Stochastic Realisation and Optimal Smoothing for Gaussian Generalised Reciprocal Processes

This paper derives stochastic realisation and opti- mal smoothing algorithms for a class of Gaussian Generalised Reciprocal Processes (GGRP). The paper exploits the interplay between reciprocal processes and Markov bridges which un- derpin the GGRP model. A forwards-backwards algorithm for stochastic realisation of GGRP is described. The form on the inverse covariance matrix for the GGRP is used, via Cholesky factorisation, to derive a similar procedure for optimal (MMSE) smoothing of GGRP observed in noise. The paper 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.