Gaussian Reciprocal Sequences from the Viewpoint of Conditionally Markov Sequences

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 CML and CMF (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.

[1]  D. Slepian.  First Passage Time for a Particular Gaussian Process , 1961 .

[2]  B. Jamison,et al.  Reciprocal Processes: The Stationary Gaussian Case , 1970 .

[3]  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.

[4]  H. Weinert,et al.  A new characterization for multivariate Gaussian reciprocal processes , 1993, IEEE Trans. Autom. Control..

[5]  Enzo Baccarelli,et al.  Recursive filtering and smoothing for reciprocal Gaussian processes with Dirichlet boundary conditions , 1998, IEEE Trans. Signal Process..

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

[7]  A. Krener,et al.  Dynamics and kinematics of reciprocal diffusions , 1993 .

[8]  Francesco Carravetta.  Nearest-neighbour modelling of reciprocal chains , 2007 .

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

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

[11]  Alessandro Beghi,et al.  Continuous-time Gauss-Markov processes with fixed reciprocal dynamics , 1997 .

[12]  Multivariate reciprocal stationary Gaussian processes , 1987 .

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

[14]  Langford B. White,et al.  Optimal Smoothing for Finite State Hidden Reciprocal Processes , 2011, IEEE Transactions on Automatic Control.

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

[16]  X. Rong Li,et al.  Gaussian Conditionally Markov Sequences: Algebraically Equivalent Dynamic Models , 2019, IEEE Transactions on Aerospace and Electronic Systems.

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

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

[19]  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.

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

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

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

[23]  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).

[24]  E. Schrödinger.  Sur la théorie relativiste de l'électron et l'interprétation de la mécanique quantique , 1932 .

[25]  S. C. Chay,et al.  On quasi-Markov random fields , 1972 .

[26]  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).

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

[28]  Arthur J. Krener,et al.  Reciprocal diffusions and stochastic differential equations of second order , 1988 .

[29]  Christian L'eonard,et al.  Reciprocal processes. A measure-theoretical point of view , 2013, 1308.0576.

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

[31]  Ruggero Frezza,et al.  Gaussian reciprocal processes and self-adjoint stochastic differential equations of second order , 1991 .

[32]  Radu Theodorescu,et al.  Representations for multivariate reciprocal Gaussian processes , 1988, IEEE Trans. Inf. Theory.

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

[34]  Arthur J. Krener,et al.  Stochastic mechanics of reciprocal diffusions , 1996 .

[35]  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).

[36]  B. Jamison.  Reciprocal processes , 1974 .

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

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

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

[40]  Langford B. White,et al.  Maximum Likelihood Sequence Estimation for Hidden Reciprocal Processes , 2013, IEEE Transactions on Automatic Control.