Stochastic processes

Stochastic processes are probabilistic models of data streams such as speech, audio and video signals, stock market prices, and measurements of physical phenomena by digital sensors such as medical instruments, GPS receivers, or seismographs. 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. These course notes introduce the theory of discrete-time multivariate stochastic processes (i.e. sequences of random vectors) that is needed for estimation and prediction. Students are assumed to have knowledge of basic probability and of matrix algebra. The course starts with a succinct review of the theory of discrete and continuous random variables and random vectors. Bayesian estimation of linear functions of multivariate normal (Gaussian) random vectors is introduced. There follows a presentation of random sequences, including discussions of convergence, ergodicity, and power spectral density. State space models of linear discrete-time dynamic systems are introduced, and their response to transient and stationary random inputs is studied. The estimation problem for linear discretetime systems with normal (i.e. Gaussian) signals is introduced and the Kalman filter algorithm is derived. Additional course materials, including exercise problems and recorded lectures, are available at the author’s home page http://www.tut.fi/~piche/stochastic