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FSI-SSPAcad. year: 2023/2024
The course provides an introduction to the theory of stochastic processes. The following topics are dealt with: types and basic characteristics, Markov chains, stationarity, autocovariance function, spectral density, examples of typical processes, parametric and nonparametric methods of decomposition of stochastic processes, identification of periodic components, ARMA processes. Students will learn the applicability of the methods for the description and prediction of the stochastic processes using suitable software on PC.
Language of instruction
Number of ECTS credits
Mode of study
Guarantor
Department
Entry knowledge
Rules for evaluation and completion of the course
Course-unit credit requirements: active participation in seminars, demonstration of basic skills in practical data analysis on PC in a project, and succesfull solution of possible written tests.
Examination: oral exam, questions are selected from a list of 3 set areas (30+30+40 points). At least a basic knowledge of the terms and their properties is required in each of the areas. Evaluation by points: excellent (90 - 100 points), very good (80 - 89 points), good (70 - 79 points), satisfactory (60 - 69 points), sufficient (50 - 59 points), failed (0 - 49 points).
Aims
The course provides students with basic knowledge of modeling stochastic processes (Markov chain, 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.
Study aids
Prerequisites and corequisites
Basic literature
Recommended literature
Elearning
Classification of course in study plans
Lecture
Teacher / Lecturer
Syllabus
Stochastic process, types.Makov chains.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.
Computer-assisted exercise
Markov chains.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.