Editing Kalman filter (section)

== Technical description and context ==
The Kalman filter is an efficient [[recursive filter]] [[estimator|estimating]] the internal state of a [[linear dynamical system|linear dynamic system]] from a series of [[statistical noise|noisy]] measurements.  It is used in a wide range of [[engineering]] and [[econometric]] applications from [[radar]] and [[computer vision]] to estimation of structural macroeconomic models,<ref>{{cite journal| author1=Ingvar Strid |author2=Karl Walentin |date=April 2009|title=Block Kalman Filtering for Large-Scale DSGE Models |journal=Computational Economics |volume=33 |pages=277–304 |url=http://www.riksbank.se/en/Press-and-published/Published-from-the-Riksbank/Other-reports/Working-Paper-Series/2008/No-224-Block-Kalman-filtering-for-large-scale-DSGE-models/|issue=3| doi=10.1007/s10614-008-9160-4|hdl=10419/81929 |citeseerx=10.1.1.232.3790 |s2cid=3042206 }}</ref><ref>{{cite web |author=Martin Møller Andreasen |year=2008 |title=Non-linear DSGE Models, The Central Difference Kalman Filter, and The Mean Shifted Particle Filter |url=https://econpapers.repec.org/paper/aahcreate/2008-33.htm }}</ref> and is an important topic in [[control theory]] and [[control system]]s engineering.  Together with the [[linear-quadratic regulator]] (LQR), the Kalman filter solves the [[linear–quadratic–Gaussian control]] problem (LQG).  The Kalman filter, the linear-quadratic regulator, and the linear–quadratic–Gaussian controller are solutions to what arguably are the most fundamental problems of control theory.

In most applications, the internal state is much larger (has more [[degrees of freedom (physics and chemistry)|degrees of freedom]]) than the few "observable" parameters which are measured.  However, by combining a series of measurements, the Kalman filter can estimate the entire internal state.

For the [[Dempster–Shafer theory]], each state equation or observation is considered a special case of a [[linear belief function]] and the Kalman filtering is a special case of combining linear belief functions on a join-tree or [[Markov chain|Markov tree]]. Additional methods include [[belief filter]]ing which use Bayes or evidential updates to the state equations.

A wide variety of Kalman filters exists by now: Kalman's original formulation - now termed the "simple" Kalman filter, the [[Kalman–Bucy filter]], Schmidt's "extended" filter, the [[#Information filter|information filter]], and a variety of "square-root" filters that were developed by Bierman, Thornton, and many others. Perhaps the most commonly used type of very simple Kalman filter is the [[phase-locked loop]], which is now ubiquitous in radios, especially [[frequency modulation]] (FM) radios, television sets, [[satellite communications]] receivers, outer space communications systems, and nearly any other [[electronics|electronic]] communications equipment.