Bayesian strategies for repulsive spatial point processes

Determinantal point processes (DPPs) have recently proved to be a useful class of models in several areas of statistics, including spatial statistics, statistical learning and telecommunications networks. They are models for repulsive (or regular, or inhibitive) point processes, in the sense that nearby points of the process tend to repel each other. We consider two ways to quantify the repulsiveness of a point process, both based on its second-order properties, and we address the question of how repulsive a stationary DPP can be. We determine the most repulsive stationary DPP, when the intensity is fixed, and for a given $R>0$ we investigate repulsiveness in the subclass of $R$-dependent stationary DPPs, that is, stationary DPPs with $R$-compactly supported kernels. Finally, in both the general case and the $R$-dependent case, we present some new parametric families of stationary DPPs that can cover a large range of DPPs, from the stationary Poisson process (the case of no interaction) to the most repulsive DPP.

Distance distributions in random networks

This paper presents a statistically based method for spotting target words in documents. The crux of the method is the representation of a word by a spatial (planar) point process evolving on a regular lattice of coordinate pairs. This is accomplished by extracting the coordinate pairs, i.e. pixel locations, where the binary bitmap values of the word are non-zero. With this representation the word is completely determined by the spatial intensity function, i.e. the unnormalized spatial probability density function, associated with the extracted set of coordinate pairs. In this work, we use a finite number of moments of the intensity function to characterize the word. Location and scale invariance is obtained by transforming the coordinate pairs to have zero mean and unit variance. Finally, optimal detection strategies are applied to the moments to make the decision. 1.

ABSTRACTIn this paper, we derive an exact formula for the covariance of two innovations computed from a spatial Gibbs point process and suggest a fast method for estimating this covariance. We show how this methodology can be used to estimate the asymptotic covariance matrix of the maximum pseudo‐likelihood estimator of the parameters of a spatial Gibbs point process model. This allows us to construct asymptotic confidence intervals for the parameters. We illustrate the efficiency of our procedure in a simulation study for several classical parametric models. The procedure is implemented in the statistical software R, and it is included in spatstat, which is an R package for analyzing spatial point patterns.

Spatio-temporal models for georeferenced unemployment data

Methods for the statistical analysis of stationary spatial point process data are now well established, methods for nonstationary processes less so. One of many sources of nonstationary point process data is a case-control study in environmental epidemiology. In that context, the data consist of a realization of each of two spatial point processes representing the locations, within a specified geographical region, of individual cases of a disease and of controls drawn at random from the population at risk. In this article, we extend work by Baddeley, Møller, and Waagepetersen (2000, Statistica Neerlandica54, 329-350) concerning estimation of the second-order properties of a nonstationary spatial point process. First, we show how case-control data can be used to overcome the problems encountered when using the same data to estimate both a spatially varying intensity and second-order properties. Second, we propose a semiparametric method for adjusting the estimate of intensity so as to take account of explanatory variables attached to the cases and controls. Our primary focus is estimation, but we also propose a new test for spatial clustering that we show to be competitive with existing tests. We describe an application to an ecological study in which juvenile and surviving adult trees assume the roles of controls and cases.

The projection method obtains non-trivial point processes from higher-dimensional Poisson point processes by constructing a random subset of the higher-dimensional space and projecting the points of the Poisson process lying in that set onto the lower-dimensional region. This paper presents a review of this method related to spatial point processes as well as some examples of its applications. The results presented here are known for sometime but were not published before. Also, we present a backward construction of general spatial pure-birth processes and spatial birth and death processes based on the projection method that leads to a perfect simulation scheme for some Gibbs distributions in compact regions.

In making inference on Spatial Point Processes often there it is the problem of knowing (or estimating) the distribution of relevant statistics. In the present paper it is proposed a method for estimating the distribution function of any strongly convergent statistic in a large class of Spatial Point Processes. The method is based on conditional simulation and doesn't depend on process' distribution. Asymptotically the statistics generated by this method of "autosimulation" have the same distribution as those generated by Montecarlo methods when the probability structure of the Process is completely known.

A common nonparametric approach to estimate the intensity function of an inhomogeneous spatial point process is through kernel smoothing. When conducting the smoothing, one typically uses events only in a local set around the point of interest. But the resulting estimator often is inconsistent, because the number of events in a fixed set is of order 1 for spatial point processes. In this article we propose a new covariate-based kernel smoothing method to estimate the intensity function. Our method defines the distance between any two points as the difference between their associated covariate values. Consequently, we determine the kernel weight for a given event of the process as a function of its new distance to the point of interest. Under some suitable conditions on the covariates and the spatial point process, we prove that our new estimator is consistent for the true intensity. To handle the situation with high-dimensional covariates, we also extend sliced inverse regression, a useful dimension-reduction tool in standard regression analysis, to spatial point processes. Simulations and an application to a real data example are used to demonstrate the usefulness of the proposed method.

Networks and point processes provide flexible tools for representing and modeling complex dependencies in data arising from various social and physical domains. Graphs, or networks, encode relational dependencies between entities, while point processes characterize temporal or spatial interactions among events.<br>In the first part of this dissertation, we consider dynamic network data (such as communication networks) in which links connecting pairs of nodes appear continuously over time. We propose latent space point process models to capture two different aspects of the data: (i) communication occurs at a higher rate between individuals with similar latent attributes (i.e., homophily); and (ii) individuals tend to reciprocate communications from others, but in a varied manner. Our framework marries ideas from point process models, including Poisson and Hawkes processes, with ideas from latent space models of static networks. We evaluate our models on several real-world datasets and show that a dual latent space model, which accounts for heterogeneity in both homophily and reciprocity, significantly improves performance in various link prediction and network embedding tasks.<br>In the second part of this dissertation, we develop nonparametric goodness-of-fit tests for discrete distributions and point processes that contain intractable normalization constants, providing the first generally applicable and computationally feasible approaches under those circumstances. Specifically, we propose and characterize Stein operators for discrete distributions, and construct a general Stein operator for point processes using the Papangelou conditional intensity function. Based on the proposed Stein operators, we establish kernelized Stein discrepancy measures for discrete distributions and point processes, which enable us to develop nonparametric goodness-of-fit tests for un-normalized density/intensity functions. We apply the kernelized Stein discrepancy tests to discrete distributions (including network models) as well as temporal and spatial point processes. Our experiments demonstrate that the proposed tests typically outperform two-sample tests based on the maximum mean discrepancy, which, unlike our goodness-of-fit tests, assume the availability of exact samples from the null model.<br>

Point pattern data are ubiquitous in both the natural environment and human activities. Spatial and spatio-temporal point processes are widely utilized to model the occurrence of random events across space and time. These models aim to uncover the underlying patterns of such events by analyzing environmental variables that influence their probabilities, and by testing, estimating and simulating spatial or spatio-temporal correlations. Despite significant advances in point process theory and methodology in the last decades, classical approaches still require continuous refinement and extension. This need arises from the distinctive characteristics of point pattern data in practical applications, as well as the increasing amount and complexity of data available in contemporary research. This thesis explores spatial and spatio-temporal modelling in the context of fire risk prediction in the Netherlands. Fire occurrences exhibit unique features that pose specific challenges in point process modelling. Four critical research questions are identified based on real applications of point process methods to the data. To address these challenges, we developed novel statistical and machine learning approaches, proposing targeted solutions that advance the state-of-the-art in the research domain.<br/><br/>

We investigate the network connectivity of Vehicular Ad Hoc Networks (VANETs) for a typical highway at the free flow state. We propose a new approach to model the spatial distribution of vehicles in a given area using the statistical properties of spatial point patterns. We consider the VANET as a homogenous Poisson point process in which the number of vehicles per unit area has a Poisson distribution. The main advantage of such an approach is its ease of translation to a practical system. Our proposed model provides an analytical result for the vehicles' nearest-neighbour distance distribution. We present simulation results to verify the obtained closed form solution of the network connectivity. Our model is verified by comparing it to real data obtained from the Bureau of Transport Statistics (BTS) NSW, Australia. The main advantage of our proposed model is reducing data acquisition in measuring of data, as the network connectivity can be studied dynamically using aerial images taken by aerial traffic monitoring systems. Another advantage is the ease with which arbitrary function for transmission coverage areas (e.g., antenna radiation pattern) can be included in the final closed form solution which shows capability and flexibility of the proposed model.

Spatial point processes in astronomy

Future mobile networks are forecast as being increasingly heterogeneous and dense. An aspect crucial to managing such networks is the existence of a flexible and effective backhaul infrastructure. Since backhaul infrastructure is essential, it becomes important to analyze the cost of its implementation. This paper aims to realize a framework to estimate deployment costs in a network which consists of users, two types of base stations, and backhaul nodes that could either be microwave or fiber optic backhaul nodes. The main contribution of this work is the derivation of a framework using spatial point processes that helps estimate the cost of deploying a backhaul node based on the number of users and base stations connected to it. The framework, along with assumptions of typical costs of various network components, is utilized to examine whether there exist an optimal number of backhaul nodes that can minimize the overall deployment cost of a network while catering to a given number of users in the area.

Modeling spatial and spatio-temporal continuous processes is an important and challenging problem in spatial statistics. Advanced Spatial Modeling with Stochastic Partial Differential Equations Using R and INLA describes in detail the stochastic partial differential equations (SPDE) approach for modeling continuous spatial processes with a Matern covariance, which has been implemented using the integrated nested Laplace approximation (INLA) in the R-INLA package. Key concepts about modeling spatial processes and the SPDE approach are explained with examples using simulated data and real applications. This book has been authored by leading experts in spatial statistics, including the main developers of the INLA and SPDE methodologies and the R-INLA package. It also includes a wide range of applications: * Spatial and spatio-temporal models for continuous outcomes * Analysis of spatial and spatio-temporal point patterns * Coregionalization spatial and spatio-temporal models * Measurement error spatial models * Modeling preferential sampling * Spatial and spatio-temporal models with physical barriers * Survival analysis with spatial effects * Dynamic space-time regression * Spatial and spatio-temporal models for extremes * Hurdle models with spatial effects * Penalized Complexity priors for spatial models All the examples in the book are fully reproducible. Further information about this book, as well as the R code and datasets used, is available from the book website at http://www.r-inla.org/spde-book. The tools described in this book will be useful to researchers in many fields such as biostatistics, spatial statistics, environmental sciences, epidemiology, ecology and others. Graduate and Ph.D. students will also find this book and associated files a valuable resource to learn INLA and the SPDE approach for spatial modeling.