ISBN: 3-540-66208-1
TITLE: Modelling Spatial Processes
AUTHOR: Tiefelsdorf, Michael
TOC:

1 Introduction 1
1.1 The Origins of the Exact Approach 2
1.2 Organization of the Monograph 6
I Underlying Concepts of Spatial Processes 2 The Regression Model 13
2.1 A Review of Ordinary Least Squares 13
2.2 Ordinary Least Squares with Correlated Disturbances 17
2.3 The Impact of Unaccounted Variables 18
2.3.1 The Two Stage Estimation Procedure 19
2.3.2 The Misspecified Regression Model 20
3 The Specification of Spatial Relations 23
3.1 Implementation of Spatial Structures 24
3.1.1 Typology of Spatial Objects 24
3.1.2 Relations between Spatial Objects 25
3.1.3 Similarity Versus Dissimilarity Relations 25
3.1.4 Functional Relationships in Relative Space 28
3.1.5 Coding Schemes of Spatial Structure Matrices 29
3.2 Some Special Spatial Structures 31
3.2.1 Regular Spatial Structures 31
3.2.2 Higher Order Link Matrices 32
3.2.3 A Link between the Contiguity Matrices and Spatial Configurations 33
3.2.4 Local Spatial Structures 34
4 Gaussian Spatial Processes 37
4.1 Blending the Disturbances with a Spatial Structure 37
4.1.1 The Internal Composition of Stochastic Spatial Processes 38
4.1.2 Constraining Assumptions on Spatial Stochastic Processes 40
4.1.3 Excursion: Non-Hierarchical Aggregation of Spatial Objects 41
4.2 Definition and Properties of Gaussian Spatial Processes 42
4.2.1 The Autoregressive Spatial Process 43
4.2.2 The Moving Average Process 46
4.2.3 Excursion: The Common Factor Constraint and the Spatial Lag Model 47
4.3 Testing for Gaussian Spatial Processes 47
4.3.1 Test Principles for Spatial Autocorrelation in Regression Residuals 49
4.3.2 Statistical Properties of the Moran's I Test 51
II Development and Properties of Moran's I's Distribution
5 The Characteristic Function 59
5.1 General Properties of the Characteristic Function 59
5.2 The Inversion Theorem 61
5.3 Distribution of a Weighted Sum of 2-Distributed Variables 63
5.4 Extensions and Comments 66
6 Numerical Evaluation of Imhof's Formula 67
6.1 Influential Weights in Imhof's Formula 67
6.2 A Numerical Example 67
6.3 The Behavior of the Integrand at the Origin 68
6.4 Derivation of an Upper Truncation Point 69
6.5 Numerical Algorithms to Approximate Imhof's Formula 70
6.5.1 Equally Spaced Step-Size 71
6.5.2 Variable Spaced Step-Size 72
7 The Exact Distribution of Moran's I 75
7.1 Derivation of the Distribution under the Influence of a Spatial Process 75
7.1.1 Transformation into Independent Random Variables 76
7.1.2 Transformation of Moran's I into a Simple Quadratic Form 77
7.1.3 Determination of the Weights 77
7.1.4 Discussion 78
7.2 Simplifications under the Assumption of Spatial Independence 79
7.2.1 Independence of the Eigenvalue Spectrum from the Denominator 80
7.2.2 Shifts in the Eigenvalue Spectrum in Dependence of Moran's I's Observed Value 82
7.2.3 Bounding Distributions of Moran's I to Obtain Independence from the Projection Matrix 83
8 The Shape of Moran's I Exact Distribution 89
8.1 The Feasible Range of Moran's I 89
8.2 Specific Eigenvalue Spectra and Moran's I's Distribution 90
8.2.1 Small Irregular Spatial Tessellations 91
8.2.2 Large Regular Spatial Tessellations 93
9 The Moments of Moran's I 99
9.1 The Moments under the Assumption of Spatial Independence 99
9.1.1 The Theorem by Pitman 99
9.1.2 The Expectations of Moran's I Numerator and Denominator 100
9.1.3 Eigenvalue or Trace Expressions for the Central Moments 101
9.2 The Conditional Moments of Moran's I 102
9.3 Correlation between Different Spatial Structures 105
9.4 Some Applications Using Moran's I's Theoretical Moments 106
9.4.1 Approximations of Moran's I's Distribution by its Moments 106
9.4.2 Potential Applications of Moran's I's Conditional Expectation 108
III An Empirical Application of Moran's I's Exact Conditional Distribution
10 The Basic Ecological Model and Spatial Setting 117
10.1 The Setting of the Regression Model 118
10.2 The Spatial Setting 122
10.2.1 Rationale for an Underlying Spatial Process in Disease Rates 123
10.2.2 Approximation of a Hierarchical Migratory Graph 124
10.2.3 Testing for Inherent Spatial Autocorrelation 128
11 A Spatial Analysis of the Cancer Data 129
11.1 Linking Moran's I to an Autoregressive Spatial Process 129
11.2 Exploratory Analysis by means of Local Moran's I_i 131
11.2.1 Local Moran's Ii as Ratio of Quadratic Forms 132
11.2.2 The Conditional Distribution of Local Moran's I_i 133
11.2.3 Correlation between Local Moran's I_i 136
11.2.4 Impact of the Local Regression Residuals 138
11.3 The Identification of a Prevailing Spatial Process 140
11.3.1 The Power Function of Global Moran's I 141
11.3.2 Identification of a Spatial Process by a Non-Nested Test 142
11.4 Discussion 144
12 Conclusions 147
References 153
Index 163
END
