Course detail

Analysis of Experiment and Forecasting

FSI-SEP-AAcad. year: 2024/2025

The course provides an introduction to the theory of stochastic processes. The following topics are dealt with: types and basic characteristics, queueing theory, Markov chains, stationarity, autocovariance function, decomposition of stochastic processes, ARMA processes. 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

English

Number of ECTS credits

5

Mode of study

Not applicable.

Guarantor

doc. RNDr. Libor Žák, Ph.D.

Department

Institute of Mathematics (ÚM)

Entry knowledge

Rudiments of probability theory and mathematical statistics, liner models.

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 make students familiar with the principles of the theory of stochastic processes and models used for their analysis as well as with estimation algorithms of their parameters. At seminars, students apply theoretical procedures on simulated or real data 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 (Queueing theory, Poisson processes, Markov chains, decomposition, ARMA) and ways to estimate their assorted characteristics in order to describe the mechanism of the process behavior on the basis of its observations. 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)
Tijms, H.C. A First Course in Stochastic Models, John Wiley & Sons, 2003. 478 p. ISBN:9780471498803 (EN)
Shortle, J.F., Thompson, J.M., Gross, D., Harris, C.M. Fundamentals of Queueing Theory, 5th ed. John Wiley & Sons, 2018. 576 p. ISBN: 978-1-118-94352-6  (EN)

Recommended reading

Not applicable.

Classification of course in study plans

  • Programme N-LAN-A Master's 1 year of study, summer semester, compulsory

  • Programme C-AKR-P Lifelong learning

    specialization CLS , 1 year of study, summer semester, elective

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

doc. RNDr. Libor Žák, Ph.D.

Syllabus

1. Stochastic process, types.
2. Introduction to Queueing Theory, Birth-and-Death Processes
3. Poisson Process and M/M/1 Model
4. Markov chains I.
5. Markov chains II.
6. Design of experiments.
7. Strict and weak stationarity.
8. Autocorrelation function. Sample autocorrelation function.
9. Decomposition model (additive, multiplicative), trend estimation (polynomial regression, linear filters)
10. Trend estimation in model with seasonality. Randomness tests.
11. Linear processes.
12. ARMA(p,q) processes, causality, invertibility, partial autocorrelation function.
13. Best linear prediction, Yule-Walker system, Durbin-Levinson, and Innovations algorithm

Computer-assisted exercise

26 hod., compulsory

Teacher / Lecturer

doc. RNDr. Libor Žák, Ph.D.

Syllabus

1. Input, storage, and visualization of data, simulation of stochastic processes, queueing systems especially.
2. Poisson processes.
3. Markov chains.
4. Design of experiments
5. Moment characteristics of a stochastic process.
6. Detecting heteroscedasticity. Transformations stabilizing variance (power and Box-Cox transform).
7. Use of linear regression model on time series decomposition.
8. Denoising by means of linear filtration (moving average)
9. Filtering by means of stepwise polynomial regression, exponential smoothing.
10. Randomness tests.
11. Simulation, identification, parameters estimate, and verification for ARMA model.
12. Prediction of process. Testing significance of (partial) correlations.
13. Tutorials on student projects