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We define a general notion of entropy in elementary, algebraic terms. Based on that, weak forms of a scalar product and a distance measure are derived. We give basic properties of these quantities, generalize the Cauchy-Schwarz inequality, and relate our approach to the theory of scoring rules. Many supporting examples illustrate our approach and give new perspectives on established notions, such as the likelihood, the Kullback-Leibler divergence, the uncorrelatedness of random variables, the scalar product itself, the Tichonov regularization and the mutual information., Comment: 35 pages

So far, the pseudo cross-variogram is primarily used as a tool for the structural analysis of multivariate random fields. Mainly applying recent theoretical results on the pseudo cross-variogram, we use it as a cornerstone in the construction of valid covariance models for multivariate random fields. In particular, we extend known univariate constructions to the multivariate case, and generalize existing multivariate models. Furthermore, we provide a general construction principle for conditionally negative definite matrix-valued kernels, which we use to reinterpret previous modeling proposals.

Principal Component Analysis (PCA) is a well known procedure to reduce intrinsic complexity of a dataset, essentially through simplifying the covariance structure or the correlation structure. We introduce a novel algebraic, model-based point of view and provide in particular an extension of the PCA to distributions without second moments by formulating the PCA as a best low rank approximation problem. In contrast to hitherto existing approaches, the approximation is based on a kind of spectral representation, and not on the real space. Nonetheless, the prominent role of the eigenvectors is here reduced to define the approximating surface and its maximal dimension. In this perspective, our approach is close to the original idea of Pearson (1901) and hence to autoencoders. Since variable selection in linear regression can be seen as a special case of our extension, our approach gives some insight, why the various variable selection methods, such as forward selection and best subset selection, cannot be expected to coincide. The linear regression model itself and the PCA regression appear as limit cases.

Pseudo-variograms appear naturally in the context of multivariate Brown-Resnick processes, and are a useful tool for analysis and prediction of multivariate random fields. We give a necessary and sufficient criterion for a matrix-valued function to be a pseudo-variogram, and further provide a Schoenberg-type result connecting pseudo-variograms and multivariate correlation functions. By means of these characterizations, we provide extensions of the popular univariate space-time covariance model of Gneiting to the multivariate case.

With the rise of deep learning models in the field of computer vision, new possibilities for their application in industrial processes proves to return great benefits. Nevertheless, the actual fit of machine learning for highly standardised industrial processes is still under debate. This paper addresses the challenges on the industrial realization of the AI tools, considering the use case of Laser Beam Welding quality control as an example. We use object detection algorithms from the TensorFlow object detection API and adapt them to our use case using transfer learning. The baseline models we develop are used as benchmarks and evaluated and compared to models that undergo dataset scaling and hyperparameter tuning. We find that moderate scaling of the dataset via image augmentation leads to improvements in intersection over union (IoU) and recall, whereas high levels of augmentation and scaling may lead to deterioration of results. Finally, we put our results into perspective of the underlying use case and evaluate their fit.

Stochastic models of point patterns in space and time are widely used to issue forecasts or assess risk, and often they affect societally relevant decisions. We adapt the concept of consistent scoring functions and proper scoring rules, which are statistically principled tools for the comparative evaluation of predictive performance, to the point process setting, and place both new and existing methodology in this framework. With reference to earthquake likelihood model testing, we demonstrate that extant techniques apply in much broader contexts than previously thought. In particular, the Poisson log-likelihood can be used for theoretically principled comparative forecast evaluation in terms of cell expectations. We illustrate the approach in a simulation study and in a comparative evaluation of operational earthquake forecasts for Italy., Comment: 40 pages, 14 figures

Sup-normalized spectral functions form building blocks of max-stable and Pareto processes and therefore play an important role in modeling spatial extremes. For one of the most popular examples, the Brown-Resnick process, simulation is not straightforward. In this paper, we generalize two approaches for simulation via Markov Chain Monte Carlo methods and rejection sampling by introducing new classes of proposal densities. In both cases, we provide an optimal choice of the proposal density with respect to sampling efficiency. The performance of the procedures is demonstrated in an example., Comment: 11 pages, 2 figures

The Mat\'ern hard-core processes are classical examples for point process models obtained from (marked) Poisson point processes. Points of the original Poisson process are deleted according to a dependent thinning rule, resulting in a process whose points have a prescribed hard-core distance. We present a new model which encompasses recent approaches. It generalizes the underlying point process, the thinning rule and the marks attached to the original process. The new model further reveals several connections to mixed moving maxima processes, e.g. a process of visible storm centres., Comment: 16 pages, 3 figures

We present a model of a random field on a topological space $M$ that unifies well-known models such as the Poisson hyperplane tessellation model, the random token model, and the dead leaves model. In addition to generalizing these submodels from $\mathbb{R}^d$ to other spaces such as the $d$-dimensional unit sphere $\mathbb{S}^d$, our construction also extends the classical models themselves, e.g. by replacing the Poisson distribution by an arbitrary discrete distribution. Moreover, the method of construction directly produces an exact and fast simulation procedure. By investigating the covariance structure of the general model we recover various explicit correlation functions on $\mathbb{R}^d$ and $\mathbb{S}^d$ and obtain several new ones.

A new approach for evaluating time-trends in extreme values accounting also for spatial dependence is proposed. Based on exceedances over a space-time threshold, estimators for a trend function and for extreme value parameters are given, leading to a homogenization procedure for then applying stationary extreme value processes. Extremal dependence over space is further evaluated through variogram analysis including anisotropy. We detect significant inhomogeneities and trends in the extremal behaviour of daily precipitation data over a time period of 84 years and from 68 observational weather stations in North-West Germany. We observe that the trend is not monotonous over time in general. Asymptotic normality of the estimators under maximum domain of attraction conditions are proven., Comment: 24 pages, 5 figures

The integration of physical relationships into stochastic models is of major interest e.g. in data assimilation. Here, a multivariate Gaussian random field formulation is introduced, which represents the differential relations of the two-dimensional wind field and related variables such as streamfunction, velocity potential, vorticity and divergence. The covariance model is based on a flexible bivariate Mat\'ern covariance function for streamfunction and velocity potential. It allows for different variances in the potentials, non-zero correlations between them, anisotropy and a flexible smoothness parameter. The joint covariance function of the related variables is derived analytically. Further, it is shown that a consistent model with non-zero correlations between the potentials and positive definite covariance function is possible. The statistical model is fitted to forecasts of the horizontal wind fields of a mesoscale numerical weather prediction system. Parameter uncertainty is assessed by a parametric bootstrap method. The estimates reveal only physically negligible correlations between the potentials. In contrast to the numerical estimator, the statistical estimator of the ratio between the variances of the rotational and divergent wind components is unbiased., Comment: 23 pages, 6 figures Submitted to Journal of the Atmospheric Sciences

We introduce a class of spatial stochastic processes in the max-domain of attraction of familiar max-stable processes. The new class is based on Cox processes and comprises models with short range dependence. We show that statistical inference is possible within the given framework, at least under some reasonable restrictions., Comment: 29 pages, 11 Figures

We provide sufficient conditions of P\'olya type which guarantee the positive definiteness of a $2\times 2$-matrix-valued function in $\mathbb{R}$ and $\mathbb{R}^3$. Several bivariate covariance models have been proposed in literature, where all components of the covariance matrix are of the same parametric family, such as the bivariate Mat\'{e}rn model. Based on the P\'olya type conditions, we introduce two novel bivariate parametric covariance models of this class, the powered exponential (or stable) covariance model and the generalized Cauchy covariance model. Both models allow for flexible smoothness, variance, scale, and cross-correlation parameters. The smoothness parameters are in $(0, 1]$. Additionally, the bivariate generalized Cauchy model allows for distinct long range parameters. We also show that the univariate spherical model can be generalized to the bivariate case within the above class only in a trivial way. In a data example on the content of copper and zinc in the top soil of Swiss Jura we compare the bivariate powered exponential model to the traditional linear model of coregionalization and the bivariate Mat\'{e}rn model.

A theorem of the Kolmogorov--Chentsov type is proved for random fields on a Riemannian manifold.

The extremal coefficient function (ECF) of a max-stable process $X$ on some index set $T$ assigns to each finite subset $A\subset T$ the effective number of independent random variables among the collection $\{X_t\}_{t\in A}$. We introduce the class of Tawn-Molchanov processes that is in a 1:1 correspondence with the class of ECFs, thus also proving a complete characterization of the ECF in terms of negative definiteness. The corresponding Tawn-Molchanov process turns out to be exceptional among all max-stable processes sharing the same ECF in that its dependency set is maximal w.r.t. inclusion. This entails sharp lower bounds for the finite dimensional distributions of arbitrary max-stable processes in terms of its ECF. A spectral representation of the Tawn-Molchanov process and stochastic continuity are discussed. We also show how to build new valid ECFs from given ECFs by means of Bernstein functions., Comment: Published at http://dx.doi.org/10.3150/13-BEJ567 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

Semi-regular sequences over $\mathbb{F}_2$ are sequences of homogeneous elements of the algebra $ B^{(n)}=\mathbb{F}_2[X_1,...,X_n]/(X_1^2,...,X_n^2) $, which have as few relations between them as possible. They were introduced in order to assess the complexity of Gr\"obner basis algorithms such as ${\bf F}_4, {\bf F}_5$ for the solution of polynomial equations. Despite the experimental evidence that semi-regular sequences are common, it was unknown whether there existed semi-regular sequences for all $n$, except in extremely trivial situations. We prove some results on the existence and non-existence of semi-regular sequences. In particular, we show that if an element of degree $d$ in $B^{(n)}$ is semi-regular, then we must have $n\leq 3d$. Also, we show that if $d=2^t$ and $n=3d$ there exits a semi-regular element of degree $d$ establishing that the bound is sharp for infinitely many $n$. Finally, we generalize the result of non-existence of semi-regular elements to the case of sequences of a fixed length $m$.

A simple variogram model with two parameters is presented that includes the power variogram for the fractional Brownian motion, a modified De Wijsian model, the generalized Cauchy model and the multiquadrics model. One parameter controls the smoothness of the process. The other parameter allows for a smooth parametrization between stationary and intrinsically stationary second order processes in a Gaussian framework, or between mixing and non-ergodic max-stable processes when modeling spatial extremes by a Brown-Resnick process.

For a stochastic process $\{X_t\}_{t \in T}$ with identical one-dimensional margins and upper endpoint $\tau_{\text{up}}$ its tail correlation function (TCF) is defined through $\chi^{(X)}(s,t) = \lim_{\tau \to \tau_{\text{up}}} P(X_s > \tau \,\mid\, X_t > \tau )$. It is a popular bivariate summary measure that has been frequently used in the literature in order to assess tail dependence. In this article, we study its realization problem. We show that the set of all TCFs on $T \times T$ coincides with the set of TCFs stemming from a subclass of max-stable processes and can be completely characterized by a system of affine inequalities. Basic closure properties of the set of TCFs and regularity implications of the continuity of $\chi$ are derived. If $T$ is finite, the set of TCFs on $T \times T$ forms a convex polytope of $\lvert T \rvert \times \lvert T \rvert$ matrices. Several general results reveal its complex geometric structure. Up to $\lvert T \rvert = 6$ a reduced system of necessary and sufficient conditions for being a TCF is determined. None of these conditions will become obsolete as $\lvert T \rvert\geq 3$ grows., Comment: 42 pages, 7 Tables

The tail correlation function (TCF) is one of the most popular bivariate extremal dependence measures that has entered the literature under various names. We study to what extent the TCF can distinguish between different classes of well-known max-stable processes and identify essentially different processes sharing the same TCF., Comment: 31 pages, 4 Tables, 5 Figures

To improve the forecasts of weather extremes, we propose a joint spatial model for the observations and the forecasts, based on a bivariate Brown-Resnick process. As the class of stationary bivariate Brown-Resnick processes is fully characterized by the class of pseudo cross-variograms, we contribute to the theorical understanding of pseudo cross-variograms refining the knowledge of the asymptotic behaviour of all their components and introducing a parsimonious, but flexible parametric model. Both findings are of interest in classical geostatistics on their own. The proposed model is applied to real observation and forecast data for extreme wind gusts at 119 stations in Northern Germany., Comment: 22 pages, 4 figures

The normalized spectral representation of a max-stable process on a compact set is the unique representation where all spectral functions share the same supremum. Among the class of equivalent spectral representations of a process, the normalized spectral representation plays a distinctive role as a solution of two optimization problems in the context of an efficient simulation of max-stable processes. Our approach has the potential of considerably reducing the simulation time of max-stable processes., Comment: 32 pages

The expected level of linkage disequilibrium (LD) in a finite ideal population at equilibrium is of relevance for many applications in population and quantitative genetics. Several recursion formulae have been proposed during the last decades, whose derivations mostly contain heuristic parts and therefore remain mathematically questionable. We propose a more justifiable approach, including an alternative recursion formula for the expected LD. Since the exact formula depends on the distribution of allele frequencies in a very complicated manner, we suggest an approximate solution and analyze its validity extensively in a simulation study. Compared to the widely used formula of Sved, the proposed formula performs better for all parameter constellations considered. We then analyze the expected LD at equilibrium using the theory on discrete-time Markov chains based on the linear recursion formula, with equilibrium being defined as the steady-state of the chain, which finally leads to a formula for the effective population size N_e. An additional analysis considers the effect of non-exactness of a recursion formula on the steady-state, demonstrating that the resulting error in expected LD can be substantial. In an application to the HapMap data of two human populations we illustrate the dependency of the N_e-estimate on the distribution of minor allele frequencies (MAFs), showing that the estimated N_e can vary by up to 30% when a uniform instead of a skewed distribution of MAFs is taken as a basis to select SNPs for the analyses. Our analyses provide new insights into the mathematical complexity of the problem studied., Comment: 32 pages, 11 Figures, formatting edits compared to v1 (pdf-links)

For a non-stationary or non-ergodic marked point process (MPP) on $\R^d$, the definition of averages becomes ambiguous as the process might have a different stochastic behavior in different realizations (non-ergodicity) or in different areas of the observation window (non-stationarity). We investigate different definitions for the moments, including a new hierarchical definition for non-ergodic MPPs, and embed them into a family of weighted mean marks. We point out examples of application in which different weighted mean marks all have a sensible meaning. Further, asymptotic properties of the corresponding estimators are investigated as well as optimal weighting procedures., Comment: 25 pages

This paper provides the basis for new methods of inference for max-stable processes \xi\ on general spaces that admit a certain incremental representation, which, in important cases, has a much simpler structure than the max-stable process itself. A corresponding peaks-over-threshold approach will incorporate all single events that are extreme in some sense and will therefore rely on a substantially larger amount of data in comparison to estimation procedures based on block maxima. Conditioning a process \eta\ in the max-domain of attraction of \xi\ on being extremal, several convergence results for the increments of \eta\ are proved. In a similar way, the shape functions of mixed moving maxima (M3) processes can be extracted from suitably conditioned single events \eta. Connecting the two approaches, transformation formulae for processes that admit both an incremental and an M3 representation are identified., Comment: 29 pages

Estimation of extreme-value parameters from observations in the max-domain of attraction (MDA) of a multivariate max-stable distribution commonly uses aggregated data such as block maxima. Since we expect that additional information is contained in the non-aggregated, single "large" observations, we introduce a new approach of inference based on a multivariate peaks-over-threshold method. We show that for any process in the MDA of the frequently used H\"usler-Reiss model or its spatial extension, the Brown-Resnick process, suitably defined conditional increments asymptotically follow a multivariate Gaussian distribution. This leads to computationally efficient estimates of the H\"usler-Reiss parameter matrix. Further, the results enable parametric inference for Brown-Resnick processes. A simulation study compares the performance of the new estimators to other commonly used methods. As an application, we fit a non-isotropic Brown-Resnick process to the extremes of 12 year data of daily wind speed measurements., Comment: 26 pages, 7 figures

We focus on two dependency quantities of a max-stable random field $X$ on some space $T$: the extremal coefficient function $\theta$ which we define on finite sets of $T$ and the extremal correlation function $\chi(s,t)=\lim_{x \uparrow \infty} \PP(X_s \geq x \mid X_t \geq x)$. We fully characterize extremal coefficient functions $\theta$ by a property called complete alternation and construct a corresponding max-stable random field. Simple properties and consequences concerning the convex geometry of extremal coefficients are derived. We study how the continuity of $X$, $\theta$ and $\chi$ are linked to each other, and we show that extremal correlation functions $\chi$ allow for convex combinations in general, and for products and pointwise limits if the resulting function is continuous. These are operations which are well-known for positive definite functions, but the latter are non-trivial for extremal correlation functions. Finally, we regard some additional implications, when the random field $X$ on $T=\mathbb{R}^d$ is stationary., Comment: 25 pages, 2 figures

Let $X_{i,n},n\in \mathbb{N},1\leq i\leq n$, be a triangular array of independent $\mathbb{R}^d$-valued Gaussian random vectors with correlation matrices $\Sigma_{i,n}$. We give necessary conditions under which the row-wise maxima converge to some max-stable distribution which generalizes the class of H\"{u}sler-Reiss distributions. In the bivariate case, the conditions will also be sufficient. Using these results, new models for bivariate extremes are derived explicitly. Moreover, we define a new class of stationary, max-stable processes as max-mixtures of Brown-Resnick processes. As an application, we show that these processes realize a large set of extremal correlation functions, a natural dependence measure for max-stable processes. This set includes all functions $\psi(\sqrt{\gamma(h)}),h\in \mathbb{R}^d$, where $\psi$ is a completely monotone function and $\gamma$ is an arbitrary variogram., Comment: Published at http://dx.doi.org/10.3150/13-BEJ560 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

This paper deals with the question of conditional sampling and prediction for the class of stationary max-stable processes which allow for a mixed moving maxima representation. We develop an exact procedure for conditional sampling using the Poisson point process structure of such processes. For explicit calculations we restrict ourselves to the one-dimensional case and use a finite number of shape functions satisfying some regularity conditions. For more general shape functions approximation techniques are presented. Our algorithm is applied to the Smith process and the Brown-Resnick process. Finally, we compare our computational results to other approaches. Here, the algorithm for Gaussian processes with transformed marginals turns out to be surprisingly competitive., Comment: 35 pages; version accepted for publication in Extremes. The final publication is available at http://link.springer.com

We study the behavior of a real-valued and unobservable process (Y_t) under an extreme event of a related process (X_t) that is observable. Our analysis is motivated by the well-known GARCH model which represents two such sequences, i.e. the observable log returns of an asset as well as the hidden volatility process. Our results complement the findings of Segers (2007) and Smith (1992) for a single time series. We show that under suitable assumptions their concept of a tail chain as a limiting process is also applicable to our setting. Furthermore, we discuss existence and uniqueness of a limiting process under some weaker assumptions. Finally, we apply our results to the GARCH(1,1) case.

Modelling spatio-temporal processes has become an important issue in current research. Since Gaussian processes are essentially determined by their second order structure, broad classes of covariance functions are of interest. Here, a new class is described that merges and generalizes various models presented in the literature, in particular models in Gneiting (J. Amer. Statist. Assoc. 97 (2002) 590--600) and Stein (Nonstationary spatial covariance functions (2005) Univ. Chicago). Furthermore, new models and a multivariate extension are introduced., Comment: Published in at http://dx.doi.org/10.3150/09-BEJ226 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

We prove that a stationary max--infinitely divisible process is mixing (ergodic) iff its dependence function converges to 0 (is Cesaro summable to 0). These criteria are applied to some classes of max--infinitely divisible processes., Comment: 15 pages

We aim to link random fields and marked point processes and therefore introduce a new class of stochastic processes which are defined on a random set in R^d. Unlike for random fields, the mark covariance function of a marked random set is in general not positive definite. This implies that in many situations the use of simple geostatistical methods appears to be questionable. Surprisingly, for a special class of processes based on Gaussian random fields, we do have positive definiteness for the corresponding mark covariance function and mark correlation function.

Let $W_i,i\in{\mathbb{N}}$, be independent copies of a zero-mean Gaussian process $\{W(t),t\in{\mathbb{R}}^d\}$ with stationary increments and variance $\sigma^2(t)$. Independently of $W_i$, let $\sum_{i=1}^{\infty}\delta_{U_i}$ be a Poisson point process on the real line with intensity $e^{-y} dy$. We show that the law of the random family of functions $\{V_i(\cdot),i\in{\mathbb{N}}\}$, where $V_i(t)=U_i+W_i(t)-\sigma^2(t)/2$, is translation invariant. In particular, the process $\eta(t)=\bigvee_{i=1}^{\infty}V_i(t)$ is a stationary max-stable process with standard Gumbel margins. The process $\eta$ arises as a limit of a suitably normalized and rescaled pointwise maximum of $n$ i.i.d. stationary Gaussian processes as $n\to\infty$ if and only if $W$ is a (nonisotropic) fractional Brownian motion on ${\mathbb{R}}^d$. Under suitable conditions on $W$, the process $\eta$ has a mixed moving maxima representation., Comment: Published in at http://dx.doi.org/10.1214/09-AOP455 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Fractal behavior and long-range dependence have been observed in an astonishing number of physical systems. Either phenomenon has been modeled by self-similar random functions, thereby implying a linear relationship between fractal dimension, a measure of roughness, and Hurst coefficient, a measure of long-memory dependence. This letter introduces simple stochastic models which allow for any combination of fractal dimension and Hurst exponent. We synthesize images from these models, with arbitrary fractal properties and power-law correlations, and propose a test for self-similarity., Comment: 8 pages, 2 figures