Smoothing problem (stochastic processes): Difference between revisions

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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).

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

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 definedwith 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 convoluiton is the answer. However, in laer 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:

1. Convolution: The smoothing in the sense of '''convolution''' (egis simpler. For example, moving average, low-pass filtering, convolution with a kernel, or blurring using Laplace filters in [[image processing]]). It is simpleraoften a [[filter design]] problem. 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.

It is one of the main problems definedsolved 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.