First- and Second-Order Properties of Spatiotemporal Point Patterns in the Space-Time and Frequency Domains

Abstract

Point processes are common in many physical applications found in engineering and biology. These processes can be observed in one-dimension as a time series or two-dimensions as a spatial point pattern with extensive amounts of literature devoted to their analyses. However, if the observed process is a hybrid of spatial and temporal point process, very few practical methods exist. In such cases, practitioners often remove the temporal component and analyze the spatial dependencies. This marginal spatial analysis may lead to misleading results if time is an important factor in the process.

In this dissertation we extend the current analysis of spatial point patterns to include a temporal dimension. First- and second-order intensity measures for analyzing spatiotemporal point patterns are explicitly defined. Estimation of first-order intensities are examined using 3-dimensional smoothing techniques.

Conditions for weak stationarity are provided so that subsequent second-order analysis can be conducted. We consider second-order analysis of spatiotemporal point patterns first in the space-time domain through an extension of Ripley's Κ-function. An alternative analysis is given in the frequency domain though construction of a spatiotemporal periodogram.

The methodology provided is tested through simulation of spatiotemporal point patterns and by analysis of a real data set. The biological application concerns the estimation of the homerange of groups of the endangered red-cockaded woodpecker in the Fort Bragg area of North Carolina. Monthly or bimonthly point patterns of the bird distribution are analyzed and integrated over a 23 month period.