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Course Catalog 2012-2013
MAT-51266 Stochastic Processes, 6 cr |
Additional information
Pre-recorded lectures in English are available on-line; contact teaching consists entirely of tutorial sessions.
Suitable for postgraduate studies
Person responsible
Robert Piche
Lessons
| Study type | P1 | P2 | P3 | P4 | Summer | Implementations | Lecture times and places |
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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 | Edition, availability, ... | Examination material | Language |
| Other online content | course home page | RP | Lecture slides, lecture recordings, course notes, problems | English | |||
| Online book | Stochastic Processes | Robert Piché | English |
Prerequisites
| Course | Mandatory/Advisable | Description |
| MAT-20501 Todennäköisyyslaskenta | Mandatory | |
| MAT-31096 Matrix Algebra 1 | Mandatory |
Additional information about prerequisites
The prerequisites are basic courses in Matrix Analysis and in Probability Theory.
Prerequisite relations (Requires logging in to POP)
Correspondence of content
| Course | Corresponds course | Description |
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More precise information per implementation
| Implementation | Description | Methods of instruction | Implementation |
| No lectures; instead, students self-study pre-recorded lectures that are available on-line. Exercise sessions are held once a week. |