Functional maximum-likelihood estimation of ARH(p) models

In this paper the problem of functional filtering of an autoregressive Hilbertian (ARH) process, affected by additive Hilbertian noise, is addressed when the functional parameters defining the ARH(p) equation are unknown. The maximum-likelihood estimation of such parameters is obtained from the implementation of an expectation-maximization algorithm. Specifically, a finite-dimensional matrix approximation of the state equation is considered from its diagonalization in terms of the spectral decomposition of the functional parameters involved (Principal-Oscillation-Pattern-based diagonalization). The Expectation step and maximization step are then computed from the forward Kalman filtering followed by a backward Kalman smoothing recursion in terms of the Fourier coefficients associated with such a decomposition.