Accession Number : AD0715945

Title :   Consideration of Bias in Linear Estimation.

Descriptive Note : Technical rept.,

Corporate Author : STANFORD UNIV CALIF STANFORD ELECTRONICS LABS

Personal Author(s) : Stepner,David E.

Report Date : OCT 1970

Pagination or Media Count : 126

Abstract : The Kalman formulation of the minimum mean square linear estimation problem prescribes that the initial state x(0) be a random variable with known distribution and that x primed(0), the estimate of the initial state, is to be E braces x(0), the mean of that distribution. This report begins with the premise that x(0) is a random variable but with unknown distribution. The immediate result is that the initial state estimate cannot fail to be biased, unlike the Kalman estimate. Two examples are given which illustrate the fact that if one then insists upon zero bias at some later time, one cannot also simultaneously minimize error variance. It is proposed that the alternate performance criterion of minimizing an arbitrary sum of error variance and outer product of error mean be used and, starting from first principles, a necessary and sufficient condition is derived for the optimal estimator. An estimator is found which satisfies this condition and an equivalent recursive form is obtained. It turns out that this form is exactly like the Kalman equations except for the interpretation of x primed(0) and the fact that the initial condition required for the Riccati equation is the arbitrary sum of the initial error covariance and the outer product of error mean. (Mean square error is just a special case of this more general presentation). This latter initial condition cannot be found exactly and, for the special case of x(0) treated as an unknown nonrandom parameter, three methods are presented for obtaining an approximate initializing value for the Riccati equation. (Author)

Descriptors :   (*INFORMATION THEORY, STATISTICAL ANALYSIS), (*CONTROL SYSTEMS, MATHEMATICAL MODELS), RANDOM VARIABLES, STOCHASTIC PROCESSES, MATRICES(MATHEMATICS), RECURSIVE FUNCTIONS, PROBABILITY DENSITY FUNCTIONS, INTEGRALS, DIFFERENTIAL EQUATIONS, THESES

Subject Categories : Statistics and Probability
      Cybernetics

Distribution Statement : APPROVED FOR PUBLIC RELEASE