In this talk we introduce a class of statistics for spatial data that is observed on an irregular set of locations. Our aim is to obtain a unified framework for inference and the statistics we consider include both parametric and nonparametric estimators of the spatial covariance function, Whittle likelihood estimation, goodness of fit tests and a test for second order spatial stationarity. To ensure that the statistics are computationally feasible they are defined within the Fourier domain, and in most cases can be expressed as a quadratic form of a discrete Fourier-type transform of the spatial data. Evaluation of such statistic is computationally tractable, requiring $O(nb)$ operations, where $b$ are the number Fourier frequencies used in the definition of the statistic (which varies according to the application) and $n$ is the sample size. The asymptotic sampling properties of the statistics are derived using mixed spatial asymptotics, where the number of locations grows at a faster rate than the size of the spatial domain and under the assumption that the spatial random field is stationary and the irregular design of the locations are independent, identically distributed random variables. We show that there are quite intriguing differences in the behaviour of the statistic when the spatial process is Gaussian and non-Gaussian. In particular, the choice of the number of frequencies $b$ in the construction of the statistic depends on whether the spatial process is Gaussian or not. If time permits we describe how the results can also be used in variance estimation. And if we still have time some simulations and real data will be presented.[-]

In time series analysis there is an apparent dichotomy between time and frequency domain methods. The aim of this paper is to draw connections between frequency and time domain methods. Our focus will be on reconciling the Gaussia likelihood and the Whittle likelihood. We derive an exact, interpretable, bound between the Gaussian and Whittle likelihood of a second order stationary time series. The derivation is based on obtaining the transformation which is biorthogonal to the discrete Fourier transform of the time series. Such a transformation yields a new decomposition for the inverse of a Toeplitz matrix and enables the representation of the Gaussian likelihood within the frequency domain. We show that the difference between the Gaussian and Whittle likelihood is due to the omission of the best linear predictions outside the domain of observation in the periodogram associated with the Whittle likelihood. Based on this result, we obtain an approximation for the difference between the Gaussian and Whittle likelihoods in terms of the best fitting, finite order autoregressive parameters. These approximations are used to define two new frequency domain quasi-likelihoods criteria. We show these new criteria yield a better approximation of the spectral divergence criterion, as compared to both the Gaussian and Whittle likelihoods. In simulations, we show that the proposed estimators have satisfactory finite sample properties.[-]

Anomaly detection in random fields is an important problem in many applications including the detection of cancerous cells in medicine, obstacles in autonomous driving and cracks in the construction material of buildings. Scan statistics have the potential to detect local structure in such data sets by enhancing relevant features. Frequently, such anomalies are visible as areas with different expected values compared to the background noise where the geometric properties of these areas may depend on the type of anomaly. Such geometric properties can be taken into account by combinations and contrasts of sample means over differently-shaped local windows. For example, in 2D image data of concrete both cracks, which we aim to detect, as well as integral parts of the material (such as air bubbles or gravel) constitute areas with different expected values in the image. Nevertheless, due to their different geometric properties we can define scan statistics that enhance cracks and at the same time discard the integral parts of the given concrete. Cracks can then be detected using asuitable threshold for appropriate scan statistics. 9 In order to derive such thresholds, we prove weak convergence of the scan statistics towards a functional of a Gaussian process under the null hypothesis of no anomalies. The result allows for arbitrary (but fixed) dimension, makes relatively weak assumptions on the underlying noise, the shape of the local windows and the combination of finitely-many of such windows. These theoretical findings are accompanied by some simulations as well as applications to semi-artifical 2D-images of concrete.[-]

In this talk, I will present mixing properties for a broad class of Poisson count time series satisfying a certain contraction condition. Using specific coupling techniques, we obtain absolute regularity at a geometric rate not only for stationary Poisson-GARCH processes but also for models with an explosive trend. Easily verifiable sufficient conditions for absolute regularity can be deduced from our general results for a variety of models including classical (log-)linear models.[-]

Multivariate extreme value distributions are a common choice for modelling multivariate extremes. In high dimensions, however, the construction of flexible and parsimonious models is challenging. We propose to combine bivariate extreme value distributions into a Markov random field with respect to a tree. Although in general not an extreme value distribution itself, this Markov tree is attracted by a multivariate extreme value distribution. The latter serves as a tree-based approximation to an unknown extreme value distribution with the given bivariate distributions as margins. Given data, we learn an appropriate tree structure by Prim's algorithm with estimated pairwise upper tail dependence coefficients or Kendall's tau values as edge weights. The distributions of pairs of connected variables can be fitted in various ways. The resulting tree-structured extreme value distribution allows for inference on rare event probabilities, as illustrated on river discharge data from the upper Danube basin.[-]

Statistical modelling of complex dependencies in extreme events requires meaningful sparsity structures in multivariate extremes. In this context two perspectives on conditional independence and graphical models have recently emerged: One that focuses on threshold exceedances and multivariate pareto distributions, and another that focuses on max-linear models and directed acyclic graphs. What connects these notions is the exponent measure that lies at the heart of each approach. In this work we develop a notion of conditional independence defined directly on the exponent measure (and even more generally on measures that explode at the origin) that extends recent work of Engelke and Hitz (2019), who had been confined to homogeneous measures with density. We prove easier checkable equivalent conditions to verify this new conditional independence in terms of a reduction to simple test classes, probability kernels and density factorizations. This provides a pathsway to graphical modelling among general multivariate (max-)infinitely distributions. Structural max-linear models turn out to form a Bayesian network with respect to our new form of conditional independence.[-]

We discuss the problem of positive-semidefinite extension: extending a partially specified covariance kernel from a subdomain Ω of a rectangular domain I x I to a covariance kernel on the entire domain I x I. For a broad class of domains Ω called serrated domains, we present a complete theory. Namely, we demonstrate that a canonical completion always exists and can be explicitly constructed. We characterise all possible completions as suitable perturbations of the canonical completion, and determine necessary and sufficient conditions for a unique completion to exist. We interpret the canonical completion via the graphical model structure it induces on the associated Gaussian process. Furthermore, we show how the determination of the canonical completion reduces to the solution of a system of linear inverse problems in the space of Hilbert-Schmidt operators, and derive rates of convergence when the kernel is to be empirically estimated. We conclude by providing extensions of our theory to more general forms of domains, and by demonstrating how our results can be used in statistical inverse problems associated with stochastic processes.[-]