Reduced Palm distribution generating function for a spatial point process

The generating function of a marginal distribution of the reduced Palm distribution of a spatial point process is considered. It serves as a bivariate summary function, providing more information than some other popular univariate summary functions, such as the reduced second-moment function and the nearest-neighbour distance distribution function. Simulation confirmed that the new summary function is more informative when applied to patterns that exhibit both clustering and regularity on the same scale of observation.     

A Sequential Point Process Model and Bayesian Inference for Spatial Point Patterns with Linear Structures

Abstract. We introduce a flexible spatial point process model for spatial point patterns exhibiting linear structures, without incorporating a latent line process. The model is given by an underlying sequential point process model. Under this model, the points can be of one of three types: a ‘background point’ an ‘independent cluster point’ or a ‘dependent cluster point’. The background and independent cluster points are thought to exhibit ‘complete spatial randomness’, whereas the dependent cluster points are likely to occur close to previous cluster points. We demonstrate the flexibility of the model for producing point patterns with linear structures and propose to use the model as the likelihood in a Bayesian setting when analysing a spatial point pattern exhibiting linear structures. We illustrate this methodology by analysing two spatial point pattern datasets (locations of bronze age graves in Denmark and locations of mountain tops in Spain).     

A Stochastic Geometry Model for Functional Magnetic Resonance Images

In functional magnetic resonance imaging, spatial activation patterns are commonly estimated using a non-parametric smoothing approach. Significant peaks or clusters in the smoothed image are subsequently identified by testing the null hypothesis of lack of activation in every volume element of the scans. A weakness of this approach is the lack of a model for the activation pattern; this makes it difficult to determine the variance of estimates, to test specific neuroscientific hypotheses or to incorporate prior information about the brain area under study in the analysis. These issues may be addressed by formulating explicit spatial models for the activation and using simulation methods for inference. We present one such approach, based on a marked point process prior. Informally, one may think of the points as centres of activation, and the marks as parameters describing the shape and area of the surrounding cluster. We present an MCMC algorithm for making inference in the model and compare the approach with a traditional non-parametric method, using both simulated and visual stimulation data. Finally we discuss extensions of the model and the inferential framework to account for non-stationary responses and spatio-temporal correlation.     

NON‐PARAMETRIC BAYESIAN INFERENCE FOR INHOMOGENEOUS MARKOV POINT PROCESSES

With reference to a specific dataset, we consider how to perform a flexible non‐parametric Bayesian analysis of an inhomogeneous point pattern modelled by a Markov point process, with a location‐dependent first‐order term and pairwise interaction only. A priori we assume that the first‐order term is a shot noise process, and that the interaction function for a pair of points depends only on the distance between the two points and is a piecewise linear function modelled by a marked Poisson process. Simulation of the resulting posterior distribution using a Metropolis–Hastings algorithm in the ‘conventional’ way involves evaluating ratios of unknown normalizing constants. We avoid this problem by applying a recently introduced auxiliary variable technique. In the present setting, the auxiliary variable used is an example of a partially ordered Markov point process model.     

A hierarchical point process with application to storm cell modelling

In environmetrics, interest often centres around the development of models and methods for making inference on observed point patterns assumed to be generated by latent spatial or spatio‐temporal processes, which may have a hierarchical structure. In this research, motivated by the analysis of spatio‐temporal storm cell data, we generalize the Neyman–Scott parent–child process to account for hierarchical clustering. This is accomplished by allowing the parents to follow a log‐Gaussian Cox process thereby incorporating correlation and facilitating inference at all levels of the hierarchy. This approach is applied to monthly storm cell data from the Bismarck, North Dakota radar station from April through August 2003 and we compare these results to simpler cluster processes to demonstrate the advantages of accounting for both levels of correlation present in these hierarchically clustered point patterns. The Canadian Journal of Statistics 47: 46–64; 2019 © 2019 Statistical Society of Canada     

Modelling Aggregation on the Large Scale and Regularity on the Small Scale in Spatial Point Pattern Datasets

We consider a dependent thinning of a regular point process with the aim of obtaining aggregation on the large scale and regularity on the small scale in the resulting target point process of retained points. Various parametric models for the underlying processes are suggested and the properties of the target point process are studied. Simulation and inference procedures are discussed when a realization of the target point process is observed, depending on whether the thinned points are observed or not. The paper extends previous work by Dietrich Stoyan on interrupted point processes.     

Contrast Estimation for Parametric Stationary Determinantal Point Processes

We study minimum contrast estimation for parametric stationary determinantal point processes. These processes form a useful class of models for repulsive (or regular, or inhibitive) point patterns and are already applied in numerous statistical applications. Our main focus is on minimum contrast methods based on the Ripley's K‐function or on the pair correlation function. Strong consistency and asymptotic normality of theses procedures are proved under general conditions that only concern the existence of the process and its regularity with respect to the parameters. A key ingredient of the proofs is the recently established Brillinger mixing property of stationary determinantal point processes. This work may be viewed as a complement to the study of Y. Guan and M. Sherman who establish the same kind of asymptotic properties for a large class of Cox processes, which in turn are models for clustering (or aggregation).     

Flexible clustering via hidden hierarchical Dirichlet priors

The Bayesian approach to inference stands out for naturally allowing borrowing information across heterogeneous populations, with different samples possibly sharing the same distribution. A popular Bayesian nonparametric model for clustering probability distributions is the nested Dirichlet process, which however has the drawback of grouping distributions in a single cluster when ties are observed across samples. With the goal of achieving a flexible and effective clustering method for both samples and observations, we investigate a nonparametric prior that arises as the composition of two different discrete random structures and derive a closed-form expression for the induced distribution of the random partition, the fundamental tool regulating the clustering behavior of the model. On the one hand, this allows to gain a deeper insight into the theoretical properties of the model and, on the other hand, it yields an MCMC algorithm for evaluating Bayesian inferences of interest. Moreover, we single out limitations of this algorithm when working with more than two populations and, consequently, devise an alternative more efficient sampling scheme, which as a by-product, allows testing homogeneity between different populations. Finally, we perform a comparison with the nested Dirichlet process and provide illustrative examples of both synthetic and real data.     

Curve prediction and clustering with mixtures of Gaussian process functional regression models

Shi, Wang, Murray-Smith and Titterington (Biometrics 63:714–723, 2007) proposed a Gaussian process functional regression (GPFR) model to model functional response curves with a set of functional covariates. Two main problems are addressed by their method: modelling nonlinear and nonparametric regression relationship and modelling covariance structure and mean structure simultaneously. The method gives very good results for curve fitting and prediction but side-steps the problem of heterogeneity. In this paper we present a new method for modelling functional data with ‘spatially’ indexed data, i.e., the heterogeneity is dependent on factors such as region and individual patient’s information. For data collected from different sources, we assume that the data corresponding to each curve (or batch) follows a Gaussian process functional regression model as a lower-level model, and introduce an allocation model for the latent indicator variables as a higher-level model. This higher-level model is dependent on the information related to each batch. This method takes advantage of both GPFR and mixture models and therefore improves the accuracy of predictions. The mixture model has also been used for curve clustering, but focusing on the problem of clustering functional relationships between response curve and covariates, i.e. the clustering is based on the surface shape of the functional response against the set of functional covariates. The model is examined on simulated data and real data.     

Leverage and Influence Diagnostics for Spatial Point Processes

Abstract. For a spatial point process model fitted to spatial point pattern data, we develop diagnostics for model validation, analogous to the classical measures of leverage and influence in a generalized linear model. The diagnostics can be characterized as derivatives of basic functionals of the model. They can also be derived heuristically (and computed in practice) as the limits of classical diagnostics under increasingly fine discretizations of the spatial domain. We apply the diagnostics to two example datasets where there are concerns about model validity.     

State estimation for aoristic models

Aoristic data can be described by a marked point process in time in which the points cannot be observed directly but are known to lie in observable intervals, the marks. We consider Bayesian state estimation for the latent points when the marks are modeled in terms of an alternating renewal process in equilibrium and the prior is a Markov point process. We derive the posterior distribution, estimate its parameters and present some examples that illustrate the influence of the prior distribution. The model is then used to estimate times of occurrence of interval censored crimes.     

The nested joint clustering via Dirichlet process mixture model

This article focuses on the clustering problem based on Dirichlet process (DP) mixtures. To model both time invariant and temporal patterns, different from other existing clustering methods, the proposed semi-parametric model is flexible in that both the common and unique patterns are taken into account simultaneously. Furthermore, by jointly clustering subjects and the associated variables, the intrinsic complex shared patterns among subjects and among variables are expected to be captured. The number of clusters and cluster assignments are directly inferred with the use of DP. Simulation studies illustrate the effectiveness of the proposed method. An application to wheal size data is discussed with an aim of identifying novel temporal patterns among allergens within subject clusters.     

Bayesian clustering and product partition models

Summary. We present a decision theoretic formulation of product partition models (PPMs) that allows a formal treatment of different decision problems such as estimation or hypothesis testing and clustering methods simultaneously. A key observation in our construction is the fact that PPMs can be formulated in the context of model selection. The underlying partition structure in these models is closely related to that arising in connection with Dirichlet processes. This allows a straightforward adaptation of some computational strategies—originally devised for nonparametric Bayesian problems—to our framework. The resulting algorithms are more flexible than other competing alternatives that are used for problems involving PPMs. We propose an algorithm that yields Bayes estimates of the quantities of interest and the groups of experimental units. We explore the application of our methods to the detection of outliers in normal and Student t regression models, with clustering structure equivalent to that induced by a Dirichlet process prior. We also discuss the sensitivity of the results considering different prior distributions for the partitions.     

A functional marked point process model for lupus data

Modelling for marked point processes is an important problem, but has received remarkably little attention in the statistical literature. The authors developed a marked point process model that incorporates the use of functional data analysis in a joint estimation of the frequency function of the point process and the intensity of the mark, with application to data from 22 lupus patients consisting of times of flares in symptom severity combined with a quantitative assessment of the severity. The data indicate that a rapid decrease in drug dose is significantly associated with a decrease in flare frequency. Experiments with simulated data designed to model the actual data further support this conclusion. The Canadian Journal of Statistics 40: 517–529; 2012 © 2012 Statistical Society of Canada     

TESTING FOR CONTINUOUS LOCAL MARTINGALES USING THE CROSSING TREE

We present statistical tests for the continuous martingale hypothesis; that is, for whether an observed process is a continuous local martingale, or equivalently a continuous time‐changed Brownian motion. Our technique is based on the concept of the crossing tree. Simulation experiments are used to assess the power of the tests, which is generally higher than that of recently proposed tests using the estimated quadratic variation (i.e. realized volatility). In particular, the crossing tree shows significantly higher power with shorter data sets. We then show results from applying the methodology to five high‐frequency currency exchange rate data sets from 2003. For four of them we show that at small time‐scales (less than 15 minutes or so) the continuous martingale hypothesis is rejected, but not so at larger time‐scales. For the fifth, the hypothesis is rejected at small time‐scales and at some moderate time‐scales, but not all.     

Generalized linear models to study spatial distribution of tree species in Argentinean arid Chaco

This work adapts some generalized linear models in order to study the spatial pattern of an important tree species. The classical multivariate Ising model, which incorporates the dependence on neighbour individuals in a regular lattice, was adapted by setting a Poisson regression with an extra variation parameter to fit over-dispersion. Because the spatial pattern is only evident to a special reference scale, plots were sampled at two different scales. Two individual presence-absence matrices were analysed for each case through over-dispersion Poisson regression and log-linear models, including binary indicators for a neighbour in the four directions in the linear predictor. The results showed that the species, in the adult stage, has a spatial distribution in patches having no more than two adult individuals.     

Modelling hierarchical clustered censored data with the hierarchical Kendall copula

This article proposes a new model for right‐censored survival data with multi‐level clustering based on the hierarchical Kendall copula model of Brechmann (2014) with Archimedean clusters. This model accommodates clusters of unequal size and multiple clustering levels, without imposing any structural conditions on the parameters or on the copulas used at various levels of the hierarchy. A step‐wise estimation procedure is proposed and shown to yield consistent and asymptotically Gaussian estimates under mild regularity conditions. The model fitting is based on multiple imputation, given that the censoring rate increases with the level of the hierarchy. To check the model assumption of Archimedean dependence, a goodness‐of test is developed. The finite‐sample performance of the proposed estimators and of the goodness‐of‐fit test is investigated through simulations. The new model is applied to data from the study of chronic granulomatous disease. The Canadian Journal of Statistics 47: 182–203; 2019 © 2019 Statistical Society of Canada     

Bayesian inference for non-Gaussian Ornstein–Uhlenbeck stochastic volatility processes

Summary.  We develop Markov chain Monte Carlo methodology for Bayesian inference for non-Gaussian Ornstein–Uhlenbeck stochastic volatility processes. The approach introduced involves expressing the unobserved stochastic volatility process in terms of a suitable marked Poisson process. We introduce two specific classes of Metropolis–Hastings algorithms which correspond to different ways of jointly parameterizing the marked point process and the model parameters. The performance of the methods is investigated for different types of simulated data. The approach is extended to consider the case where the volatility process is expressed as a superposition of Ornstein–Uhlenbeck processes. We apply our methodology to the US dollar–Deutschmark exchange rate.     

Scale-space theory: a basic tool for analyzing structures at different scales

An inherent property of objects in the world is that they only exist as meaningful entities over certain ranges of scale. If one aims to describe the structure of unknown real-world signals, then a multi-scale representation of data is of crucial importance. This paper gives a tutorial review of a special type of multi-scale representation—linear scale-space representation—which has been developed by the computer vision community to handle image structures at different scales in a consistent manner. The basic idea is to embed the original signal into a one-parameter family of gradually smoothed signals in which the fine-scale details are successively suppressed. Under rather general conditions on the type of computations that are to be performed at the first stages of visual processing, in what can be termed 'the visual front-end', it can be shown that the Gaussian kernel and its derivatives are singled out as the only possible smoothing kernels. The conditions that specify the Gaussian kernel are, basically, linearity and shift invariance, combined with different ways of formalizing the notion that structures at coarse scales should correspond to simplifications of corresponding structures at fine scales-they should not be accidental phenomena created by the smoothing method. Notably, several different ways of choosing scale-space axioms give rise to the same conclusion. The output from the scale-space representation can be used for a variety of early visual tasks; operations such as feature detection, feature classification and shape computation can be expressed directly in terms of (possibly non-linear) combinations of Gaussian derivatives at multiple scales. In this sense the scale-space representation canserve as a basis for early vision. During the last few decades, a number of other approaches to multiscale representations have been developed, which are more or less related to scale-space theory, notably the theories of pyramids, wavelets and multi grid methods.Despite their qualitative differences, the increasing propularity of each of these approaches indicates that the crucial notion of scale is increasingly appreciated by the computer.vision community and by researchers in other related fields. An interesting similarity to biological vision is that the scale-space operators closely resemble receptive field profiles registered in neurophysiological studies of the mam- malian retina and visual cortex.