Stochastic Processes

• 26 hrs lectures
• 13 hrs exercises

The course provides students with basic knowledge of modeling stochastic processes (decomposition, ARMA) and ways to estimate their assorted characteristics in order to describe the mechanism of the process behavior on the basis of its sample path. Students learn basic methods used for real data evaluation.

The course objective is to make students familiar with the principles of the theory of stochastic processes and models used for the analysis of time series 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.

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

• Ljung, L. System Identification-Theory For the User. 2nd ed. PTR Prentice Hall : Upper Saddle River, 1999.
• Hamilton, J.D. Time series analysis. Princeton University Press, 1994. xiv, 799 s. ISBN 0-691-04289-6.

• 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.
• Brockwell, P.J. - Davis, R.A. Time series: Theory and Methods. 2-nd edition 1991. New York: Springer. ISBN 978-1-4419-0319-8.

Stochastic process, types.
Strict and weak stationarity.
Autocorrelation function. Sample autocorrelation function.
Decomposition model (additive, multiplicative), variance stabilization, trend estimation in model without seasonality: (polynomial regression, linear filters)
Trend estimation in model with seasonality. Randomness tests.
Linear processes.
ARMA(1,1) processes. Asymptotic properties of the sample mean and autocorrelation function.
Best linear prediction in ARMA(1,1), Durbin-Levinson, and Innovations algorithm.
ARMA(p,q) processes, causality, invertibility, partial autocorrelation function.
Spectral density function (properties).
Identification of periodic components: periodogram, periodicity tests.
Best linear prediction, Yule-Walker system of equations, prediction error.
ARIMA processes and nonstationary stochastic processes.

Input, storage, and visualization of data, simulation of stochastic processes.
Moment characteristics of a stochastic process.
Detecting heteroscedasticity. Transformations stabilizing variance (power and Box-Cox transform).
Use of linear regression model on time series decomposition.
Estimation of polynomial degree for trend and separation of periodic components.
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.
Filtering by means of stepwise polynomial regression, exponential smoothing.
Randomness tests.
Simulation, identification, parameters estimate, and verification for ARMA model.
Prediction of process.
Testing significance of (partial) correlations.
Identification of periodic components, periodogram, and testing.
Tutorials on student projects.

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.

The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical topics presented in lectures.

Attendance at seminars is compulsory whereas the teacher decides on the compensation for absences.