Design of permutation order statistic filters through group colorings

In this paper, we develop a group theoretic method for reducing the class cardinality of Permutation (P) filters. P filters are a recently introduced class of nonlinear filters that have their roots in permutation theory and are based on the mapping x/spl rarr/x/sup r/, where x and x/sup r/ are temporal-ordered and rank-ordered observation vectors, respectively. While P filters contain as a proper subset a large body of nonlinear filters proposed to date, and have shown to be effective at estimating nonstationary processes in non-Gaussian environments, the large number of parameters required to define a P filter limits the filter's practical window size. Also, the information required to formulate an acceptable estimate is often only a subset of that contained in the permutation mapping x/spl rarr/x/sup r/. To reduce in a flexible and systematic way the information contained in the permutation mapping, the concept of coloring is introduced. A coloring is an equivalence relation applied to rank-ordering, temporal-ordering, or both temporal- and rank-ordering. These equivalences lead to the definitions of Rank colored Permutation (RP), Temporal colored Permutation (TP), and Temporal-Rank colored Permutation (TRP) filters, respectively. The rank- and temporal-order equivalences are shown to induce equivalences on the group of permutations. The induced permutation equivalences form a quantization of the permutation group. A distance bound on this quantization is derived as well as the cardinalities of RP, TP, and TRP filter classes. Additionally, symmetry constraints are formally developed and the number of filters in an arbitrarily constrained class counted. Several simulations are provided showing the effect of coloring and demonstrating the performance of colored P filters.