Gaussian Reciprocal Sequences from the Viewpoint of Conditionally Markov Sequences

Abstract

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.

Gaussian Conditionally Markov Sequences: Dynamic Models and Representations of Reciprocal and Other Classes

“…Studying the evolution of other sequences belonging to more than one CM class, for example CM L ∩ [k 1 , N ]-CM F , is also useful for studying reciprocal sequences. The NG reciprocal sequence is equivalent to CM L ∩ CM F [18]. Proposition 3.5 below presents a dynamic model of CM L ∩ [k 1 , N ]-CM F sequences, based on which one can see a full spectrum of models from a CM L sequence to a reciprocal sequence.…”

“…can be also modeled by a CM L model (16)- (17). By the Markov property, parameters of (16) are obtained as (12)-(14) based on (18 (19)) (i.e., the same transition (18)), but [x k ] can have any joint distribution of the states at the endpoints. In other words, [x k ] can model any origin and destination.…”

“…In this paper, we address this issue and make it clear. Including reciprocal sequences as a special case ( [17], [18]), CM sequences are more general for trajectory modeling with waypoints and/or a destination [1], for example, CM L sequences for trajectory modeling with destination information. However, guidelines for parameter design of a CM L model are lacking.…”

“…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.…”

“…As a result, for estimation of the current state, prior information (density) of the next state is required. However, such information is not available.We presented a different dynamic model of NG reciprocal sequence (called reciprocal CM L model) in [18] from the CM viewpoint. That model has a good structure for trajectory modeling with a destination.…”