Spatial Autocorrelation and Spatial Filtering: Gaining Understanding Through Theory and Scientific Visualization

1 Introduction. - 1.1 Scientific Visualization. - 1.2 What Is Spatial Autocorrelation? - 1.3 Selected Visualization Tools: An Overview. - 1.3.1 Graphical Portrayals of Spatial Autocorrelation. - 1.4 The Sample Georeferenced Datasets. - 1.4.1 Selected Interval/Ratio Datasets. - 1.4.2 Selected Counts Datasets. - 1.4.3 Selected Binomial Datasets. - 2 Salient Properties of Geographic Connectivity Underlying Spatial Autocorrelation. - 2.1 Eigenfunctions Associated with Geographic Connectivity Matrices. - 2.1.1 Eigenvalue Decompositions. - 2.1.2 Eigenvectors Associated with Geographic Connectivity Matrices. - 2.1.3 The Maximum MC Value (MCmax). - 2.1.4 Moments of Eigenvalue Distributions. - 2.2 Generalized Eigenvalue Frequency Distributions. - 2.2.1 The Extreme Eigenvalues of Matrices C and W. - 2.2.2 Spectrum Results for Matrices C and W. - 2.2.3 Spectrum Results for Matrix (I - 11T/n)C(I - 11T/n). - 2.3 The Auto-Gaussian Jacobian Term Normalizing Factor. - 2.3.1 Simplification of the Auto-Gaussian 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. - 2.5 Remarks and Discussion. - 3 Sampling Distributions Associated with Spatial Autocorrelation. - 3.1 Samples as Random Permutations of Values across Locations on a Man: Randomization. - 3.2 Simple Random Samples at Each Location on a Map: Unconstrained Selection. - 3.3 Samples as Ordered Random Drawings from a Parent Frequency Distribution: Extending the Permutation Perspective. - 3.3.1 The Samnling Distribution fnr MC. - 3.3.2 The Distribution of p for an Auto-normal SAR Model. - 3.4 Samples as Outcomes of a Multivariate Drawing: Extending the Simple Random Samnling Persnective. - 3.4.1 The Auto-normal Model: ML Estimation. - 3.4.2 The Auto-logistic/binomial Model. - 3.4.3 Embedding Spatial Autocorrelation through the Mean Response. - 3.5 Effective Sample Size. - 3.5.1 Estimates Based upon a Single Mean Response. - 3.5.2 Estimates Based upon Multiple Mean Responses. - 3.5.3 Estimates Based upon a Difference of Means for Correlated (Paired) Samples. - 3.5.4 Relationships between Effective Sample Size and the Configuration of Sample Points. - 3.6. Remarks and Discussion. - 4 Spatial Filtering. - 4.1 Eigenvector-based Spatial Filtering. - 4.1.1 Map Patterns Depicted by Eigenvectors of Matrix (I-? C)T(I-? C). - 4.1.2 Similarities with Conventional PCA. - 4.1.3 Orthogonality and Uncorrelatedness of the Eigenvectors. - 4.1.4 Linear Combinations of Eigenvectors of Matrix (I - 11T/n)C(I - 11T/n). - 4.2 Coefficients for Single and Linear Combinations of Distinct Map Patterns. - 4.2.1 Decomposition of Regressor and Regressand Attribute Variables. - 4.2.2 The Sampling Distributions of y? and r. - 4.3 Eigenvector Selection Criteria. - 4.3.1 The Auto-normal Model. - 4.3.2 The Auto-logistic/binomial Model. - 4.3.3 The Auto-Poisson Model. - 4.3.4 The Case of Negative Spatial Autocorrelation. - 4.4 Regression Analysis: Standard Errors Based upon Simulation Experiments and Resampling. - 4.4.1 Simulating Error for Georeferenced Data. - 4.4.2 Bootstrapping Georeferenced Data. - 4.5 The MC Local Statistic and Illuminating Diagnostics. - 4.5.1 The MCis. - 4.5.2 Diagnostics Based upon Eigenvectors of Matrix (I-11T/n)C(I-11T/n). - 4.6 Remarks and Discussion. - 5 Spatial Filtering Applications: Selected Interval/Ratio Datasets. - 5.1 Geographic Distributions of Settlement Size in Peru. - 5.2 The Geographic Distribution of Lyme Disease in Georgia. - 5.3 The Geographic Distribution or Biomass in the Hign Peak District. - 5.4 The Geographic Distribution of Agricultural and Topographic Variables in Puerto Rico. - 5.5 Remarks and Discussion. - 5.5.1 Relationship between the SAR and Eigenvector Spatial Filtering Specifications. - 5.5.2 Computing Back-transformations. - 6 Spatial Filtering Applications: Selected Counts Datasets. - 6.1 Geographic Distributions of Settlement Counts in Pennsylvania. - 6.2 The Geographic Distribution of Farms in Loiza, Puerto Rico. - 6.3 The Geographic Distribution of Volcanoes in Uganda. - 6.4 The Geographic Distribution of Cholera Deaths in London. - 6.5 The Geographic Distribution of Drumlins in Ireland. - 6.6 Remarks and Discussion. - 7 Spatial Filtering Applications: Selected Percentage Datasets. - 7.1 The Geographic Distribution of the Presence/Absence of Plant Disease in an Agricultural Field. - 7.2 The Geographic Distribution of Plant Disease in an Agricultural Field. - 7.3 The Geographic Distribution of Blood Group A in Eire. - 7.4 The Geographic Distribution of Urbanization across the Island of Puerto Rico. - 7.5 Remarks and Discussion. - 8 Concluding Comments. - 8.1 Spatial Filtering versus Spatial Autoregression. - 8.2 Some Numerical Issues in Spatial Filtering. - 8.2.1 Covariation of Spatial Filter and SAR Spatial Autocorrelation Measures. - 8.2.2 Exploding Georeferenced Data with a Spatial Filter When Maps Have Holes or Gaps: Estimating Missing Data Values. - 8.2.3 Rotation and Theoretical Eigenvectors Given by Theorem 2.5 for Regular Square Tessellations Forming Rectangular Regions. - 8.2.4 Effective Sample Size Revisited. - 8.3 Stepwise Selection of Eigenvectors for an Auto-Poisson Model. - 8.4 Binomial and Poisson Overdispersion. - 8.5 Future Research: What Next? - List of Symbols. - List of Tables. - List of Figures. - References. - Author Index. - Place Index.