Fuzzy time-rank relations and order statistics

The rank ordering of samples is widely used in robust statistics and robust signal processing. Advances in these areas have focused on utilizing joint time-rank (TR) information. The TR information utilized to date is that resulting from a binary, or crisp, relation between the marginal time and rank ordering of samples. This crisp relation, while powerful, contains no information on sample values or spread. This paper generalizes the TR relation through fuzzy set theory. This generalization includes information on sample spread and leads to the concepts of fuzzy TR relations, fuzzy time and rank ordered samples, and fuzzy time and rank indices. These concepts are developed and analyzed through the derivation of fundamental properties. It is shown that the fuzzy TR relations, samples, and indices contain their crisp (standard) counterparts as special cases. These fuzzy generalizations constitute powerful tools that can be exploited in the design of signal processing algorithms.