Masaharu Tanemura

Estimation of interaction potentials of spatial point patterns through the maximum likelihood procedure

SummaryA homogeneous spatial point pattern is regarded as one of thermal equilibrium configurations whose points interact on each other through a certain pairwise potential. Parameterizing the potential function, the likelihood is then defined by the Gibbs canonical ensemble. A Monte Carlo simulation method is reviewed to obtain equilibrium point patterns which correspond to a given potential function. An approximate log likelihood function for gas-like patterns is derived in order to compute the maximum likelihood estimates efficiently. Some parametric potential functions are suggested, and the Akaike Information Criterion is used for model selection. The feasibility of our procedure is demonstrated by some computer experiments. Using the procedure, some real data are investigated.

On Parameter Estimation for Pairwise Interaction Point Processes

Summary Pairwise interaction point processes form a useful class of models for spatial point patterns, especially patterns for which the spatial distribution of points is more regular than for a homogeneous planar Poisson process. Several authors have proposed methods for estimating the parameters of a pairwise interaction point process. However, there appears to be no general theory which provides grounds for preferring a particular method, nor have any extensive empirical comparisons been published. In this paper, we review three general methods of estimation which have been proposed in the literature and present the results of a comparative simulation study of the three methods.

Geometrical models of territory. I. Models for synchronous and asynchronous settlement of territories.

Abstract There is often the case in which it can be assumed that the condition of a habitat of territorial animals is uniform and that the habitat is divided into territories without any gap. If this were the case, the density and the spatial pattern of territories will be mainly affected by the interaction between individuals and by the settling process of territories. This paper presents geometrical models of the establishment of territories, where it is supposed that the repulsive forces act among individuals and that all individuals are identical in strength. In the case of synchronous settlement of territories, individuals will adjust the position of their territorial centres until a stable set of boundaries is obtained. A model of this process is constructed under the assumption that each territory is regarded as a Voronoi polygon. In the case of asynchronous settlement, on the contrary, the adjustment of centres will not occur. This settling process is studied as a problem of random sequential packing of circles with a fixed radius. The models for respective settling processes explain well the existing data of observations.

On random complete packing by discs

SummaryA computational algorithm for random complete packing by discs is proposed. Monte Carlo simulations using this algorithm give the value 0.5473 for random packing density of discs. It greatly improves the Solomons result, 04756.

On the pattern of space division by territories

SummaryIn this paper the tesselation of territories is discussed. When a mass of animals with teritoriality is introduced at random simultaneously into a finite two-dimensional (2D) region, territories are gradually formed and finally settle to a steady state. A model calculation for this process is carried out, where a Voronoi polygon is assumed as a territory. Comparisons are made between the model calculation and Barlows observation on an artificial population of mouthbreeder fish. Differences between 1D and 2D case are also discussed.

Modelling heterogeneous space–time occurrences of earthquakes and its residual analysis

Earthquake intensities are modelled as a function of previous activity whose specific form is based on established empirical laws in seismology, but whose parameter values can vary from place to place. This model is used for characterizing regional features of seismic activities in and around Japan, and also for exploring regions where the actual seismicity rate systematically deviates from that of the modelled rate. Copyright 2003 Royal Statistical Society.

Estimation of interaction potentials of marked spatial point patterns through the maximum likelihood method

The likelihood procedure in estimating interaction potentials of spatial point patterns is developed for a set of point locations with attached marks, where the potential functions depend on the marks. Approximate log likelihood functions are derived under the assumption that the point patterns do not deviate much from the Poisson pattern in some sense. Some methods examining this assumption are provided. Analyses of ecological data sets are discussed through a model selection procedure.

Likelihood estimation of soft-core interaction potentials for Gibbsian point patterns

The likelihood method is developed for the analysis of socalled regular point patterns. Approximating the normalizing factor of Gibbs canonical distribution, we simultaneously estimate two parameters, one for the scale and the other which measures the softness (or hardness), of repulsive interactions between points. The approximations are useful up to a considerably high density. Some real data are analyzed to illustrate the utility of the parameters for characterizing the regular point pattern.

Spatial data analysis by epidermal Langerhans cells reveals an elegant system

Langerhans cells are dendritic cells situated in the mammalian epidermis. In human epidermis, the concentration is between 460 and 1000 mm(-2). Langerhans cells fulfill an essential role in skin immune responses. Numerous scientific reports on Langerhans cells have appeared, but with no systematic research on the pattern of the spatial distributions. On the contrary, in certain fields, a spatial distribution is an important theme, and spatial data analysis has a long history. We hypothesized that epidermal Langerhans cells were set in the best formation for their immuno-surveillance by a sophisticated mechanism. To prove this hypothesis, we have imported spatial data analysis into the study of epidermal Langerhans cells. Here, we show that the distribution is completely regular; the pattern of Voronoi divisions fits the territories; the random packing model simulates their bone marrow derivation; a repulsive interaction is demonstrated and a repulsive potential function is estimated. Spatial data analysis-based computer simulation will be a new method of Langerhans cell study. In addition, this procedure shows promise for future distribution research of certain cells.

Summary of Contributed Papers to Volume 3

In this paper the informational approach to statistics based on an informational interpretation of log likelihood of a statistical model is contrasted with the conventional Fisherian framework. It is pointed out that for the recognition of practical importance of statistical science more emphasis should be placed on the development of new models than the refinement of mathematical analysis of existing procedures, although the latter activity should never be ignored. This is based on the observation that it is the contribution of statistics to the development of new hypotheses for the interpretation of data that creates the importance of the subject. It is argued that the informational approach prepares a basis for the rapid expansion of the area of application of statistics by providing a theoretical foundation for the development and evaluation of new statistical models in diverse fields.