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Course Catalog 2010-2011
MAT-51266 Stochastic Processes, 6 cr |
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 continuous and 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 parameters of systems with random inputs
Content
| Content | Core content | Complementary knowledge | Specialist knowledge |
| 1. | deterministic dynamic systems: linear time-invariant continuous-time state space models (differential equations), LTI discrete-time state space models (difference equations) models; response; stability | solution uniqueness; sampling; observability; transfer function models; Lyapunov equation; frequency response: z-transform, Laplace transform, Fourier transform; system analysis and simulation using Matlab | |
| 2. | probability: vector random variables, joint distribution, conditional distribution, multivariate gaussian density, functions of vector random variables, expectation, covariance, conditional expectation | moment generating function; characteristic function; Chernoff bound; Chebyshev inequality; Schwarz inequality | |
| 3. | random sequences: convergence (sure, almost sure, mean square, stochastic, distribution), Markov chains, Markov sequence, Brownian motion, Wiener process, stationarity, ergodicity; response of LTI discrete-time system to random input | central limit theorem; laws of large numbers; Poisson process; Chapman-Kolmogorov matrix equation; Lyapunov equation; martingale; simulations with Matlab | |
| 4. | random processes: mean-square stochastic calculus (continuity, differentiation, integration); power spectral density; white noise; stationarity; ergodicity; Fourier transform of random process | Karhunen-Loève expansion | |
| 5. | estimation: Bayesian estimation of parameters in linear multivariate gaussian model, Kalman filter, | Kalman filter special questions: information form, noiseless measurements, square root filter; hidden Markov models; particle filters; implementing and simulating filters in Matlab | navigation (GPS, inertial, Doppler) |
Evaluation criteria for the course
exam, or exam + homework.
Assessment scale:
Numerical evaluation scale (1-5) will be used on the course
Partial passing:
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 |
Prerequisites
| Course | Mandatory/Advisable | Description |
| MAT-20501 Todennäköisyyslaskenta | Mandatory | |
| MAT-31096 Matrix Algebra 1 | Mandatory |
Prerequisite relations (Requires logging in to POP)
Correspondence of content
| Course | Corresponds course | Description |
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Additional information
Suitable for postgraduate studies
More precise information per implementation
| Implementation | Description | Methods of instruction | Implementation |
| Lectures Mondays and Tuesdays 2-4pm in Td308 during periods 1-2 in 2010. |