http://swrc.ontoware.org/ontology#Thesis
Bayesian Estimation of Repulsive Interaction Potential Models for Spatial Point Patterns
en
岡部 正浩
オカベ マサヒロ
OKABE Masahiro
総研大甲第992号
A spatial Point pattern is a set of locations of points (objects), irregularly distributed within a designated region and presumed to have been generated by some form of stochastic mechanism (Diggle (2003)). Each point is considered as a particle, individual of animals or plants,and so on. During afew decades, the methods of statisticalanalysis for spatial polnt Patterns have been developed: various diagnostic statistics and graphs have been studied by uslng the second-order and the nearest-neighbor distance methods (Ripley (1977, 1979a, 2004), Besag (1977), Diggle (1979, 2003)); by clumping indices based on the quadrate methods (David and Moore (1954), Morisita (1954), Lloyd (1967)) or based on the distance methods (Hopkins and Skellam (1954)). Modelling spatial polnt Patterns for which interactions exist between individuals has been studied by some authors (see for example Mat'ern (1960), Ripley (1977), Ogata and Tanemura (1981, 1984)). Spatial point patterns are generally classified into three types: completely random, clustered (aggregated) and regular. lf we observe a point patten where a certain repulsive force is actlng between individuals,the pattem is called a regular type. For example, if territorial animals or plants live in a habitat, a certain spacng out among them happens. Besides if birds fly in fomation or fish swim in shoals, inhibitions between the individuals realize. In other case, if few or many nanoneter- or micrometer-sized dust particles are immersed in a plasma,then the particles with charge form two-or three-dimensional dust Coulomb crystals. h outer space,the crystals can be observed such asthe ionosphere or commentary tails, etc. h the laboratory,the circularSymJneby of the confining potentialandtheinteraction withthe nutualrepulsion lead to dust Coulonb crystals, which canbe observed bythe naked eyes throughthe CCD camerawith applyingthe laser light. Then the behavior of charged dust particles andthe structtqe of the crystals has beeninvestigated (e.g.Melzer et al･ (1994), Nitter (1996), Juanet al. (1998), Thomasand Watson (1999), Lai and Lin (1999)).<br /> Many regularpolnt PatternSare Observedinnature,then wewish to study the stochastic mechanism of the regularpattems.lmthis thesis, weare particularly interestedinthe interaction between individuals and it will be interestlng to describe this certain spacing out by a repulsive interaction potential. Then we consider these interactions between individuals by repulsive interaction potential models.<br /> We assume that a given regular polnt Pattern is in equilibrium under a certain repulsive interaction potential ina finite two-dimensional region. It is known that such an equilibrium point Pattern is statistically represented by the Gibbs distribution. The likelihood of parameters which characterize the interaction potential can be described by the Gibbs distribution for a given equilibrium point pattern. Since the form of the normalizing factor of the Gibbs distribution is a high multiplicity of integral, it is very difficult to obtain the likelihood function in principle. For this reason, Bayesian analysis for these spatial point patterns has been hardly studied. Then, we use the useful approximate log-likelihood (Ogata and Tanemura (1989)), which will be described in Chapter 3.2,and consider our Bayesian estimation of various regular point pattens. Bayesian inference may help us to sensitively estimate parameters of the interaction potentials･ The essential characteristic of Bayesian methods is their explicit use of probability for quantifying uncertaintyin inferences based on statisticaldata analysis (Gelman et al. (2004)). Because of the development of recent computational methodology, the complex posterior density can be simulated by using MCMC (Markov chain Monte Carlo) methods.<br /> In this thesis, our main purpose is as follows. For a point Pattern of repulsive by interacting points in a finite two-dimensional region, we propose a method to obtain the posterior density of the parameters of the parameterized interaction potential functions by uslng MCMC methods. There, the effective approximate log-likelihood for the models (Ogata and Tanemura (1989)) plays an important role in the Metropolis-Hastings algorithm. Then two types of prior densities corresponding to the parameters of the repulsive interaction potential models are considered. Jumping (proposal) densities with similar type as prior density are applied in Markov chain simulations. Our Bayesian inference is confirmed by applying to various simulated equilibrium polnt Pattems Which are generated from MCMC of the Soft-Core models for the cases of large and relatively small number of points. In order to obtain posterior inference for realdata sets, we consider the fitting of posterior densities to some paranebic functions.<br /> Moreover, MCMC convergence of iterative simulation is als investigated in detail. In the thesis, the approach of single long run is adopted. After a long time iterative simulation have been run in the Metropolis-Hastings algoritlm,there are followlng important problems: when should we begin and finish sampling?, i.e. when does the run begin to reach stationary and when should we terminate the run? To solve these problems, we evaluate the burn-in and the stopplng time of our single long run based on independent simulated multiple short runs with various starting points (Gelman and Rubin (1992), Cowles and Carlin(1996), Gelman et al. (2004)), which will be remarked in Chapte5.3and 8.2.<br /> The layout of the thesis is as follows. In Chapter 2, a log-likelihood of parameters for equilibrium point patten is given. In Chapter 3,the repulsive interaction potential models (soft-Core potential models) with two parameters and their effective approximate log-likelihood are introduced. In Chapter 4,the fundamentals of Bayesian inference for the soft-Core models are described. In Chapter 5,the Metropolis-Hastings algorithm for Bayesian inference, its jumping rule and assessment of the convergence (the burn-in and the stopping time)from iterative simulation are stated. In Chapter 6, firstly, our Bayesian estimation procedure is applied to various simulated equilibrium polnt pattens which are generated by MCMC methods of the Soft-Core models for the cases of large and relatively small number of points. Then MCMC convergence is evaluated and the comparison of marginal posterior densities of parameters under two types of the prior densities is also shown. In Chapter 7, four real data sets are illustrated. Then as a preliminary analysis, we classify the type of distribution of each point pattern. In Chapter 8, the results of our Bayesian estimation of the Soft-Core models for these real data sets are shown. There, the assessment of MCMC convergence is investigated in detail. In order to obtain posterior inference from iterative simulation, parametric fitting of the generalized gamma distribution to marginal posterior densities is considered. To examine the validity of our results, the L-statistics for observed data is compared graphically with the envelopes of simulated point patterns for the posterior mode of model parameters. We then make reference to the literature of Okabe and Tanemura (2006). Finally, in chapter 9, some concluding remarks are given.