State Space Realisations and Optimal Smoothing for Gaussian Generalized Reciprocal Processes

This technical note derives stochastic realisation and optimal smoothing algorithms for a class of Gaussian Generalised Reciprocal Processes (GGRP). The note exploits the interplay be- tween reciprocal processes and Markov bridges which underpin the GGRP model. A forwards-backwards algorithm for stochastic realisation of GGRP is described. The form on the inverse covari- ance matrix for the GGRP is used, via Cholesky factorisation, to derive a procedure for optimal (MMSE) smoothing of GGRP observed in noise. The note demonstrates that the associated smoothing error is also a GGRP with known covariance which may be used to assess the performance of smoothing as a function of the model parameters. A numerical example is provided to illustrate the performance of the MMSE smoother compared to those derived from compatible Markov and Reciprocal model based algorithms.