Reduced-complexity filtering for partially observed nearly completely decomposable Markov chains

This paper provides a systematic method of obtaining reduced-complexity approximations to aggregate filters for a class of partially observed nearly completely decomposable Markov chains. It is also shown why an aggregate filter adapted from Courtois' (1977) aggregation scheme has the same order of approximation as achieved by the algorithm proposed in this paper. This algorithm can also be used systematically to obtain reduced-complexity approximations to the full-order fitter as opposed to algorithms adapted from other aggregation schemes. However, the computational savings in computing the full-order filters are substantial only when the large scale Markov chain has a large number of weakly interacting blocks or "superstates" with small individual dimensions. Some simulations are carried out to compare the performance of our algorithm with algorithms adapted from various other aggregation schemes on the basis of an average approximation error criterion in aggregate (slow) filtering. These studies indicate that the algorithms adapted from other aggregation schemes may become ad hoc under certain circumstances. The algorithm proposed in this paper however, always yields reduced-complexity filters with a guaranteed order of approximation by appropriately exploiting the special structures of the system matrices.