Multi-spectral decomposition of functional autoregressive models

Functional data models provides a suitable framework for the statistical analysis of several environmental phenomena involving continuous time evolution and/or spatial variation. The functional autoregressive model of order p,  p ≥ 1, (FAR(p)) extends to the infinite-dimensional space context the classical autoregressive model AR(p) (see, for example, Mourid T (1993) Processus autorégressiifs d’ordre supérieur. Acad Sci t.317(Sér. I):1167–1172). In this paper, we derive a multidimensional diagonalization of the functional parameters (operators) involved in the FAR(p), p > 1, formulation. The functional state equation is then transformed into a discrete system of scalar state equations. The decomposition obtained is optimal regarding information on spatiotemporal interaction affecting the evolution of the spatial behavior of the process of interest. For functional prediction and filtering, we implement the Kalman filter equations from the diagonal version derived for FAR(p) models. This work has been supported in part by projects MTM2005-08597 of the DGI, MEC, and P05-FQM-00990, P06-FQM-02271 of the Andalousian CICE, Spain.