The subject of modelling and application of stochastic processes is too vast to be exhausted in a single volume. In this book, attention is focused on a small subset of this vast subject. The primary emphasis is on realization and approximation of stochastic systems. Recently there has been considerable interest in the stochastic realization problem, and hence, an attempt has been made here to collect in one place some of the more recent approaches and algorithms for solving the stochastic realiza­ tion problem. Various different approaches for realizing linear minimum-phase systems, linear nonminimum-phase systems, and bilinear systems are presented. These approaches range from time-domain methods to spectral-domain methods. An overview of the chapter contents briefly describes these approaches. Also, in most of these chapters special attention is given to the problem of developing numerically ef­ ficient algorithms for obtaining reduced-order (approximate) stochastic realizations. On the application side, chapters on use of Markov random fields for modelling and analyzing image signals, use of complementary models for the smoothing problem with missing data, and nonlinear estimation are included. Chapter 1 by Klein and Dickinson develops the nested orthogonal state space realization for ARMA processes. As suggested by the name, nested orthogonal realizations possess two key properties; (i) the state variables are orthogonal, and (ii) the system matrices for the (n + l)st order realization contain as their "upper" n-th order blocks the system matrices from the n-th order realization (nesting property).

1. Nested Orthogonal Realizations for Linear Prediction of Arma Processes.- 2. q-Markov Covariance Equivalent Realizations.- 3. Reduced-Order Modelling of Stochastic Processes with Applications to Estimation.- 4. Generalized Principal Components Analysis and its Application in Approximate Stochastic Realization.- 5. Finite-Data Algorithms for Approximate Stochastic Realization.- 6. Model Reduction Via Balancing, and Connections with Other Methods.- 7. The Scattering Matrix Associated with a Stationary Stochastic Process: System Theoretic Properties and Role in Realization.- 8. Realization and Reduction of S.I.S.O. Nonminimum Phase Stochastic Systems.- 9. On Stochastic Bilinear Systems.- 10. Markov Random Fields for Image Modelling and Analysis.- 11. Smoothing with Blackouts.- 12. Stochastic Bilinear Models and Estimators with Nonlinear Observation Feedback.