arXiv subjects

Sebastian Engelke

Publications and source records attributed to Sebastian Engelke.

54 records · Page 3Linked to original sources

arXiv2019

Extreme Value Theory for Open Set Classification -- GPD and GEV Classifiers

Classification tasks usually assume that all possible classes are present during the training phase. This is restrictive if the algorithm is used over a long time and possibly encounters samples from unknown classes. The recently introduced extreme value machine, a classifier motivated by extreme value theory, addresses this problem and achieves competitive performance in specific cases. We show that this algorithm can fail when the geometries of known and unknown classes differ. To overcome this problem, we propose two new algorithms relying on approximations from extreme value theory. We show the effectiveness of our classifiers in simulations and on the LETTER and MNIST data sets.

stat.ML↗

arXiv2019

Extremal dependence of random scale constructions

A bivariate random vector can exhibit either asymptotic independence or dependence between the largest values of its components. When used as a statistical model for risk assessment in fields such as finance, insurance or meteorology, it is crucial to understand which of the two asymptotic regimes occurs. Motivated by their ubiquity and flexibility, we consider the extremal dependence properties of vectors with a random scale construction $(X_1,X_2)=R(W_1,W_2)$, with non-degenerate $R>0$ independent of $(W_1,W_2)$. Focusing on the presence and strength of asymptotic tail dependence, as expressed through commonly-used summary parameters, broad factors that affect the results are: the heaviness of the tails of $R$ and $(W_1,W_2)$, the shape of the support of $(W_1,W_2)$, and dependence between $(W_1,W_2)$. When $R$ is distinctly lighter tailed than $(W_1,W_2)$, the extremal dependence of $(X_1,X_2)$ is typically the same as that of $(W_1,W_2)$, whereas similar or heavier tails for $R$ compared to $(W_1,W_2)$ typically result in increased extremal dependence. Similar tail heavinesses represent the most interesting and technical cases, and we find both asymptotic independence and dependence of $(X_1,X_2)$ possible in such cases when $(W_1,W_2)$ exhibit asymptotic independence. The bivariate case often directly extends to higher-dimensional vectors and spatial processes, where the dependence is mainly analyzed in terms of summaries of bivariate sub-vectors. The results unify and extend many existing examples, and we use them to propose new models that encompass both dependence classes.

math.PR↗

arXiv2018

Synergy Effect between Convolutional Neural Networks and the Multiplicity of SMILES for Improvement of Molecular Prediction

In our study, we demonstrate the synergy effect between convolutional neural networks and the multiplicity of SMILES. The model we propose, the so-called Convolutional Neural Fingerprint (CNF) model, reaches the accuracy of traditional descriptors such as Dragon (Mauri et al. [22]), RDKit (Landrum [18]), CDK2 (Willighagen et al. [43]) and PyDescriptor (Masand and Rastija [20]). Moreover the CNF model generally performs better than highly fine-tuned traditional descriptors, especially on small data sets, which is of great interest for the chemical field where data sets are generally small due to experimental costs, the availability of molecules or accessibility to private databases. We evaluate the CNF model along with SMILES augmentation during both training and testing. To the best of our knowledge, this is the first time that such a methodology is presented. We show that using the multiplicity of SMILES during training acts as a regulariser and therefore avoids overfitting and can be seen as ensemble learning when considered for testing.

cs.LG↗

arXiv2017

Extremal Behavior of Aggregated Data with an Application to Downscaling

The distribution of spatially aggregated data from a stochastic process $X$ may exhibit a different tail behavior than its marginal distributions. For a large class of aggregating functionals $\ell$ we introduce the $\ell$-extremal coefficient that quantifies this difference as a function of the extremal spatial dependence in $X$. We also obtain the joint extremal dependence for multiple aggregation functionals applied to the same process. Explicit formulas for the $\ell$-extremal coefficients and multivariate dependence structures are derived in important special cases. The results provide a theoretical link between the extremal distribution of the aggregated data and the corresponding underlying process, which we exploit to develop a method for statistical downscaling. We apply our framework to downscale daily temperature maxima in the south of France from a gridded data set and use our model to generate high resolution maps of the warmest day during the 2003 heatwave.

stat.ME↗

arXiv2017

Bayesian inference for multivariate extreme value distributions

Statistical modeling of multivariate and spatial extreme events has attracted broad attention in various areas of science. Max-stable distributions and processes are the natural class of models for this purpose, and many parametric families have been developed and successfully applied. Due to complicated likelihoods, the efficient statistical inference is still an active area of research, and usually composite likelihood methods based on bivariate densities only are used. Thibaud et al. (2016, Ann. Appl. Stat., to appear) use a Bayesian approach to fit a Brown--Resnick process to extreme temperatures. In this paper, we extend this idea to a methodology that is applicable to general max-stable distributions and that uses full likelihoods. We further provide simple conditions for the asymptotic normality of the median of the posterior distribution and verify them for the commonly used models in multivariate and spatial extreme value statistics. A simulation study shows that this point estimator is considerably more efficient than the composite likelihood estimator in a frequentist framework. From a Bayesian perspective, our approach opens the way for new techniques such as Bayesian model comparison in multivariate and spatial extremes.

stat.ME↗

arXiv2017

Asymptotic properties of the maximum likelihood estimator for multivariate extreme value distributions

Max-stable distributions and processes are important models for extreme events and the assessment of tail risks. The full, multivariate likelihood of a parametric max-stable distribution is complicated and only recent advances enable its use. The asymptotic properties of the maximum likelihood estimator in multivariate extremes are mostly unknown. In this paper we provide natural conditions on the exponent function and the angular measure of the max-stable distribution that ensure asymptotic normality of the estimator. We show the effectiveness of this result by applying it to popular parametric models in multivariate extreme value statistics and to the most commonly used families of spatial max-stable processes.

math.ST↗

arXiv2017

Robust bounds in multivariate extremes

Extreme value theory provides an asymptotically justified framework for estimation of exceedance probabilities in regions where few or no observations are available. For multivariate tail estimation, the strength of extremal dependence is crucial and it is typically modeled by a parametric family of spectral distributions. In this work we provide asymptotic bounds on exceedance probabilities that are robust against misspecification of the extremal dependence model. They arise from optimizing the statistic of interest over all dependence models within some neighborhood of the reference model. A certain relaxation of these bounds yields surprisingly simple and explicit expressions, which we propose to use in applications. We show the effectiveness of the robust approach compared to classical confidence bounds when the model is misspecified. The results are further applied to quantify the effect of model uncertainty on the Value-at-Risk of a financial portfolio.

math.PR↗

arXiv2016

Statistical regionalization for estimation of extreme river discharges

Regionalization methods have long been used to estimate high return levels of river discharges at ungauged locations on a river network. In these methods, the recorded discharge measurements of a group of similar, gauged, stations is used to estimate high quantiles at the target catchment that has no observations. This group is called the region of influence and its similarity to the ungauged location is measured in terms of physical and meteorological catchment attributes. We develop a statistical method for estimation of high return levels based on regionalizing the parameters of a generalized extreme value distribution. The region of influence is chosen in an optimal way, ensuring similarity and in-group homogeneity. Our method is applied to discharge data from the Rhine basin in Switzerland, and its performance at ungauged locations is compared to that of classical regionalization methods. For gauged locations we show how our approach improves the estimation uncertainty for long return periods by combining local measurements with those from the region of influence.

stat.AP↗

arXiv2016

Generalized Pickands constants and stationary max-stable processes

Pickands constants play a crucial role in the asymptotic theory of Gaussian processes. They are commonly defined as the limits of a sequence of expectations involving fractional Brownian motions and, as such, their exact value is often unknown. Recently, Dieker and Yakir (2014) derived a novel representation of Pickands constant as a simple expected value that does not involve a limit operation. In this paper we show that the notion of Pickands constants and their corresponding Dieker-Yakir representations can be extended to a large class of stochastic processes, including general Gaussian and Lévy processes. We furthermore provide a link to spatial extreme value theory and show that Pickands-type constants coincide with certain constants arising in the study of max-stable processes with mixed moving maxima representations.

math.PR↗

arXiv2016

Extremes on river networks

Max-stable processes are the natural extension of the classical extreme-value distributions to the functional setting, and they are increasingly widely used to estimate probabilities of complex extreme events. In this paper we broaden them from the usual situation in which dependence varies according to functions of Euclidean distance to situations in which extreme river discharges at two locations on a river network may be dependent because the locations are flow-connected or because of common meteorological events. In the former case dependence depends on river distance, and in the second it depends on the hydrological distance between the locations, either of which may be very different from their Euclidean distance. Inference for the model parameters is performed using a multivariate threshold likelihood, which is shown by simulation to work well. The ideas are illustrated with data from the upper Danube basin.

stat.ME↗

arXiv2015

A characterization of the normal distribution using stationary max-stable processes

Consider the max-stable process $η(t) = \max_{i\in\mathbb N} U_i \rm{e}^{\langle X_i, t\rangle - κ(t)}$, $t\in\mathbb{R}^d$, where $\{U_i, i\in\mathbb{N}\}$ are points of the Poisson process with intensity $u^{-2}\rm{d} u$ on $(0,\infty)$, $X_i$, $i\in\mathbb{N}$, are independent copies of a random $d$-variate vector $X$ (that are independent of the Poisson process), and $κ: \mathbb{R}^d \to \mathbb{R}$ is a function. We show that the process $η$ is stationary if and only if $X$ has multivariate normal distribution and $κ(t)-κ(0)$ is the cumulant generating function of $X$. In this case, $η$ is a max-stable process introduced by R. L. Smith.

math.PR↗

arXiv2015

Max-stable processes and stationary systems of Lévy particles

We study stationary max-stable processes $\{η(t)\colon t\in\mathbb R\}$ admitting a representation of the form $η(t)=\max_{i\in\mathbb N}(U_i+ Y_i(t))$, where $\sum_{i=1}^{\infty} δ_{U_i}$ is a Poisson point process on $\mathbb R$ with intensity ${\rm e}^{-u} {\rm d} u$, and $Y_1,Y_2,\ldots$ are i.i.d.\ copies of a process $\{Y(t)\colon t\in\mathbb R\}$ obtained by running a Lévy process for positive $t$ and a dual Lévy process for negative $t$. We give a general construction of such Lévy-Brown-Resnick processes, where the restrictions of $Y$ to the positive and negative half-axes are Lévy processes with random birth and killing times. We show that these max-stable processes appear as limits of suitably normalized pointwise maxima of the form $M_n(t)=\max_{i=1,\ldots,n} ξ_i(s_n+t)$, where $ξ_1,ξ_2,\ldots$ are i.i.d.\ Lévy processes and $s_n$ is a sequence such that $s_n\sim c \log n$ with $c>0$. Also, we consider maxima of the form $\max_{i=1,\ldots,n} Z_i(t/\log n)$, where $Z_1,Z_2,\ldots$ are i.i.d.\ Ornstein--Uhlenbeck processes driven by an $α$-stable noise with skewness parameter $β=-1$. After a linear normalization, we again obtain limiting max-stable processes of the above form. This gives a generalization of the results of Brown and Resnick [Extreme values of independent stochastic processes, J.\ Appl.\ Probab., 14 (1977), pp.\ 732--739] to the totally skewed $α$-stable case.

math.PR↗

arXiv2015

Exact simulation of max-stable processes

Max-stable processes play an important role as models for spatial extreme events. Their complex structure as the pointwise maximum over an infinite number of random functions makes simulation highly nontrivial. Algorithms based on finite approximations that are used in practice are often not exact and computationally inefficient. We will present two algorithms for exact simulation of a max-stable process at a finite number of locations. The first algorithm generalizes the approach by \citet{DM-2014} for Brown--Resnick processes and it is based on simulation from the spectral measure. The second algorithm relies on the idea to simulate only the extremal functions, that is, those functions in the construction of a max-stable process that effectively contribute to the pointwise maximum. We study the complexity of both algorithms and prove that the second procedure is always more efficient. Moreover, we provide closed expressions for their implementation that cover the most popular models for max-stable processes and extreme value copulas. For simulation on dense grids, an adaptive design of the second algorithm is proposed.

stat.ME↗

arXiv2015

Maxima of independent, non-identically distributed Gaussian vectors

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 $Σ_{i,n}$. We give necessary conditions under which the row-wise maxima converge to some max-stable distribution which generalizes the class of Hüsler-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 $ψ(\sqrt{γ(h)}),h\in \mathbb{R}^d$, where $ψ$ is a completely monotone function and $γ$ is an arbitrary variogram.

math.PR↗

arXiv2014

A Lévy process on the real line seen from its supremum and max-stable processes

We consider a process $Z$ on the real line composed from a Lévy process and its exponentially tilted version killed with arbitrary rates and give an expression for the joint law of $Z$ seen from its supremum, the supremum $\overline Z$ and the time $T$ at which the supremum occurs. In fact, it is closely related to the laws of the original and the tilted Lévy processes conditioned to stay negative and positive. The result is used to derive a new representation of stationary particle systems driven by Lévy processes. In particular, this implies that a max-stable process arising from Lévy processes admits a mixed moving maxima representation with spectral functions given by the conditioned Lévy processes.

math.PR↗

arXiv2013

Extremal behavior of squared Bessel processes attracted by the Brown-Resnick process

The convergence of properly time-scaled and normalized maxima of independent standard Brownian motions to the Brown-Resnick process is well-known in the literature. In this paper, we study the extremal functional behavior of non-Gaussian processes, namely squared Bessel processes and scalar products of Brownian motions. It is shown that maxima of independent samples of those processes converge weakly on the space of continuous functions to the Brown-Resnick process.

math.PR↗

arXiv2012

Estimation of Huesler-Reiss distributions and Brown-Resnick processes

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 Hüsler-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 Hüsler-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.

stat.ME↗

arXiv2012

Representations of max-stable processes based on single extreme events

This paper provides the basis for new methods of inference for max-stable processes ξ 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 η in the max-domain of attraction of ξ on being extremal, several convergence results for the increments of η are proved. In a similar way, the shape functions of mixed moving maxima (M3) processes can be extracted from suitably conditioned single events η. Connecting the two approaches, transformation formulae for processes that admit both an incremental and an M3 representation are identified.

math.PR↗