# Seminar: Stochastic Geometry and its Applications

## Seminar Supervisor

Prof. Dr. Evgeny Spodarev

Dr. Vitalii Makogin

## Date and Place

It will be a one-day block-seminar in June.

## Prerequisites

The level of difficulty in this seminar is varying between the different topics. The audience is at least supposed to be familiar with basic probability, statistics, basic analysis and measure theory. We ensure the participants, that most of the 'beyond' knowledge will be learned during the seminar.

## Intended Audience

Bachelor and Master Students majoring in any mathematical course of studies.

## Content

##### Dimension of data and its estimation

Assume that we have data represented as vectors of dimension d. The data is embedded in Rd, but this does not necessarily imply that its actual dimension is d. For example, the sample of directions saved as vectors in R2 belongs actually to the interval [0,2π] with dimension one. Roughly speaking, the dimensionality of a data set is the minimum number m of free variables needed to represent the data without information loss. The statistical estimation of an unknown right value of m is the main task of our seminar. Moreover, there are sample paths of stochastic processes and geometrical objects with fractal structure which can have non-integer dimension (see Figure 1). Therefore, we consider also the theory for of Hausdorff dimension.

Dimension estimation is a part of the larger problem in a big data analysis: dimensionality reduction, that is the process of reducing the number of random variables under consideration. If m is significantly smaller than given d, then processing of the data is extremely faster and the further analysis is much more correct.

###### The topics of the seminar are covered by three main blocks
• Introduction to geometry: fractals, manifolds, Riemannian manifolds, Hausdorff measures and dimensions.
• Statistical dimension estimation of: manifolds, fractals, trajectories of random processes. Maximum likelihood estimation of intrinsic dimension.
• Big Data: dimension reduction, manifold learning, principal component analysis, projections techniques.

## Registration

To register for the seminar, please write an E-Mail to vitalii.makogin@uni-ulm.de until 30th April 2019. In the e-mail please give your name, matriculation number, your programme of studies and subjects you have taken in the area of Probability or Statistics.

## Criteria to pass the seminar

Each student is supposed to give a talk. Those who give a (good) talk together with written summary will pass the seminar. Talks will be held in German or English.