Model Design and Representations of CM Sequences
This work is incremental, offering detailed parameter design methods for CM sequences, which are mathematical tools used in modeling problems.
The paper tackles the parameter design problem for conditionally Markov (CM) sequences, specifically reciprocal and CM_L/CM_F models, by showing that reciprocal CM_L models can be induced by Markov models and that CM_L/CM_F sequences can be represented as sums of Markov sequences and uncorrelated vectors, providing a basis for parameter design.
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.