A fast algorithm for signal extraction, influence and cross-validation in state space models

Abstract

SUMMARY A fast algorithm is developed for computing the conditional mean and variance of the signal given the observations in a signal plus noise model. The resulting recursions can be applied immediately to provide new and efficient formulae for smoothing part or all of the state vector. The ideas of studentized residuals and leverage from regression analysis are generalized to state space models, and the new algorithm is used to compute the various measures. The results are also applied to obtain a new efficient algorithm for polynomial spline smoothing. Suppose observations are generated by a Gaussian signal plus noise process, with the signal described by a state space model. Such a model for the signal occurs often in practice, for example when the signal is the output of a stochastic difference or differential equation. This paper presents a new algorithm for signal extraction, that is, the computa- tion of the conditional mean and variance of the signal given the observations. The usual method for signal extraction is to use the Kalman filter, see, for example, Anderson & Moore (1979, p. 105), followed by a smoothing step using an algorithm such as the fixed interval smoothing algorithm (Anderson & Moore, 1979, p. 187). Our approach also uses the Kalman filter, but for the smoothing step introduces a new set of recursions which are more efficient than those for the fixed interval smoothing algorithm. As for the fixed interval smoothing algorithm, recursions for estimating the signal and its variance are carried out separately, so that considerable additional savings can be made if the signal estimate alone is required. The development uses ideas introduced by Ansley & Kohn (1987a). A remarkable property of the new recursions is that they can be applied immediately