Markov processes are widely used in modeling random phenomena/problems. However, they may not be adequate in some cases where more general processes are needed. The conditionally Markov (CM) process is a generalization of the Markov process based on conditioning. There are several classes of CM processes (one of them is the class of reciprocal processes), which provide more capability (than Markov) for modeling random phenomena. Reciprocal processes have been used in many different applications (e.g., image processing, intent inference, intelligent systems). In this paper, nonsingular Gaussian (NG) CM sequences are studied, characterized, and their dynamic models are presented. The presented results provide effective tools for studying reciprocal sequences from the CM viewpoint, which is different from that of the literature. Also, the presented models and characterizations serve as a basis for application of CM sequences, e.g., in motion trajectory modeling with destination information.

The conditionally Markov (CM) sequence contains several classes, including the reciprocal sequence. Reciprocal sequences have been widely used in many areas of engineering, including image processing, acausal systems, intelligent systems, and intent inference. In this paper, the reciprocal sequence is studied from the CM sequence point of view, which is different from the viewpoint of the literature and leads to more insight into the reciprocal sequence. Based on this viewpoint, new results, properties, and easily applicable tools are obtained for the reciprocal sequence. The nonsingular Gaussian (NG) reciprocal sequence is modeled and characterized from the CM viewpoint. It is shown that a NG sequence is reciprocal if and only if it is both CM L and CM F (two special classes of CM sequences). New dynamic models are presented for the NG reciprocal sequence. These models (unlike the existing one, which is driven by colored noise) are driven by white noise and are easily applicable. As a special reciprocal sequence, the Markov sequence is also discussed. Finally, it can be seen how all CM sequences, including Markov and reciprocal, are unified.

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 CML and CMF models, and a model of the NG reciprocal sequence, called reciprocal CML 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 CML model can be induced by a Markov model. Then, parameters of each reciprocal CML model can be obtained from those of the Markov model. Also, it is shown that a NG CML (CMF ) 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 CML (CMF ) 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 CML model to a reciprocal CML model is also presented. Some examples are presented for illustration.Gaussian CM processes were introduced in [16] based on mean and covariance functions, where the processes were assumed nonsingular on the interior of the index (time) interval. [16] considered conditioning at the first time of the CM interval.[17] extended the definition of Gaussian CM processes (presented in [16]) to the general (Gaussian/non-Gaussian) case. In [1] we presented definitions of different (Gaussian/non-Gaussian) CM processes based on conditioning at the first or the last time of the CM interval, studied (stationary/non-stationary) NG CM sequences, and presented their dynamic models and characterizations. Two of these models for two important classes of NG CM sequences (i.e., sequences being CM L or CM F over [0, N ]) are called CM L and CM F models. Applications of CM sequences for trajectory modeling in different scenarios were also discussed. In addition, [1] provided a foundation and preliminaries for studying the reciprocal sequence from the viewpoint of the CM sequence in [18].Reciprocal processes were introduced in [13] and studied in [19]-[37] and others.[19]-[23] studied reciprocal processes in a general setting. [17] made an inspiring comment that reciprocal and CM processes are related, and discussed the relationship between the Gaussian reciprocal process and the Gaussian CM process.[18] elaborated on the comment of [17] and obtained a relationship between (Gaussian/non-Gaussian) CM and reciprocal processes. It was shown in [17] that a NG continuous-time CM (including reciprocal) process can be represented in terms of a Wiener process and an uncorrelated NG vector. Following [17], [24]-[25] ob...

The conditionally Markov (CM) sequence contains different classes, including Markov, reciprocal, and so-called CML and CMF (two CM classes defined in our previous work). Markov sequences are special reciprocal sequences, and reciprocal sequences are special CML and CMF sequences. Each class has its own forward and backward dynamic models. The evolution of a CM sequence can be described by different models. For a given problem, a model in a specific form is desired or needed, or one model can be easier to apply and better than another. Therefore, it is important to study the relationship between different models and to obtain one model from another. This paper studies this topic for models of nonsingular Gaussian (NG) CML, CMF , reciprocal, and Markov sequences. Two models are probabilistically equivalent (PE) if their stochastic sequences have the same distribution, and are algebraically equivalent (AE) if their stochastic sequences are path-wise identical. A unified approach is presented to obtain an AE forward/backward CML/CMF /reciprocal/Markov model from another such model. As a special case, a backward Markov model AE to a forward Markov model is obtained. While existing results are restricted to models with nonsingular state transition matrices, our approach is not. In addition, a simple approach is presented for studying and determining Markov models whose sequences share the same reciprocal/CML model.In theory, Gaussian CM processes were introduced in [22] based on mean and covariance functions.[23] extended the definition of Gaussian CM processes (presented in [22]) to the general (Gaussian/non-Gaussian) case. In [1], we defined other (Gaussian/non-Gaussian) CM processes, studied (stationary/non-stationary) NG CM sequences, obtained dynamic models and characterizations of CM L and CM F sequences, and discussed their applications. Reciprocal processes were introduced in [24].[25]-[27] studied reciprocal processes in a general setting. Based on a valuable observation, [23] commented on the relationship between Gaussian CM and Gaussian reciprocal processes.[28] elaborated on the comment of [23] and presented a relationship between the CM process and the reciprocal process for the general (Gaussian/non-Gaussian) case.[29]- [30] presented and studied a dynamic model of NG reciprocal sequences.[28] and [31]-[32] studied reciprocal sequences from the CM viewpoint and developed dynamic models, called reciprocal CM L and reciprocal CM F models, with white dynamic noise for the NG reciprocal sequence. A characterization of the NG Markov sequence was presented in [33].[34] considered modeling and estimation of finite-state reciprocal sequences.The evolution of a Markov sequence can be modeled by a Markov, reciprocal, or CM L model 2 . Similarly, a reciprocal sequence can have a model in the form of the one in [29] or in the form of a CM L (CM F ) model of [28]. Therefore, a CM sequence can have more than one model. One model can be easier to apply than another for an application. For example, the reciprocal CM L mo...

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