# Seminar: Stochastic Geometry and its Applications

## Seminar Supervisor

Prof. Dr. Evgeny Spodarev

Dr. Vitalii Makogin

## Date and Place

It can be regular or a block seminar.  Place: Helmholtzstraße 18, E60

## 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 in any mathematical programme of studies. The "(B)" in the list of talks, means that this talk is supposed to be given by a Bachelor's Student.

## Content

##### Directional distributions

Directional distributions characterize randomness in unit-vectors (directions) or more general random elements with bounded values.

If the motion appears on the plane then directions lie on a unit circle and they are called circular data. If we study motions or orientations in 3-dimensional volumes we deal with spherical data.

The goal of the seminar is to give a probabilistic background for statistical analysis of directional data. Standard approaches lead here to meaningless conclusions. For example, consider the average direction of data set {αi}(1≤i≤N) with αi∈[0,2π] presented on the Figure. Computing the standard linear mean α°=(α1+...+αN)/N yield the angle α°=π. But from the circular plot, one can deduce that the "mean" direction shows to the right, i,e., the mean angle is 0. Therefore, we consider directional (circular) mean  for such samples.

The topics of the seminar are covered by two main blocks

• Circular data: Measures of concentration, von Mises distributions, statistical estimation

• Distributions on Spheres: Models for axial data, Von Mises-Fisher distribution

## Preliminary list of talks

Circular data:

1. Descriptive statistics, measures of concentration and dispersion (circular standard deviation), uniform distribution, projected normal distributions. (Valentin Trimcheski)
2. Characteristic functions, Limiting distributions of circular statistics. (B)
3. Von Mises distributions: characteristic function, maximum likelihood estimation, large sample asymptotic. (Markus Brachert)
4. Mixtures of von Mises distributions.
5. Tests on von Mises Distribution, multi-sample tests
6. Non-parametric tests on directional data, density estimation.
7. Change-point Problems: tests for change in mean direction.

Distributions on Spheres:

8. Spherical data: descriptive measures, mean direction, principle axes. (B)
9. Von Mises-Fisher distribution.  (B)
10. Models for axial data: Watson distribution, angular central Gaussian distribution.
11. Mixtures of angular central Gaussian distribution. Maximum likelihood estimation.
12. Measures of correlation.

Registration

To register for the seminar, please write an E-Mail to Vitalii Makogin until 30th of September 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.

## Literature

•  Mardia, K. V., & Jupp, P. E. (2009). Directional statistics (Vol. 494). John Wiley & Sons.
•  Jammalamadaka, S. R., & Sengupta, A. (2001). Topics in circular statistics (Vol. 5). World scientific.

Contact

•
• Office hours: Wednesday 4 - 5 pm
• Phone: +49 (0)731/50-23530
• Homepage