Course detail

Stochastic Processes

FSI-SSPAcad. year: 2025/2026

The course provides an introduction to the theory of stochastic processes. The following topics are dealt with: types and basic characteristics, stationarity, autocovariance function, spectral density, examples of typical processes, parametric and nonparametric methods of decomposition of stochastic processes, identification of periodic components, ARMA processes, Markov chains. Students will learn the applicability of the methods for the description and prediction of the stochastic processes using suitable software on PC.

Language of instruction

Czech

Number of ECTS credits

5

Mode of study

Not applicable.

Guarantor

doc. Mgr. Zuzana Hübnerová, Ph.D.

Department

Institute of Mathematics (ÚM)

Entry knowledge

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

Rules for evaluation and completion of the course

Course-unit credit requirements: active participation in seminars, demonstration of basic skills in practical data analysis on PC in a project, and succesfull solution of possible written tests.

Examination: oral exam, questions are selected from a list of 3 set areas (30+30+40 points). At least a basic knowledge of the terms and their properties is required in each of the areas. Evaluation by points: excellent (90 - 100 points), very good (80 - 89 points), good (70 - 79 points), satisfactory (60 - 69 points), sufficient (50 - 59 points), failed (0 - 49 points).


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

Aims

The course objective is to familiarize students with the principles of the theory of stochastic processes and models used for the analysis of time series and with estimation algorithms of their parameters. Students apply theoretical procedures on simulated or real data at seminars using suitable software. The semester is concluded with a project of analysis and prediction of a real stochastic process.

The course provides students with basic knowledge of modeling stochastic processes (decomposition, ARMA, Markov chain) and ways to estimate their assorted characteristics to describe the mechanism of the process behavior based on its sample path. Students learn basic methods used for real data evaluation.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

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. (EN)
Brockwell, P.J. - Davis, R.A. Time series: Theory and Methods. 2-nd edition 1991. New York: Springer. ISBN 978-1-4419-0319-8. (EN)
Grimmett, G., Stirzaker, D.: Probability and random processes. Oxford; New York: Oxford University Press. 2001. (EN)
Shumway, R., Stoffer, D. Time Series Analysis and Its Applications With R Examples. Springer, 2017. 978-3-319-52452-8. (EN)

Recommended reading

Cipra, Tomáš. Analýza časových řad s aplikacemi v ekonomii. 1. vyd. Praha : SNTL - Nakladatelství technické literatury, 1986. 246 s.
Hamilton, J.D. Time series analysis. Princeton University Press, 1994. xiv, 799 s. ISBN 0-691-04289-6. (EN)
Ljung, L. System Identification-Theory For the User. 2nd ed. PTR Prentice Hall : Upper Saddle River, 1999. (EN)

Classification of course in study plans

  • Programme N-MAI-P Master's 1 year of study, summer semester, compulsory

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

doc. Mgr. Zuzana Hübnerová, Ph.D.

Syllabus

Stochastic process, types.
Strict and weak stationarity.
Autocorrelation function. Sample autocorrelation function.
Decomposition model (additive, multiplicative), variance stabilization, trend estimation in model without seasonality: (polynomial regression, linear filters)
Trend estimation in model with seasonality. Randomness tests.
Linear processes.
ARMA(1,1) processes. Asymptotic properties of the sample mean and autocorrelation function.
Best linear prediction in ARMA(1,1), Durbin-Levinson, and Innovations algorithm.
ARMA(p,q) processes, causality, invertibility, partial autocorrelation function.
Spectral density function (properties).
Identification of periodic components: periodogram, periodicity tests.

Markov chains.
Best linear prediction, Yule-Walker system of equations, prediction error.
ARIMA processes and nonstationary stochastic processes.

Computer-assisted exercise

13 hod., compulsory

Teacher / Lecturer

doc. Mgr. Zuzana Hübnerová, Ph.D.

Syllabus

Input, storage, and visualization of data, simulation of stochastic processes.
Moment characteristics of a stochastic process.
Detecting heteroscedasticity. Transformations stabilizing variance (power and Box-Cox transform).
Use of linear regression model on time series decomposition.
Estimation of polynomial degree for trend and separation of periodic components.
Denoising using linear filtration (moving average): design of optimal weights preserving polynomials up to a given degree, Spencer's 15-point moving average.
Filtering utilizing stepwise polynomial regression, exponential smoothing.
Randomness tests.
Simulation, identification, parameters estimate, and verification for ARMA model.
Prediction of process.
Testing significance of (partial) correlations.
Identification of periodic components, periodogram, and testing.

Markov chains.
Tutorials on student projects.