### Course Description

# Contemporary mathematical and statistical methods in natural sciences and engineering

## Program

Ecotechnologies, third-level study programmeNanosciences and Nanotechnologies, third-level study programme

Information and Communication Technologies, third-level study programme

Sensor Technologies

## Lecturers:

prof. dr. Matjaž Omladič## Goals:

The goal of the course is to broaden the knowledge of mathematics and statistics. It is assumed that the probabilistic background needed is not deeply theoretical, but good feeling for applications is necessary. The emphasis will be on ergodic theory, which helps understanding convergence of contemporary iterative methods.

The competences of the students completing this course successfully would include understanding of basic concepts from both areas, familiarity with state-of-the art methods, and knowledge of application examples in many areas of informatics and engineering.

## Content:

Probability theory: (conditional) probability and independence, discrete and continuous random variables, mathematical expectation and covariance, multivariate joint, marginal and conditional probabilities, non-correlated and independent random variables, variance-covariance matrix, random vectors.

Introduction to statistics: some special distributions (binomial, Poisson, multivariate normal, exponential) sampling and statistics (order statistics, confidence intervals, testing of hypotheses, Pearson’s chi-squared test, Monte Carlo methods, Bootstrap method, maximum likelihood method), limit theorems (laws of large numbers, the central limit theorem).

Bayesian statistics: subjective probabilities, Bayesian procedures (prior and posterior distributions, point estimation and interval estimation, testing, Gibbs sampler).

Applications of mathematical and statistical methods to machine learning: Bayesian approach to some important distributions (Gaussian and exponential family of distributions, non-parametric methods), decision theory (decision trees, maximizing utility and minimizing regret), support vector machines (comparison to discriminant analysis), reinforcement learning, neuron networks including deep learning.

Realization of the syllabus will be adjusted to the students enrolled with respect to their previous knowledge and the program of study. Their seminar work will also be adjusted accordingly.

## Course literature:

Selected chapters from the following books:

• C. M. Bishop: Pattern Recognition and Machine Learning, Springer-Verlag, Cambridge, 2006

• N. Cristianini, J. Shawe-Taylor: An Introduction to Support Vector Machines and other kernel-based learning methods, Cambridge University Press, 2000

• G. Grimmett, D. Stirzaker: Probability and Random Processes, 3rd edition, Oxford Univ. Press, Oxford, 2001.

• R. V. Hogg, J. W. McKean, A, T. Craig, Introduction to Mathematical Statistics, Pearson Prentice Hall, Upper Saddle River, 2005.

• J. R. Norris: Markov Chains, Cambridge Univ. Press, Cambridge, 1999.

• S. I. Resnick: Adventures in Stochastic Processes, Birkhäuser, Boston, 1992.

• R. S. Sutton, A. G. Barto: Reinforcement Learning: An Introduction, MIT Press, Cambridge, MA, 1998

## Significant publications and references:

• Omladič, Matjaž; Ružić, Nina; Shock models with recovery option via the maxmin copulas. Fuzzy Sets and Systems 284 (2016), 113–128.

• Kuzma, Bojan; Omladič, Matjaž; Šivic, Klemen; Teichmann, Josef; Exotic one-parameter semigroups of endomorphisms of a symmetric cone. Linear Algebra Appl. 477 (2015), 42–75.

• Omladič, Matjaž; On maximal nilspaces of matrices. Linear Multilinear Algebra 62 (2014), no. 9, 1258–1265.

• Omladič, Matjaž; Radjavi, Heydar; Self-adjoint semigroups with nilpotent commutators. Linear Algebra Appl. 436 (2012), no. 7, 2597–2603. (

• Grunenfelder, L.; Košir, T.; Omladič, M.; Radjavi, H.; Finite groups with submultiplicative spectra. J. Pure Appl. Algebra 216 (2012), no. 5, 1196–1206

• Omladič, Matjaž; Radjavi, Heydar Nilpotent commutators and reducibility of semigroups. Linear Multilinear Algebra 57 (2009), no. 3, 307–317.

## Examination:

Homework (25%)

Seminar work (50%)

Oral defense (25%)

## Students obligations:

Homework

Seminar work

Oral defense