Smoothing problem (stochastic processes): Difference between revisions

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The '''Smoothing problem''' (not to be confused with [[smoothing]] in signal 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 or implementation that implements a solution to such problem. Please refer to the article [[Recursive Bayesian estimation]] for more information.

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==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-10: 1107619289

ISBN-13: 978-1107619289</ref>:

* Rauch–Tung–Striebel (RTS) smoother

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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 deisgn 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.

The distinction is described in the following two senses:

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2. Estimation: The '''smoothing problem''' (or Smoothing in the sense of '''estimation''') uses Bayesian and state-space models to estimate the hidden state variables. This is used in the context of World War 2 defined by people like Norbert Wiener, in (stochastic) control theory, radar, signal detection, tracking, etc. The most common use is the Kalman Smoother used with Kalman Filter, which is actually developed by Rauch. The procedure is called Kalman-Rauch recursion.

It is one of the main problems solved by [[Norbert Wiener]] .<ref name="wiener-report"/> <ref name="wiener-book"/>.

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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* [[Smoothing (disambiguation)]]

* [[Smoothing problem]]

== See Alsoalso ==

* [[Generalized filtering]]

==References==

{{Reflist}}

[[Category:Bayesian estimation]]