Discrete approximations for Markov-chain filters
Nigel J. Newton · Spiral (Imperial College London) · 1984
It is well-known that the conditional distribution of a Markov-chain signal process based on obser vations of a "signal-plus-white-noise" type obeys a vector stochastic differential equation.The usual Bayes estimates of functions of the signal can be found from this distribution, pointwise in time.Unfortunately, the differential equation mentioned above has, in general, no closed form solution, and so it is necessary to use some form of approximation scheme to produce a practical filter.We consider discrete approximation schemes based on regular partitions of a finite timeinterval and show that, under the condition that the distribution of the observations-process is Wiener-measure (in R d), the discrete-observations-conditional distribution of the approximation error (suitably normalised) can at best converge to a normal distribution with zero mean and given covariance matrix.For the more practical case, where the distribution of the observations-process is only absolutely continuous with respect to Wiener-measure (i.e. the signal-plus-white-noise case), a similar result holds.We demonstrate the existence of schemes that are efficient, in that their normalised error se quences converge to this best limit, and schemes that possess the highest possible order of conver gence (i.e.first-order) but that are not efficient.The latter methods have limit conditional distribu tions with non-zero means.We investigate various practical approximation schemes, in particular a class of schemes based on a "Taylor-series type" expansion of the conditional distribution of the signal about the points of the partition.A parallel set of conditional-distribution-limit-results apply to the normalised errors in the ap proximation of the Bayes estimates of functions of the signal process when these approximations are calculated from the corresponding approximate distributions of the signal-process.It is these ap proximate estimates that form the "output" of any practical filter.I would like to express my many thanks to all who have helped and encouraged me in the work which led to this thesis: most of all, to Dr. J. M. C. Clark my supervisor for his invaluable help and guidance in many discussions during the course of the