A Class of Convolution‐Based Models for Spatio‐Temporal Processes with Non‐Separable Covariance Structure

Abstract. In this article, we propose a new parametric family of models for real‐valued spatio‐temporal stochastic processes S ( x , t ) and show how low‐rank approximations can be used to overcome the computational problems that arise in fitting the proposed class of models to large datasets. Separable covariance models, in which the spatio‐temporal covariance function of S ( x , t ) factorizes into a product of purely spatial and purely temporal functions, are often used as a convenient working assumption but are too inflexible to cover the range of covariance structures encountered in applications. We define positive and negative non‐separability and show that in our proposed family we can capture positive, zero and negative non‐separability by varying the value of a single parameter.     

Globally intensity-reweighted estimators for K- and pair correlation functions

We introduce new estimators of the inhomogeneous K-function and the pair correlation function of a spatial point process as well as the cross K-function and the cross pair correlation function of a bivariate spatial point process under the assumption of second-order intensity-reweighted stationarity. These estimators rely on a ‘global’ normalisation factor which depends on an aggregation of the intensity function, while the existing estimators depend ‘locally’ on the intensity function at the individual observed points. The advantages of our new global estimators over the existing local estimators are demonstrated by theoretical considerations and a simulation study.     

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     

Hidden Second‐order Stationary Spatial Point Processes

In the existing statistical literature, the almost default choice for inference on inhomogeneous point processes is the most well‐known model class for inhomogeneous point processes: reweighted second‐order stationary processes. In particular, the K‐function related to this type of inhomogeneity is presented as the inhomogeneous K‐function. In the present paper, we put a number of inhomogeneous model classes (including the class of reweighted second‐order stationary processes) into the common general framework of hidden second‐order stationary processes, allowing for a transfer of statistical inference procedures for second‐order stationary processes based on summary statistics to each of these model classes for inhomogeneous point processes. In particular, a general method to test the hypothesis that a given point pattern can be ascribed to a specific inhomogeneous model class is developed. Using the new theoretical framework, we reanalyse three inhomogeneous point patterns that have earlier been analysed in the statistical literature and show that the conclusions concerning an appropriate model class must be revised for some of the point patterns.     

Parameter Estimation for Inhomogeneous Space‐Time Shot‐Noise Cox Point Processes

We consider the problem of parameter estimation for inhomogeneous space‐time shot‐noise Cox point processes. We explore the possibility of using a stepwise estimation method and dimensionality‐reducing techniques to estimate different parts of the model separately. We discuss the estimation method using projection processes and propose a refined method that avoids projection to the temporal domain. This remedies the main flaw of the method using projection processes – possible overlapping in the projection process of clusters, which are clearly separated in the original space‐time process. This issue is more prominent in the temporal projection process where the amount of information lost by projection is higher than in the spatial projection process. For the refined method, we derive consistency and asymptotic normality results under the increasing domain asymptotics and appropriate moment and mixing assumptions. We also present a simulation study that suggests that cluster overlapping is successfully overcome by the refined method.     

Non‐stationary Cross‐Covariance Models for Multivariate Processes on a Globe

Abstract. In geophysical and environmental problems, it is common to have multiple variables of interest measured at the same location and time. These multiple variables typically have dependence over space (and/or time). As a consequence, there is a growing interest in developing models for multivariate spatial processes, in particular, the cross‐covariance models. On the other hand, many data sets these days cover a large portion of the Earth such as satellite data, which require valid covariance models on a globe. We present a class of parametric covariance models for multivariate processes on a globe. The covariance models are flexible in capturing non‐stationarity in the data yet computationally feasible and require moderate numbers of parameters. We apply our covariance model to surface temperature and precipitation data from an NCAR climate model output. We compare our model to the multivariate version of the Matérn cross‐covariance function and models based on coregionalization and demonstrate the superior performance of our model in terms of AIC (and/or maximum loglikelihood values) and predictive skill. We also present some challenges in modelling the cross‐covariance structure of the temperature and precipitation data. Based on the fitted results using full data, we give the estimated cross‐correlation structure between the two variables.     

A general central limit theorem and a subsampling variance estimator for α‐mixing point processes

We establish a central limit theorem for multivariate summary statistics of nonstationary α‐mixing spatial point processes and a subsampling estimator of the covariance matrix of such statistics. The central limit theorem is crucial for establishing asymptotic properties of estimators in statistics for spatial point processes. The covariance matrix subsampling estimator is flexible and model free. It is needed, for example, to construct confidence intervals and ellipsoids based on asymptotic normality of estimators. We also provide a simulation study investigating an application of our results to estimating functions.     

Geometrically Corrected Second Order Analysis of Events on a Linear Network,with Applications to Ecology and Criminology

Abstract. We study point patterns of events that occur on a network of lines, such as road accidents recorded on a road network. Okabe and Yamada developed a ‘network K function’, analogous to Ripley's K function, for analysis of such data. However, values of the network K‐function depend on the network geometry, making interpretation difficult. In this study we propose a correction of the network K‐function that intrinsically compensates for the network geometry. This geometrical correction restores many natural and desirable properties of K, including its direct relationship to the pair correlation function. For a completely random point pattern, on any network, the corrected network K‐function is the identity. The corrected estimator is intrinsically corrected for edge effects and has approximately constant variance. We obtain exact and asymptotic expressions for the bias and variance of under complete randomness. We extend these results to an ‘inhomogeneous’ network K‐function which compensates for a spatially varying intensity of points. We demonstrate applications to ecology (webs of the urban wall spider Oecobius navus) and criminology (street crime in Chicago).     

Data Dependent Cells Chi‐Square Test With Recurrent Events

We consider a recurrent event wherein the inter‐event times are independent and identically distributed with a common absolutely continuous distribution function F. In this article, interest is in the problem of testing the null hypothesis that F belongs to some parametric family where the q‐dimensional parameter is unknown. We propose a general Chi‐squared test in which cell boundaries are data dependent. An estimator of the parameter obtained by minimizing a quadratic form resulting from a properly scaled vector of differences between Observed and Expected frequencies is used to construct the test. This estimator is known as the minimum chi‐square estimator. Large sample properties of the proposed test statistic are established using empirical processes tools. A simulation study is conducted to assess the performance of the test under parameter misspecification, and our procedures are applied to a fleet of Boeing 720 jet planes' air conditioning system failures.     

Inverse Probability of Censoring Weighted U‐statistics for Right‐Censored Data with an Application to Testing Hypotheses

Abstract. A right‐censored version of a U ‐statistic with a kernel of degree m 1 is introduced by the principle of a mean preserving reweighting scheme which is also applicable when the dependence between failure times and the censoring variable is explainable through observable covariates. Its asymptotic normality and an expression of its standard error are obtained through a martingale argument. We study the performances of our U ‐statistic by simulation and compare them with theoretical results. A doubly robust version of this reweighted U ‐statistic is also introduced to gain efficiency under correct models while preserving consistency in the face of model mis‐specifications. Using a Kendall's kernel, we obtain a test statistic for testing homogeneity of failure times for multiple failure causes in a multiple decrement model. The performance of the proposed test is studied through simulations. Its usefulness is also illustrated by applying it to a real data set on graft‐versus‐host‐disease.     

An Optimal Semiparametric Method for Two‐group Classification

In the classical discriminant analysis, when two multivariate normal distributions with equal variance–covariance matrices are assumed for two groups, the classical linear discriminant function is optimal with respect to maximizing the standardized difference between the means of two groups. However, for a typical case‐control study, the distributional assumption for the case group often needs to be relaxed in practice. Komori et al. (Generalized t ‐statistic for two‐group classification. Biometrics 2015, 71: 404–416) proposed the generalized t ‐statistic to obtain a linear discriminant function, which allows for heterogeneity of case group. Their procedure has an optimality property in the class of consideration. We perform a further study of the problem and show that additional improvement is achievable. The approach we propose does not require a parametric distributional assumption on the case group. We further show that the new estimator is efficient, in that no further improvement is possible to construct the linear discriminant function more efficiently. We conduct simulation studies and real data examples to illustrate the finite sample performance and the gain that it produces in comparison with existing methods.     

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.     

Non‐Parametric Change‐Point Tests for Long‐Range Dependent Data

Abstract. We propose a non‐parametric change‐point test for long‐range dependent data, which is based on the Wilcoxon two‐sample test. We derive the asymptotic distribution of the test statistic under the null hypothesis that no change occurred. In a simulation study, we compare the power of our test with the power of a test which is based on differences of means. The results of the simulation study show that in the case of Gaussian data, our test has only slightly smaller power minus.3pt than the ‘difference‐of‐means’ test. For heavy‐tailed data, our test outperforms the ‘difference‐of‐means’ test.     

Scalable spatio‐temporal Bayesian analysis of high‐dimensional electroencephalography data

We present a scalable Bayesian modelling approach for identifying brain regions that respond to a certain stimulus and use them to classify subjects. More specifically, we deal with multi‐subject electroencephalography (EEG) data with a binary response distinguishing between alcoholic and control groups. The covariates are matrix‐variate with measurements taken from each subject at different locations across multiple time points. EEG data have a complex structure with both spatial and temporal attributes. We use a divide‐and‐conquer strategy and build separate local models, that is, one model at each time point. We employ Bayesian variable selection approaches using a structured continuous spike‐and‐slab prior to identify the locations that respond to a certain stimulus. We incorporate the spatio‐temporal structure through a Kronecker product of the spatial and temporal correlation matrices. We develop a highly scalable estimation algorithm, using likelihood approximation, to deal with large number of parameters in the model. Variable selection is done via clustering of the locations based on their duration of activation. We use scoring rules to evaluate the prediction performance. Simulation studies demonstrate the efficiency of our scalable algorithm in terms of estimation and fast computation. We present results using our scalable approach on a case study of multi‐subject EEG data.     

One‐Way anova for Functional Data via Globalizing the Pointwise F‐test

In this paper, we propose and study a new global test, namely, GPF test, for the one‐way anova problem for functional data, obtained via globalizing the usual pointwise F‐test. The asymptotic random expressions of the test statistic are derived, and its asymptotic power is investigated. The GPF test is shown to be root‐n consistent. It is much less computationally intensive than a parametric bootstrap test proposed in the literature for the one‐way anova for functional data. Via some simulation studies, it is found that in terms of size‐controlling and power, the GPF test is comparable with two existing tests adopted for the one‐way anova problem for functional data. A real data example illustrates the GPF test.     

Second‐order analysis of marked inhomogeneous spatiotemporal point processes: Applications to earthquake data

To analyze interactions in marked spatiotemporal point processes (MSTPPs), we introduce marked second‐order reduced moment measures and K‐functions for inhomogeneous second‐order intensity‐reweighted stationary MSTPPs. These summary statistics, which allow us to quantify dependence between different mark‐based classifications of points, depend on the specific mark space and mark reference measure chosen. Unbiased and consistent minus‐sampling estimators are derived for all statistics considered, and a test for random labeling is indicated. In addition, we treat Voronoi intensity estimators for MSTPPs. These new statistics are finally employed to analyze an Andaman Sea earthquake data set.     

A two‐step proximal‐point algorithm for the calculus of divergence‐based estimators in finite mixture models

Estimators derived from the expectation‐maximization (EM) algorithm are not robust since they are based on the maximization of the likelihood function. We propose an iterative proximal‐point algorithm based on the EM algorithm to minimize a divergence criterion between a mixture model and the unknown distribution that generates the data. The algorithm estimates in each iteration the proportions and the parameters of the mixture components in two separate steps. Resulting estimators are generally robust against outliers and misspecification of the model. Convergence properties of our algorithm are studied. The convergence of the introduced algorithm is discussed on a two‐component Weibull mixture entailing a condition on the initialization of the EM algorithm in order for the latter to converge. Simulations on Gaussian and Weibull mixture models using different statistical divergences are provided to confirm the validity of our work and the robustness of the resulting estimators against outliers in comparison to the EM algorithm. An application to a dataset of velocities of galaxies is also presented. The Canadian Journal of Statistics 47: 392–408; 2019 © 2019 Statistical Society of Canada     

Space‐time cluster identification in point processes

The authors propose a new type of scan statistic to test for the presence of space‐time clusters in point processes data, when the goal is to identify and evaluate the statistical significance of localized clusters. Their method is based only on point patterns for cases; it does not require any specific knowledge of the underlying population. The authors propose to scan the three‐dimensional space with a score test statistic under the null hypothesis that the underlying point process is an inhomogeneous Poisson point process with space and time separable intensity. The alternative is that there are one or more localized space‐time clusters. Their method has been implemented in a computationally efficient way so that it can be applied routinely. They illustrate their method with space‐time crime data from Belo Horizonte, a Brazilian city, in addition to presenting a Monte Carlo study to analyze the power of their new test.     

Goodness‐of‐Fit Tests for Bivariate and Multivariate Skew‐Normal Distributions

Abstract. Goodness‐of‐fit tests are proposed for the skew‐normal law in arbitrary dimension. In the bivariate case the proposed tests utilize the fact that the moment‐generating function of the skew‐normal variable is quite simple and satisfies a partial differential equation of the first order. This differential equation is estimated from the sample and the test statistic is constructed as an L ₂ ‐type distance measure incorporating this estimate. Extension of the procedure to dimension greater than two is suggested whereas an effective bootstrap procedure is used to study the behaviour of the new method with real and simulated data.     

Non‐parametric Copula Estimation Under Bivariate Censoring

In this paper, we consider non‐parametric copula inference under bivariate censoring. Based on an estimator of the joint cumulative distribution function, we define a discrete and two smooth estimators of the copula. The construction that we propose is valid for a large range of estimators of the distribution function and therefore for a large range of bivariate censoring frameworks. Under some conditions on the tails of the distributions, the weak convergence of the corresponding copula processes is obtained in l^∞([0,1]²). We derive the uniform convergence rates of the copula density estimators deduced from our smooth copula estimators. Investigation of the practical behaviour of these estimators is performed through a simulation study and two real data applications, corresponding to different censoring settings. We use our non‐parametric estimators to define a goodness‐of‐fit procedure for parametric copula models. A new bootstrap scheme is proposed to compute the critical values.