Geographical Weighted Regression (GWR) is a new local modellingtechnique for analysing spatial analysis. This technique allowslocal as opposed to global models of relationships to be measuredand mapped. This is the first and only book on this technique,offering comprehensive coverage on this new 'hot' topic in spatialanalysis. * Provides step-by-step examples of how to use the GWR model usingdata sets and examples on issues such as house price determinants,educational attainment levels and school performance statistics * Contains a broad discussion of and basic concepts on GWR throughto ideas on statistical inference for GWR models * uniquely features accompanying author-written software thatallows users to undertake sophisticated and complex forms of GWRwithin a user-friendly, Windows-based, front-end (see book fordetails).
A. Stewart Fotheringham, Professor of Quantitative Geography, University of Newcastle. Chris Brunsdon, Senior Lecturer in Spatial Analysis, University of Newcastle. Martin Charlton, Lecturer in Geographical Information Systems, University of Newcastle.
Acknowledgements
Contents
1 Local Statistics and Local Models for Spatial Data 1
1.1 Introduction 1
1.2 Local Aspatial Statistical Methods 3
1.3 Local versus Global Spatial Statistics 6
1.4 Spatial Non-stationarity 9
1.5 Examples of Local Univariate Methods for Spatial Data Analysis 11
1.5.1 Local Forms of Point Pattern Analysis 11
1.5.2 Local Graphical Analysis 12
1.5.3 Local Filters 13
1.5.4 Local Measures of Spatial Dependency 14
1.6 Examples of Local Multivariate Methods for Spatial Data Analysis 15
1.6.1 The Spatial Expansion Method 16
1.6.2 Spatially Adaptive Filtering 17
1.6.3 Multilevel Modelling 18
1.6.4 Random Coef®cient Models 20
1.6.5 Spatial Regression Models 21
1.7 Examples of Local Methods for Spatial Flow Modelling 24
1.8 Summary 25
2 Geographically Weighted Regression:The Basics 27
2.1 Introduction 27
2.2 An Empirical Example 27
2.2.1 The Data 28
2.2.2 A Global Regression Model 28
2.2.3 Global Regression Results 34
2.3 Borough-Speci®c Calibrations of the Global Model 38
2.4 Moving Window Regression 42
2.5 Geographically Weighted Regression with Fixed Spatial Kernels 44
2.6 Geographically Weighted Regression with Adaptive Spatial Kernels 46
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2.7 The Mechanics of GWR in More Detail 52
2.7.1 The Basic Methodology 52
2.7.2 Local Standard Errors 54
2.7.3 Choice of Spatial Weighting Function 56
2.7.4 Calibrating the Spatial Weighting Function 59
2.7.5 Bias-Variance Trade-Off 62
2.8 Testing for Spatial Non-stationarity 63
2.9 Summary 64
3 Extensions to the Basic GWR Model 65
3.1 Introduction 65
3.2 Mixed GWR Models 65
3.3 An Example 68
3.4 Outliers and Robust GWR 73
3.5 Spatially Heteroskedastic Models 80
3.6 Summary 82
4 Statistical Inference and Geographically Weighted Regression 83
4.1 Introduction 83
4.2 What is Meant by `Inference' and How Does it Relate to GWR? 84
4.2.1 How Likely is it that Some Fact is True on the Basis of the Data? 85
4.2.2 Within What Interval Does Some Model Coef®cient Lie? 85
4.2.3 Which One of a Series of Potential Mathematical Models is `Best'? 86
4.3 GWR as a Statistical Model 87
4.3.1 Local Likelihood 90
4.3.2 Using Classical Inference ± Working with p-values 91
4.3.3 Testing Individual Parameter Stationarity 92
4.4 Con®dence Intervals 94
4.5 An Alternative Approach Using the AIC 95
4.6 Two Examples 97
4.6.1 Basic Estimates 97
4.6.2 Estimates of Pointwise Standard Errors 99
4.6.3 Working with the AIC 99
4.7 Summary 102
5 GWR and Spatial Autocorrelation 103
5.1 Introduction 103
5.2 The Empirical Setting 104
5.3 Local Measures of Spatial Autocorrelation using GWR 104
5.4 Residuals in Global Regression Models and in GWR 112
5.5 Local Parameter Estimates from Autoregressive and Non-Autoregressive Models 117
5.6 Spatial Regression Models and GWR 121
5.6.1 Overview 121
5.6.2 Conditional Autoregressive (CA) Models 122
5.6.3 Simultaneous Autoregressive (SA) Models 122
5.6.4 GWR, Conditional Autoregressive Models and Simultaneous Autoregressive Models 123
5.7 Summary 124
6 Scale Issues and Geographically Weighted Regression 127
6.1 Introduction 127
6.2 Bandwidth and Scale: The Example of School Performance Analysis 130
6.2.1 Introduction 130
6.2.2 The School Performance Data 131
6.2.3 Global Regression Results 133
6.2.4 Local Regression Results 134
6.3 GWR and the MAUP 144
6.3.1 Introduction 144
6.3.2 An Experiment 147
6.4 Summary 153
7 Geographically Weighted Local Statistics 159
7.1 Introduction 159
7.2 Basic Ideas 161
7.3 A Single Continuous Variable 163
7.4 Two Continuous Variables 173
7.5 A Single Binary Variable 175
7.6 A Pair of Binary Variables 177
7.7 Towards More Robust Geographically Weighted Statistics 181
7.8 Summary 183
8 Extensions of Geographical Weighting 187
8.1 Introduction 187
8.2 Geographically Weighted Generalised Linear Models 188
8.2.1 A Poisson GWGLM 190
8.2.2 A Binomial GWGLM 193
8.3 Geographically Weighted Principal Components 196
8.3.1 Local Multivariate Models 196
8.3.2 Calibrating Local Multivariate Models 198
8.3.3 Interpreting S and r 199
8.3.4 An Example 200
8.4 Geographically Weighted Density Estimation 202
8.4.1 Kernel Density Estimation 202
8.4.2 Geographically Weighted Kernels 203
8.4.3 An Example Using House Prices 203
8.5 Summary 205
9 Software for Geographically Weighted Regression 207
9.1 Introduction 207
9.2 Some Terminology 208
9.3 The Data File 208
9.4 What Do INeed to Specify? 209
9.5 Kernels 210
9.6 Choosing a Bandwidth 211
9.6.1 User-Supplied Bandwidth 211
9.6.2 Estimation by Cross-validation 212
9.6.3 Estimation by Minimising the AIC 212
9.6.4 The Golden Section Search 212
9.7 Signi®cance Tests 213
9.8 Casewise Diagnostics for GWR 214
9.8.1 Standardised Residuals 214
9.8.2 Local r-square 215
9.8.3 In¯uence Statistics 216
9.9 A Worked Example 216
9.9.1 Running GWR 2.0 on a PC 216
9.9.2 The Outputs 224
9.9.3 Running GWR 2.0 under UNIX 230
9.10 Visualising the Output 231
9.10.1 Viewing the Results in ArcView 233
9.10.2 Point Symbols 234
9.10.3 Area Symbols 236
9.10.4 Contour Plots 237
9.10.5 Pseudo-3D Display 238
9.11 Summary 239
10 Epilogue 241
10.1 Overview 241
10.2 Summarising the Book 242
10.3 Empirical Applications of GWR 243
10.4 Software Development 245
10.4.1 Embedding GWR in Larger Packages 246
10.4.2 Software Extending the Basic GWR Idea 247
10.5 Cautionary Notes 248
10.5.1 Multiple Hypothesis Testing 249
10.5.2 Locally Varying Intercepts 251
10.5.3 Interpretation of Parameter Surfaces 251
10.6 Summary 252
Bibliography 255
Index 267
Acknowledgements. Local Statistics and Local Models for Spatial Data. Geographically Weighted Regression: The Basics. Extensions to the Basic GWR Model. Statistical Inference and Geographically Weighted Regression. GWR and Spatial Autocorrelation. Scale Issues and Geographically Weighted Regression. Geographically Weighted Local Statistics. Extensions of Geographically Weighting. Software for Geographically Weighted Regression. Epilogue. Bibliography. Index.
