Statistical Analysis of Spatial and Spatio-Temporal Point Patterns

Introduction
Spatial point patterns
Sampling
Edge-effects
Complete spatial randomness
Objectives of statistical analysis
The Dirichlet tessellation
Monte Carlo tests
Software

Preliminary Testing
Tests of complete spatial randomness
Inter-event distances
Nearest neighbor distances
Point to nearest event distances
Quadrat counts
Scales of pattern
Recommendations

Methods for Sparsely Sampled Patterns
General remarks
Quadrat counts
Distance measurements
Tests of independence
Recommendations

Spatial Point Processes
Processes and summary descriptions
Second-order properties
Higher order moments and nearest neighbor distributions
The homogeneous Poisson process
Independence and random labeling
Estimation of second-order properties
Displaced amacrine cells in the retina of a rabbit
Estimation of nearest neighbor distributions
Concluding remarks

Nonparametric Methods
Estimating weighted integrals of the second-order intensity
Nonparametric estimation of a spatially varying intensity
Analyzing replicated spatial point patterns
Parametric or nonparametric methods?

Models
Contagious distributions
Poisson cluster processes
Inhomogeneous Poisson processes
Cox processes
Trans-Gaussian Cox processes
Simple inhibition processes
Markov point processes
Other constructions
Multivariate models

Model-Fitting Using Summary Descriptions
Parameter estimation using the K-function
Goodness-of-fit assessment using nearest neighbor distributions
Examples
Parameter estimation via goodness-of-fit testing

Model-Fitting Using Likelihood-Based Methods
Likelihood inference for inhomogeneous Poisson processes
Likelihood inference for Markov point processes
Likelihood inference for Cox processes
Additional reading

Point Process Methods in Spatial Epidemiology
Spatial clustering
Spatial variation in risk
Point source models
Stratification and matching
Disentangling heterogeneity and clustering

Spatio-Temporal Point Processes
Motivating examples
A classification of spatio-temporal point patterns and processes
Second-order properties
Conditioning on the past
Empirical and mechanistic models

Exploratory Analysis
Animation
Marginal and conditional summaries
Second-order properties

Empirical Models and Methods
Poisson processes
Cox processes
Log-Gaussian Cox processes
Inference
Gastro-intestinal illness in Hampshire, UK
Concluding remarks: point processes and geostatistics

Mechanistic Models and Methods
Conditional intensity and likelihood
Partial likelihood
The 2001 foot-and-mouth epidemic in Cumbria, UK
Nesting patterns of Arctic terns

References