# German-Swiss Research Group FOR 916:

Statistical Regularisation and Qualitative Constraints:

Inference Algorithms, Asymptotics and Applications

**Partner:**

**Project Managment:**

- Prof. Dr. A. Munk, Göttingen
- Prof. Dr. L. Dümbgen, Bern

**Staff:**

- Klaus Frick
- Andrea Krajina
- Thomas Rippl
- Till Sabel
- Hannes Sieling

**Description:**

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:In the first funding period we have developed asymptotic theory for locally constant functions in statistical inverse regression models and have begun to investigate the problem of pathwise volatility estimation in microstructure noise models. Based on this work we will combine and extend these methods in the second funding period to obtain shape constrained confidence bands for the volatility function itself. To this end we will develop shape constrained confidence bands for deconvolution problems in a first step. This project will be performed in cooperation with project A1, project B4 and members of the econometrics group in part A (project A3, project A4 and project A7). Our methods will be used to analyse the spot volatility of FGBL high frequency tick data sampled at a rate of a few seconds. This will be done in cooperation with M. Hoffmann (ENSAE Paris).B3 A. Munk: Statistical Multiscale Parameter Selection Strategies:Parameter selection is a final but very important step in any (statistical) regularization 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. Whereas in the first part of the project selection strategies of regularisation parameters for various regularisation methods have been investigated in the second part we will construct and investigate methods for locally adaptive statistical multiscale regularisation as a shape constraint. Theoretical analysis of the methods involves convergence rates for the resulting estimators and almost sure and distributional limits for maxima of the underlying partial sum processes. This will be applied to nanoscale fluorescence microscopy imaging of cells (in collaboration with S. Hell, MPI Biophysical Chemistry) and to the fully automatic image reconstruction in Magnetic Resonance Imaging (in collaboration with J. Frahm, MPI Biophysical Chemistry, BiomedNMR).