Nonlinear filtering of time series

Abstract

The optimal filtering problem has been formulated in the foregoing under the assumption that estimates of values of an unobserved process can be gained by using a linear filter. Such an assumption fits naturally into many applied problems, at the same time allowing success to be achieved in studies of optimal filter properties. The abandoment of the assumption of the linearity of estimates leads to a substantially more complicated filtering problem. This is so, indeed, if one wishes to obtain efficient algorithms for processing observable processes, although the filtering performance would thus be expected to be significantly improved (because of decreasing the filtering error). General results of an investigation into nonlinear filtering have been obtained on the special assumptions of properties of a partially observed process like Markovian property of the process, the‘linearity of an observation procedure’ and so on. In a number of nonlinear optimal filtering problems we do not succeed, as a rule, in restricting ourselves to some statistical moments of the stochastic partially observed process but, with rare exception, we have to deal with a posteriori distributions. (The exeptions are filtering problems for the Gaussian and conditionally Gaussian processes, since in that case under some additional conditions we are able to obtain optimal filtering recursive algorithms without explicitly utilizing distributions of the processes estimated and observed).