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Journal Article

Fixed-Point Smoothing of Scalar Diffusions II: The Error of the Optimal Smoother

Y. Steinberg, B. Z. Bobrovsky and Z. Schuss
SIAM Journal on Applied Mathematics
Vol. 61, No. 4 (Nov., 2000 - Jan., 2001), pp. 1431-1444
Stable URL: http://www.jstor.org/stable/3061836
Page Count: 14
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Fixed-Point Smoothing of Scalar Diffusions II: The Error of the Optimal Smoother
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Abstract

The problem of fixed-point smoothing of a scalar diffusion process consists of estimating the initial value of the process, given its noisy measurements as a function of time. An asymptotic expansion of the joint filtering-smoothing conditional density function is constructed in the limit of small measurements noise. The approximate optimal nonlinear fixed-point smoother of Part I [SIAM J. Appl. Math., 54 (1994), pp. 833-853] is rederived from the expansion. A detailed analysis of the conditional mean square estimation error (CMSEE) of the optimal fixed-point smoother and of its leading-order approximation is presented. It is shown that if the initial error is small, e.g., if asymptotically optimal filtering is used first, the leading-order approximation to the optimal smoother is three dimensional and thus simpler than the four-dimensional extended Kalman smoother. Furthermore, nonlinear fixed-point smoothing can reduce the CMSEE relative to that of filtering by a factor of $1\over{2}$ within smoothing time proportional to the noise-intensity parameter. If the initial error is not small, it is shown that even in the linear case the CMSEE of the optimal fixed-point smoother is asymptotically the same as that of the optimal filter, in this limit.

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