- Prof. Dr. A. Munk, Göttingen
- Prof. Dr. L. Dümbgen, Bern
- Klaus Frick
- Andrea Krajina
- Thomas Rippl
- Till Sabel
- Hannes Sieling
A basic challenge for statistics at the interface of different sciences is the development of methods for the analysis of massive data sets, complex data structures and high-dimensional predictors. The objectives of this German-Swiss research group are specific development and analysis of statistical regularization methods for complex data structures as they may occur in different fields of application. In the foreground, there are methods in which regularization is given by qualitative constraints on the structure or geometry of data models. Our basic hypothesis is that statistical regularization by qualitative constraints produces a consistent methodology for modeling of data structures which, on the one hand, is flexible enough to identify and scientifically utilize main structural features of data, but, on the other hand, specific enough to control prediction and classification error. Our research group consists of scientists who have been pursuing this objective for a long time from the perspective of different disciplines (statistics, numerical analysis, machine learning, pattern recognition, imaging, econometrics). All of us cooperated already with some members of this group for a certain time, and now we start to work all together at this ambitious project. Each of the 14 subprojects examines some aspects of this methodological target. In cooperation with other members, specific application problems are being studied which can originate directly from the research group or be brought up by associated groups. At the same time, we are working on diverse problems from the fields of systems biology, medical event-time analysis, astrophysics, atmospheric research, forest sciences, materials sciences, job market policy, biophotonics, medical image processing and empirical finance. Statistical regularization procedures with structural or qualitative constraints allow a consistent methodical perspective and solution strategy while dealing with those subject fields.
The following subprojects are assigned to the research group "Applied and Mathematical Statistics":
- B1 A. Munk: Statisitcal Inference in Inverse Problems with Qualitive Prior Information: This project aims for the development and analysis of statistical regularisation methods in the context of noisy inverse problems with specific prior information, given by shape constraints such as monotonicity, positivity and in particular on the jump behaviour of functions. Besides of convergence results, we will be concerned with distributional results, mainly of asymptotic nature, in order to obtain, confidence bounds for the reconstructed functions, and for the locations of jumps. In a second step testing procedures will be developed to determine the number of jumps in a statistical inverse problem. Our methods will be applied to measurements of the dynamic moduli of polymer solutions to determine the relaxation time spectrum (in cooperation with A.Lier, J. Hohnerkamp at the Freiburger Materialforschungszentrum), and to the estimation of ion channel activity of lipid membranes, which are measured by impedance spectroscopy (in cooperation with with C. Steinem, department of chemistry, Göttingen).
- A3 A. Munk: Statistical Multiscale Parameter Selection Strategies: Parameter selection is a final but very important step in any (statistical) regularisation process in order to determine the level of resolution of a given regularized reconstruction in a statistical inverse problem. In this project we aim for fully data driven parameter selection methods which are based on a statistical multiscale analysis of the residuals. Two major fields of application will be addressed: Construction of proper stopping criteria for iterative methods and automatic selection of smoothing, regularisation or penalisation parameters in a regularisation scheme. Theoretical analysis of the methods involves almost sure and distributional limits for maxima of partial sum processes. This will be applied to various projects within the group and to positron emission tomography in cooperation with B.Mair (Univ. of Florida).