Geographically Weighted Regression The Analysis of Spatially Varying Relationships Wiley 2002 PDF

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Y FL AM TE Team-Fly® Geographically Weighted Regression Geographically Weighted Regression the analysis of spatially varying relationships A. Stewart F otheringham Chris Brunsdon M artin Charlton University of N ewcastle, UK JOHN WILEY & SONS, LTD Copyright  2002 John Wiley & Sons Ltd, The ...

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AM FL Y TE Team-Fly®

Geographically Weighted Regression

Geographically Weighted Regression the analysis of spatially varying relationships

A. Stewart F otheringham Chris Brunsdon Martin Charlton University of N ewcastle, UK

JOHN WILEY & SONS, LTD

Copyright  2002 John Wiley & Sons Ltd, The Atrium, Southern G ate, Chichester, West Sussex PO19 8SQ, England Telephone (+ 44) 1243 779777 Email (for orders and customer service enquiries): [email protected] Visit our Home Page on www.wileyeurope.com or www.wiley.com All R ights R eserved. N o part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, D esigns and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court R oad, London W1T 4LP, UK, without the permission in writing of the Publisher. R equests to the Publisher should be addressed to the Permissions D epartment, John Wiley & Sons Ltd, The Atrium, Southern G ate, Chichester, West Sussex PO19 8SQ, England, or emailed to [email protected], or faxed to (+ 44) 1243 770571. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the Publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought. Other W iley Editorial Offices John Wiley & Sons Inc., 111 R iver Street, H oboken, N J 07030, U SA Jossey-Bass, 989 Market Street, San F rancisco, CA 94103-1741, U SA Wiley-VCH Verlag GmbH, Boschstr. 12, D -69469 Weinheim, G ermany John Wiley & Sons Australia Ltd, 33 Park R oad, M ilton, Queensland 4064, Australia John Wiley & Sons (Asia) Pte Ltd, 2 Clementi Loop # 02-01, Jin Xing D istripark, Singapore 129809 John Wiley & Sons Canada Ltd, 22 Worcester R oad, Etobicoke, Ontario, Canada M 9W 1L1 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 0-471-49616-2 Typeset in 10/12pt Times by Kolam Information Services Pvt. Ltd, Pondicherry, India Printed and bound in Great Britain by Antony R owe Ltd, Chippenham, Wiltshire This book is printed on acid-free paper responsibly manufactured from sustainable forestry in which at least two trees are planted for each one used for paper production.

A S F: T o Barbara, Iain and N eill CB: T o Francis and A nne M EC: T o T ed and A vril

Contents Acknowledgements

xi

1

Local Statistics and Local Models for Spatial Data 1.1 Introduction 1.2 Local Aspatial Statistical M ethods 1.3 Local versus Global Spatial Statistics 1.4 Spatial Non-stationarity 1.5 Examples of Local Univariate Methods for Spatial Data Analysis 1.5.1 Local F orms of Point Pattern Analysis 1.5.2 Local Graphical Analysis 1.5.3 Local F ilters 1.5.4 Local M easures of Spatial D ependency 1.6 Examples of Local Multivariate Methods for Spatial Data Analysis 1.6.1 The Spatial Expansion M ethod 1.6.2 Spatially Adaptive F iltering 1.6.3 M ultilevel M odelling 1.6.4 R andom Coefficient M odels 1.6.5 Spatial R egression Models 1.7 Examples of Local Methods for Spatial F low M odelling 1.8 Summary

1 1 3 6 9 11 11 12 13 14 15 16 17 18 20 21 24 25

2

Geographically Weighted Regression: The Basics 2.1 Introduction 2.2 An Empirical Example 2.2.1 The Data 2.2.2 A Global R egression M odel 2.2.3 Global R egression R esults 2.3 Borough-Specific Calibrations of the Global M odel 2.4 M oving Window R egression 2.5 G eographically Weighted R egression with F ixed Spatial K ernels 2.6 Geographically Weighted R egression with Adaptive Spatial Kernels

27 27 27 28 28 34 38 42 44 46

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Contents

2.7

The M echanics of G WR in M ore D etail 2.7.1 The Basic M ethodology 2.7.2 Local Standard Errors 2.7.3 Choice of Spatial Weighting F unction 2.7.4 Calibrating the Spatial Weighting F unction 2.7.5 Bias-Variance Trade-Off 2.8 Testing for Spatial Non-stationarity 2.9 Summary

52 52 54 56 59 62 63 64

3

Extensions to the Basic GWR Model 3.1 Introduction 3.2 M ixed G WR M odels 3.3 An Example 3.4 Outliers and R obust G WR 3.5 Spatially H eteroskedastic M odels 3.6 Summary

65 65 65 68 73 80 82

4

Statistical Inference and Geographically Weighted Regression 4.1 Introduction 4.2 What is M eant by ‘Inference’ and H ow D oes it R elate to G WR ? 4.2.1 How Likely is it that Some F act is True on the Basis of the D ata? 4.2.2 Within What Interval Does Some M odel Coefficient Lie? 4.2.3 Which One of a Series of Potential Mathematical M odels is ‘Best’? 4.3 GWR as a Statistical Model 4.3.1 Local Likelihood 4.3.2 Using Classical Inference – Working with p-values 4.3.3 Testing Individual Parameter Stationarity 4.4 Confidence Intervals 4.5 An Alternative Approach Using the AIC 4.6 Two Examples 4.6.1 Basic Estimates 4.6.2 Estimates of Pointwise Standard Errors 4.6.3 Working with the AIC 4.7 Summary

83 83 84

5

GWR and Spatial Autocorrelation 5.1 Introduction 5.2 The Empirical Setting 5.3 Local M easures of Spatial Autocorrelation using GWR 5.4 R esiduals in Global R egression Models and in GWR 5.5 Local Parameter Estimates from Autoregressive and Non-Autoregressive Models 5.6 Spatial R egression M odels and G WR

85 85 86 87 90 91 92 94 95 97 97 99 99 102 103 103 104 104 112 117 121

Contents

ix

121 122 122

Scale Issues and Geographically Weighted Regression 6.1 Introduction 6.2 Bandwidth and Scale: The Example of School Performance Analysis 6.2.1 Introduction 6.2.2 The School Performance D ata 6.2.3 Global R egression R esults 6.2.4 Local R egression R esults 6.3 GWR and the M AU P 6.3.1 Introduction 6.3.2 An Experiment 6.4 Summary

127 127 130 130 131 133 134 144 144 147 153

7

Geographically Weighted Local Statistics 7.1 Introduction 7.2 Basic Ideas 7.3 A Single Continuous Variable 7.4 Two Continuous Variables 7.5 A Single Binary Variable 7.6 A Pair of Binary Variables 7.7 Towards M ore R obust Geographically Weighted Statistics 7.8 Summary

159 159 161 163 173 175 177 181 183

8

Extensions of Geographical Weighting 8.1 Introduction 8.2 Geographically Weighted Generalised Linear Models 8.2.1 A Poisson GWGLM 8.2.2 A Binomial GWGLM 8.3 Geographically Weighted Principal Components 8.3.1 Local M ultivariate M odels 8.3.2 Calibrating Local Multivariate M odels 8.3.3 Interpreting Σ and ρ 8.3.4 An Example 8.4 Geographically Weighted Density Estimation 8.4.1 Kernel Density Estimation 8.4.2 Geographically Weighted Kernels 8.4.3 An Example U sing H ouse Prices 8.5 Summary

187 187 188 190 193 196 196 198 199 200 202 202 203 203 205

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5.6.1 Overview 5.6.2 Conditional Autoregressive (CA) Models 5.6.3 Simultaneous Autoregressive (SA) M odels 5.6.4 G WR , Conditional Autoregressive M odels and Simultaneous Autoregressive Models 5.7 Summary

Team-Fly®

123 124

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Contents

9

10

Software for Geographically Weighted Regression 9.1 Introduction 9.2 Some Terminology 9.3 The Data F ile 9.4 What D o I N eed to Specify? 9.5 Kernels 9.6 Choosing a Bandwidth 9.6.1 User-Supplied Bandwidth 9.6.2 Estimation by Cross-validation 9.6.3 Estimation by M inimising the AIC 9.6.4 The Golden Section Search 9.7 Significance Tests 9.8 Casewise Diagnostics for GWR 9.8.1 Standardised R esiduals 9.8.2 Local r-square 9.8.3 Influence Statistics 9.9 A Worked Example 9.9.1 R unning GWR 2.0 on a PC 9.9.2 The Outputs 9.9.3 R unning GWR 2.0 under UNIX 9.10 Visualising the Output 9.10.1 Viewing the R esults in ArcView 9.10.2 Point Symbols 9.10.3 Area Symbols 9.10.4 Contour Plots 9.10.5 Pseudo-3D Display 9.11 Summary

207 207 208 208 209 210 211 211 212 212 212 213 214 214 215 216 216 216 224 230 231 233 234 236 237 238 239

Epilogue 10.1 Overview 10.2 Summarising the Book 10.3 Empirical Applications of G WR 10.4 Software Development 10.4.1 Embedding GWR in Larger Packages 10.4.2 Software Extending the Basic GWR Idea 10.5 Cautionary Notes 10.5.1 M ultiple Hypothesis Testing 10.5.2 Locally Varying Intercepts 10.5.3 Interpretation of Parameter Surfaces 10.6 Summary

241 241 242 243 245 246 247 248 249 251 251 252

Bibliography

255

Index

267

Acknowledgements The UK maps in this book are based on copyright digital map data owned and supplied by Bartholomew Ltd and are used with permission. Some of the maps are also based on census data provided with the support of the ESR C and JISC and use boundary material which is the copyright of the Crown and the ED-LIN E Consortium. The U S census data and boundaries were obtained from CensusCD + M aps, a product of Geolytics Inc. The authors are grateful for the enlightened attitude of the US Government in making spatial data relatively freely available. Throughout the book we make extensive use of house price data that has been supplied by the Nationwide Building Society to the University of Newcastle upon Tyne and we are extremely grateful for their generosity. Dr R obin F lowerdew of the Department of Geography at the University of St Andrews supplied the school performance data used in Chapter 5 as part of a conference on local modelling with spatial data. A number of people also deserve credit for assisting with various aspects of this book. Ann R ooke applied her usual superlative cartographic skills to some of the figures. Stamatis Kalogirou wrote the Visual Basic front end to the GWR software and Barbara F otheringham did a very professional job of helping to proofread the manuscript. We would also like to thank Sally Wilkinson, Lyn R oberts and Keily Larkins at John Wiley & Sons, Ltd for their encouragement, patience, assistance and good nature during the various evolutionary stages of this book. F urther, we acknowledge a great debt to the reviewers of both the initial book proposal and an earlier version of the finished product for their strong support and useful insights. The book is far better for their comments. F inally, we make the usual disclaimer that any errors remaining in the book are the sole responsibility of the authors – apologies for not catching them all!

xii

Acknowledgements

This publication contains maps based on copyright digital map data owned and supplied by Bartholomew Ltd and is used with permission. This applies to: F igures 2.1, 2.2, 2.4, 2.9, 2.12, 2.14–2.19, 5.1, 6.2–6.7, 7.1–7.6, 7.8–7.12, 8.1–8.3. D ata taken from Bartholomews. F igures 2.1, 2.2, 2.9, 2.12, 2.13–2.19, 5.1. Data taken from Bartholomews and UKBorders. F igures 2.2, 5.1. D ata taken from Bartholomews, U K Borders and N ationwide Building Society. Maps are based on data provided with the support of the ESR C and JISC and use boundary material which is copyright of the Crown and the ED LIN E consortium. This applies to: F igures 2.1–2.3, 2.5–2.7, 2.9, 2.12, 2.14–2.19, 3.1, 3.3–3.5, 3.8, 3.9, 3.11, 3.12, 5.1–5.4, 5.6–5.10, 6.8–6.13. Table 2.1 is calculated from data supplied by the N ationwide Building Society to the University of Newcastle upon Tyne.

1 Local Statistics and Local Models for Spatial Data

1.1 Introduction Imagine reading a book on the climate of the U nited States which contained only data averaged across the whole country, such as mean annual rainfall, mean annual number of hours of sunshine, and so forth. M any would feel rather short-changed with such a lack of detail. We would suspect, quite rightly, that there is a great richness in the underlying data on which these averages have been calculated; we would probably want to see these data, preferably drawn on maps, in order to appreciate the spatial variations in climate that are hidden in the reported averages. Indeed, the averages we have been presented with may be practically useless in telling us anything about climate in any particular part of the U nited States. It is known, for instance, that parts of the north-western United States receive a great deal more precipitation than parts of the Southwest and that F lorida receives more hours of sunshine in a year than New York. In fact, it might be the case that not a single weather station in the country has the characteristics depicted by the mean climatic statistics. The average values in this scenario can be termed global observations: in the absence of any other information, they are assumed to represent the situation in every part of the study region. The individual data on which the averages are calculated can be termed local observations: they describe the situation at the local level.1 1

There is at least one other slightly different definition of ‘local’ and ‘global’ in the literature. Thioulouse et al. (1995) define a local statistic as one which is calculated on pairs of points or areas which are adjacent and a global statistic as one calculated over all possible pairs of points or areas. Their use of the term ‘local’, however, is not the same as used throughout this book because it still produces a global model; it merely separates the model applications into different spatial regimes.

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Geographically Weighted Regression

Only if there is little or no variation in the local observations do the global observations provide any reliable information on the local areas within the study area. As the spatial variation of the local observations increases, the reliability of the global observation as representative of local conditions decreases. While the above scenario might appear rather ludicrous (surely no one would publish a book containing average climatic data without describing at least some of the local data?), consider a second scenario which is much more plausible and indeed describes a methodology which is exceedingly common in spatial analysis. Suppose we had data on house prices and their determinants across the whole of England and that we wanted to model house price as a function of these determinants (such models are often referred to as hedonic price models and an example of the calibration of these models is provided in Chapter 2). Typically, we might run a regression of house prices on a set of structural attributes of each house, such as the age and floor area of the house; a set of neighbourhood attributes, such as crime rate or unemployment rate; and a set of locational attributes, such as distance to a major road or to a certain school. The output from this regression would be a set of parameter estimates, each estimate reflecting the relationship between house price and a particular attribute of the house. It would be quite usual to publish the results of such an analysis in the form of a table describing the parameter estimates for each attribute and commenting on their sign and magnitude, possibly in relation to some a priori set of hypotheses. In fact this is the standard approach of the vast majority of empirical analyses of spatial data. However, the parameter estimates in this second scenario are global statistics and are possibly just as inadequate at representing local conditions as are the average climatic data described above. Each parameter estimate describes the average relationship between house price and a particular attribute of the house across the study region (in this case, the whole of England). This average relationship might not be representative of the situation in any particular part of England and may hide some very interesting and important local differences in the determinants of house prices. F or example, suppose one of the determinants of house prices in our model is the age of the house and the global parameter estimate is close to zero. Superficially this would be interpreted as indicating that house prices are relatively independent of the age of the property. H owever, it might well be that there are contrasting relationships in different parts of the study area which tend to cancel each other out in the calculation of the global parameter estimate. F or example, in rural parts of England, old houses might have character and appeal, thus generating higher prices than newer houses, ceteris paribus, whereas in urban areas, older houses, built to low standards to house workers in rapidly expanding cities at the middle of the nineteenth century, might be in poor condition and have substantially lower prices than newer houses. This local variation in the relationship between house price and age of the house would be completely lost if all that is reported is the global parameter estimate. It would be far more informative to produce a set of local statistics, in this case local parameter estimates, and to map these than simply to rely on the assumption that a single global estimate will be an accurate representation of all parts of the study area. The only difference between the examples of the US climate and English house prices presented above is that the first describes the representation of spatial data,

Local Statistics and Local Models for Spatial Data

3

whereas the second describes the representation of spatial relationships. It would seem that while we generally find it unhelpful to report solely global observations on spatial data, we are quite happy to accept global statements of spatial relationships. Indeed, as hinted at above, journals and textbooks in a variety of disciplines dealing with spatial data are filled with examples of global forms of spatial analysis. Local forms of spatial analysis or spatial models are very rare exceptions to the overwhelming tide of global forms of analysis that dominates the literature. In this book, through a series of examples and discussions, we hope to convince the reader of the value of local forms of spatial analysis and spatial modelling, and in particular, the value of one form of local modelling which we term Geographically W eighted R egression ( GW R ) . We hope to show that in many instances undertaking a global spatial analysis or calibrating a global spatial model can be as misleading as describing precipitation rates across the USA with a single value.

1.2 Local Aspatial Statistical Methods Spatial data contain both attribute and locational information: aspatial data contain only attribute information. F or instance, data on the manufacturing output of firms graphed against the number of their employees are aspatial, whereas the numbers of people suffering from a certain type of disease in different parts of a country are spatial. Unemployment rates measured for one location over different time periods are aspatial but unemployment rates at different locations are spatial and the spatial component of the data might be very useful in understanding why the rates vary. The difference between aspatial and spatial data is important because many statistical techniques developed for aspatial data are not valid for spatial data. The latter have unique properties and problems that necessitate a different set of statistical techniques and modelling approaches (for more on this, see F otheringham et al. 20...