Smoothing problem (stochastic processes): Difference between revisions

Revision as of 11:22, 22 May 2018

Revision as of 04:03, 13 March 2019

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The '''Smoothing problem''' (not to be confused with [[smoothing]] in [[statistics]], [[image processing]] and other contexts) refers to [[Recursive Bayesian estimation]] also known as [[Bayes filter]] is the problem of [[density estimation|estimating]] an unknown [[probability density function]] recursively over time using incremental incoming measurements. It is one of the main problems defined by [[Norbert Wiener]]

<ref name="wiener-report">1942, ''Extrapolation, Interpolation and Smoothing of Stationary Time Series''. A war-time classified report nicknamed "the yellow peril" because of the color of the cover and the difficulty of the subject. Published postwar 1949 [[MIT Press]]. http://www.isss.org/lumwiener.htm])</ref><ref name="wiener-book">Wiener, Norbert (1949). Extrapolation, Interpolation, and Smoothing of Stationary Time Series. New York: Wiley. {{ISBN|0-262-73005-7}}.</ref>

.<ref name="wiener-book">Wiener, Norbert (1949). Extrapolation, Interpolation, and Smoothing of Stationary Time Series. New York: Wiley. {{ISBN|0-262-73005-7}}.</ref>

A '''smoother''' is an algorithm or implementation that implements a solution to such problem. Please refer to the article [[Recursive Bayesian estimation]] for more information.

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Smoothing (estimation) and smoothing (convolution) can mean totally different, but sound like they are apparently similar. The concepts are different and are used in almost different historical contexts. The '''requirements''' are very different.

Note that initially, the Wiener's filter was just a convolution, but the later developments were different: one was estimation and the other one was filter deisgndesign in the sense of design of a convolution filter. This is a source of confusion.

Both the smoothing problem (in sense of estimation) and the filtering problem (in sense of estimation) are often confused with smoothing and filtering in other contexts (especially non-stochastic signal processing, often a name of various types of convolution). These names are used in the context of World War 2 with problems framed by people like [[Norbert Wiener]].<ref name="wiener-report"/><ref name="wiener-book" /> One source of confusion is the [[Wiener Filter]] is in form of a simple convolution. However, in Wiener's filter, two time-series are given. When the filter is defined, a straigtforward convolution is the answer. However, in later developments such as Kalman filtering, the nature of filtering is different to convolution and it deserves a different name.