Gaussian Conditionally Markov Sequences: Dynamic Models and Representations of Reciprocal and Other Classes
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 <inline-formula><tex-math notation="LaTeX">$CM_L$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$CM_F$</tex-math></inline-formula> models, and a model of the NG reciprocal sequence, called reciprocal <inline-formula><tex-math notation="LaTeX">$CM_L$</tex-math></inline-formula> 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 <inline-formula><tex-math notation="LaTeX">$CM_L$</tex-math></inline-formula> model can be induced by a Markov model. Then, parameters of each reciprocal <inline-formula><tex-math notation="LaTeX">$CM_L$</tex-math></inline-formula> model can be obtained from those of the Markov model. Also, it is shown that an NG <inline-formula><tex-math notation="LaTeX">$CM_L$</tex-math></inline-formula> (<inline-formula><tex-math notation="LaTeX">$CM_F$</tex-math></inline-formula>) sequence can be represented by a sum of an NG Markov sequence and an uncorrelated NG vector. This (necessary and sufficient) representation provides a basis for designing parameters of a <inline-formula><tex-math notation="LaTeX">$CM_L$</tex-math></inline-formula> (<inline-formula><tex-math notation="LaTeX">$CM_F$</tex-math></inline-formula>) 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 <inline-formula><tex-math notation="LaTeX">$CM_L$</tex-math></inline-formula> model to a reciprocal <inline-formula><tex-math notation="LaTeX">$CM_L$</tex-math></inline-formula> model is also presented. Some examples are presented for illustration.