Course details

Stochastic Processes

SSP FME BUT SSP Acad. year 2018/2019 Summer semester 4 credits

Current academic year

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.


Language of instruction



Credit+Examination (written+oral)

Time span

26 hrs lectures, 13 hrs pc labs

Assessment points

51 exam, 49 projects




Generic 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.

Learning objectives

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.

Prerequisite kwnowledge and skills

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

Study literature

  • Cipra, T.: Analýza časových řad s aplikacemi v ekonomii. 1. vyd. Praha: SNTL - Nakladatelství technické literatury, 1986. 246 s.
  • Brockwell, P.J., Davis, R.A.: Time series: Theory and Methods. 2nd edition 1991. Hardcover: Corr. 6th printing, 1998. Springer Series in Statistics. ISBN 0-387-97429-6.
  • Hamilton, J.D.: Time series analysis. Princeton University Press, 1994. xiv, 799 s. ISBN 0-691-04289-6.
  • Anděl, J.: Statistická analýza časových řad. Praha: SNTL, 1976.
  • Ljung, L.: System Identification-Theory For the User. 2nd ed., PTR Prentice Hall: Upper Saddle River, 1999.
  • Brockwell, P.J., Davis, R.A.: Introduction to time series and forecasting. 2nd ed., New York: Springer, 2002. xiv, 434 s. ISBN 0-387-95351-5.

Syllabus of lectures

  1. Stochastic processes, trajectories, examples, classification of stochastic processes.
  2. Consistent system of distribution functions, strict and weak stationarity.
  3. Momentum characteristics: the mean value, autocorrelation and partial autocorrelation, spectral density.
  4. Poisson processes.
  5. Statistical analysis of Poisson processes.
  6. Markov processes.
  7. Birth and death processes.
  8. Markov strings, transition probabilities, properties.
  9. Homogeneous Markov strings, state classification and stationary probabilities.
  10. Time series, stationarity, ergodicity.
  11. Trend estimation and methods of prediction.
  12. AR and MA processes.
  13. ARMA processes.

Syllabus of computer exercises

  1. Statistical software Statistica, Statgraphics, Matlab.
  2. Reading and visualizing data. Simulation.
  3. Descriptional statistics of time series.
  4. Momentum characteristics of stochastic processes.
  5. Selected properties of Poisson processes: practical usage.
  6. Real-life examples of Poisson processes, applications in the theory of reliability, defect analyzis.
  7. Markov processes: examples, models of queues, looking for limit state probabilities.
  8. Yule's birth processes: computing state probabilities, examples of applications on processes of growth and death.
  9. Markov strings: practical examples, construction of matrices of transition probabilities, computation of state probabilities for homogeneous strings.
  10. Practical examples of state classification, computation of stationary probabilities.
  11. Analysis of time series, trend estimation.
  12. Computing autocorrelation and partial autocorrelation functions, AR(1) and MA(1) processes.
  13. Model identification, computing predictions using up-to-date software.

Controlled instruction

Attendance at seminars is controlled and the teacher decides on the compensation for absences.

Exam prerequisites

Active participation in seminars, demonstration of basic skills in practical data analysis on PC, evaluation is based on the result of personal project.

Course inclusion in study plans

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