B.C. Levy, R. Frezza, and A.J. Krener. Modeling and estimation of discrete-time Gaussian reciprocal processes. IEEE Trans. Automatic Control, 35:1013--1023, September 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....169] Another method that deserves mention is that developed in [351] using the so called Bethe tree approximation. In this approximation, the computation of statistics at a specific individual node is approximated by replacing the original graph by a tree rooted at that node, where each See [4, 5, 258, 141, 210, 260] for a methodology applied to time series, 101] for analogous results for MR trees, and [337] for results for discrete state graphical models on single loops. path away from the node in the original graph is replaced by a path in the tree. If the path in the original graph contains a loop and ....

B.C. Levy, R. Frezza, and A.J. Krener. Modeling and estimation of discrete-time Gaussian reciprocal processes. IEEE Trans. Automatic Control, 35:1013--1023, September 1990.

....effective bandwidth are related by BW =4N , the results obtained make it possible to calculate the effective bandwidth of virtually every periodic analytic function by performing a maximization over a strip in the complex plane, followed by a minimization over the positive real axis. REFERENCES [1] J. Rissanen, Modeling by shortest data description, Automatica, vol. 14, pp. 465 471, 1978. 2] L. Knockaert, Maximum a posteriori maximum entropy order determination, IEEE Trans. Signal Processing, vol. 45, pp. 1553 1559, June 1997. 3] J. D. Gibson, Principles of Digital and Analog ....

....the usual self adjoint second order noncausal difference model [1, Sec. III.A] 4, Sec. II.B] 9, Sec. III.E] M0(k)x(k) 0M (k)x(k 1)0M T ( k01)x(k 0 1) e(k) k2T 0 j[M 1;N01] 1) where fM0 (k) k 2T 0 g;fM (k) M kN01g are sequences of (n2n) matrices. The so called conjugate process [1] fe(k) k 2 T 0 g of (1) is bi orthogonal to fx(k)g (i.e. Efe(k)x T (s)g = Iffi(k; s) k 2 T 0 ;s 2 T) 1 and its autocorrelation function (acf) is given in [1, 3.4) 9, 53) and [4, 27) RGP s (which are also known as noncausal one dimensional (1D) Gauss Markov random fields, ....

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B. C. Levy, R. Frezza, and A. J. Krener, "Modeling and estimation of discrete-time Gaussian reciprocal processes," IEEE Trans. Automat. Contr., vol. 5, pp. 1013--1023, Sept. 1990.

....the same reciprocal description. In this paper we study how these Markov processes are related to each other. We will limit our discussion to the Gaussian context, where an interesting characterization of reciprocal processes is available. In fact, it has been shown by Krener, Levy and Frezza [12], 10] that all the Gaussian reciprocal processes in a given class can be obtained varying the boundary conditions of a self adjoint, second order stochastic boundary value problem (SBVP) The main goal of this paper is to show how to construct a Markov process belonging to a given reciprocal ....

....the covariance function R(t; s) via (2.1a) or using the stochastic differential equation (2.5a) In both cases, it is necessary to impose some boundary conditions to describe completely the process x(t) over the whole interval I. This can be accomplished satisfactorily in more than one way. In [12], in the discrete time context, it has been shown that, besides Dirichlet boundary conditions of the kind (2.1b) 2.5b) it is possible to assign cyclic boundary conditions to the 6 MARKOV PROCESSES WITH FIXED RECIPROCAL DYNAMICS second order model describing the process. This set of conditions ....

B. C. Levy, R. Frezza, and A. J. Krener. Modeling and estimation of discrete-time Gaussian reciprocal processes, IEEE Trans. Automat. Control 35 (1990), 1013--1023.

....of standard Kalman filter algorithms. Using this formalism, an approach to derive an information Kalman filter for a descriptor system is developed in this thesis. Also, the Markovianity condition for a Gaussian random field, expressed in terms of the elements of the associated information matrix [44], is used to relate MRF modeling and the spatially local approximations for the information Kalman filter algorithms developed in this thesis. The approximate Kalman filtering techniques should be applicable to general large scale, multi dimensional filtering problems. Specifically, the ....

....dynamic representation to (3.29) 3. 30) can be obtained using the estimation error process e f j b f Gamma f as L e f = e i: It can be shown that the mean covariance pair of the driving noise e i is given by e i ( 0; L ) This dynamic model is shown to be a Markov Random Field (MRF) [44], which has a multi dimensional extension of the Markovian property for causal processes [77] Interpreting the information matrix L as an MRF model for the estimation error e f can be quite useful, as it connects our ML estimation formulations directly with other important formulations in ....

B. C. Levy, R. Frezza, and A. J. Krener. Modeling and estimation of discrete-time Gaussian reciprocal processes. to appear in IEEE Transactions on Automatic Control, 1990.

....the necessary framework for applications such as those mentioned above. This paper is organized as follows. In Section 2 we briefly review the class of multiscale stochastic models of interest here and the scale recursive estimation algorithm associated with them. In 4 More generally, Levy et al. [12] have recently shown that the smoothing error processes associated with the class of Gaussian reciprocal processes, which contains the class of Gauss Markov processes, are themselves Gaussian reciprocal. See also [13] for similar results corresponding to 2 D Gauss Markov random fields. Section 3 ....

B. Levy, R. Frezza, and A. Krener, "Modeling and estimation of discrete time Gaussian reciprocal processes," IEEE Transactions on Automatic Control, vol. 35, pp. 1013--1023, 1990.

.... 2 R is said to be reciprocal 3 (or bilateral Markov, two sided Markov or non causal Markov) if it has the property that the probability distribution of a state in any open interval (T 1 ; T 2 ) conditioned on the boundary states z(T 1 ) z(T 2 ) is independent of states outside of the interval [43, 81]. That is, for t 2 (T 1 ; T 2 ) p z(t)jz( 2(T 1 ;T 2 ) c(Z t jZ ; 2 (T 1 ; T 2 ) c ) 2 Obvious modifications of the neighborhood set must be made for the root node at the top of the tree, which has no parent, and the nodes at the finest level of the tree, which have no offspring. 3 ....

B. C. Levy, R. Frezza, and A. J. Krener. "Modeling and estimation of discrete time Gaussian reciprocal processes". IEEE Transactions on Automatic Control, 35:1013--1023, 1990.

....real line. More precisely, a stochastic process z t ; t 2 R is reciprocal 9 if it has the property that the conditional probability distribution of a state in any open interval (T 1 ; T 2 ) conditioned on the states outside of this interval, depends only on the boundary states z T 1 ; z T 2 [22, 32]. That is, for t 2 (T 1 ; T 2 ) p z t jz ; 2(T 1 ;T 2 ) c(Z t jZ ; 2 (T 1 ; T 2 ) c ) p z t jz T 1 ;z T 2 (Z t jZ T 1 ; Z T 2 ) 6) 8 The process is Markov only if m(s) 1= L 1) In this case, the values of the process at any level in the tree are independent of one another. ....

B. C. Levy, R. Frezza, and A. J. Krener. "Modeling and estimation of discrete time Gaussian reciprocal processes". IEEE Transactions on Automatic Control, 35:1013-- 1023, 1990.

....N ] Gamma [l; m] given x(l) and x(m) For a more rigorous definition see [11] Reciprocal processes generalize Markov processes since a Markov process is reciprocal while the converse, in general, is not true, see [11] for an example of a process which is reciprocal and not Markov. It is known [12, 13, 6] that a discrete time Gaussian reciprocal process x(k) under the assumption that the covariance of [ x(k Gamma 1) x(k 1) T is full rank, satisfies a nearest neighbor model like the following m 0 (k)x(k) Gamma m Gamma (k)x(k Gamma 1) Gamma m (k)x(k 1) k) 2.1) where (k) is a ....

....methods like the Lanczos algorithm or power and inverse iteration [8] The algorithm also serves to identify Markov models. Note first that the Markov processes are a subclass of the reciprocal processes, and the model corresponds to a Jacobi matrix (a subclass of the periodic Jacobi matrices) In [13], it was shown that a Markov process x(k) satisfying the model Omega x = Bw (2.12) C.F. BORGES AND R. FREZZA where Omega = 2 6 6 6 6 6 6 4 1 Gammaa(1) 1 Gammaa(2) Gammaa(n Gamma 1) 1 3 7 7 7 7 7 7 5 (2.13) B = diag(b(k) 2.14) and w = Theta w(0) w(1) Delta Delta ....

B. Levy, R. Frezza, and A. Krener. Modeling and estimation of discrete time Gaussian reciprocal processes, IEEE Trans. Automat. Contr., AC--35 (1990), 1013--1023.

....shown [10] 11] that the problem of changing the final density of a Markov process can be formulated as a stochastic optimal control problem. Starting with Krener s work [12] a significant amount of attention has focused on developing dynamical models for reciprocal processes. It was shown in [13], 14] that Gaussian reciprocal processes whose covariance is uniformly positive definite over the interval of interest admit self adjoint second order models driven by locally correlated noise, where the noise correlation structure is totally determined by the model dynamics. In this paper, given ....

....admit a stochastic optimal control interpretation, but an alternative, more general, interpretation is given in terms of an estimation problem. The paper is organized as follows. In Section 2 we briefly review the properties of the second order models of Gaussian reciprocal processes introduced in [13]. In Section 3 it is shown how to construct a Markov process with given second order model and end point marginal densities. In Section 4 we consider the problem of changing the end point density of a Markov process, while remaining in the same reciprocal class. A stochastic GAUSS MARKOV ....

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B. C. Levy, R. Frezza, and A. J. Krener. "Modeling and estimation of discrete-time Gaussian reciprocal processes," IEEE Trans. Automat. Control 35 (Sept. 1990), 1013--1023.

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