Stochastic Processes (FSI-SSP)

Academic year 2018/2019
Supervisor: doc. RNDr. Vítězslav Veselý, CSc.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: Czech
Course type: theoretical basis, applied basis
Aims of the course unit:
The course objective is to make students familiar with principles of theory 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.
Learning outcomes and competences:
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 time series observations. Students learn basic methods used for real data evaluation.
Prerequisites:
Rudiments of the differential and integral calculus, probability theory and mathematical statistics.
Course contents:
The course provides the introduction to the theory of stochastic processes. The following topics are dealt with: types and basic characteristics, covariation function, spectral density, stationarity, examples of typical processes, time series and evaluating, parametric and nonparametric methods, identification of periodical components, ARMA processes. Applications of methods for elaboration of project time series evaluation and prediction supported by the computational system MATLAB.
Teaching methods and criteria:
The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical topics presented in lectures.
Assesment methods and criteria linked to learning outcomes:
Graded course-unit credit requirements: active participation in seminars, demonstration of basic skills in practical data analysis on PC, evaluation is based on the oral exam and outcome of an individual data analysis project.
Controlled participation in lessons:
Attendance at seminars is compulsory whereas the teacher decides on the compensation for absences.
Type of course unit:
    Lecture  13 × 2 hrs. optionally                  
    Computer-assisted exercise  13 × 1 hrs. compulsory                  
Course curriculum:
    Lecture 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).
    Computer-assisted exercise 1. Counting, 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.
Literature - fundamental:
1. 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.
Literature - recommended:
1. Ljung, L. System Identification-Theory For the User. 2nd ed. PTR Prentice Hall : Upper Saddle River, 1999.
2. Hamilton, J.D. Time series analysis. Princeton University Press, 1994. xiv, 799 s. ISBN 0-691-04289-6.
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
M2A-P full-time study M-MAI Mathematical Engineering -- Cr,Ex 4 Compulsory 2 1 S
IT-MGR-2 full-time study MMI Management and Information Technologies -- Cr,Ex 4 Elective 1 0 S
IT-MGR-2 full-time study MBI Bioinformatics and Biocomputing -- Cr,Ex 4 Elective 1 0 S
IT-MGR-2 full-time study MSK Computer Networks and Communication -- Cr,Ex 4 Elective 1 0 S
IT-MGR-2 full-time study MMM Mathematical Methods in Information Technology -- Cr,Ex 4 Compulsory-optional 1 0 S
IT-MGR-2 full-time study MBS Information Technology Security -- Cr,Ex 4 Elective 1 0 S
IT-MGR-2 full-time study MPV Computer and Embedded Systems -- Cr,Ex 4 Elective 1 0 S