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Graduiertenkolleg Statistische Modellbildung

 

Research projects

in the funding period starting in January 2009

 

A     Data Collection for Modelling

There is a close connection between experimental design and modelling. Depending on the model used, very different experimental designs can be sensible. The two projects in the sec­tion consider different aspects of robustness of experimental designs against model variations in crossover studies.

 

  • A1: Experimental design to identify models in crossover studies (Prof. Dr. Joachim Kunert)

    In crossover designs it often happens that optimal designs under one model for carryover ef­fects perform rather poorly under another model. The project tries to determine designs that can identify which structure of carryover effects is present and still allow estimating the main effects with sufficient efficiency.

  • A2: Optimal Crossover Designs for the comparison of treatments with a control (Prof. Dr. Joachim Kunert)

    The project tries to extend Kushner's method of finding optimal crossover designs to the situation where the experimenter is only interested in the comparison of treatments with a control treatment.

 

B     Basic Principles of Modelling

The projects in this area consider principles of modelling from a viewpoint of mathematical statistics, like e.g. the Bayes approach, robustness or general concepts of selection of estima­tors.

 

  • B1: Nonparametric Bayesian regression using shape constraints (Prof. Dr. Katja Ickstadt)

    Motivated by applications in clinical trials, where relationships between two variables such as dose and response or time and concentration often are unimodal or monotone, we extend nonparametric Bayesian regression models taking into account such structural prior knowledge. A suitable Markov chain Monte Carlo scheme will be developed to analyse these models efficiently.

  • B2: Statistical measurement models and generalized statistical inference (Prof. Dr. Joachim Hartung, Dr. Guido Knapp)

    In statistical measurement models, the concepts of generalized p-values and generalized confidence intervals may be used when, for instance, nuisance parameters prevent exact statistical inference on the parameters of interest. This projects aims to develop the above mentioned concepts for regressions models with variance-covariance matrices of ANOVA-type with unbalanced sample sizes and heteroscedastic error variances.

  • B3: Multi-criteria optimization of correlated quality characteristics by means of desirability indices (Prof. Dr. Claus Weihs)

    The project plans to derive or approximate the distribution of desirability indices for correlated quality characteristics in order to avoid bias in optimization results by optimizing a realistic expectation of the index. Formulations of desirability indices not used in the literature might be necessary to adequately model joint desirability in the correlated case.

  • B4: Regularization methods for robust variable selection in linear models (Prof. Dr. Ursula Gather, PD Dr. Sonja Kuhnt)

    An alternative to classical variable selection methods for the linear model based on t-tests or AIC is to penalize a suitable norm of the coefficient vector in order to reduce the influence of variables or to eliminate them completely. The best known examples are Ridge Regression and the more recently proposed LASSO as well as the Dantzig Selector. This project investi­gates robustness of these methods with respect to outliers and model assumptions and devel­ops robust alternatives.

  • B5: Robust classification (Prof. Dr. Ursula Gather, PD Dr. Sonja Kuhnt)

    Parametric classification methods like Linear and Quadratic Discriminant Analysis or Logis­tic Regression can be used if assumptions about the underlying class densities or the likeli­hood ratios are justified. Since violations of these assumptions may result in great losses of performance, robustification of these methods is necessary. This project aims to assess the robustness of existing methods and to propose new robust alternatives, especially for prob­lems with more than two classes or time-dependent data.

  • B6: Robustness of statistical methods against violation of independence (Prof. Dr. Ursula Gather, PD Dr. Sonja Kuhnt)

    This project analyses the vulnerability of statistical methods with respect to violations of the usually made independence assumptions. It proposes alternatives that are more robust to de­pendencies in the data. This includes both asymptotic results as well as the investigation of finite-sample behaviour using simulations, where even the task of generating random vari­ables, which are subject to certain dependence conditions, is not yet solved.

 

C     Empirical Modelling

This section collects all projects which investigate models for specific, yet typical situations. Each project deals with a question that arises in a particular field of application of statistical methods.

 

  • C1: Dimension reduction for high-dimensional genetic measurements with gene group tests (Prof. Dr. Jörg Rahnenführer)

    For a meaningful interpretation of microarray experiments modern methods identify relevant biological processes or functions by scoring the statistical significance of predefined functional gene groups. We develop methods for increasing the power of this approach by modelling dependencies between the large numbers of resulting test statistics.

  • C2: Statistical models for the dependence of survival times from complex genetic markers (Prof. Dr. Jörg Rahnenführer)

    In cancer research, modelling development and progression of tumors helps in improving diagnosis and therapy decisions. The goal of this project is to develop statistical methods for classifying patients according to survival times based on tumor markers that are estimated from genetic measurements of the tumor.

  • C4: Statistical modelling of music: From generation to perception (Prof. Dr. Claus Weihs)

    In this project the whole process of generation, resonance, spatial propagation, and perception of music sounds will be modelled. The transformations of such models will be monitored from generation up to perception. Aim of the project is the characterization of the distortion of perception of music relative to the generated sound in various instrument-room-situations.

  • C5: Modelling signal transduction networks (Prof. Dr. Roland Fried, Prof. Dr. Katja Ickstadt)

    In order to understand cellular signal transduction, mathematical models such as systems of differential equations have been employed as well as stochastic approaches such as Bayesian networks. However, so far the stochastic models do not allow for modelling feedback. In our project we will introduce graph-based methods including feedback components.

  • C6: Modelling spatial effects of cellular signals (Prof. Dr. Katja Ickstadt)

    Spatial effects such as gradients or clustering, e.g., clustering of molecules such as GTPase Ras in the plasma membrane, as well as stochastic effects play an important role in signal transduction of the cell. In this project we will adapt spatial statistical models such as hierarchical Poisson/gamma models or cluster models in order to study these spatial and stochastic effects.

  • C7: Financial contagion in international capital markets (Prof. Dr. Walter Krämer)

    Financial crises which originate in a particular country tend to spread around the world ac­cording to mechanisms which are not yet fully understood. The present project will develop statistical tests to discriminate between structural changes and the propagation of shocks in a given statistical model.

  • C8: Time varying dependence in returns of risky assets (Prof. Dr. Walter Krämer)

    Returns of risky assets tend to be more highly correlated in economic downturns that in eco­nomic upturns. This project will try to find out whether this is an artefact of conditioning or whether the return dependence truly changes.

  • C9: Constructing rating models with empirical processes (not to attend at the moment)

    Rating migration matrices from an inhomogeneous Markov process are estimated non-para­metrically with the Aalen-Johansen estimator. For relevant deviation from the homogeneity, a Cox-regression supports semi-parametric stress scenarios based on the time-dependent covariates gross domestic product growth, London interbank offer rate, and stock index.

  • C10: The impact of estimation noise on portfolio credit risk (not to attend at the moment)

    The regulatory capital determinants loss given default, probability of default and correlation are to be determined by estimation. For sparse data, compared to the parameter dimension of an estimation model, the distribution of a parameter estimate effects the risk measures loss variance, credit value-at-risk and capital. Real portfolios assess the vulnerability of a model.

  • C11: Modelling of mesothelioma incidence following exposure to fibres (Prof. Dr. Joachim Kunert)

    An important problem in controlling the cancerogenic potential of synthetic mineral fibres is the extension of the results from studies with animals (generally rats) to humans. The project tries to discuss how relevant models can be found for this extension.

 

D     Algorithms for Modelling

This section combines projects with a focus on appropriate algorithms. Either new algorithms are developed, or the performance of existing algorithms is explored, to see how well they perform their tasks under various model assumptions.

 

  • D1: Robust time series analysis (Prof. Dr. Roland Fried)

    The sensitivity of classical techniques for time series analysis against violations of the underlying modelling assumptions is well-known. Many robust alternatives have been suggested, but so far proposals how to combine them for a systematic and integrated analysis from the very beginning are lacking, with very few exceptions. In this project we will compare and complement existing approaches to robust time series analysis, with the final aim of developing an integrated toolbox for robust time series analysis.

  • D2: Problem specific optimization of the ECOC bisection of classes for multi-class problems (Prof. Dr. Claus Weihs)

    Multi-class problems are often solved by summarizing class probabilities of various binary classification problems derived from class combinations (ECOC). This project aims at deriving problem specific ECOC bisections of classes optimal with regard to runtime and class assignment.

  • D3: Clustering- and classification methods for spectral analyses (JProf. Dr. Uwe Ligges)

    Although spectral analysis is nowadays frequently used in statistical practice (analyses of images, music, mass spectroscopy, econometrics …), not much work has been done to adapt and improve clustering and classification methods on features that are derived from spectral analysis. This is the aim of the project, that means for example adapting classification methods to spectral features or finding appropriate distance measures for cluster analysis.

  • D4: Numerical properties of algorithms in statistical learning methods (JProf. Dr. Uwe Ligges)

    On the way from statistical models to statistical algorithms one has to consider numerical properties given the implementation happens in some programming language for a digital computer. Important topics are speed / efficiency and accuracy / precision and finding some compromises between speed and accuracy. The aim is to analyze numerical properties of modern statistical learning methods and improve them if possible.

 

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