Recursive filtering and smoothing for Gaussian reciprocal processes with continuous indices

In this paper, we study continuous index Gaussian reciprocal processes (Grp's) (or two-point boundary valued processes) with Dirichlet boundary conditions. Our main contributions are 1) deriving first order white noise driven representations from the second order correlated noise driven representations of Grp's given by Krener, Frezza, and Levy [1]; 2) deriving Kalman-Bucy like recursive filtering equations for Grp's with continuous indices; and 3) deriving recursive smoothing equations for Grp's with continuous indices.

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

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

[3]  Arthur J. Krener,et al.  Least squares smoothing of nonlinear systems , 2005 .

[4]  C. Striebel,et al.  On the maximum likelihood estimates for linear dynamic systems , 1965 .

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

[6]  Alan S. Willsky,et al.  Linear estimation of boundary value stochastic processes--Part II: 1-D smoothing problems , 1984 .

[7]  A. Bagchi,et al.  Smoothing and likelihood ratio for Gaussian boundary value processes , 1989 .

[8]  H. Akaike,et al.  Comment on "An innovations approach to least-squares estimation, part I: Linear filtering in additive white noise" , 1970 .

[9]  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.

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

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

[12]  T. Kailath,et al.  An innovations approach to least-squares estimation--Part II: Linear smoothing in additive white noise , 1968 .

[13]  A. Willsky,et al.  Linear estimation of boundary value stochastic processes-- Part I: The role and construction of complementary models , 1984 .

[14]  I. N. Sneddon,et al.  Boundary value problems , 2007 .

[15]  José M. F. Moura,et al.  Gauss-Markov random fields (CMrf) with continuous indices , 1997, IEEE Trans. Inf. Theory.