Models and Representations of Gaussian Reciprocal and Conditionally Markov Sequences

Reza Rezaie, X. Rong Li

Published 2018-11-19, updated 2019-12-03Version 2

Conditionally Markov (CM) sequences are powerful mathematical tools for modeling random phenomena. There are several classes of CM sequences one of which is the reciprocal sequence. Reciprocal sequences have been widely used in many areas including image processing, intelligent systems, and acausal systems. To use them in application, we need not only their applicable dynamic models, but also some general approaches to designing parameters of dynamic models. Dynamic models governing two important classes of nonsingular Gaussian (NG) CM sequences (called $CM_L$ and $CM_F$ models), and a dynamic model governing the NG reciprocal sequence (called reciprocal $CM_L$ model) were presented in our previous work. In this paper, these models are studied in more detail and general approaches are presented for their parameter design. It is shown that every reciprocal $CM_L$ model can be induced by a Markov model and parameters of the reciprocal $CM_L$ model can be obtained from those of the Markov model. Also, it is shown how NG CM sequences can be represented in terms of an NG Markov sequence and an independent NG vector. This representation provides a general approach for parameter design of $CM_L$ and $CM_F$ models. In addition, it leads to a better understanding of CM sequences, including the reciprocal sequence.