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{{technical|date=November 2017}}
The '''smoothing problem''' (not to be confused with [[smoothing]] in [[statistics]], [[image processing]] and other contexts) 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 {{Webarchive|url=https://web.archive.org/web/20150816041622/http://www.isss.org/lumwiener.htm |date=2015-08-16 }}</ref><ref name="wiener-book">{{cite book |last=Wiener |first=Norbert |author-link=Norbert Wiener |year=1949 |title=Extrapolation, Interpolation, and Smoothing of Stationary Time Series: With Engineering Applications |url=https://direct.mit.edu/books/oa-monograph/4361/Extrapolation-Interpolation-and-Smoothing-of |publisher=[[MIT Press]] |isbn=9780262257190}}</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.
A smoother is often a two-pass process, composed of forward and backward passes. Consider doing estimation (prediction/retrodiction) about an ongoing process (e.g. tracking a missile) based on incoming observations. When new observations arrive, estimations about past needs to be updated to have a smoother (more accurate) estimation of the whole estimated path until now (taking into account the newer observations). Without a backward pass (for [[retrodiction]]), the sequence of predictions in an online filtering algorithm does not look smooth. In other words, retrospectively, it is as if we are using future observations for improving estimation of a point in past, when those observations about future points become available. Note that time of estimation (which determines which observations are available) can be different to the time of the point that the prediction is about (that is subject to prediction/retrodiction). The observations about later times can be used to update and improved the estimations about earlier times. Doing so leads to smoother-looking estimations (retrodiction) about the whole path.
Some variants include
▲===Example smoothers ===
▲Some variants include <ref name="Sarkka-book">Simo Särkkä. Bayesian Filtering and Smoothing. Publisher: Cambridge University Press (5 Sept. 2013)
Language: English
{{ISBN
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* Rauch–Tung–Striebel (RTS) smoother
* Gaussian smoothers (e.g., extended Kalman smoother or sigma-point smoothers) for non-linear state-space models.
* Particle smoothers
== The confusion in terms and the relation between Filtering and Smoothing problems==
{{Cleanup section|reason=this section needs reorganization and also needs additional citations.|date=December 2021}}
The terms Smoothing and Filtering are used for four concepts that may initially be confusing: Smoothing (in two senses: estimation and convolution), and Filtering (again in two senses: estimation and convolution).
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 defined by people like [[Norbert Wiener]] <ref name="wiener-report"/><ref name="wiener-book" />. They are distinct in the following two senses:▼
Smoothing (estimation) and smoothing (convolution) despite being labelled with the same name in English language, can mean totally different mathematical procedures. The requirements of problems they solve are different. These concepts are distinguished by the context (signal processing versus estimation of stochastic processes).
The historical reason for this confusion is that initially, the Wiener's suggested a "smoothing" filter that was just a convolution. Later on his proposed solutions for obtaining a smoother estimation separate developments as two distinct concepts. One was about attaining a smoother estimation by taking into account past observations, and the other one was smoothing using filter design (design of a convolution filter).
▲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
The distinction is described in the following two senses:
1. Convolution: The smoothing in the sense of
2. Estimation: The
It is one of the main problems
Most importantly, in the Filtering problem (sense 2) the information from observation up to the time of the current sample is used. In smoothing (also sense 2) all observation samples (from future) are used. Filtering is causal but smoothing is batch processing of the same problem, namely, estimation of a time-series process based on serial incremental observations.
But the usual and more common smoothing and filtering (in the sense of 1.) do not have such distinction because there is no distinction between hidden and observable.
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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.
== See
* [[Filtering problem]]
*
* [[Kalman filter]],
* [[Generalized filtering]]
* [[Smoothing
==References==
{{Reflist}}
[[Category:Bayesian estimation]]
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