180 1.28400 1.15326 0.13074 0.374463 0.608617 0.180005 0.000810 181 1.28400 1.33079 -0.04679 -0.127490 0.818037 0.094127 0.000044 182 1.28400 1.04186 0.24214 0.675894 0.674100 0.136669 0.001902 183 1.28400 1.18185 0.10215 0.274987 0.798520 0.071766 0.000154

174

184 1.28400 1.28141 0.00259 0.007499 0.786926 0.198019 0.000000 185 1.28400 1.17446 0.10954 0.294372 0.809849 0.068535 0.000168 186 1.28400 1.29174 -0.00774 -0.021018 0.735794 0.087173 0.000001 187 1.28400 1.23150 0.05250 0.139642 0.820428 0.049362 0.000027 188 1.30300 1.12334 0.17966 0.504491 0.597262 0.146900 0.001153 189 1.30300 1.28292 0.02008 0.053258 0.821894 0.043682 0.000003 190 1.30300 1.21304 0.08996 0.241245 0.694037 0.064615 0.000106 191 1.30300 1.17132 0.13168 0.357704 0.608085 0.088446 0.000327 192 1.30300 1.33345 -0.03045 -0.082657 0.743446 0.087285 0.000017 193 1.30300 1.35105 -0.04805 -0.133768 0.798479 0.132040 0.000072 194 1.32300 1.28515 0.03785 0.101885 0.839818 0.071404 0.000021 195 1.32300 1.14732 0.17568 0.480678 0.677214 0.101461 0.000686 196 1.32300 1.25238 0.07062 0.195491 0.675752 0.122254 0.000140 197 1.32300 1.17201 0.15099 0.412604 0.738293 0.099232 0.000493 198 1.34500 1.42424 -0.07924 -0.227903 0.822330 0.186885 0.000314 199 1.34500 1.24662 0.09838 0.279751 0.729264 0.168079 0.000416 200 1.34500 1.26645 0.07855 0.213490 0.650501 0.089412 0.000118 201 1.34500 1.24047 0.10453 0.292942 0.814971 0.143459 0.000378 202 1.34500 1.35997 -0.01497 -0.042276 0.674137 0.156786 0.000009 203 1.34500 1.21381 0.13119 0.353594 0.768624 0.074061 0.000263 204 1.34500 1.29771 0.04729 0.135567 0.725241 0.181541 0.000107 205 1.36900 1.16489 0.20411 0.565815 0.841342 0.124607 0.001199 206 1.36900 1.39717 -0.02817 -0.077662 0.840887 0.115211 0.000021

175

207 1.36900 1.11178 0.25722 0.702138 0.743894 0.097273 0.001397 208 1.36900 1.17013 0.19887 0.530679 0.761359 0.055300 0.000434 209 1.36900 1.25668 0.11232 0.299534 0.823435 0.054207 0.000135 210 1.39700 1.10124 0.29576 0.782776 0.820044 0.039711 0.000667 211 1.39700 1.22922 0.16778 0.451760 0.808829 0.072190 0.000418 212 1.39700 0.84934 0.54766 1.472592 0.720629 0.069631 0.004270 213 1.39700 1.26923 0.12777 0.359181 0.651229 0.148774 0.000593 214 1.39700 1.18367 0.21333 0.588035 0.801630 0.114688 0.001178 215 1.39700 1.12474 0.27226 0.739041 0.797857 0.087077 0.001370 216 1.39700 1.33522 0.06178 0.166694 0.823924 0.076116 0.000060 217 1.42900 1.12974 0.29926 0.833579 0.664333 0.133033 0.002805 218 1.42900 1.25118 0.17782 0.501493 0.804292 0.154307 0.001207 219 1.42900 1.00613 0.42287 1.123735 0.726982 0.047444 0.001655 220 0.28700 0.26124 0.02576 0.079535 0.850542 0.294399 0.000069 221 0.28700 0.30498 -0.01798 -0.059683 0.920128 0.389359 0.000060 222 0.30500 0.47227 -0.16727 -0.557098 0.768737 0.393587 0.005299 223 0.32200 0.57480 -0.25280 -0.814179 0.830844 0.351465 0.009451 224 0.32200 0.55854 -0.23654 -0.648008 0.850814 0.103735 0.001279 225 0.36900 0.71409 -0.34509 -0.937150 0.857049 0.087889 0.002226 226 0.36900 0.52757 -0.15857 -0.430325 0.855016 0.086589 0.000462 227 0.41200 0.56893 -0.15693 -0.426987 0.845337 0.091387 0.000482 228 0.42500 0.49765 -0.07265 -0.223106 0.930259 0.286780 0.000527 229 0.42500 0.49765 -0.07265 -0.223106 0.930259 0.286780 0.000527

176

230 0.45100 0.84360 -0.39260 -1.134807 0.687280 0.194858 0.008199 231 0.47600 0.67899 -0.20299 -0.539792 0.858309 0.048772 0.000393 232 0.48800 0.89868 -0.41068 -1.089281 0.731721 0.043822 0.001431 233 0.53500 0.93885 -0.40385 -1.081278 0.779224 0.061620 0.002020 234 0.56900 0.83217 -0.26317 -0.721020 0.712755 0.103848 0.001585 235 0.58000 0.65708 -0.07708 -0.215084 0.742512 0.136132 0.000192 236 0.59100 0.64544 -0.05444 -0.152404 0.739263 0.141835 0.000101 237 0.60100 0.59679 0.00421 0.012053 0.742655 0.178600 0.000001 238 0.60100 0.70109 -0.10009 -0.276204 0.742769 0.116696 0.000265 239 0.64400 0.68400 -0.04000 -0.114803 0.790470 0.183212 0.000078 240 0.67400 0.89615 -0.22215 -0.598263 0.744418 0.072468 0.000736 241 0.80500 0.95162 -0.14662 -0.398164 0.726858 0.087791 0.000401 242 0.80500 0.88254 -0.07754 -0.220868 0.914936 0.170936 0.000265 243 0.84600 0.87429 -0.02829 -0.078469 0.833583 0.125492 0.000023 244 0.85600 1.14156 -0.28556 -0.757828 0.690604 0.044870 0.000710 245 0.94800 0.83509 0.11291 0.303940 0.747000 0.071703 0.000188 246 0.69500 0.74091 -0.04591 -0.124877 0.640982 0.090633 0.000041 247 0.71500 0.78157 -0.06657 -0.179802 0.640689 0.077774 0.000072 248 0.79500 0.78254 0.01246 0.033626 0.640904 0.076684 0.000002 249 0.75500 0.72122 0.03378 0.089741 0.645545 0.046755 0.000010 250 0.78500 0.71751 0.06749 0.178139 0.624687 0.034498 0.000030 251 0.82500 0.71084 0.11416 0.301589 0.632708 0.036165 0.000090 252 0.63300 0.66181 -0.02881 -0.077974 0.639512 0.081999 0.000014

177

253 0.20100 0.65180 -0.45080 -1.224142 0.633744 0.087774 0.003793 254 0.73500 0.74419 -0.00919 -0.024246 0.614826 0.032918 0.000001 255 0.75500 0.80012 -0.04512 -0.122135 0.620120 0.081900 0.000035 256 0.65400 0.73630 -0.08230 -0.218860 0.600476 0.048770 0.000065 257 0.78500 0.69756 0.08744 0.239390 0.599623 0.102493 0.000172 258 0.86600 0.79867 0.06733 0.188038 0.584103 0.137666 0.000148 259 0.72500 0.70500 0.02000 0.054115 0.563185 0.081568 0.000007 260 0.72500 0.66654 0.05846 0.162066 0.584103 0.124837 0.000099 261 0.87600 0.83568 0.04032 0.123088 0.602427 0.278111 0.000154 262 0.84600 0.79991 0.04609 0.160841 0.604063 0.447600 0.000551 263 0.69500 0.70525 -0.01025 -0.027820 0.560708 0.086633 0.000002 264 0.67400 0.69942 -0.02542 -0.069038 0.561936 0.087996 0.000012 265 0.78500 0.76992 0.01508 0.042346 0.552292 0.147253 0.000008 266 0.78500 0.80946 -0.02446 -0.067238 0.741646 0.110002 0.000015 267 0.66400 0.80291 -0.13891 -0.370496 0.682105 0.054341 0.000208 268 0.79500 0.80151 -0.00651 -0.017297 0.657349 0.047819 0.000000 269 0.79500 0.78489 0.01011 0.026923 0.654365 0.052357 0.000001 270 0.74500 0.85379 -0.10879 -0.297931 0.675882 0.103146 0.000269 271 0.80500 0.77906 0.02594 0.069686 0.716278 0.067961 0.000009 272 0.83500 0.76101 0.07399 0.199287 0.657828 0.072787 0.000082 273 0.78500 0.77888 0.00612 0.016345 0.716799 0.055981 0.000000 274 0.47600 0.74208 -0.26608 -0.723307 0.580254 0.089684 0.001356 275 0.78500 0.70075 0.08425 0.226019 0.579898 0.065409 0.000094

178

276 0.79500 0.71816 0.07684 0.206107 0.575249 0.065117 0.000078 277 0.73500 0.72717 0.00783 0.021171 0.576303 0.080861 0.000001 278 0.69500 0.68976 0.00524 0.014217 0.581041 0.086378 0.000001 279 0.77500 0.77768 -0.00268 -0.007283 0.572403 0.086314 0.000000 280 0.69500 0.70438 -0.00938 -0.025191 0.580277 0.067531 0.000001 281 0.69500 0.70298 -0.00798 -0.021507 0.574328 0.073557 0.000001 282 0.55800 0.86507 -0.30707 -0.925049 0.693489 0.258774 0.007859 283 0.64400 0.75643 -0.11243 -0.304494 0.664374 0.082958 0.000221 284 0.65400 0.75378 -0.09978 -0.270212 0.664827 0.082748 0.000173 285 0.50000 0.62146 -0.12146 -0.361268 0.680922 0.239699 0.001082 286 0.65400 0.71521 -0.06121 -0.181926 0.683706 0.238495 0.000273 287 0.69500 0.62615 0.06885 0.202615 0.727544 0.223208 0.000310 ** Results written to .e00 file Predictions from this model... Obs Y(i) Yhat(i) Res(i) X(i) Y(i) 1 0.907 1.065 -0.158 475834.000 5899694.000 F 2 1.047 1.061 -0.014 471169.000 5897743.000 F 3 1.133 1.134 -0.001 473455.000 5896147.000 F 4 1.284 1.314 -0.030 461889.000 5914076.000 F 5 1.159 1.068 0.091 521980.000 5869677.000 F 6 0.464 0.564 -0.100 487624.000 5859745.000 F 7 0.866 0.875 -0.009 552510.000 5932735.000 F 8 0.917 1.039 -0.122 486874.000 5918154.000 F 9 0.601 0.741 -0.140 535076.000 5884827.000 F

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10 0.991 1.058 -0.067 540711.000 5915664.000 F 11 1.024 0.977 0.047 474060.000 5911421.000 F 12 1.233 1.178 0.055 473732.000 5890924.000 F 13 1.133 1.279 -0.146 475183.000 5911679.000 F 14 1.284 1.062 0.222 477854.000 5878965.000 F 15 1.133 1.212 -0.079 471873.000 5892719.000 F 16 1.024 1.187 -0.163 474672.000 5918127.000 F 17 0.142 0.285 -0.143 477770.000 5984722.000 F 18 0.247 0.228 0.019 470783.000 5954257.000 F 19 0.322 0.197 0.125 472008.000 5974489.000 F 20 0.524 0.730 -0.206 451717.000 5902884.000 F 21 0.546 0.669 -0.123 452645.000 5903401.000 F 22 0.569 0.884 -0.315 490125.000 5883222.000 F 23 0.580 0.796 -0.216 493591.000 5851947.000 F 24 0.591 0.773 -0.182 490326.000 5852573.000 F 25 0.591 0.815 -0.224 560731.000 5820630.000 F 26 0.601 0.772 -0.171 568523.000 5792324.000 F 27 0.623 0.937 -0.314 481960.000 5857700.000 F 28 0.633 0.892 -0.259 499918.000 5856340.000 F 29 0.654 0.915 -0.261 468711.000 5952123.000 F 30 0.664 0.562 0.102 481801.000 5858918.000 F 31 0.685 0.914 -0.229 468709.000 5951966.000 F 32 0.685 0.859 -0.174 520909.000 5957111.000 F

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33 0.695 0.982 -0.287 596463.000 5819066.000 F 34 0.735 0.942 -0.207 592896.000 5848049.000 F 35 0.745 0.945 -0.200 506450.000 5858194.000 F 36 0.765 0.892 -0.127 506074.000 5854142.000 F 37 0.765 0.915 -0.150 614635.000 5812821.000 F 38 0.765 0.747 0.018 520790.000 5957984.000 F 39 0.805 0.932 -0.127 518018.000 5854674.000 F 40 0.835 0.933 -0.098 468331.000 5869068.000 F 41 0.846 0.892 -0.046 591568.000 5809062.000 F 42 0.856 0.918 -0.062 571748.000 5830597.000 F 43 0.866 1.072 -0.206 523041.000 5908129.000 F 44 0.866 0.931 -0.065 544375.000 5934799.000 F 45 0.866 0.937 -0.071 526528.000 5946130.000 F 46 0.866 0.985 -0.119 496302.000 5944915.000 F 47 0.866 0.810 0.056 567932.000 5824376.000 F 48 0.886 0.951 -0.065 507817.000 5857327.000 F 49 0.886 0.889 -0.003 489396.000 5853494.000 F 50 0.896 0.902 -0.006 489900.000 5860355.000 F 51 0.896 1.299 -0.403 495922.000 5926853.000 F 52 0.896 1.042 -0.146 485132.000 5948330.000 F 53 0.907 1.110 -0.203 476539.000 5899389.000 F 54 0.907 0.883 0.024 515373.000 5929898.000 F 55 0.917 1.106 -0.189 494112.000 5889680.000 F

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56 0.917 0.801 0.116 585998.000 5814162.000 F 57 0.917 0.941 -0.024 600678.000 5842008.000 F 58 0.927 1.121 -0.194 515880.000 5890328.000 F 59 0.927 1.123 -0.196 470833.000 5922593.000 F 60 0.927 1.167 -0.240 534968.000 5913130.000 F 61 0.927 0.998 -0.071 466244.000 5960459.000 F 62 0.927 1.004 -0.077 598723.000 5801219.000 F 63 0.938 0.783 0.155 543361.000 5796594.000 F 64 0.948 1.008 -0.060 477279.000 5895246.000 F 65 0.948 0.946 0.002 589680.000 5807961.000 F 66 0.948 0.522 0.426 614360.000 5817911.000 F 67 0.959 1.027 -0.068 467079.000 5893412.000 F 68 0.959 0.939 0.020 575628.000 5817832.000 F 69 0.970 0.951 0.019 505547.000 5858777.000 F 70 0.970 1.075 -0.105 520480.000 5869132.000 F 71 0.980 1.077 -0.097 509987.000 5901712.000 F 72 0.991 0.845 0.146 521682.000 5916236.000 F 73 0.991 0.980 0.011 473102.000 5923245.000 F 74 0.991 1.139 -0.148 512990.000 5883657.000 F 75 0.991 1.193 -0.202 520659.000 5962924.000 F 76 1.002 1.110 -0.108 519670.000 5853860.000 F 77 1.002 0.945 0.057 493153.000 5860545.000 F 78 1.002 1.054 -0.052 480047.000 5987182.000 F

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79 1.013 1.055 -0.042 498988.000 5879930.000 F 80 1.013 1.049 -0.036 450396.000 5886641.000 F 81 1.013 1.135 -0.122 473898.000 5920609.000 F 82 1.013 1.062 -0.049 549992.000 5889169.000 F 83 1.013 1.062 -0.049 484522.000 5881912.000 F 84 1.013 1.175 -0.162 488420.000 5903090.000 F 85 1.024 0.926 0.098 524782.000 5916603.000 F 86 1.024 1.046 -0.022 521525.000 5856391.000 F 87 1.036 1.215 -0.179 457992.000 5911867.000 F 88 1.036 1.016 0.020 489567.000 5889658.000 F 89 1.036 1.024 0.012 514045.000 5915462.000 F 90 1.036 1.275 -0.239 514601.000 5926018.000 F 91 1.036 1.163 -0.127 474325.000 5893796.000 F 92 1.036 1.001 0.035 467534.000 5869108.000 F 93 1.047 0.940 0.107 512789.000 5852735.000 F 94 1.047 1.205 -0.158 525619.000 5878476.000 F 95 1.047 0.879 0.168 490587.000 5848923.000 F 96 1.059 1.087 -0.028 540759.000 5927235.000 F 97 1.059 1.080 -0.021 519735.000 5855874.000 F 98 1.059 1.213 -0.154 535174.000 5924281.000 F 99 1.059 1.146 -0.087 480102.000 5880037.000 F 100 1.071 0.939 0.132 498825.000 5883789.000 F 101 1.071 0.907 0.164 487966.000 5860322.000 F

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102 1.083 1.097 -0.014 506910.000 5884821.000 F 103 1.083 1.005 0.078 507448.000 5858343.000 F 104 1.083 1.158 -0.075 518760.000 5855891.000 F 105 1.083 1.069 0.014 570829.000 5835824.000 F 106 1.095 1.063 0.032 519133.000 5855521.000 F 107 1.095 1.029 0.066 597649.000 5850491.000 F 108 1.095 1.240 -0.145 552851.000 5854297.000 F 109 1.107 1.133 -0.026 469502.000 5891661.000 F 110 1.107 1.125 -0.018 495687.000 5883083.000 F 111 1.107 1.235 -0.128 503721.000 5886259.000 F 112 1.107 1.008 0.099 456630.000 5911293.000 F 113 1.107 1.042 0.065 511734.000 5867281.000 F 114 1.107 1.059 0.048 579914.000 5846517.000 F 115 1.120 1.261 -0.141 553147.000 5886136.000 F 116 1.120 1.164 -0.044 528624.000 5878941.000 F 117 1.120 1.164 -0.044 523106.000 5935335.000 F 118 1.120 1.120 0.000 485983.000 5944523.000 F 119 1.133 1.011 0.122 457531.000 5911151.000 F 120 1.133 1.043 0.090 470033.000 5872308.000 F 121 1.133 1.034 0.099 532139.000 5871301.000 F 122 1.133 1.039 0.094 526176.000 5911536.000 F 123 1.133 1.247 -0.114 523872.000 5881187.000 F 124 1.133 0.998 0.135 595028.000 5849512.000 F

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125 1.133 1.004 0.129 470423.000 5967484.000 F 126 1.133 1.185 -0.052 605119.000 5843613.000 F 127 1.146 1.044 0.102 459886.000 5910288.000 F 128 1.146 1.085 0.061 595889.000 5850680.000 F 129 1.159 1.031 0.128 495095.000 5882008.000 F 130 1.159 1.255 -0.096 497441.000 5893595.000 F 131 1.159 1.295 -0.136 458125.000 5906224.000 F 132 1.173 1.275 -0.102 460135.000 5915016.000 F 133 1.173 1.254 -0.081 551100.000 5883782.000 F 134 1.173 1.078 0.095 546266.000 5903227.000 F 135 1.173 1.168 0.005 523545.000 5874740.000 F 136 1.173 1.182 -0.009 553288.000 5918498.000 F 137 1.173 1.043 0.130 471663.000 5846463.000 F 138 1.173 1.088 0.085 579280.000 5846647.000 F 139 1.187 1.143 0.044 465250.000 5914996.000 F 140 1.187 1.088 0.099 470251.000 5905375.000 F 141 1.187 1.184 0.003 532644.000 5877606.000 F 142 1.202 1.096 0.106 527382.000 5857503.000 F 143 1.202 0.808 0.394 477697.000 5869570.000 F 144 1.202 1.181 0.021 493986.000 5886873.000 F 145 1.202 1.157 0.045 542436.000 5897303.000 F 146 1.202 1.251 -0.049 455921.000 5906534.000 F 147 1.202 1.093 0.109 468567.000 5885441.000 F

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148 1.202 1.216 -0.014 523624.000 5877135.000 F 149 1.202 0.907 0.295 497085.000 5914408.000 F 150 1.202 1.099 0.103 495733.000 5881688.000 F 151 1.202 0.987 0.215 538181.000 5925366.000 F 152 1.217 1.077 0.140 478251.000 5878760.000 F 153 1.217 1.137 0.080 516471.000 5890667.000 F 154 1.217 1.063 0.154 526588.000 5889480.000 F 155 1.217 1.170 0.047 534000.000 5894235.000 F 156 1.217 1.244 -0.027 532591.000 5878425.000 F 157 1.217 1.365 -0.148 526964.000 5857612.000 F 158 1.217 1.250 -0.033 459777.000 5909469.000 F 159 1.233 1.094 0.139 515088.000 5855195.000 F 160 1.233 1.063 0.170 481742.000 5906858.000 F 161 1.233 1.100 0.133 526965.000 5847464.000 F 162 1.233 1.124 0.109 495423.000 5916712.000 F 163 1.233 0.903 0.330 505526.000 5918308.000 F 164 1.233 1.211 0.022 462148.000 5912632.000 F 165 1.233 1.226 0.007 542793.000 5913364.000 F 166 1.233 0.902 0.331 505073.000 5861880.000 F 167 1.249 1.105 0.144 491519.000 5885910.000 F 168 1.249 1.293 -0.044 496839.000 5894952.000 F 169 1.249 1.278 -0.029 496784.000 5893942.000 F 170 1.249 1.219 0.030 550947.000 5880539.000 F

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171 1.249 1.198 0.051 547957.000 5888284.000 F 172 1.249 1.314 -0.065 514196.000 5924260.000 F 173 1.249 1.065 0.184 534777.000 5932501.000 F 174 1.266 1.172 0.094 461883.000 5911517.000 F 175 1.266 1.057 0.209 526865.000 5857046.000 F 176 1.266 1.213 0.053 539475.000 5891908.000 F 177 1.266 1.292 -0.026 517441.000 5927980.000 F 178 1.266 1.101 0.165 460684.000 5907645.000 F 179 1.266 1.276 -0.010 460274.000 5906718.000 F 180 1.284 1.153 0.131 525254.000 5878015.000 F 181 1.284 1.331 -0.047 460213.000 5914680.000 F 182 1.284 1.042 0.242 525469.000 5913451.000 F 183 1.284 1.182 0.102 550911.000 5883251.000 F 184 1.284 1.281 0.003 473419.000 5924810.000 F 185 1.284 1.174 0.110 480256.000 5927156.000 F 186 1.284 1.292 -0.008 509003.000 5916838.000 F 187 1.284 1.232 0.052 460467.000 5909613.000 F 188 1.303 1.123 0.180 523622.000 5877089.000 F 189 1.303 1.283 0.020 460255.000 5907939.000 F 190 1.303 1.213 0.090 518366.000 5919376.000 F 191 1.303 1.171 0.132 525092.000 5877747.000 F 192 1.303 1.333 -0.030 507825.000 5917599.000 F 193 1.303 1.351 -0.048 489553.000 5921996.000 F

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194 1.323 1.285 0.038 452199.000 5902846.000 F 195 1.323 1.147 0.176 524522.000 5891657.000 F 196 1.323 1.252 0.071 526256.000 5912815.000 F 197 1.323 1.172 0.151 475635.000 5900001.000 F 198 1.345 1.424 -0.079 494276.000 5901012.000 F 199 1.345 1.247 0.098 529286.000 5926505.000 F 200 1.345 1.266 0.079 510663.000 5892598.000 F 201 1.345 1.240 0.105 478769.000 5928312.000 F 202 1.345 1.360 -0.015 508504.000 5893343.000 F 203 1.345 1.214 0.131 498952.000 5925111.000 F 204 1.345 1.298 0.047 510426.000 5915549.000 F 205 1.369 1.165 0.204 452603.000 5901223.000 F 206 1.369 1.397 -0.028 452388.000 5901728.000 F 207 1.369 1.112 0.257 507402.000 5921856.000 F 208 1.369 1.170 0.199 504303.000 5918953.000 F 209 1.369 1.257 0.112 459170.000 5909301.000 F 210 1.397 1.101 0.296 460637.000 5909605.000 F 211 1.397 1.229 0.168 466007.000 5911426.000 F 212 1.397 0.849 0.548 533585.000 5920100.000 F 213 1.397 1.269 0.128 524393.000 5903997.000 F 214 1.397 1.184 0.213 499086.000 5901960.000 F 215 1.397 1.125 0.272 488358.000 5934119.000 F 216 1.397 1.335 0.062 459154.000 5908450.000 F

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217 1.429 1.130 0.299 509168.000 5894881.000 F 218 1.429 1.251 0.178 497357.000 5893877.000 F 219 1.429 1.006 0.423 513849.000 5927684.000 F 220 0.287 0.261 0.026 614927.000 5818900.000 F 221 0.287 0.305 -0.018 574051.000 5846747.000 F 222 0.305 0.472 -0.167 496416.000 5929413.000 F 223 0.322 0.575 -0.253 491758.000 5907612.000 F 224 0.322 0.559 -0.237 609061.000 5817020.000 F 225 0.369 0.714 -0.345 591952.000 5815021.000 F 226 0.369 0.528 -0.159 600216.000 5816713.000 F 227 0.412 0.569 -0.157 592130.000 5799584.000 F 228 0.425 0.498 -0.073 550883.000 5850570.000 F 229 0.425 0.498 -0.073 550883.000 5850570.000 F 230 0.451 0.844 -0.393 534373.000 5911017.000 F 231 0.476 0.679 -0.203 596258.000 5819019.000 F 232 0.488 0.899 -0.411 535223.000 5925387.000 F 233 0.535 0.939 -0.404 494895.000 5923619.000 F 234 0.569 0.832 -0.263 516414.000 5923253.000 F 235 0.580 0.657 -0.077 540311.000 5931151.000 F 236 0.591 0.645 -0.054 531617.000 5930793.000 F 237 0.601 0.597 0.004 540392.000 5931268.000 F 238 0.601 0.701 -0.100 540239.000 5931418.000 F 239 0.644 0.684 -0.040 494151.000 5919558.000 F

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240 0.674 0.896 -0.222 556694.000 5909174.000 F 241 0.805 0.952 -0.147 528423.000 5925989.000 F 242 0.805 0.883 -0.078 474542.000 5973502.000 F 243 0.846 0.874 -0.028 481939.000 5911466.000 F 244 0.856 1.142 -0.286 542431.000 5894850.000 F 245 0.948 0.835 0.113 547693.000 5934840.000 F 246 0.695 0.741 -0.046 470202.000 5811066.000 F 247 0.715 0.782 -0.067 469754.000 5811752.000 F 248 0.795 0.783 0.012 469858.000 5811696.000 F 249 0.755 0.721 0.034 473309.000 5806960.000 F 250 0.785 0.718 0.067 462205.000 5817444.000 F 251 0.825 0.711 0.114 465116.000 5815359.000 F 252 0.633 0.662 -0.029 469628.000 5811162.000 F 253 0.201 0.652 -0.451 466299.000 5813677.000 F 254 0.735 0.744 -0.009 459132.000 5819699.000 F 255 0.755 0.800 -0.045 460850.000 5818389.000 F 256 0.654 0.736 -0.082 446467.000 5826754.000 F 257 0.785 0.698 0.087 443692.000 5829018.000 F 258 0.866 0.799 0.067 441128.000 5836892.000 F 259 0.725 0.705 0.020 436108.000 5843478.000 F 260 0.725 0.667 0.058 440853.000 5834496.000 F 261 0.876 0.836 0.040 450781.000 5825347.000 F 262 0.846 0.800 0.046 449767.000 5826351.000 F

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263 0.695 0.705 -0.010 435235.000 5844277.000 F 264 0.674 0.699 -0.025 435911.000 5843740.000 F 265 0.785 0.770 0.015 434587.000 5846512.000 F 266 0.785 0.809 -0.024 440806.000 5860825.000 F 267 0.664 0.803 -0.139 452724.000 5850930.000 F 268 0.795 0.802 -0.007 454137.000 5845894.000 F 269 0.795 0.785 0.010 454425.000 5845380.000 F 270 0.745 0.854 -0.109 453813.000 5848090.000 F 271 0.805 0.779 0.026 445023.000 5858494.000 F 272 0.835 0.761 0.074 454964.000 5845508.000 F 273 0.785 0.779 0.006 445120.000 5858447.000 F 274 0.476 0.742 -0.266 438357.000 5839577.000 F 275 0.785 0.701 0.084 439105.000 5833930.000 F 276 0.795 0.718 0.077 437114.000 5839217.000 F 277 0.735 0.727 0.008 437745.000 5840840.000 F 278 0.695 0.690 0.005 439702.000 5834782.000 F 279 0.775 0.778 -0.003 435781.000 5836600.000 F 280 0.695 0.704 -0.009 439532.000 5837124.000 F 281 0.695 0.703 -0.008 437227.000 5840135.000 F 282 0.558 0.865 -0.307 437613.000 5856516.000 F 283 0.644 0.756 -0.112 476525.000 5812977.000 F 284 0.654 0.754 -0.100 476699.000 5813003.000 F 285 0.500 0.621 -0.121 483951.000 5806811.000 F

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286 0.654 0.715 -0.061 484460.000 5806343.000 F 287 0.695 0.626 0.069 491651.000 5803462.000 F ********************************************************** * ANOVA * ********************************************************** Source SS DF MS F OLS Residuals 12.1 4.00 GWR Improvement 5.7 34.01 0.1682 GWR Residuals 6.3 248.99 0.0254 6.6119 ********************************************************** * PARAMETER 5-NUMBER SUMMARIES * ********************************************************** Label Minimum Lwr Quartile Median Upr Quartile Maximum -------- ------------- ------------- ------------- ------------- ------------- Intrcept -9.579215 -6.712800 -5.898440 -2.977145 -0.563489 El -0.001058 -0.000290 -0.000077 0.000044 0.001227 Gr -0.000583 0.009305 0.013353 0.014580 0.017842 Wt 0.005045 0.017060 0.027391 0.035639 0.048835 <------------------ LOWER -----------------><------------------ UPPER -----------------> Label Far Out Outer Fence Outside Inner Fence Inner Fence Outside Outer Fence Far Out -------- ------- ------------- ------- ------------- ------------- ------- ------------- ------- Intrcept 0 -17.919766 0 -12.316283 2.626338 0 8.229821 0 El 0 -0.001290 8 -0.000790 0.000543 19 0.001043 3 Gr 0 -0.006520 1 0.001393 0.022493 0 0.030405 0

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Wt 0 -0.038678 0 -0.010809 0.063508 0 0.091377 0 ************************************************* * * * Test for spatial variability of parameters * * * ************************************************* Tests based on the Monte Carlo significance test procedure due to Hope [1968,JRSB,30(3),582-598] Parameter P-value ---------- ------------------ Intercept 0.00000 *** El 0.00000 *** Gr 0.00000 *** Wt 0.00000 *** *** = significant at .1% level ** = significant at 1% level * = significant at 5% level Program terminates normally at: Tue Apr 25 11:15:37 2006

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APPENDIX D: LIST OF ACRONYMNS

AIC: Akaike information criterion (AIC) is a statistical model fit measure. It quantifies

the relative goodness-of-fit of various derived statistical models, give a sample

dataset. The preferred model is that with the lowest AIC value.

DEM: Digital elevation model (DEM) is a representation of the topography of the Earth

or another surface in digital format, by coordinates and numerical descriptions of

altitude. DEMs are used often in geographic information systems for calculating

slope, angle of incidence, view shade, hill shade and aspect.

GWR: Geographically weighted regression (GWR) is a method of analyzing spatially

varying relationships. This usually involves fitting a model to predict the values

of one variable (response or dependent variable) from a set of one or more

independent (predictor) variables.

Moran’ I: Moran's I tests for global spatial autocorrelation in a give dataset. Positive

spatial autocorrelation means that nearby areas have similar rates, indicating

spatial clustering. Nearby areas have similar rates when their populations and

exposures are alike. When rates in nearby areas are similar, Moran's I will be

large and positive. When rates are dissimilar, Moran's I will be negative.

NDVI: Normalized difference vegetation index (NDVI) is an index that provides a

standardized method of comparing vegetation greenness between satellite images.

NDVI can be used as an indicator of relative biomass and greenness.

OLS: Ordinary least square (OLS) is usual method of estimation for the regression

model. In the linear regression model, the dependent variable is assumed to be a

linear function of one or more independent variables plus an error introduced to

account for all other factors.

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R: Correlation coefficient (R) indicates the strength and direction of a linear

relationship between two random variables in probability theory and statistics.

R2: Coefficient of determination (R2) is a statistic that is widely used to determine

how well a regression fits. It represents the fraction of variability in response that

can be explained by the variability in predictor variables. In the simple linear

regression case, R2 is simply the square of the correlation coefficient.

SAR: Spatial autoregression model (SAR) is a generalization of the linear regression

model to account for spatial autocorrelation. It has been successfully used to

analyze spatial datasets exhibiting strong spatial autocorrelation.

SR: Simple ratio (SR) index is the simplest form of vegetation index is a ratio between

near infrared and red reflectance. For healthy living vegetation, this ratio will be

high due to the inverse relationship between vegetation brightness in the red and

infrared regions of the spectrum

TCT: Tasseled cap transformation (TCT) is one of the available methods for enhancing

spectral information content of Landsat TM data. It especially optimizes data

viewing for vegetation studies. Tasseled Cap index is calculated from data of the

related six TM bands.

TM: Thematic mapper (TM) is an electromechanical sensor onboard the series of

Landsat satellites (4 and 5). A Thematic Mapper image has data recorded in 7

spectral bands of the electromagnetic spectrum.

VI: Vegetation index (VI) is a quantitative measure used to measure biomass or

vegetative vigor, usually formed from combinations of several spectral bands,

whose values are added, divided, or multiplied in order to yield a single value that

indicates the amount or vigor of vegetation.

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VIF: Variance inflation factor (VIF) is a common way for detecting multicollinearity. A

general rule is that the VIF should not exceed 10.