Nonsingular Gaussian Conditionally Markov Sequences

Rezaie, Reza; Li, X. Rong

doi:10.1109/wnyipw.2018.8576382

Cited by 14 publications

(49 citation statements)

References 29 publications

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“…A sequence [x k ] is [k 1 , k 2 ]-CM c , c ∈ {k 1 , k 2 } (i.e., CM over [k 1 , k 2 ]) iff conditioned on the state at time k 1 (or k 2 ), the sequence is Markov over [k 1 + 1, k 2 ] ([k 1 , k 2 − 1]). The above definition is equivalent to the following lemma [1].…”

Section: Definitions and Notationsmentioning

confidence: 99%

“…By Lemma 3.1, the set of reciprocal sequences modeled by a reciprocal CM L model contains Markov and non-Markov sequences (depending on the parameters of the boundary condition). So, a sequence modeled by a reciprocal CM L model and a boundary condition determined as in the proof of Lemma 3.1 (i.e., satisfying (11)) is actually a Markov sequence whose C −1 is (block) tri-diagonal (i.e., (1) with D 0 = · · · = D N −2 = 0). Given this C −1 , we can obtain parameters of Markov model (15)…”

Section: Reciprocal Sequencesmentioning

confidence: 99%

“…Parameters of a CM L model of a CM L sequence are unique [1]. Comparing (54)-(57) and (58)-(61), it can be seen that the parametersŨ k,…”

Section: Representations Of CM and Reciprocal Sequencesmentioning

confidence: 99%

See 2 more Smart Citations

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

Rezaie¹,

Li²

2020

IEEE Trans. Signal Process.

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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...

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Section: Definitions and Notationsmentioning

confidence: 99%

Section: Reciprocal Sequencesmentioning

confidence: 99%

See 1 more Smart Citation

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

Rezaie¹,

Li²

2020

IEEE Trans. Signal Process.

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“…In other words, inside and outside are independent given the boundaries. A sequence is CM F (CM L ) over [k 1 , k 2 ] iff conditioned on the state at time k 1 (k 2 ), the sequence is Markov over [k 1 + 1, k 2 ] ([k 1 , k 2 − 1]) [1]. The set of CM sequences is very large and it includes many classes.…”

Section: Introductionmentioning

confidence: 99%

“…But the covariance of a singular sequence is not invertible. Also, proofs of models presented in [1] and [15]- [16] are based on the nonsingularity of sequences and do not work for the singular case. In this paper, we use different ideas and approaches to obtain dynamic models and characterizations for the general singular/nonsingular case.…”

Section: Introductionmentioning

confidence: 99%

Gaussian Conditionally Markov Sequences: Singular/Nonsingular

Rezaie

2020

IEEE Trans. Automat. Contr.

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Most existing results about modeling and characterizing Gaussian Markov, reciprocal, and conditionally Markov (CM) processes assume nonsingularity of the processes. This assumption makes the analysis easier, but restricts application of these processes. This paper studies, models, and characterizes the general (singular/nonsingular) Gaussian CM (including reciprocal and Markov) sequence. For example, to our knowledge, there is no dynamic model for the general (singular/nonsingular) Gaussian reciprocal sequence in the literature. We obtain two such models from the CM viewpoint. As a result, the significance of studying reciprocal sequences from the CM viewpoint is demonstrated. The results of this paper unify singular and nonsingular Gaussian CM (including reciprocal and Markov) sequences and provide tools for their application. An application of CM sequences in trajectory modeling with a destination is discussed and illustrative examples are presented.

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Gaussian conditionally Markov sequences: Modeling and characterization

Rezaie

2021

Automatica

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Nonsingular Gaussian Conditionally Markov Sequences

Cited by 14 publications

References 29 publications

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

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

Gaussian Conditionally Markov Sequences: Singular/Nonsingular

Gaussian conditionally Markov sequences: Modeling and characterization

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