Smoothing problem (stochastic processes): Difference between revisions

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== Relation between Filtering and Smoothing problems==

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

Both the smoothing problemsproblem (in sense of estimation) and the filtering problemsproblem (in sense of estimation) are often confused with smoothing and filtering in other contexts (especially non-stochastic signal processing). 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:

Smoothing1. Convolution: The smoothing in the sense of '''convolution''' (eg, meansmoving average, low-pass filtering, convolution with a kernel, or blurring (using Laplace filters in [[image processing]]) is simpler. Especially non-stochastic and non-Bayesian signal processing, without any hidden variables.

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.

In the Filtering problem the information from observation up to the time of the current sample is used. In smoothing all observation samples are used (from future). Filtering is causal but smoothing is batch processing of the same problem, namely, estimation of a time-series process based on serial incremental observations.

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>.

InMost 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 are used (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.

The distinction between Smoothing (estimation) and Filtering (estimation):

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

== See Also==