Exploring Spatial Scale in Geography

Exploring Spatial Scale in Geography provides a conceptual and practical guide to issues of spatial scale in all areas of the physical and social sciences. Scale is at the heart of geography and other spatial sciences. Whether dealing with geomorphological processes, population movements or meteorology, a consideration of spatial scale is vital. Exploring Spatial Scale in Geography takes a practical approach with a core focus on real world problems and potential solutions. Links are made to appropriate software environments with an associated website providing access to guidance material which outlines how particular problems can be approached using popular GIS and spatial data analysis software. This book offers alternative definitions of spatial scale, presents approaches for exploring spatial scale and makes use of a wide variety of case studies in the physical and social sciences to demonstrate key concepts, making it a key resource for anyone who makes use of geographical information. Exploring Spatial Scale in Geography 3 Contents 9 Preface 15 Acknowledgements 17 About the Companion Website 19 1 Introduction 21 1.1 The purpose of the book 21 1.1.1 What this book adds 23 1.1.2 Scales of analysis and alternative definitions 23 1.3 Case studies and examples 25 1.4 Why is spatial scale important? 25 1.5 Structure of the book 26 1.6 Further reading 26 References 27 2 Scale in Spatial Data Analysis: Key Concepts 29 2.1 Definitions of spatial scale 29 2.2 Spatial autocorrelation and spatial dependence 31 2.3 Scale dependence 33 2.4 Scale and data models 34 2.5 Spatial scales of inquiry 34 2.6 Scale and spatial data analysis 34 2.7 Scale and neighbourhoods 35 2.8 Scale and space 36 2.9 Scale, spatial data analysis and physical processes 43 2.10 Scale, spatial data analysis and social processes 45 2.11 Summary 46 2.12 Further reading 46 References 46 3 The Modifiable Areal Unit Problem 49 3.1 Basic concepts 49 3.2 Scale and zonation effects 49 3.3 The ecological fallacy 52 3.4 The MAUP and univariate statistics 54 3.4.1 Case study: segregation in Northern Ireland 55 3.4.2 Spatial approaches to segregation 58 3.5 Geographical weighting and the MAUP 58 3.6 The MAUP and multivariate statistics 59 3.6.1 Case study: population variables in Northern Ireland 60 3.7 Zone design 61 3.8 Summary 62 3.9 Further reading 62 References 62 4 Measuring Spatial Structure 65 4.1 Basic concepts 65 4.2 Measures of spatial autocorrelation 65 4.2.1 Neighbourhood size 67 4.2.2 Spatial autocorrelation and kernel size 67 4.2.3 Spatial autocorrelation and lags 70 4.2.4 Local measures 70 4.2.5 Global and local and spatial scale 71 4.3 Geostatistics and characterising spatial structure 73 4.3.1 The theory of regionalised variables 74 4.4 The variogram 77 4.4.1 Bias in variogram estimation 79 4.5 The covariance function and correlogram 79 4.6 Alternative measures of spatial structure 80 4.7 Measuring dependence between variables 83 4.8 Variograms of risk 84 4.9 Variogram clouds and h-scatterplots 84 4.10 Variogram models 85 4.11 Fitting variogram models 88 4.12 Variogram case study 90 4.12 Variogram case study 90 4.13 Anisotropy and variograms 94 4.13.1 Variogram surfaces 94 4.13.2 Geometric and zonal anisotropy 95 4.14 Variograms and non-stationarity 97 4.14.1 Variograms and long-range trends 97 4.14.2 Variogram non-stationarity 99 4.15 Space–time variograms 102 4.16 Software 103 4.17 Other methods 103 4.18 Point pattern analysis 104 4.18.1 Spatial dependence and point patterns 105 4.18.2 Local function 111 4.18.3 Cross function 112 4.19 Summary 117 4.20 Further reading 117 References 117 5 Scale and Multivariate Data 123 5.1 Regression frameworks 124 5.2 Spatial scale and regression 124 5.3 Global regression 125 5.4 Spatial regression 125 5.5 Regression and spatial data 126 5.5.1 Generalised least squares 126 5.5.2 Spatial autoregressive models 127 5.5.3 Spatially lagged dependent variable models and spatial error models case study 129 5.6 Local regression and spatial scale 131 5.6.1 Spatial expansion method 131 5.6.2 Geographically weighted regression 132 5.6.3 Scale and GWR 135 5.6.4 GWR case study: fixed bandwidths 135 5.6.5 GWR case study: variable bandwidths 136 5.6.6 Bayesian spatially varying coefficient process models 138 5.7 Multilevel modelling 139 5.7.1 Case study 145 5.8 Spatial structure of multiple variables 149 5.9 Multivariate analysis and spatial scale 150 5.10 Summary 151 5.11 Further reading 151 References 151 6 Fractal Analysis 155 6.1 Basic concepts 155 6.2 Measuring fractal dimension 158 6.2.1 Walking-divider method 159 6.2.2 Box-counting method 160 6.2.3 Variogram method 162 6.3 Fractals and spatial structure 162 6.3.1 Case study: fractal of land surfaces 163 6.3.2 Case study: local fractal 166 6.3.3 Fractals and topographic form 169 6.4 Other applications of fractal analysis 172 6.4.1 Fractals and remotely sensed imagery 172 6.4.2 Fractals and urban form 173 6.5 How useful is the fractal model in geography? 175 6.6 Summary 175 6.7 Further reading 175 References 175 7 Scale and Gridded Data: Fourier and Wavelet Transforms 179 7.1 Basic concepts 179 7.2 Fourier transforms 180 7.2.1 Continuous Fourier transform 180 7.2.2 Discrete Fourier transform 181 7.2.3 Fast Fourier transform 183 7.2.4 FFT case study 183 7.2.5 Spectral analysis and the covariance function 185 7.2.6 Spectral analysis case study 187 7.3 Wavelet transforms 188 7.3.1 Continuous wavelet transforms 189 7.3.2 Discrete wavelet transforms 190 7.3.3 The Haar basis functions 191 7.3.4 Other basis functions 192 7.3.5 Fast wavelet transform 193 7.3.6 Two-dimensional wavelet transforms 194 7.4 Wavelet analysis applications and other issues 200 7.5 Summary 200 7.6 Further reading 200 References 201 8 Areal Interpolation 203 8.1 Basic concepts 203 8.2 Areal weighting 204 8.3 Using additional data 206 8.3.1 Types of secondary data sources for mapping populations 212 8.4 Surface modelling 213 8.4.1 Population surface case study 215 8.5 Other approaches to changing support 216 8.6 Summary 217 8.7 Further reading 218 References 218 9 Geostatistical Interpolation and Change of Support 221 9.1 Basic concepts 221 9.2 Regularisation 221 9.2.1 Regularisation with an irregular support 224 9.3 Variogram deconvolution 225 9.3.1 Variogram deconvolution for irregular supports 226 9.3.2 Variography and change of support 228 9.4 Kriging 230 9.4.1 Punctual kriging 230 9.4.2 Poisson kriging 232 9.4.3 Factorial kriging 233 9.4.4 Factorial kriging case study 235 9.4.5 Kriging in the presence of a trend 235 9.4.6 Cokriging 242 9.4.7 Kriging with an external drift and other techniques 242 9.4.8 Interpreting the kriging variance 243 9.4.9 Cross-validation 243 9.4.10 Conditional simulation 244 9.4.11 Comparison of kriging approaches 244 9.5 Kriging and change of support 246 9.5.1 Block kriging 246 9.5.2 Area-to-point kriging 247 9.5.3 Case study 249 9.6 Assessing uncertainty and optimal sampling design 251 9.6.1 Nested sampling 251 9.6.2 Assessing optimal sampling design 252 9.6.3 Optimal spatial resolution 255 9.6.4 Other approaches to optimal sampling design 256 9.7 Summary 256 9.8 Further reading 256 References 256 10 Summary and Conclusions 261 10.1 Overview of key concepts and methods 261 10.2 Problems and future directions 263 10.3 Summary 265 References 265 Index 267 Supplemental Images 274

Exploring Spatial Scale in Geography provides a conceptual and practical guide to issues of spatial scale in all areas of the physical and social sciences. Scale is at the heart of geography and other spatial sciences. Whether dealing with geomorphological processes, population movements or meteorology, a consideration of spatial scale is vital.

Exploring Spatial Scale in Geography takes a practical approach with a core focus on real world problems and potential solutions. Links are made to appropriate software environments with an associated website providing access to guidance material which outlines how particular problems can be approached using popular GIS and spatial data analysis software.

This book offers alternative definitions of spatial scale, presents approaches for exploring spatial scale and makes use of a wide variety of case studies in the physical and social sciences to demonstrate key concepts, making it a key resource for anyone who makes use of geographical information.

This Book Offers Alternative Definitions Of Spatial Scale, Presents Approaches For Exploring Spatial Scale, And Makes Use Of A Wide Variety Of Case Studies In The Physical And Social Sciences To Demonstrate Key Concepts. It Consists Of Three Integrated Strands. The First Is Conceptual, Outlining Some Definitions Of Spatial Scale And Debating The Meaning And Value Of Concepts Of Scale. The Second Strand Outlines Methods For Exploration Of Spatial Scale Including Standard Measures Of Spatial Autocorrelation, Fractals, Wavelets, Multilevel Models And Geostatistical Measures. The Third And Final Strand Demonstrates The Application Of These Concepts And Methods To Real World Case Studies. --