Cluster Algorithms for Spatial Point Processes with Symmetric and Shift Invariant Interactions

Abstract

In this article, Swendsen–Wang–Wolff algorithms are extended to simulate spatial point processes with symmetric and stationary interactions. Convergence of these algorithms is considered. Some further generalizations of the algorithms are discussed. The ideas presented in this article can also be useful in handling some large and complicated systems.     

Spatial Regression Models for Extremes

Meteorological data are often recorded at a number of spatial locations. This gives rise to the possibility of pooling data through a spatial model to overcome some of the limitations imposed on an extreme value analysis by a lack of information. In this paper we develop a spatial model for extremes based on a standard representation for site-wise extremal behavior, combined with a spatial latent process for parameter variation over the region. A smooth, but possibly non-linear, spatial structure is an intrinsic feature of the model, and difficulties in computation are solved using Markov chain Monte Carlo inference. A simulation study is carried out to illustrate the potential gain in efficiency achieved by the spatial model. Finally, the model is applied to data generated from a climatological model in order to characterize the hurricane climate of the Gulf and Atlantic coasts of the United States.     

Spatial Point Process Models of Defensive Strategies: Detecting Changes

The study of stochastic processes can take many forms. Theoretical properties are important to ensure consistent model definition. Statistical inference on unknown parameters is equally important but can be difficult. This is principally because many of the standard assumptions for proving consistency and asymptotic normality of estimators involve independence and homogeneity. In the case where inference is concerned with detecting change in a spatial process from one time point to another, a statistical-computing approach can be rewarding. Regardless of the complexity of the stochastic process, if simulating from it is relatively easy, then detecting change is possible using a Monte Carlo approach. The methodology is applied in a military scenario, where a country’s defensive posture changes as a function of its perceived threat. For tactical-decision purposes, it is extremely important to know whether the country’s perceived threat level has changed.     

A J-Function for Marked Point Patterns

We propose a new summary statistic for marked point patterns. The underlying principle is to compare the distance from a marked point to the nearest other marked point in the pattern to the same distance seen from an arbitrary point in space. Information about the range of interaction can be inferred, and the statistic is well-behaved under random mark allocation. We develop a range of Hanisch style kernel estimators to tackle the problems of exploding tail variance earlier associated with J-function plug-in estimators, and carry out an exploratory analysis of a forestry data set.     

层次分析法一致性指标的临界值研究

层次分析法中一致性检验的显著性水平(即判断矩阵偏离一致性的程度)问题在已有研究文献中尚未得到很好的解决。论文在现有一致性指标CI基础上,从随机过程的视角,采用蒙特卡罗模拟法构造一致性指标的随机样本,根据样本确定了显著性水平为0.05和0.01情形下对应的一致性指标的临界值,为评价和调整判断矩阵提供了理论标准。同时,本研究能够确定判断矩阵偏离一致性的程度,并给出决策制定的显著性水平。     

A Computationally Efficient Projection-Based Approach for Spatial Generalized Linear Mixed Models

Inference for spatial generalized linear mixed models (SGLMMs) for high-dimensional non-Gaussian spatial data is computationally intensive. The computational challenge is due to the high-dimensional random effects and because Markov chain Monte Carlo (MCMC) algorithms for these models tend to be slow mixing. Moreover, spatial confounding inflates the variance of fixed effect (regression coefficient) estimates. Our approach addresses both the computational and confounding issues by replacing the high-dimensional spatial random effects with a reduced-dimensional representation based on random projections. Standard MCMC algorithms mix well and the reduced-dimensional setting speeds up computations per iteration. We show, via simulated examples, that Bayesian inference for this reduced-dimensional approach works well both in terms of inference as well as prediction; our methods also compare favorably to existing “reduced-rank” approaches. We also apply our methods to two real world data examples, one on bird count data and the other classifying rock types. Supplementary material for this article is available online.     

A Computational Approach for Full Nonparametric Bayesian Inference Under Dirichlet Process Mixture Models

Widely used parametric generalized linear models are, unfortunately, a somewhat limited class of specifications. Nonparametric aspects are often introduced to enrich this class, resulting in semiparametric models. Focusing on single or k-sample problems, many classical nonparametric approaches are limited to hypothesis testing. Those that allow estimation are limited to certain functionals of the underlying distributions. Moreover, the associated inference often relies upon asymptotics when nonparametric specifications are often most appealing for smaller sample sizes. Bayesian nonparametric approaches avoid asymptotics but have, to date, been limited in the range of inference. Working with Dirichlet process priors, we overcome the limitations of existing simulation-based model fitting approaches which yield inference that is confined to posterior moments of linear functionals of the population distribution. This article provides a computational approach to obtain the entire posterior distribution for more general functionals. We illustrate with three applications: investigation of extreme value distributions associated with a single population, comparison of medians in a k-sample problem, and comparison of survival times from different populations under fairly heavy censoring.     

Forest of stochastic meshes: A new method for valuing high-dimensional swing options

Swing options generalize American-style options as they allow the holder multiple exercise rights and control over the exercise amounts. In this work, we replace the standard (binomial) trees in the forest of trees algorithm with stochastic meshes, yielding the forest of stochastic meshes; a simulation-based method for valuing high-dimensional swing options. This new method handles general price processes and payoffs, produces high- and low-biased consistent estimators and a true option price confidence interval.     

Goodness-of-fit tests for copulas: A review and a power study

Many proposals have been made recently for goodness-of-fit testing of copula models. After reviewing them briefly, the authors concentrate on “blanket tests”, i.e., those whose implementation requires neither an arbitrary categorization of the data nor any strategic choice of smoothing parameter, weight function, kernel, window, etc. The authors present a critical review of these procedures and suggest new ones. They describe and interpret the results of a large Monte Carlo experiment designed to assess the effect of the sample size and the strength of dependence on the level and power of the blanket tests for various combinations of copula models under the null hypothesis and the alternative. To circumvent problems in the determination of the limiting distribution of the test statistics under composite null hypotheses, they recommend the use of a double parametric bootstrap procedure, whose implementation is detailed. They conclude with a number of practical recommendations.