Spatial Autocorrelation and Spatial Filtering : Gaining Understanding Through Theory and Scientific Visualization (Advances in Spatial Science) (2003. XIV, 247 p. w. 150 figs.)

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Spatial Autocorrelation and Spatial Filtering : Gaining Understanding Through Theory and Scientific Visualization (Advances in Spatial Science) (2003. XIV, 247 p. w. 150 figs.)

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Full Description


Scientific visualization may be defined as the transformation of numerical scientific data into informative graphical displays. The text introduces a nonverbal model to subdisciplines that until now has mostly employed mathematical or verbal-conceptual models. The focus is on how scientific visualization can help revolutionize the manner in which the tendencies for (dis)similar numerical values to cluster together in location on a map are explored and analyzed. In doing so, the concept known as spatial autocorrelation - which characterizes these tendencies - is further demystified.

Table of Contents

Preface                                            VII
1 Introduction 1 (32)
1.1 Scientific Visualization 2 (1)
1.2 What Is Spatial Autocorrelation? 3 (3)
1.3 Selected Visualization Tools: An Overview 6 (6)
1.3.1 Graphical Portrayals of Spatial 8 (4)
Autocorrelation
1.4 The Sample Georeferenced Datasets 12 (21)
1.4.1 Selected Interval/Ratio Datasets 14 (9)
1.4.2 Selected Counts Datasets 23 (5)
1.4.3 Selected Binomial Datasets 28 (5)
2 Salient Properties of Geographic Connectivity 33 (32)
Underlying Spatial Autocorrelation
2.1 Eigenfunctions Associated with Geographic 35 (11)
Connectivity Matrices
2.1.1 Eigenvalue Decompositions 36 (2)
2.1.2 Eigenvectors Associated with 38 (1)
Geographic Connectivity Matrices
2.1.3 The Maximum MC Value (MCmax) 38 (3)
2.1.4 Moments of Eigenvalue Distributions 41 (5)
2.2 Generalized Eigenvalue Frequency 46 (7)
Distributions
2.2.1 The Extreme Eigenvalues of Matrices C 46 (2)
and W
2.2.2 Spectrum Results for Matrices C and W 48 (3)
2.2.3 Spectrum Results for Matrix (I 51 (2)
-11T/n)C(I -11T/)
2.3 The Auto-Gaussian Jacobian Term 53 (5)
Normalizing Factor
2.3.1 Simplification of the Auto-Gaussian 56 (2)
Jacobian Term Based upon Matrix W for a
Regular Square Tessellation and the Rook's
Definition of Connectivity
2.4 Eigenfunctions Associated with the GR 58 (2)
2.5 Remarks and Discussion 60 (5)
3 Sampling Distributions Associated with 65 (26)
Spatial Autocorrelation
3.1 Samples as Random Permutations of Values 66 (3)
across Locations on a Map: Randomization
3.2 Simple Random Samples at Each Location on 69 (1)
a Map: Unconstrained Selection
3.3 Samples as Ordered Random Drawings from a 70 (4)
Parent Frequency Distribution: Extending the
Permutation Perspective
3.3.1 The Sampling Distribution for MC 71 (1)
3.3.2 The Distribution of P for an 72 (2)
Auto-normal SAR Model
3.4 Samples as Outcomes of a Multivariate 74 (8)
Drawing: Extending the Simple Random Sampling
Perspective
3.4.1 The Auto-normal Model: ML Estimation 74 (2)
3.4.2 The Auto-logistic/binomial Model 76 (5)
3.4.3 Embedding Spatial Autocorrelation 81 (1)
through the Mean Response
3.5 Effective Sample Size 82 (6)
3.5.1 Estimates Based upon a Single Mean 83 (1)
Response
3.5.2 Estimates Based upon Multiple Mean 84 (2)
Responses
3.5.3 Estimates Based upon a Difference of 86 (1)
Means for Correlated (Paired) Samples
3.5.4 Relationships between Effective 87 (1)
Sample Size and the Configuration of Sample
Points
3.6. Remarks and Discussion 88 (3)
4 Spatial Filtering 91 (40)
4.1 Eigenvector-based Spatial Filtering 92 (15)
4.1.1 Map Patterns Depicted by Eigenvectors 92 (1)
of Matrix (I - ρC)T(I - ρC)
4.1.2 Similarities with Conventional PCA 93 (10)
4.1.3 Orthogonality and Uncorrelatedness of 103(2)
the Eigenvectors
4.1.4 Linear Combinations of Eigenvectors 105(2)
of Matrix (I -11T/n)C(I -11T/n)
4.2 Coefficients for Single and Linear 107(6)
Combinations of Distinct Map Patterns
4.2.1 Decomposition of Regressor and 108(2)
Regressand Attribute Variables
4.2.2 The Sampling Distributions of y and r 110(3)
4.3 Eigenvector Selection Criteria 113(8)
4.3.1 The Auto-normal Model 113(1)
4.3.2 The Auto-logistic/binomial Model 114(4)
4.3.3 The Auto-Poisson Model 118(1)
4.3.4 The Case of Negative Spatial 119(2)
Autocorrelation
4.4 Regression Analysis: Standard Errors 121(4)
Based upon Simulation Experiments and
Resampling
4.4.1 Simulating Error for Georeferenced 121(2)
Data
4.4.2 Bootstrapping Georeferenced Data 123(2)
4.5 The MC Local Statistic and Illuminating 125(3)
Diagnostics
4.5.1 The MCis 126(1)
4.5.2 Diagnostics Based upon Eigenvectors 126(2)
of Matrix (I -11T/n)C(I -11T/n)
4.6 Remarks and Discussion 128(3)
5 Spatial Filtering Applications: Selected 131(22)
Interval/Ratio Datasets
5.1 Geographic Distributions of Settlement 131(4)
Size in Peru
5.2 The Geographic Distribution of Lyme 135(5)
Disease in Georgia
5.3 The Geographic Distribution of Biomass in 140(3)
the High Peak District
5.4 The Geographic Distribution of 143(5)
Agricultural and Topographic Variables in
Puerto Rico
5.5 Remarks and Discussion 148(5)
5.5.1 Relationship between the SAR and 149(1)
Eigenvector Spatial Filtering Specifications
5.5.2 Computing Back-transformations 150(3)
6 Spatial Filtering Applications: Selected 153(24)
Counts Datasets
6.1 Geographic Distributions of Settlement 154(6)
Counts in Pennsylvania
6.2 The Geographic Distribution of Farms in 160(3)
Loiza, Puerto Rico
6.3 The Geographic Distribution of Volcanoes 163(1)
in Uganda
6.4 The Geographic Distribution of Cholera 164(3)
Deaths in London
6.5 The Geographic Distribution of Drumlins 167(5)
in Ireland
6.6 Remarks and Discussion 172(5)
7 Spatial Filtering Applications: Selected 177(16)
Percentage Datasets
7.1 The Geographic Distribution of the 178(2)
Presence/Absence of Plant Disease in an
Agricultural Field
7.2 The Geographic Distribution of Plant 180(2)
Disease in an Agricultural Field
7.3 The Geographic Distribution of Blood 182(2)
Group A in Eire
7.4 The Geographic Distribution of 184(4)
Urbanization across the Island of Puerto Rico
7.5 Remarks and Discussion 188(5)
8 Concluding Comments 193(18)
8.1 Spatial Filtering versus Spatial 194(1)
Autoregression
8.2 Some Numerical Issues in Spatial Filtering 195(10)
8.2.1 Covariation of Spatial Filter and SAR 195(1)
Spatial Autocorrelation Measures
8.2.2 Exploding Georeferenced Data with a 196(3)
Spatial Filter When Maps Have Holes or
Gaps: Estimating Missing Data Values
8.2.3 Rotation and Theoretical Eigenvectors 199(3)
Given by Theorem 2.5 for Regular Square
Tessellations Forming Rectangular Regions
8.2.4 Effective Sample Size Revisited 202(3)
8.3 Stepwise Selection of Eigenvectors for an 205(2)
Auto-Poisson Model
8.4 Binomial and Poisson Overdispersion 207(1)
8.5 Future Research: What Next? 208(3)
List of Symbols 211(8)
List of Tables 219(4)
List of Figures 223(4)
References 227(6)
Author Index 233(2)
Place Index 235(4)
Subject Index 239