Stochastic Processes (FSI-SSP)

The course objective is to familiarize students with the principles of the theory of stochastic processes and models used for the analysis of time series and with estimation algorithms of their parameters. Students apply theoretical procedures on simulated or real data at seminars 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 (decomposition, ARMA, Markov chain) and ways to estimate their assorted characteristics to describe the mechanism of the process behavior based on its sample path. Students learn basic methods used for real data evaluation.

The course provides an introduction to the theory of stochastic processes. The following topics are dealt with: types and basic characteristics, stationarity, autocovariance function, spectral density, examples of typical processes, parametric and nonparametric methods of decomposition of stochastic processes, identification of periodic components, ARMA processes, Markov chains. Students will learn the applicability of the methods for the description and prediction of the stochastic processes using suitable software on PC.

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.

Markov chains.
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 using linear filtration (moving average): design of optimal weights preserving polynomials up to a given degree, Spencer's 15-point moving average.
Filtering utilizing 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.

Markov chains.
Tutorials on student projects.

Grimmett, G., Stirzaker, D.: Probability and random processes. Oxford; New York: Oxford University Press. 2001.