A Generalization of the Kalman Model Based on Gauss-Markov Theory

The Kalman filter is generalized to cover state-space models in which the variance of the observation error depends on the state vector. Derivations of the filter yielding minimum mean squared error linear estimators and associated error covariance matrices are obtained from two differing viewpoints: linear Bayes theory and Gauss-Markov theory. The results are applied to a model for which ${y_t: t = 1, 2, ldots, n}$ follow a Poisson distribution with corresponding intensities ${ heta_t: t = 1, 2, ldots, n}$ that are assumed to follow an autoregressive process of order 1, namely $ heta_t - ar{ heta} = ho( heta_{t - 1} - ar{ heta}) + w_t$. The steady-state generalized Kalman filter algorithm in the case for which $ ho = 1$ gives a generalization of exponential smoothing for a Poisson process with time-varying intensity.