Research Group "Stochastic Algorithms and Nonparametric Statistics"

Song Song (HU Berlin)

Flexible Factor Modelling in Time and Space

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Elmar Diederichs (WIAS)

The Development of Sparse NonGaussian Component Analysis

Abstract:As a meanwhile classical tool for dimension reduction Sparse NonGaussian Component Analysis (SNGCA) has some earlier and less ancestor versions: Non Gaussian Component Analysis for example is an unsupervised method of extracting a linear structure from a high dimensional data based on estimating a low-dimensional non-Gaussian data component. This talk will give an introduction in the basic ideas behind this development of SNGCA into its current approach using the powerfull means of semidefinite relaxation. The new procedure differs significantly from the earlier proposals and it improves the method efficiency and sensitivity to a broad variety of deviations from normality and decreases the computational effort.

Fabrizio Durante (Freie Universität Bozen)

Tail dependence and copula

Abstract:The concept of copula represents a way for describing the dependence among the components of a given random vector. As such, it has gained a lot of popularity during the last years, especially in view of its possible applications to finance, insurance and natural sciences. According to Sklar's Theorem, a variety of multivariate distribution functions can be constructed by putting together univariate marginal distribution functions and a suitable copula, expressing the association among the variables of interest. In particular, copulas allow to construct flexible multivariate models that exhibit various kinds of dependencies in the tails of their distributions, a feature of great interest in risk management. In this talk, we present some recent results concerning the use (and some misuses) of copulas for capturing tail dependencies. In particular, we will show how a multivariate distribution function having a specific tail behavior can be obtained by considering special copula constructions. Moreover, we introduce and discuss the concept of threshold copula, emphasizing its possible application to the detection of (spatial) contagion between two financial markets.

Andreas Christmann (Bayreuth)

Some Recent Results on Support Vector Machines

Abstract:Support vector machines (SVMs) play an important role in modern statistical learn- ing theory and are successfully applied to solve hard real life problems even for high- dimensional and complex data sets. SVMs are based on regularized empirical risk minimization. The talk gives a short introduction to SVMs and their goals. Some recent results on consistency and robustness properties of SVMs will be given in the second part of the talk. The success of SVMs partially results from the now proven fact that an SVM is the solution of a well-posed mathematical problem in Hadamard’s sense: for any data set there exists a unique solution which depends in a continuous way on the data. The learning properties and the statistical robustness of SVMs depend on the choice of the loss function and on the kernel used to de%Gfi%@ne the reproducing kernel Hilbert space of functions. Leading examples of SVMs are based on the following loss functions: the hinge and the logistic loss function for classi%Gfi%@cation, the %GΗ«%@-insensitive loss function or Huber’s loss function for regression, and the pinball loss function for quantile regression. A classical kernel is the Gaussian radial basis function kernel.

Denis Belomestny (WIAS, Berlin)

Convergence rates of simulation based algorithms for optimal stopping problems

Abstract:Convergence rates of simulation based algorithms for optimal stopping problems -Abstract: In this talk we consider simulation-based optimization algorithms for solving discrete time optimal stopping problems. Using large deviation theory for the increments of empirical processes, we derive optimal convergence rates for the value function estimate and show that they can not be improved in general. The rates derived provide a guide to the choice of the number of simulated paths needed in optimization step, which is crucial for the good performance of any simulation-based optimization algorithm. Finally, we present a numerical example of solving optimal stopping problem arising in finance that illustrates our theoretical findings.

Dennis Kristensen (Columbia NY)

Stochastic Demand and Revealed Preference

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Leonid Pastur (Kharkiv, Ukraine)

Limiting laws for the eigenvalue statistics of large random matrices

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Korbinian Strimmer (Universität Leipzig)

High-dimensional feature selection by decorrelation: application in genomics and proteomics

Abstract:We present a novel approach to feature selection and variable importance using "cat" and "car" scores. These are defined as correlation-adjusted versions of t-scores and marginal correlations, thus take account of correlation among predictors, and are applicable to binary and continuous response, respectively. "cat" and "car" scores follow naturally from a reformulation of the linear model, with a canonical decomposition of the coefficient of determination, natural incorporation of grouping structures, and a simple link to classical (AIC, Cp, etc.) and adaptive (FDR) model selection criteria. We propose a shrinkage estimator for "cat" and "car" scores and apply them to feature selection in high-dimensional gene expression and proteomics data. Furthermore, we highlight some connections between FDR (false discovery rates), FNDR (false non-discovery rates), and the recently proposed approach of "Higher Criticism" methodologies. tba

Natalia Bochkina (University of Edinburgh)

Bayesian wavelet estimators: optimality and a priori assumptions

Abstract:We consider Bayesian wavelet estimators in the context of nonparametric regression. We discuss different choices of prior which lead to optimal performance, considering in particular the pointwise optimality in l_p norm of Bayes factor wavelet estimators. However, as it was shown by Cai (2008), adaptive coefficient-by-coefficient estimators considered above cannot achieve the global optimal rate without a log factor. We discuss extensions of the considered Bayesian models for wavelet coefficients that pool information across the coefficients, some of them known to achieve the optimal global rate exactly, and show that they also achieve the optimal local rate. Bayesian wavelet modelling is usually done in the domain of wavelet coefficients. We discuss how a priori assumptions in wavelet domain transfer to the function domain for the considered estimators.

Christoph Rothe (Toulouse)

Analyzing Counterfactual Distributions

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Jelena Bradic (Princeton University)

Penalized Composite Quasi-Likelihood for Ultrahigh-Dimensional Variable Selection.

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Richard Samworth (Cambridge)

Maximum likelihood estimation of a multidimensional log-concave density

Abstract:We show that if $X_1,...,X_n$ are a random sample from a density $f$ in $\mathbb{R}^d$, then with probability one there exists a unique log-concave maximum likelihood estimator $\hat{f}_n$ of $f$. The use of this estimator is attractive because, unlike kernel density estimation, the estimator is fully automatic, with no smoothing parameters to choose. We exhibit an iterative algorithm for computing the estimator and show how the method can be combined with the EM algorithm to fit finite mixtures of log-concave densities. Applications to classification, clustering and functional estimation problems will be discussed. The talk will be illustrated with pictures from the R package LogConcDEAD. I will also discuss recent theoretical results on the performance of the estimator, which will cover both the case where the true density is log-concave, and when this model is misspecified. These results will be applied to study both linear and isotonic regression problems. Co-authors: Madeleine Cule (University of Cambridge), Lutz Duembgen (University of Bern), Robert Gramacy (University of Cambridge), Dominic Schuhmacher (University of Bern) and Michael Stewart (University of Sydney).