Spatial Statistics and Modeling

Preface

Contents

Abbreviations and notation

1 Second-order spatial models and geostatistics

1.1 Some background in stochastic processes

1.2 Stationary processes

1.2.1 Definitions and examples

1.2.2 Spectral representation of covariances

1.3 Intrinsic processes and variograms

1.3.1 Definitions, examples and properties

1.3.2 Variograms for stationary processes

1.3.3 Examples of covariances and variograms

1.3.4 Anisotropy

1.4 Geometric properties: continuity, differentiability

1.4.1 Continuity and differentiability: the stationary case

1.5 Spatial modeling using convolutions

1.5.1 Continuous model

1.5.2 Discrete convolution

1.6 Spatio-temporal models

1.7 Spatial autoregressive models

1.7.1 Stationary MA and ARMA models

1.7.2 Stationary simultaneous autoregression

1.7.3 Stationary conditional autoregression

1.7.4 Non-stationary autoregressive models on finite networks S

1.7.5 Autoregressive models with covariates

1.8 Spatial regression models

1.9 Prediction when the covariance is known

1.9.1 Simple kriging

1.9.2 Universal kriging

1.9.3 Simulated experiments

Exercises

2 Gibbs-Markov random fields on networks

2.1 Compatibility of conditional distributions

2.2 Gibbs random fields on S

2.2.1 Interaction potential and Gibbs specification

2.2.2 Examples of Gibbs specifications

2.3 Markov random fields and Gibbs random fields

2.3.1 Definitions: cliques, Markov random field

2.3.2 The Hammersley-Clifford theorem

2.4 Besag auto-models

2.4.1 Compatible conditional distributions and auto-models

2.4.2 Examples of auto-models

2.5 Markov random field dynamics

2.5.1 Markov chain Markov random field dynamics

2.5.2 Examples of dynamics

Exercises

3 Spatial point processes

3.1 Definitions and notation

3.1.1 Exponential spaces

3.1.2 Moments of a point process

3.1.3 Examples of point processes

3.2 Poisson point process

3.3 Cox point process

3.3.1 log-Gaussian Cox process

3.3.2 Doubly stochastic Poisson point process

3.4 Point process density

3.4.1 Definition

3.4.2 Gibbs point process

3.5 Nearest neighbor distances for point processes

3.5.1 Palm measure

3.5.2 Two nearest neighbor distances for X

3.5.3 Second-order reduced moments

3.6 Markov point process

3.6.1 The Ripley-Kelly Markov property

3.6.2 Markov nearest neighbor property

3.6.3 Gibbs point process on Rd

Exercises

4 Simulation of spatial models

4.1 Convergence of Markov chains

4.1.1 Strong law of large numbers and central limit theorem for a homogeneous Markov chain

4.2 Two Markov chain simulation algorithms

4.2.1 Gibbs sampling on product spaces

4.2.2 The Metropolis-Hastings algorithm

4.3 Simulating a Markov random field on a network

4.3.1 The two standard algorithms

4.3.2 Examples

4.3.3 Constrained simulation

4.3.4 Simulating Markov chain dynamics

4.4 Simulation of a point process

4.4.1 Simulation conditional on a fixed number of points

4.4.2 Unconditional simulation

4.4.3 Simulation of a Cox point process

4.5 Performance and convergence of MCMC methods

4.5.1 Performance of MCMC methods

4.5.2 Two methods for quantifying rates of convergence

4.6 Exact simulation using coupling from the past

4.6.1 The Propp-Wilson algorithm

4.6.2 Two improvements to the algorithm

4.7 Simulating Gaussian random fields on SRd

4.7.1 Simulating stationary Gaussian random fields

4.7.2 Conditional Gaussian simulation

Exercises

5 Statistics for spatial models

5.1 Estimation in geostatistics

5.1.1 Analyzing the variogram cloud

5.1.2 Empirically estimating the variogram

5.1.3 Parametric estimation for variogram models

5.1.4 Estimating variograms when there is a trend

5.1.5 Validating variogram models

5.2 Autocorrelation on spatial networks

5.2.1 Moran's index

5.2.2 Asymptotic test of spatial independence

5.2.3 Geary's index

5.2.4 Permutation test for spatial independence

5.3 Statistics for second-order random fields

5.3.1 Estimating stationary models on bold0mu mumu ZZunitsZZZZd

5.3.2 Estimating autoregressive models

5.3.3 Maximum likelihood estimation

5.3.4 Spatial regression estimation

5.4 Markov random field estimation

5.4.1 Maximum likelihood

5.4.2 Besag's conditional pseudo-likelihood

5.4.3 The coding method

5.4.4 Comparing asymptotic variance of estimators

5.4.5 Identification of the neighborhood structure of a Markov random field

5.5 Statistics for spatial point processes

5.5.1 Testing spatial homogeneity using quadrat counts

5.5.2 Estimating point process intensity

5.5.3 Estimation of second-order characteristics

5.5.4 Estimation of a parametric model for a point process

5.5.5 Conditional pseudo-likelihood of a point process

5.5.6 Monte Carlo approximation of Gibbs likelihood

5.5.7 Point process residuals

5.6 Hierarchical spatial models and Bayesian statistics

5.6.1 Spatial regression and Bayesian kriging

5.6.2 Hierarchical spatial generalized linear models

Exercises

A Simulation of random variables

A.1 The inversion method

A.2 Simulation of a Markov chain with a finite number of states

A.3 The acceptance-rejection method

A.4 Simulating normal distributions

B Limit theorems for random fields

B.1 Ergodicity and laws of large numbers

B.1.1 Ergodicity and the ergodic theorem

B.1.2 Examples of ergodic processes

B.1.3 Ergodicity and the weak law of large numbers in L2

B.1.4 Strong law of large numbers under L2 conditions

B.2 Strong mixing coefficients

B.3 Central limit theorem for mixing random fields

B.4 Central limit theorem for a functional of a Markov random field

C Minimum contrast estimation

C.1 Definitions and examples

C.2 Asymptotic properties

C.2.1 Convergence of the estimator

C.2.2 Asymptotic normality

C.3 Model selection by penalized contrast

C.4 Proof of two results in Chapter 5

C.4.1 Variance of the maximum likelihood estimator for Gaussian regression

C.4.2 Consistency of maximum likelihood for stationary Markov random fields

D Software

References

Index