State-Space Realizations and Optimal Smoothing for Gaussian Generalized Reciprocal Processes

This technical note derives stochastic realization andoptimal smoothing algorithms for a class of Gaussian generalizedreciprocal processes (GGRP). The note exploits the interplay betweenreciprocal processes and Markov bridges which underpinthe GGRP model. A forward-backward algorithm for stochastic realizationof GGRP is described. The form on the inverse covariancematrix for the GGRP is used, via Cholesky factorization, to derivea procedure for optimal (MMSE) smoothing of GGRP observed innoise. The note demonstrates that the associated smoothing erroris also a GGRP with known covariance which may be used toassess the performance of smoothing as a function of the modelparameters. A numerical example is provided to illustrate the performanceof the MMSE smoother compared to those derived fromcompatible Markov and reciprocal model-based algorithms.