M0122 Random Processes II

Faculty of Science
Spring 2008 - for the purpose of the accreditation
Extent and Intensity
2/0/0. 2 credit(s) (fasci plus compl plus > 4). Type of Completion: zk (examination).
Teacher(s)
RNDr. Marie Forbelská, Ph.D. (lecturer)
Guaranteed by
prof. RNDr. Ivanka Horová, CSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Contact Person: RNDr. Marie Forbelská, Ph.D.
Prerequisites
M9121 Time Series I
Algebra: matrix calculus, vector spaces. Selected topics from Mathematical Analysis: linear functional spaces with norm and/or inner product, power and Laurent series - basic properties, multiplication. Probability and statistics: random variables and random vectors, their distribution, moment characteristics, independence, linear regression, hypotheses testing. Computer skill: working knowledge of the numerical computing environment MATLAB.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
An advanced course of techniques for modeling of preferably discrete-time random processes. Standard contents involves modeling based on BJM (Box-Jenkins methodology), in particular AR, MA and ARMA modeling of stationary time series as well as ARIMA and SARIMA models for time series exhibiting nonstationarity in the mean. The exposition is preceded by a general introduction to the theory of discrete linear time invarinat (LTI) systems, their description via impulse response and transfer function, the notion of a recursive LTI system, conditions for causality and stability. Afterwards ARMA models can be transparently presented as a stochastic analogy of a recusrsive LTI system. The above standard material may be modified according to the specialization of the students in the current academic year. Exercises to the lecture are located in a computer lab and use MATLAB computing environment allowing the students to get the basic practical skill. They can run demo scripts related to individual topics of the lectures as well as fit models to simulated and real data using a variety of universal procedures. The implemented algorithms are fully transparent to the students and yield unlimited opportunity for their creativity.
Syllabus
  • Linear systems: definition, linear and cyclic convolution, causality and stability, impulse and frequency response, FIR and IIR linear systems.
  • The best linear prediction: Hilbert space \( L^2(\Omega,\cal{A},P) \), the best linear prediction as orthogonal projection, Durbin-Levinson algorithm, partial autocorrelation function.
  • Box-Jenkins methodology (BJM): the series \( Y_t = \sum_{j=-\infty}^{\infty}\psi_j X_{t-j} \), the general convergence theorem and its application to a stationary process including the computation of its mean and autocovariance function, general principles for modeling unkonown system.
  • ARMA processes as a special case of BJM: causality and invertibility, methods for the computation of the coefficients of the causal and inverted representation and of the autocovariance function of an ARMA$(p,q)$ process.
  • Searching an ARMA model: AR and MA models as a more simple case, identification, parameter estimation and verification, asymptotic properties of estimates.
  • SARIMA processes as a special case of BJM: ARIMA models as a more simple case, identification, parameter estimation and verification.
  • Note: Computer-aided exercises are supported by the system MATLAB.
Literature
  • BROCKWELL, Peter J. and Richard A. DAVIS. Time series :theory and methods. 2nd ed. New York: Springer-Verlag, 1991, xvi, 577 s. ISBN 0-387-97429-6. info
  • CIPRA, Tomáš. Analýza časových řad s aplikacemi v ekonomii. 1. vyd. Praha: Alfa, Státní nakladatelství technické literatury, 1986, 246 s., ob. info
  • ANDĚL, Jiří. Statistická analýza časových řad. Praha: SNTL, 1976. info
  • HAMILTON, James Douglas. Time series analysis. Princeton, N.J.: Princeton University Press, 1994, xiv, 799 s. ISBN 0-691-04289-6. info
  • LJUNG, Lennart. System identification :theory for the user. London: Prentice-Hall, 1987, 519 s. ISBN 0-13-881640-9. info
  • BROCKWELL, Peter J. and Richard A. DAVIS. ITSM for windows : a user's guide to time series modelling and forecasting. New York: Springer-Verlag, 1994, ix, 118. ISBN 0387943374. info
Assessment methods (in Czech)
Výuka: přednáška + cvičení ve formě počítačového praktika. Zápočet: zpracování individuálního projektu. Zkouška: souhrnná včetně látky "Náhodné procesy I", ústní s písemnou přípravou.
Language of instruction
Czech
Further comments (probably available only in Czech)
The course is taught annually.
The course is taught: every week.
Teacher's information
http://www.math.muni.cz/~vesely/educ_cz.html#cas_rady
The course is also listed under the following terms Spring 2011 - only for the accreditation, Spring 2003, Spring 2004, Spring 2005, Spring 2006, Spring 2007, Spring 2008, Spring 2009, Spring 2010, Spring 2011, Spring 2012, spring 2012 - acreditation, Spring 2013, Spring 2014, Spring 2015, Spring 2016, Spring 2017, spring 2018, Spring 2019, Spring 2020, Spring 2023, Spring 2024, Spring 2025.