Gaussian Conditionally Markov Sequences: Algebraically Equivalent Dynamic Models

The conditionally Markov (CM) sequence contains different classes, including Markov, reciprocal, and so-called <inline-formula><tex-math notation="LaTeX">$\text{CM}_L$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$\text{CM}_F$</tex-math></inline-formula> (two CM classes defined in our previous work). Markov sequences are special reciprocal sequences, and reciprocal sequences are special <inline-formula><tex-math notation="LaTeX">$\text{CM}_L$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$\text{CM}_F$</tex-math></inline-formula> 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 article studies this topic for models of nonsingular Gaussian <inline-formula><tex-math notation="LaTeX">$\text{CM}_L$</tex-math></inline-formula>, <inline-formula><tex-math notation="LaTeX">$\text{CM}_F$</tex-math></inline-formula>, reciprocal, and Markov sequences. Two models are <italic>probabilistically equivalent (PE)</italic> if their stochastic sequences have the same distribution and are <italic>algebraically equivalent (AE)</italic> if their stochastic sequences are pathwise identical. A unified approach is presented to obtain an AE forward/backward <inline-formula><tex-math notation="LaTeX">$\text{CM}_L$</tex-math></inline-formula>/<inline-formula><tex-math notation="LaTeX">$\text{CM}_F$</tex-math></inline-formula>/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/<inline-formula><tex-math notation="LaTeX">$\text{CM}_L$</tex-math></inline-formula> model.

[1]  Reza Rezaie.  Gaussian Conditionally Markov Sequences: Theory with Application , 2019 .

[2]  George Stamatescu,et al.  Track Extraction With Hidden Reciprocal Chains , 2018, IEEE Transactions on Automatic Control.

[3]  Reciprocal Processes: A Stochastic Analysis Approach , 2014 .

[4]  Simon J. Godsill,et al.  Bayesian Intent Prediction in Object Tracking Using Bridging Distributions , 2015, IEEE Transactions on Cybernetics.

[5]  D. Simon,et al.  Kalman filtering with state equality constraints , 2002 .

[6]  X. Rong Li,et al.  Destination-Directed Trajectory Modeling and Prediction Using Conditionally Markov Sequences , 2018, 2018 IEEE Western New York Image and Signal Processing Workshop (WNYISPW).

[7]  Bashar I. Ahmad,et al.  Intelligent Interactive Displays in Vehicles with Intent Prediction: A Bayesian framework , 2017, IEEE Signal Processing Magazine.

[8]  A. Willsky,et al.  On the fixed-interval smoothing problem † , 1981 .

[9]  A. Chiuso,et al.  Reciprocal realization and modeling of textured images , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[10]  X. Rong Li,et al.  Models and Representations of Gaussian Reciprocal and Conditionally Markov Sequences , 2018, ICVISP.

[11]  José M. F. Moura,et al.  Telescoping Recursive Representations and Estimation of Gauss–Markov Random Fields , 2009, IEEE Transactions on Information Theory.

[12]  Thomas Kailath,et al.  Correction to 'A Further Note on Backwards Markovian Models' , 1979, IEEE Trans. Inf. Theory.

[13]  Thomas Kailath,et al.  Discrete-time complementary models and smoothing , 1989 .

[14]  Yan Liang,et al.  Constrained Dynamic Systems: Generalized Modeling and State Estimation , 2017, IEEE Transactions on Aerospace and Electronic Systems.

[15]  José M. F. Moura,et al.  Recursive filtering and smoothing for discrete index gaussian reciprocal processes , 2009, 2009 43rd Annual Conference on Information Sciences and Systems.

[16]  D. Fraser,et al.  The optimum linear smoother as a combination of two optimum linear filters , 1969 .

[17]  Langford B. White,et al.  Modelling and Estimation for Finite State Reciprocal Processes , 2012, IEEE Transactions on Automatic Control.

[18]  Bernard C. Levy,et al.  Discrete-time Gauss-Markov processes with fixed reciprocal dynamics , 1997 .

[19]  Giovanni Conforti,et al.  Reciprocal Class of Jump Processes , 2014, 1406.7495.

[20]  Enzo Baccarelli,et al.  Recursive filtering and smoothing for reciprocal Gaussian processes-pinned boundary case , 1995, IEEE Trans. Inf. Theory.

[21]  Demetrios G. Lainiotis,et al.  General backwards Markov models , 1976 .

[22]  X. Rong Li,et al.  Gaussian Conditionally Markov Sequences: Singular/Nonsingular , 2019, IEEE Transactions on Automatic Control.

[23]  Lennart Ljung,et al.  Backwards Markovian models for second-order stochastic processes (Corresp.) , 1976, IEEE Trans. Inf. Theory.

[24]  J. McFadden,et al.  Certain Properties of Gaussian Processes and Their First‐Passage Times , 1965 .

[25]  T. Kailath,et al.  Direct approach to two-filter smoothing formulas† , 1984 .

[26]  X. Rong Li,et al.  Gaussian Conditionally Markov Sequences: Dynamic Models and Representations of Reciprocal and Other Classes , 2020, IEEE Transactions on Signal Processing.

[27]  X. Rong Li,et al.  Gaussian Reciprocal Sequences from the Viewpoint of Conditionally Markov Sequences , 2018, ICVISP.

[28]  LI X.RONG,et al.  Survey of maneuvering target tracking. Part I. Dynamic models , 2003 .

[29]  X. Rong Li,et al.  Nonsingular Gaussian Conditionally Markov Sequences , 2018, 2018 IEEE Western New York Image and Signal Processing Workshop (WNYISPW).

[30]  John B. Thomas,et al.  Some comments on conditionally Markov and reciprocal Gaussian processes , 1981, IEEE Trans. Inf. Theory.

[31]  X. Rong Li,et al.  Trajectory Modeling and Prediction with Waypoint Information Using a Conditionally Markov Sequence , 2018, 2018 56th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[32]  Francesca P. Carli,et al.  Modelling and Simulation of Images by Reciprocal Processes , 2008, Tenth International Conference on Computer Modeling and Simulation (uksim 2008).

[33]  Paolo Braca,et al.  Modeling vessel kinematics using a stochastic mean-reverting process for long-term prediction , 2016, IEEE Transactions on Aerospace and Electronic Systems.

[34]  George A. F. Seber,et al.  A matrix handbook for statisticians , 2007 .

[35]  Vikram Krishnamurthy,et al.  Detection of Anomalous Trajectory Patterns in Target Tracking via Stochastic Context-Free Grammars and Reciprocal Process Models , 2013, IEEE Journal of Selected Topics in Signal Processing.

[36]  Thiagalingam Kirubarajan,et al.  State Estimation With Trajectory Shape Constraints Using Pseudomeasurements , 2019, IEEE Transactions on Aerospace and Electronic Systems.

[37]  Vikram Krishnamurthy,et al.  Spatiotemporal trajectory models for metalevel target tracking , 2015, IEEE Aerospace and Electronic Systems Magazine.

[38]  Langford B. White,et al.  Normalized Optimal Smoothers for a Class of Hidden Generalized Reciprocal Processes , 2017, IEEE Transactions on Automatic Control.

[39]  D. Simon.  Kalman filtering with state constraints: a survey of linear and nonlinear algorithms , 2010 .

[40]  G. Sidhu,et al.  New smoothing algorithms based on reversed-time lumped models , 1976 .

[41]  B. Levy,et al.  Some new smoother implementations for discrete-time gaussian reciprocal processes , 1991 .

[42]  Langford B. White,et al.  Destination-aware target tracking via syntactic signal processing , 2011, 2011 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[43]  Joseph J. LaViola,et al.  On Kalman Filtering With Nonlinear Equality Constraints , 2007, IEEE Transactions on Signal Processing.

[44]  A. Krener,et al.  Modeling and estimation of discrete-time Gaussian reciprocal processes , 1990 .