Course detail

Stochastic Processes

FIT-SSPAcad. year: 2013/2014

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

Czech

Number of ECTS credits

Mode of study

Not applicable.

Guarantor

doc. RNDr. Vítězslav Veselý, CSc.

Department

Faculty of Mechanical Engineering (FSI)

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 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.

Co-requisites

Not applicable.

Planned learning activities and teaching methods

The course uses teaching methods in form of Lecture - 2 teaching hours per week, Computer exercise - 1 teaching hour per week.

Assesment methods and criteria linked to learning outcomes

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

Course curriculum

    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.

Work placements

Not applicable.

Aims

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.

Specification of controlled education, way of implementation and compensation for absences

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

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Not applicable.

Recommended reading

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.

Classification of course in study plans

  • Programme IT-MSC-2 Master's

    branch MBS , 0 year of study, winter semester, elective
    branch MMI , 0 year of study, winter semester, elective
    branch MMM , 0 year of study, winter semester, compulsory-optional
    branch MPV , 0 year of study, winter semester, elective
    branch MBI , 0 year of study, winter semester, elective
    branch MSK , 0 year of study, winter semester, elective

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

doc. RNDr. Vítězslav Veselý, CSc.

Syllabus

  1. Stochastický proces, trajektorie, příklady, klasifikace stochastických procesů.
  2. Konzistentní systém distribučních funkcí, striktní a slabá stacionarita.
  3. Momentové charakteristiky: střední hodnota, autokorelační a parciální autokorelační funkce, spektrální hustota.
  4. Poissonův proces.
  5. Statistická analýza Poissonova procesu.
  6. Markovské procesy.
  7. Procesy zrodu a zániku.
  8. Markovské řetězce, pravděpodobnosti přechodů, vlastnosti.
  9. Homogenní Markovovy řetězce, klasifikace stavů a stacionární pravděpodobnosti.
  10. Časové řady, stacionarita, ergodicita.
  11. Odhady trendu a metody predikce.
  12. AR a MA procesy.
  13. ARMA procesy.

Exercise in computer lab

13 hod., optionally

Teacher / Lecturer

doc. RNDr. Vítězslav Veselý, CSc.

Syllabus

  1. Statistický software Statistica, Statgraphics, Matlab.
  2. Načítání a vizualizace dat. Simulace.
  3. Popisná statistika časové řady.
  4. Momentové charakteristiky stochastického procesu.
  5. Vybrané vlastností Poissonova procesu - praktické užití.
  6. Reálné úlohy na Poissonův proces, aplikace v teorii spolehlivosti, analýza poruchovosti.
  7. Markovský proces - příklady, modely hromadné obsluhy, hledání limitních pravděpodobností stavů.
  8. Yuleův proces růstu - výpočet pravděpodobností stavů, úlohy na aplikace procesu růstu a zániku
  9. Markovské řetězce - praktické příklady, sestavení matice pravděpodobností přechodu, výpočet pravděpodobností stavů pro homogenní řetězec.
  10. Praktické určení klasifikace stavů, výpočet stacionárních pravděpodobností.
  11. Metoda klouzavých součtů pro časovou řadu, exponenciální vyrovnávání, odhady trendu.
  12. Výpočet autokorelační funkce a parciální autokorelační, proces AR(1) a MA(1).
  13. Identifikace modelu, výpočet predikce s využitím výpočetního software.