Stochastic Processes

IT-MSC-2MMM-Compulsory-Elective - group M
Completion:accreditation+exam (written&verbal)
Type of
Hour/semLecturesSem. ExercisesLab. exercisesComp. exercisesOther
Guarantee:Veselý Vítězslav, doc. RNDr., CSc., UM OSO
Lecturer:Veselý Vítězslav, doc. RNDr., CSc., UM OSO
Faculty:Faculty of Mechanical Engineering BUT
Department:Department of Mathematics, section of Statistics and Optimalization FME BUT
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.
  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.
Knowledge and skills required for the course:
  Rudiments of the differential and integral calculus, probability theory and mathematical statistics.
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.
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.
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.