Multivariate Cauchy Estimator with Scalar Measurement and Process Noises

Abstract

The conditional mean estimator for a $n$-state linear system with additive Cauchy measurement and process noises is developed. It is shown that although the Cauchy densities, which model the initial state, the process noise and the measurement noise have undefined first moments and an infinite second moment, the probability density function conditioned on the measurement history does have a finite conditional mean and conditional variance. For the multivariable system state, the characteristic function of the unnormalized conditional probability density function is sequentially propagated through measurement updates and dynamic state propagation, while expressing the resulting characteristic function in a closed analytical form. Once the characteristic function of the unnormalized conditional probability density function is obtained, the probability density function of the measurement history, the conditional mean, and conditional variance are easily computed from the characteristic function and its continuous first and second derivatives, evaluated at the origin in the spectral variables' domain. These closed form expressions yield the sequential state estimator. A three-state dynamic system example demonstrates numerically the performance of the Cauchy estimator.