Model Design and Representations of CM Sequences

TL;DR

This paper explores the detailed structure and parameter design of CM sequences, especially NG reciprocal sequences, showing how they relate to Markov models and providing new representations for their modeling.

Contribution

It introduces new methods for parameter design of NG CM sequences and reciprocal sequences, linking them to Markov models and offering simplified representations.

Findings

Reciprocal CM_L models can be induced by Markov models.

NG CM sequences can be represented as a sum of a Markov sequence and an uncorrelated NG vector.

A simple representation of NG reciprocal sequences reveals key properties.

Abstract

Conditionally Markov (CM) sequences are powerful mathematical tools for modeling problems. One class of CM sequences is the reciprocal sequence. In application, we need not only CM dynamic models, but also know how to design model parameters. Models of two important classes of nonsingular Gaussian (NG) CM sequences, called $CM_L$ and $CM_F$ models, and a model of the NG reciprocal sequence, called reciprocal $CM_L$ model, were presented in our previous works and their applications were discussed. In this paper, these models are studied in more detail, in particular their parameter design. It is shown that every reciprocal $CM_L$ model can be induced by a Markov model. Then, parameters of each reciprocal $CM_L$ model can be obtained from those of the Markov model. Also, it is shown that a NG $CM_L$ ($CM_F$) sequence can be represented by a sum of a NG Markov sequence and an uncorrelated NG vector. This (necessary and sufficient) representation provides a basis for designing parameters of a $CM_L$ ($CM_F$) model. From the CM viewpoint, a representation is also obtained for NG reciprocal sequences. This representation is simple and reveals an important property of reciprocal sequences. As a result, the significance of studying reciprocal sequences from the CM viewpoint is demonstrated. A full spectrum of dynamic models from a $CM_L$ model to a reciprocal $CM_L$ model is also presented. Some examples are presented for illustration.