Smoothing problem (stochastic processes): Difference between revisions

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== 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}}

ThereThe terms Smoothing and Filtering are used for four termsconcepts that causemay initially be confusionconfusing: Smoothing (in two senses: estimation and convolution), and Filtering (again in two senses: estimation and convolution).

Smoothing (estimation) and smoothing (convolution) despite being labelled with the same name in English language, can mean totally different, butmathematical soundprocedures. likeThe theyrequirements areof apparentlyproblems similar.they The conceptssolve are different. andThese concepts are useddistinguished inby almostthe differentcontext historical(signal contexts.processing Theversus requirementsestimation areof verystochastic differentprocesses).

NoteThe historical reason for this confusion is that initially, the Wiener's suggested a "smoothing" filter that was just a convolution,. butLater theon his proposed solutions for obtaining a smoother estimation laterseparate developments wereas different:two onedistinct concepts. One was about attaining a smoother estimation by taking into account past observations, and the other one was smoothing using filter design 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 straightforward 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.