Spatial and spatial-temporal point process analysis Open Access

Wang, Ming (2013)

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Spatial-temporal point pattern data are increasingly available including one-dimensional, directionless observations observed along a line. Our motivating example involves sea turtle nesting data with space and time-specific emergence locations along Juno Beach, Palm Beach County, Florida for the years 1998-2000. Our objectives are to assess spatial and temporal heterogeneity in sea turtle nesting patterns, and detect possible effects due to a 990-foot fishing pier constructed in year 1998-1999. We mainly focus on one-dimensional spatial and spatial-temporal point processes to conduct statistical inference through non-parametric and parametric methods, and yield insights about the first-order and second-order properties of point processes as well as space-time interaction.

A Log-Gaussian Cox Process (LGCP), a Cox point process with the logarithm of intensity function following a Gaussian Process, provides a flexible framework for modeling heterogeneous spatial point processes. The pair correlation function (PCF) plays a vital role in characterizing second-order spatial dependency in LGCPs and delivers key inputs on spatial association structures, yet empirical estimation of the PCF remains challenging, even more so for spatial point processes in one dimension (points along a line). We consider two common edge-correction approaches during nonparametric estimation of the PCF, and evaluate their performance via simulation for one-dimensional spatial data. We also provide a novel algorithm to estimate the PCF based on theoretical derivation combined with finite sample simulation of the kth (k<=4) moment, revealing useful information for optimal spatial designs.

To assess local variations in sea turtle nesting density, we develop a novel hierarchical Bayesian non-parametric model based on Dirichlet processes. Autoregressive temporal dependencies are incorporated in a three-level hierarchical structure. This model allows the potential for time-evolving mixed components/weights across groups. We compare our model with the existing models, e.g., Dirichlet process mixture models, hierarchical Dirichlet process models, and dynamic hierarchical Dirichlet process models, to show its advantage via simulation and real data application to our motivating example.

Table of Contents

1 Introduction

1.1 Background

1.2 First and Second-order properties

1.2.1 Intensity

1.2.2 K-function and L-function

1.2.3 Pair correlation function (PCF)

1.3 Cox process: Log-Gaussian Cox process

1.4 Motivating Example

2 Evaluation of non-parametric pair correlation fucntions for spatial Log-Gaussian Cox processes

2.1 Introduction

2.2 Methods

2.3 Statistical inference

2.4 Simulation

2.5 Conclusion and Discussion

2.6 Appendix

2.6.1 The bias of g(r)

2.6.2 The variance of g(r)

3 Spatial-temporal point pattern analysis of sea turtle nesting and emergence locations

3.1 Introduction

3.2 Methods

3.2.1. Data preprocssing

3.2.2. Notation

3.2.3. Spatial-temporal Intensity

3.2.4. Spatial-temporal K-function and PCF

3.2.5. Space-time clustering and interaction

3.3 Application to the sea turtle nesting data

3.3.1. Spatial-temporal point pattern analysis

3.3.2. Log-Gaussian Cox Processes

3.4 Conclusion and Discussion

4 Non-parametric Bayeisan modeling for density estimation of sea turtle nesting locations along Juno Beach in Florida

4.1 Introduction and motivation data

4.2 Dirichlet process mixture model

4.2.1 Model

4.2.2 Algorithm and Inference

4.3 Hierarchical Dirichlet process model

4.3.1 Model

4.3.2 Algorithm and Inference

4.4 Hierarchical Dirichlet process autoregressive model

4.4.1 Model

4.4.2 Algorithm and Inference

4.5 Simulation

4.6 Results

4.7 Conclusion and Discussion

5 Summary and future research

6 Appendix

6.1 R program for Chapter 2

6.2 R program for Chapter 3

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