Models and Representations of Gaussian Reciprocal and Conditionally Markov Sequences

Conditionally Markov (CM) sequences are powerful mathematical tools for modeling random phenomena. There are several classes of CM sequences one of which is the reciprocal sequence. Reciprocal sequences have been widely used in many areas including image processing, intelligent systems, and acausal systems. To use them in application, we need not only their applicable dynamic models, but also some general approaches to designing parameters of dynamic models. Dynamic models governing two important classes of nonsingular Gaussian (NG) CM sequences (called CML and CMF models), and a dynamic model governing the NG reciprocal sequence (called reciprocal CML model) were presented in our previous work. In this paper, these models are studied in more detail and general approaches are presented for their parameter design. It is shown that every reciprocal CML model can be induced by a Markov model and parameters of the reciprocal CML model can be obtained from those of the Markov model. Also, it is shown how NG CM sequences can be represented in terms of a NG Markov sequence and an independent NG vector. This representation provides a general approach for parameter design of CML and CMF models. In addition, it leads to a better understanding of CM sequences, including the reciprocal sequence.

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

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

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

[4]  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.

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

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

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

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

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

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

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

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

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

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

[15]  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).

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

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

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

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

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

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

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

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

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

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

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

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

[30]  Francesca P. Carli,et al.  A Maximum Entropy Solution of the Covariance Extension Problem for Reciprocal Processes , 2011, IEEE Transactions on Automatic Control.

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

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

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

[34]  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).

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

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

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

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

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

[40]  Simon J. Godsill,et al.  Destination inference using bridging distributions , 2015, 2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

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

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

[43]  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).