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 every 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. It also makes a key fact of CM sequences explicit. 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. Moreover, models for intersections of CM classes are presented and their application is pointed out. Some examples are presented for illustration.

中文翻译：

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倒数和其他高斯条件马尔可夫序列的动态模型和表示

条件马尔可夫 (CM) 序列是用于建模问题的强大数学工具。一类CM序列是倒数序列。在应用中，我们不仅需要CM动态模型，还要知道如何设计模型参数。我们之前的工作中提出了两类重要的非奇异高斯 (NG) CM 序列模型，称为 $CM_L$ 和 $CM_F$ 模型，以及称为倒数 $CM_L$ 模型的 NG 倒数序列模型，它们的应用是讨论。在本文中，更详细地研究了这些模型，特别是它们的参数设计。结果表明，每个倒数$CM_L$模型都可以由马尔可夫模型诱导。然后，可以从马尔可夫模型的参数中获得每个倒数$CM_L$模型的参数。还，结果表明，NG $CM_L$ ( $CM_F$ ) 序列可以由一个 NG 马尔可夫序列和一个不相关的 NG 向量之和来表示。这种（必要且充分的）表示为设计 $CM_L$ ( $CM_F$ ) 模型的参数提供了基础。它还明确了 CM 序列的一个关键事实。从 CM 的角度来看，还获得了 NG 互易序列的表示。这种表示很简单，并且揭示了互易序列的一个重要性质。结果，证明了从 CM 的角度研究互易序列的重要性。还介绍了从 $CM_L$ 模型到互惠 $CM_L$ 模型的全方位动态模型。此外，提出了CM类的交集模型并指出了它们的应用。提供了一些示例以供说明。