Modèles de Markov Triplet et filtrage de Kalman

Kalman filtering enables to estimate a multivariate unobservable process x = {x_n}_n∈ℕ from an observed multivariate process y = {y_n}_n∈ℕ. 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 "Pairwise" 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}_n∈_ℕ 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.