3. Conditional Expectations and Discrete-Time Kalman Filtering
Society for Industrial and Applied Mathematics eBooks · 2008
From the concepts presented in Chapters 1 and 2, conditional expectation and its special case of conditional probability were defined, where the average of a random variable is constructed given that certain events have occurred. Here, the notion of state estimation is first introduced. A simplifying example, where a Gauss-Markov process is assumed, is used to illustrate the theory and has significant practical applications. This leads to a conditional mean state error in discrete time, popularly called the discrete-time Kalman filter. This importance class of stochastic estimation problems has ramifications for the estimation and control theory presented in the remainder of this book. 3.1 Minimum Variance Estimation The thought may have crossed your mind that conditional expectation is an odd subject for a book chapter. While we have hopefully convinced you that it is quite an interesting topic, we will admit that we have an ulterior motive, which is to use it to introduce stochastic estimation. Consider the static parameter estimation problem, z=h (x)+υ. 3.1 Our aim is to determine the n × 1 vector, x, given an m × 1 measurement vector, z, that is corrupted by an m × 1 random vector, υ, where h(·) is a known function. Once again, we have switched to linear systems-styled notation, where lowercase letters represent vectors and uppercase letters represent matrices. It will be our claim that the conditional probability density, fx|z, contains all of the information that we need to solve any estimation problem. The type of estimate that we get depends on how we choose to use fx|z. The maximum a posteriori estimate, x^MAP, is obtained by maximizing fx|z. Conceptually, this can be understood to be the peak value, or mode, of fx|z. The minimax estimate, denoted xMM, is found by taking the median, or midpoint, of fx|z. Finally, the minimum variance estimate is found by taking the mean, or center of mass, of fx|z.