Mechanistic Spatio-Temporal Point Process Models for Marked Point Processes, With a View to Forest Stand Data

Abstract

We show how a spatial point process, where to each point there is associated a random quantitative mark, can be identified with a spatio-temporal point process specified by a conditional intensity function. For instance, the points can be tree locations, the marks can express the size of trees, and the conditional intensity function can describe the distribution of a tree (i.e., its location and size) conditionally on the larger trees. This enable us to construct parametric statistical models which are easily interpretable and where maximum-likelihood-based inference is tractable.

Spatiotemporal Data Warehousing

“…Such processes may be considered when the underlying point process arises as a consequence of environmental variation in intensity that cannot be described by available explanatory variables: such unobserved environmental variations are thus defined through the stochastic intensity. A widely used class of Cox process models is the log-Gaussian class (Møller et al 1998), extended to spatiotemporal setting in and Brix and Møller (2001). For this class, .s; t / D exp.Z.s; t //, where Z.s; t/ is a spatio-temporal Gaussian random field.…”

“…where H t is the history of the process up to time t and ds and dt are infinitesimal spatial and temporal regions containing s and t . See for instance Fishman and Snyder (1976) for a pioneer description of mechanistic spatio-temporal point process models or Møller et al (2014) for a very recent one in the context of marked point processes. Inference of mechanistic models is usually performed by likelihood or partial likelihood methods .…”

Cox Processes Associated with Spatial Copula Observed through Stratified Sampling

“…In case of observations corresponding to the realization of a point process, the available data can be either point spatial locations or counts of points in spatial units [1,2]. These authors developed the mathematical theory of point processes which are used in many areas for describing event occurrences, like earthquakes, accidents, pest infestations, disease appearances, neuronal spikes, and many other situations [3][4][5][6][7]. Thus, a wide range of scientific applications can be cited, for example: ecology, epidemiology, geology, forestry, neurophysiology.…”

Functional marked point processes: a natural structure to unify spatio-temporal frameworks and to analyse dependent functional data

“…Note that by construction, all components F i have the same marginal distributions. Under such a setup, a so-called binomial point process (Møller and Waagepetersen 2004;van Lieshout 2000) would yield the classical FDA setup mentioned above. Note that the idea of analysing point patterns (collections of points) with attached functions has already been noted in the literature (Comas 2009;Delicado et al 2010).…”