Smoothing problem (stochastic processes): Difference between revisions

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{{technical|date=November 2017}}

The '''Smoothingsmoothing 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> A '''smoother''' is an algorithm that implements a solution to this problem, typically based on [[recursive Bayesian estimation]]. The smoothing problem is closely related to the [[filtering problem]], both of which are studied in Bayesian smoothing theory.

<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>

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.

The [[Smoothing problem]] and [[Filtering problem]] are often considered a closely related pair of problems. They are studied in Bayesian smoothing theory.

Note: Not to be confused with blurring and smoothing using methods such as moving average. See [[smoothing]].

==Example smoothers ==

==ExampleExamples of smoothers ==

Some variants include:<ref name="Sarkka-book">Simo Särkkä. Bayesian Filtering and Smoothing. Publisher: Cambridge University Press (5 Sept. 2013)

Language: English

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== The confusion in terms and the relation between Filtering and Smoothing problems==

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There are four terms that cause confusion: Smoothing (in two senses: estimation and convolution), and Filtering (again in two senses: estimation and convolution).

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In smoothing all observation samples are used (from future). Filtering is causal, whereas smoothing is batch processing of the given data. Filtering is the estimation of a (hidden) time-series process based on serial incremental observations.

=== RelatedSee conceptsalso ===

* [[Filter (disambiguation)|Filtering]] (disambiguation)

* [[Filtering problem]]

* Not to be confused with [[Filter (signal processing)]]

* [[Kalman filter]], mosta famouswell-known filtering algorithm inrelated theboth senseto ofthe 'filtering problem' and 'the smoothing problem'.

* [[Smoothing]] (not to be confused with the Smoothing problem)

* [[Smoothing (disambiguation)]]

== See also ==

* [[Generalized filtering]]

* [[Smoothing (disambiguation)]]

==References==