Modelling Spatial Patterns

Abstract

Spatial point processes may be analysed at two levels. Quadrat and distance methods were designed for the sampling of a population in the field. In this paper we consider those situations in which a map of a spatial pattern has been produced at some cost and we wish to extract the maximum possible information. We review the stochastic models which have been proposed for spatial point patterns and discuss methods by which the fit of such a model can be tested. Certain models are shown to be the equilibrium distributions of spatial-temporal stochastic processes. The theory is illustrated by several case studies.

Spatio-temporal Modelling of Weeds by Shot-noiseG Cox processes

“…where ''6 ¼" means sum over x 6 ¼ y, and 1=w xy is the proportion of the circumference of the circle with centre x and radius kx À yk lying in W (Ripley, 1977), and where k is the Epanečnikov kernel. The bandwidth of k can be chosen using the guidelines in Stoyan and Stoyan (1994).…”

Canopy gaps and establishment patterns of spruce (Picea abies (L.) Karst.) in two old-growth coniferous forests in central Sweden

“…This transformation stabilizes the sampling variances of the estimates (see Besag in the discussion of Ripley (1977)) and gives a function whose expectation under the null hypothesis is zero for all distances t. Negative values of K*(t) indicate a regular pattern and positive values an aggregated pattern. A plot of K*(t) against t therefore reveals spatial pattern at various scales.…”

Parametric Procedures in the Analysis of Replicated Pairwise Interaction Point Patterns

“…This can be done either graphically, by means of the well known K or L functions (Ripley, 1977(Ripley, , 1988Diggle, 1983) or formally, by means of Monte Carlo methods. In a first step, we use for each of the 12 þ 9 þ 10 patterns individually the L test as described in Ripley (1988) or Stoyan and Stoyan (1994).…”

“…To simulate the point patterns, we used the conditional simulation algorithm proposed by Ripley (1977Ripley ( , 1981 with a random updating scheme, which defines a discrete-time Markov process in which births and deaths alternate. This algorithm can be viewed as single point updating Metropolis-Hastings algorithm with birth-death proposals.…”

“…We might, therefore, consider modelling the cell centre positions parametrically. Pairwise interaction processes (Ripley, 1977;Diggle et al, 1994) can incorporate varying degrees of spatial regularity and provide reasonable empirical models. Maximum pseudo-likelihood (Besag, 1977) provides a computationally straightforward method of parameter estimation for such models, and can be adapted to incorporate replications.…”