MAT-60406 Stochastic Processes, 5 cr

Additional information

No implementations during academic year 2015-2016.
Suitable for postgraduate studies
Will not be lectured year 2015-2016

Person responsible

Simo Ali-Löytty

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Requirements

Exam, or exam and homework
Completion parts must belong to the same implementation

Learning Outcomes

Stochastic (i.e. random) processes are probabilistic models of information streams such as speech, audio and video signals, stock market prices, data from medical instruments, the motion of a GPS receiver, and many more. A solid understanding of the mathematical basis of these models is essential for understanding phenomena and processing information in many branches of science and engineering including physics, communications, signal processing, automation, and structural dynamics. In this course, we focus on linear stochastic system theory for estimation and prediction. After studying this course, the student can compute the response of linear discrete-time systems with random inputs; derive the Kalman filter and apply it to estimate random state parameters in simplified versions of practical engineering problems; demonstrate his/her understanding of the underlying theory by proving theorems, deriving formulas, devising counterexamples, and solving computational problems; write short Matlab programs to analyse, simulate and estimate the state parameters of systems with random inputs

Content

Content Core content Complementary knowledge Specialist knowledge

1. Random variables and vectors: pmf, pdf, cdf, independence, expectation, characteristic function, conditional rv, correlation matrix, covariance matrix, uncorrelated rv, conditional expectation, marginal rv Chebyshev inequality, transformation, Bienayme's identity, Cauchy-Schwartz inequality, moments & cf derivatives, marginal cf binomial rv, Poisson rv, principal components, Hilbert space of finite-variance rv's, binary transmission channel, Borel set

2. Multivariate normal (gaussian) rv: characteristic function, affine transformation, marginal rv, conditional rv; Bayesian estimation of linear model parameters degenerate (singular) rv; generation of correlated random samples in Matlab, recursive estimation, weighted least squares matrix inversion lemma, periodogram

3. random sequences: autocorrelation and autocovariance, stationary rs, wise-sense stationary rs, iid sequence, random walk, autoregressive sequence; convergence (almost sure, mean square, stochastic, distribution); conditions for ergodicity in mean and in correlation Cauchy-Schwartz inequality for cross correlation; binomial sequence; asymptotically wss; central limit theorem; laws of large numbers; Cauchy-type convergence criteria de Moivre-Laplace approximation, proofs of conditions for ergodicity in mean and in correlation

4. univariate Fourier series, convolution, Parseval; power spectral density, non-negativity of psd multivariate Fourier series, Plancherel, Wiener-Khinchine theorem (psd as a limit of discrete Fourier transforms), multivariate psd proof of Wiener-Khinchine theorem

5. linear time-invariant discrete-time state space model: impulse response, eigenvalue stability criterion, transfer function, random-input response, steady-state white-noise response covariance and psd; Kalman filter: derivation, optimality Lyapunov equation stability criterion, ARMA filter; generation of stationary system response in Matlab; shaping filters via spectral factorisation, steady-state Kalman filter proofs of stability theorems; spectral factorisation in Matlab; Kalman filter stability conditions

Study material

Type Name Author ISBN URL Additional information Examination material

Other online content course home page RP Lecture slides, lecture recordings, course notes, problems No

Online book Stochastic Processes Robert Piché Yes

Prerequisites

Course Mandatory/Advisable Description

MAT-02500 Todennäköisyyslaskenta Mandatory

MAT-02550 Tilastomatematiikka Mandatory

MAT-60006 Matrix Algebra Advisable

Additional information about prerequisites
The prerequisites are basic courses in Matrix Analysis and in Probability Theory.

Correspondence of content

Course Corresponds course Description

MAT-60406 Stochastic Processes, 5 cr MAT-51266 Stochastic Processes, 6 cr

Last modified 24.03.2015

Content	Core content	Complementary knowledge	Specialist knowledge
1.	Random variables and vectors: pmf, pdf, cdf, independence, expectation, characteristic function, conditional rv, correlation matrix, covariance matrix, uncorrelated rv, conditional expectation, marginal rv	Chebyshev inequality, transformation, Bienayme's identity, Cauchy-Schwartz inequality, moments & cf derivatives, marginal cf	binomial rv, Poisson rv, principal components, Hilbert space of finite-variance rv's, binary transmission channel, Borel set
2.	Multivariate normal (gaussian) rv: characteristic function, affine transformation, marginal rv, conditional rv; Bayesian estimation of linear model parameters	degenerate (singular) rv; generation of correlated random samples in Matlab, recursive estimation, weighted least squares	matrix inversion lemma, periodogram
3.	random sequences: autocorrelation and autocovariance, stationary rs, wise-sense stationary rs, iid sequence, random walk, autoregressive sequence; convergence (almost sure, mean square, stochastic, distribution); conditions for ergodicity in mean and in correlation	Cauchy-Schwartz inequality for cross correlation; binomial sequence; asymptotically wss; central limit theorem; laws of large numbers; Cauchy-type convergence criteria	de Moivre-Laplace approximation, proofs of conditions for ergodicity in mean and in correlation
4.	univariate Fourier series, convolution, Parseval; power spectral density, non-negativity of psd	multivariate Fourier series, Plancherel, Wiener-Khinchine theorem (psd as a limit of discrete Fourier transforms), multivariate psd	proof of Wiener-Khinchine theorem
5.	linear time-invariant discrete-time state space model: impulse response, eigenvalue stability criterion, transfer function, random-input response, steady-state white-noise response covariance and psd; Kalman filter: derivation, optimality	Lyapunov equation stability criterion, ARMA filter; generation of stationary system response in Matlab; shaping filters via spectral factorisation, steady-state Kalman filter	proofs of stability theorems; spectral factorisation in Matlab; Kalman filter stability conditions

Type	Name	Author	ISBN	URL	Additional information	Examination material
Other online content	course home page	RP			Lecture slides, lecture recordings, course notes, problems	No
Online book	Stochastic Processes	Robert Piché				Yes

Course	Mandatory/Advisable	Description
MAT-02500 Todennäköisyyslaskenta	Mandatory
MAT-02550 Tilastomatematiikka	Mandatory
MAT-60006 Matrix Algebra	Advisable

Course	Corresponds course	Description
MAT-60406 Stochastic Processes, 5 cr	MAT-51266 Stochastic Processes, 6 cr