We present a hierarchical Bayesian framework for non-homogeneous pairwise interaction Gibbs point process models, where the global and local effect functions are modeled via basis function expansions. We further propose a testing procedure in order to assess complete spatial randomness. The proposed methodology is exemplified through two real benchmark data examples involving water striders and forest fires.

It is common in nature to see aggregation of objects in space. Exploring the mechanism associated with the locations of such clustered observations can be essential to understanding the phenomenon, such as the source of spatial heterogeneity, or comparison to other event generating processes in the same domain. Log-Gaussian Cox processes (LGCPs) represent an important class of models for quantifying aggregation in a spatial point pattern. However, implementing likelihood-based Bayesian inference for such models presents many computational challenges, particularly in high dimensions. In this paper, we propose a novel likelihood-free inference approach for LGCPs using the recently developed BayesFlow approach, in which invertible neural networks are employed to approximate the posterior distribution of parameters of interest. BayesFlow is a neural simulation-based method based on "amortized" posterior estimation. That is, after an initial training procedure, fast feed-forward operations allow rapid posterior inference for any data within the same model family. Comprehensive numerical studies validate the reliability of the framework and show that BayesFlow achieves substantial computational gain in repeated application, especially for two-dimensional LGCPs. We demonstrate the utility and robustness of the method by applying it to two distinct oral microbial biofilm images.

We propose and study a novel collection of signed measures, which will be called Taylor measures. Stochastic versions of the new measures are also defined and studied. We illustrate, through examples, how the deterministic and stochastic versions of the proposed Taylor measures emerge as a unifying framework that includes many concepts from mathematics and probability theory as special cases.

We generalize Taylor's theorem by introducing a stochastic formulation based on an underlying Poisson point process model. We utilize this approach to propose a novel non-linear regression framework and perform statistical inference of the model parameters. Theoretical properties of the proposed estimator are also proven, including its convergence, uniformly almost surely, to the true function. The theory is presented for the univariate and multivariate cases, and we exemplify the proposed methodology using several examples via simulations and an application to stock market data.

We introduce and study a new class of Cox point processes, based on random mixture models of exponential family components for the intensity function of the underlying Poisson process. We investigate theoretical properties of the proposed probability distributions of the point process, as well as provide procedures for parameter estimation using a classical and Bayesian approach. We illustrate the richness of the new models through examples, simulations and real data applications.

We develop a new approach to creating covariance functions for Gaussian random fields via point processes on the complex plane. We present two approaches to construct valid covariance functions by exploiting Bochner’s theorem and then modeling the characteristic function of a covariance function. In particular, we use a complex point process (CPP) to model the Fourier coefficients and illustrate how to estimate the covariance function of a Gaussian random field model from data. We further illustrate our construction approaches and compare several algorithms via simulations. The methods are exemplified via applications to real-life research data in wheat yields and earthquake studies.

In this paper we study point processes on the complex plane and illustrate their uses in several statistical areas, where the quantities of interest requiring estimation involve Fourier expansions. In particular, for any problem where we can describe a quantity in terms of its Fourier expansion, we propose modeling the coefficients of the expansion using a point process on the complex plane. We utilize the Poisson complex point process and model its intensity function using log-linear and mixture models. The proposed models are exemplified via applications to general density approximation, via modeling of the characteristic function, and time series analysis, via modeling of the spectral density.

To model spatial point patterns with discrete time stamps a flexible spatio-temporal area-interaction point process is proposed. In particular, this model is suitable for describing the dependency between point patterns over time, when the new point pattern arises from the previous point pattern. A hierarchical model is also implemented in order to incorporate the underlying evolution process of the model parameters. For parameter estimation, a double Metropolis-Hastings within Gibbs sampler is used. The performance of the estimation algorithm is evaluated through a simulation study. Finally, the point pattern forecasting procedure is demonstrated through a simulation study and an application to United States natural caused wildfire data from 2002 to 2019.

Crucial aspects of applying approximate Bayesian computation (ABC) for Gibbs point processes are the choice of summary statistic and method of constructing the discrepancy measure. In this paper, we present a comparative study of ABC for Gibbs point processes based on various summary statistics and different approaches of constructing the discrepancy measure. We also demonstrate the issue of identifiability of the parameter values for Gibbs point processes and provide a solution for parameter estimation. We further propose robust choices for the discrepancy measures for different point processes through an intensive simulation study. The ABC algorithm, with all of the tested discrepancy measures, is also applied to the Swedish pines data and Chicago crime data to illustrate the feasibility of the proposed approaches.

We present a flexible hierarchical Bayesian model and develop a comprehensive Bayesian decision theoretic framework for point process theory. We closely investigate the commonly used point process model for independent events, the Poisson process, using a finite mixture of exponential family components to model the intensity function. We employ a Bayesian hierarchical framework for parameter estimation and illustrate the Bayesian computations involved. We demonstrate the effectiveness of the Bayes rule under the Kullback–Leibler and Hellinger loss functions and compare them with the usual estimator, the posterior mean under squared error loss. The methodology is exemplified through simulations and a motivating application involving estimation of the intensity surface of homicide incidents in Chicago during 2015.

In this paper, we introduce a class of local divergences between two probability distributions and illustrate its usefulness in model selection. Explicit expressions of the proposed local divergences are derived when the underlying distributions are members of the exponential family of distributions or they are described by multivariate normal models. In addition, a local model selection criterion, termed the local divergence information criterion (LDiv.IC), is proposed. Simulations and applications are presented in order to study and exemplify the performance of the proposed criterion.

The design problem associated with robust downlink beamforming in multicast, multigroup, multicell wireless systems is addressed. The channel state information (CSI) of users is assumed to be imperfect and the uncertainty of CSI is modelled using the Frobenius norm. The objective is to optimise the signal-to-interference-plus-noise ratio over all users with a constraint on the maximum total transmitted power. This was achieved through a robust solution using the successive convex approximation (SCA) method. The beamforming problem is treated as a bi-convex problem, which is solved using the iterate-alternative convex technique. Here, the CSI uncertainty is addressed using a convex package through the non-monotone spectral projected gradient method and the beamforming vector is extracted using the SCA method. Also, the authors offer the required condition to extract the beamform vector using the SCA method through a suboptimal solution that always addressed before using different beamforming methods. Their simulation results examine all proposed system parameters in order to show convergence and feasibility of the solution. They also compare the solution with a suboptimal solution and the quality of service method for imperfect CSI in downlink beamforming. Numerical results show that the robust solution achieves the best power efficiency for practical solutions.

In this paper, the spectrum sensing problem is addressed via using multiple antennas in cognitive radio systems when the noise and the primary user signal are independent. The problem of optimal detection is considered, and an estimation framework is presented where the primary user signal is a complex zero-mean Gaussian distribution. This paper discusses the generalized likelihood ratio detector (GLRT) when the sample size is a finite number and derive the corresponding GLRT detector of the spectrum sensing problem in a multiantenna framework by utilizing the statistics of the received signal, the channel information and the prior information of the noise. The scenario involves the assumption that the noise variance is unknown while the channel matrix gain is known for the secondary user (SU). The simulation results show that the proposed Bayesian GLRT detector is optimal even under a limited number of samples and more importantly also clarify that the proposed detector outperforms other the state-of-the-art detectors.

We review a rich class of point process models, Cox point processes, and illustrate the necessity of more than one observation (point patterns) in performing parameter estimation. Furthermore, we introduce a new Cox point process model by treating the intensity function of the underlying Poisson point process as a random mixture of normal components. The behaviour and performance of the new model are compared with those of popular Cox point process models. The new model is exemplified with an application that involves a single point pattern corresponding to earthquake events in California, USA.

This paper describes the package sppmix for the statistical environment R. The sppmix package implements classes and methods for modeling spatial point patterns using inhomogeneous Poisson point processes, where the intensity surface is assumed to be a multiple of a finite additive mixture of normal components and the number of components is a finite, fixed or random integer. Extensions to the marked inhomogeneous Poisson point processes case are also presented. We provide an extensive suite of R functions that can be used to simulate, visualize and model point patterns, estimate the parameters of the models, assess convergence of the algorithms and perform model selection and checking in the proposed modeling context. In addition, several approaches have been implemented in order to handle the standard label switching issue which arises in any modeling approach involving mixture models. We adapt a hierarchical Bayesian framework in order to model the intensity surfaces and have implemented two major algorithms in order to estimate the parameters of the mixture models involved: the data augmentation and the birth–death Markov chain Monte Carlo (DAMCMC and BDMCMC). We used C++ (via the Rcpp package) in order to implement the most computationally intensive algorithms.

The aim of this paper is to propose procedures that test statistical hypotheses locally, that is, assess the validity of a model in a specific domain of the data. In this context, the one and two sample problems will be discussed. The proposed tests are based on local divergences which are defined in such a way as to quantify the divergence between probability distributions locally, in a specific area of the joint domain of the underlined models. The theoretical results are exemplified using simulations and two real datasets.

A broad class of local divergences between two probability measures or between the respective probability distributions is proposed in this paper. The introduced local divergences are based on the classic Csiszár \(\phi \)-divergence and they provide with a pseudo-distance between two distributions on a specific area of their common domain. The range of values of the introduced class of local divergences is derived and explicit expressions of the proposed local divergences are also derived when the underlined distributions are members of the exponential family of distributions or they are described by multivariate normal models. An application is presented to illustrate the behavior of local divergences.

This paper introduces the use of Bayesian full Procrustes shape analysis in object-oriented meteorological applications. In particular, the Procrustes methodology is used to generate mean forecast precipitation fields from a set of ensemble forecasts. This approach has advantages over other ensemble averaging techniques in that it can produce a forecast that retains the morphological features of the precipitation structures and present the range of forecast outcomes represented by the ensemble. The production of the ensemble mean avoids the problems of smoothing that result from simple pixel or cell averaging, while producing credible sets that retain information on ensemble spread. Also in this paper, the full Bayesian Procrustes scheme is used as an object verification tool for precipitation forecasts. This is an extension of a previously presented Procrustes shape analysis based verification approach into a full Bayesian format designed to handle the verification of precipitation forecasts that match objects from an ensemble of forecast fields to a single truth image. The methodology is tested on radar reflectivity nowcasts produced in the Warning Decision Support System — Integrated Information (WDSS-II) by varying parameters in the K-means cluster tracking scheme.