Course detail
Stochastic Processes
FSI-SSP Acad. year: 2020/2021 Summer semester
The course provides the introduction to the theory of stochastic processes. The following topics are dealt with: types and basic characteristics, covariance function, spectral density, stationarity, examples of typical processes, time series and their evaluation, parametric and nonparametric methods, identification of periodic components, ARMA processes. Applications of methods for elaboration of project time series evaluation and prediction supported by the computational system MATLAB.
Language of instruction
Czech
Number of ECTS credits
4
Supervisor
Department
Learning outcomes of the course unit
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.
Prerequisites
Rudiments of the differential and integral calculus, probability theory and mathematical statistics.
Planned learning activities and teaching methods
The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focussed 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 written or oral exam and outcome of an individual data analysis project.
Aims
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.
Specification of controlled education, way of implementation and compensation for absences
Attendance at seminars is compulsory whereas the teacher decides on the compensation for absences.
The study programmes with the given course
Programme IT-MGR-2: Information Technology, Master's
branch MBI: Bioinformatics and Biocomputing, elective
Programme IT-MGR-2: Information Technology, Master's
branch MBS: Information Technology Security, elective
Programme M2A-P: Applied Sciences in Engineering, Master's
branch M-MAI: Mathematical Engineering, compulsory
Programme IT-MGR-2: Information Technology, Master's
branch MMI: Management and Information Technologies, elective
Programme IT-MGR-2: Information Technology, Master's
branch MMM: Mathematical Methods in Information Technology, compulsory-optional
Programme IT-MGR-2: Information Technology, Master's
branch MPV: Computer and Embedded Systems, elective
Programme IT-MGR-2: Information Technology, Master's
branch MSK: Computer Networks and Communication, elective
Type of course unit
Lecture
26 hours, optionally
Teacher / Lecturer
Syllabus
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
13 hours, compulsory
Teacher / Lecturer
Syllabus
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.