Stochastic Processes

• 26 hrs lectures
• 13 hrs exercises

The course provides students with basic knowledge of modelling of stochastic processes (decomposition, ARMA) and ways of estimate calculation of their assorted characteristics in order to describe the mechanism of the process behaviour on the basis of its sample path. Students learn basic methods used for real data evaluation.

The course objective is to make students familiar with principles of the theory of stochastic processes and models used for analysis of time series as well as with estimation algorithms of their parameters. At seminars students practically apply theoretical procedures on simulated or real data using the software MATLAB. Result is a project of analysis and prediction of real time series.

Rudiments of the differential and integral calculus, probability theory and mathematical statistics.

• Ljung, L. System Identification-Theory For the User. 2nd ed. PTR Prentice Hall : Upper Saddle River, 1999.
• Hamilton, J.D. Time series analysis. Princeton University Press, 1994. xiv, 799 s. ISBN 0-691-04289-6.

• Brockwell, P.J. - Davis, R.A. Introduction to time series and forecasting. 3rd ed. New York: Springer, 2016. 425 s. ISBN 978-3-319-29852-8.
• Brockwell, P.J. - Davis, R.A. Time series: Theory and Methods. 2-nd edition 1991. New York: Springer. ISBN 978-1-4419-0319-8.

1. Stochastic process, types, trajectory, examples.
2. Consistent system of distribution functions, strict and weak stacionarity.
3. Moment characteristics: mean and autocorrelation function.
4. Spectral density function (properties).
5. Decomposition model (additive, multiplicative), variance stabilization.
6. Identification of periodic components: periodogram, periodicity tests.
7. Methods of periodic components separation.
8. Methods of trend estimation: polynomial regression, linear filters, splines.
9. Tests of randomness.
10.Best linear prediction, Yule-Walker system of equations, prediction error.
11.Partial autocorrelation function, Durbin-Levinson and Innovations algorithm.
12.Linear systems and convolution, causality, stability, response.
13.ARMA processes and their special cases (AR and MA process).

1. Input, storage and visualization of data, moment characteristics of stochastic process.
2. Simulating time series with some typical autocorrelation functions: white noise, coloured noise with correlations at lag one, exhibiting linear trend and/or periodicities.
3. Detecting heteroscedasticity. Transformations stabilizing variance (power and Box-Cox transform).
4. Identification of periodic components, periodogram, and testing.
5. Use of linear regression model on time series decomposition.
6. Estimation of polynomial degree for trend and separation of periodic components.
7. Denoising by means of linear filtration (moving average): design of optimal weights preserving polynomials up to a given degree, Spencer's 15-point moving average.
8. Filtering by means of stepwise polynomial regression.
9. Filtering by means of exponential smoothing.
10.Randomness tests.
11.Simulation, identification, parameters estimate and verification for ARMA model.
12.Testing significance of (partial) correlations.
13.Tutorials on student projects.

Graded course-unit credit requirements: active participation in seminars, demonstration of basic skills in practical data analysis on PC, evaluation is based on the written or oral exam and outcome of an individual data analysis project.

The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focussed on practical topics presented in lectures.

Attendance at seminars is compulsory whereas the teacher decides on the compensation for absences.