Triplet Markov models and Kalman filtering

Kalman filtering enables to estimate a multivariate unobservable process x = {x(n)}(nis an element ofN) from an observed multivariate process y = {y(n)}(nis an element ofN). It admits a lot of applications, in particular in signal processing. In its classical framework, it is based on a dynamic stochastic model in which x satisfies a linear evolution equation and the conditional law of y given x is given by the laws p(y(n)\x(n)). In this Note, we propose two successive generalizations of the classical model. The first one, which leads to the "Pair-wise" model, consists in assuming that the evolution equation of x is indeed satisfied by the pair (x, y). We show that the new model is strictly more general than the classical one, and yet still enables Kalman-like filtering. The second one, which leads to the "Triplet" model, consists in assuming that the evolution equation of x is satisfied by a triplet (x, r, y), in which r = {r(n)}(nis an element ofN) is an (artificial) auxiliary process. We show that the Triplet model is strictly more general than the Pairwise one, and yet still enables Kalman filtering. To cite this article: F. Desbouvries, W. Pieczynski, C. R. Acad. Sci. Paris, Ser. 1336 (2003). (C) 2003 Academie des sciences/Editions scientifiques et medicales Elsevier SAS. All rights reserved.