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.

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.

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.

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.

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.","subitem_description_type":"Other"}]},"item_1_description_7":{"attribute_name":"学位記番号","attribute_value_mlt":[{"subitem_description":"総研大甲第992号","subitem_description_type":"Other"}]},"item_1_select_14":{"attribute_name":"所蔵","attribute_value_mlt":[{"subitem_select_item":"有"}]},"item_1_select_8":{"attribute_name":"研究科","attribute_value_mlt":[{"subitem_select_item":"複合科学研究科"}]},"item_1_select_9":{"attribute_name":"専攻","attribute_value_mlt":[{"subitem_select_item":"15 統計科学専攻"}]},"item_1_text_10":{"attribute_name":"学位授与年度","attribute_value_mlt":[{"subitem_text_value":"2006"}]},"item_creator":{"attribute_name":"著者","attribute_type":"creator","attribute_value_mlt":[{"creatorNames":[{"creatorName":"OKABE, Masahiro","creatorNameLang":"en"}],"nameIdentifiers":[{}]}]},"item_files":{"attribute_name":"ファイル情報","attribute_type":"file","attribute_value_mlt":[{"accessrole":"open_date","date":[{"dateType":"Available","dateValue":"2016-02-17"}],"displaytype":"simple","filename":"甲992_要旨.pdf","filesize":[{"value":"325.1 kB"}],"format":"application/pdf","licensetype":"license_11","mimetype":"application/pdf","url":{"label":"要旨・審査要旨","url":"https://ir.soken.ac.jp/record/775/files/甲992_要旨.pdf"},"version_id":"458882d7-384e-436a-9f8f-176ad3a6602e"},{"accessrole":"open_date","date":[{"dateType":"Available","dateValue":"2016-02-17"}],"displaytype":"simple","filename":"甲992_本文.pdf","filesize":[{"value":"4.0 MB"}],"format":"application/pdf","licensetype":"license_11","mimetype":"application/pdf","url":{"label":"本文","url":"https://ir.soken.ac.jp/record/775/files/甲992_本文.pdf"},"version_id":"b74f21f5-bc65-4c72-8f80-4d63fbc5e722"}]},"item_language":{"attribute_name":"言語","attribute_value_mlt":[{"subitem_language":"eng"}]},"item_resource_type":{"attribute_name":"資源タイプ","attribute_value_mlt":[{"resourcetype":"thesis","resourceuri":"http://purl.org/coar/resource_type/c_46ec"}]},"item_title":"Bayesian Estimation of Repulsive Interaction Potential Models for Spatial Point Patterns","item_titles":{"attribute_name":"タイトル","attribute_value_mlt":[{"subitem_title":"Bayesian Estimation of Repulsive Interaction Potential Models for Spatial Point Patterns"},{"subitem_title":"Bayesian Estimation of Repulsive Interaction Potential Models for Spatial Point Patterns","subitem_title_language":"en"}]},"item_type_id":"1","owner":"1","path":["17"],"pubdate":{"attribute_name":"公開日","attribute_value":"2010-02-22"},"publish_date":"2010-02-22","publish_status":"0","recid":"775","relation_version_is_last":true,"title":["Bayesian Estimation of Repulsive Interaction Potential Models for Spatial Point Patterns"],"weko_creator_id":"1","weko_shared_id":1},"updated":"2023-06-20T15:59:50.889596+00:00"}