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{{Short description|Algorithm that estimates unknowns from a series of measurements over time}}
{{Use American English|date=March 2018}}
[[File:Basic concept of Kalman filtering.svg|class=skin-invert-image|thumb|400px|The Kalman filter keeps track of the estimated state of the system and the [[variance]] or uncertainty of the estimate. The estimate is updated using a [[state transition]] model and measurements. <math>\hat{x}_{k\mid k-1}</math> denotes the estimate of the system's state at time step ''k'' before the ''k''-th measurement ''y''<sub>''k''</sub> has been taken into account; <math>P_{k \mid k-1}</math> is the corresponding uncertainty.]]

In [[statistics]] and [[control theory]], '''Kalman filtering''' (also known as '''linear quadratic estimation''') is an [[algorithm]] that uses a series of measurements observed over time, including [[statistical noise]] and other inaccuracies, to produce estimates of unknown variables that tend to be more accurate than those based on a single measurement, by estimating a [[joint probability distribution]] over the variables for each time-step. The filter is constructed as a mean squared error minimiser, but an alternative derivation of the filter is also provided showing how the filter relates to maximum likelihood statistics.<ref>{{Cite web |last=Lacey |first=Tony |date= |title=Chapter 11 Tutorial: The Kalman Filter |url=https://web.mit.edu/kirtley/kirtley/binlustuff/literature/control/Kalman%20filter.pdf}}</ref> The filter is named after [[Rudolf E. Kálmán]].

Kalman filtering<ref>{{cite journal | url=https://journal.ump.edu.my/mekatronika/article/view/4991 | doi=10.15282/mekatronika.v1i2.4991 | title=A Three-bar Truss Design using Single-solution Simulated Kalman Filter Optimizer | journal=Mekatronika | date=15 July 2019 | volume=1 | issue=2 | pages=98–102 | last1=Fauzi | first1=Hilman | last2=Batool | first2=Uzma | s2cid=222355496 | doi-access=free }}</ref> has numerous technological applications. A common application is for [[guidance, navigation, and control]] of vehicles, particularly aircraft, spacecraft and ships [[Dynamic positioning|positioned dynamically]].<ref name="ZarchanMusoff2000">{{cite book|author1=Paul Zarchan|author2=Howard Musoff|title=Fundamentals of Kalman Filtering: A Practical Approach|url=https://books.google.com/books?id=AQxRAAAAMAAJ|year=2000|publisher=American Institute of Aeronautics and Astronautics, Incorporated|isbn=978-1-56347-455-2}}</ref> Furthermore, [https://journal.ump.edu.my/mekatronika/article/view/4990 Kalman filtering] is much applied in [[time series]] analysis tasks such as [[signal processing]] and [[econometrics]]. Kalman filtering is also important for robotic [[motion planning]] and control,<ref>{{Cite journal |last1=Lora-Millan |first1=Julio S. |last2=Hidalgo |first2=Andres F. |last3=Rocon |first3=Eduardo |date=2021 |title=An IMUs-Based Extended Kalman Filter to Estimate Gait Lower Limb Sagittal Kinematics for the Control of Wearable Robotic Devices |journal=IEEE Access |volume=9 |pages=144540–144554 |doi=10.1109/ACCESS.2021.3122160 |s2cid=239938971 |issn=2169-3536|doi-access=free |bibcode=2021IEEEA...9n4540L |hdl=10261/254265 |hdl-access=free }}</ref><ref>{{Cite journal |last1=Kalita |first1=Diana |last2=Lyakhov |first2=Pavel |date=December 2022 |title=Moving Object Detection Based on a Combination of Kalman Filter and Median Filtering |journal=Big Data and Cognitive Computing |language=en |volume=6 |issue=4 |page=142 |doi=10.3390/bdcc6040142 |issn=2504-2289|doi-access=free }}</ref> and can be used for [[trajectory optimization]].<ref>{{Cite book|last1=Ghysels|first1=Eric|title=Applied Economic Forecasting using Time Series Methods|last2=Marcellino|first2=Massimiliano|publisher=Oxford University Press|year=2018|isbn=978-0-19-062201-5|location=New York, NY|page=419|oclc=1010658777}}</ref> Kalman filtering also works for modeling the [[central nervous system]]'s control of movement. Due to the time delay between issuing motor commands and receiving [[sensory feedback]], the use of Kalman filters<ref>{{cite journal | url=https://journal.ump.edu.my/mekatronika/article/view/4990 | doi=10.15282/mekatronika.v1i2.4990 | title=Design of an Helical Spring using Single-solution Simulated Kalman Filter Optimizer | journal=Mekatronika | date=15 July 2019 | volume=1 | issue=2 | pages=93–97 | last1=Azzam | first1=M. Abdullah | last2=Batool | first2=Uzma | last3=Fauzi | first3=Hilman | s2cid=221855079 | doi-access=free }}</ref> provides a realistic model for making estimates of the current state of a motor system and issuing updated commands.<ref>{{cite journal|title=Computational principles of movement neuroscience|journal=Nature Neuroscience |doi=10.1038/81497 |pmid=11127840 |volume=3|pages=1212–7 |year=2000|last1=Wolpert|first1=Daniel|last2=Ghahramani|first2=Zoubin|s2cid=736756 }}</ref>

The algorithm works via a two-phase process: a ''prediction phase'' and an ''update phase''. In the prediction phase, the Kalman filter produces estimates of the current [[state variable]]s, including their uncertainties. Once the outcome of the next measurement (necessarily corrupted with some error, including random noise) is observed, these estimates are updated using a [[Weighted mean|weighted average]], with more weight given to estimates with greater certainty. The algorithm is [[Recursion (computer science)|recursive]]. It can operate in [[real-time Control System|real time]], using only the present input measurements and the state calculated previously and its uncertainty matrix; no additional past information is required.

Optimality of Kalman filtering assumes that errors have a [[Normal distribution|normal (Gaussian)]] distribution. In the words of [[Rudolf E. Kálmán]], "The following assumptions are made about random processes: Physical random phenomena may be thought of as due to primary random sources exciting dynamic systems. The primary sources are assumed to be independent gaussian random processes with zero mean; the dynamic systems will be linear."<ref>{{cite journal|last1=Kalman|first1=R. E.|year=1960|title=A New Approach to Linear Filtering and Prediction Problems|url=https://semanticscholar.org/paper/922b5eaa5ca03b12d9842b7b84e0e420ccd2feee|journal=Journal of Basic Engineering|volume=82|pages=35–45|doi=10.1115/1.3662552|s2cid=1242324}}</ref> Regardless of Gaussianity, however, if the process and measurement covariances are known, then the Kalman filter is the best possible ''[[Linearity|linear]]'' estimator in the [[Minimum mean square error|minimum mean-square-error sense]],<ref>{{cite journal|last1=Humpherys|first1=Jeffrey|year=2012|title= A Fresh Look at the Kalman Filter|journal= SIAM Review|volume=54|issue=4|pages=801–823|doi=10.1137/100799666}}</ref> although there may be better nonlinear estimators. It is a common misconception (perpetuated in the literature) that the Kalman filter cannot be rigorously applied unless all noise processes are assumed to be Gaussian.<ref>{{cite journal |last1=Uhlmann |first1=Jeffrey |last2=Julier |first2=Simon |year=2022 |title=Gaussianity and the Kalman Filter: A Simple Yet Complicated Relationship |url=https://jci.uniautonoma.edu.co/2022/2022-1_2.pdf |journal=Journal de Ciencia e Ingeniería |volume=14 |issue=1 |pages=21–26 |doi=10.46571/JCI.2022.1.2|s2cid=251143915 }} See Uhlmann and Julier for roughly a dozen instances of this misconception in the literature.</ref>

Extensions and [[generalization]]s of the method have also been developed, such as the [[extended Kalman filter]] and the [[#Unscented Kalman filter|unscented Kalman filter]] which work on [[nonlinear system]]s. The basis is a [[hidden Markov model]] such that the [[state space]] of the [[latent variable]]s is [[State-space representation|continuous]] and all latent and observed variables have Gaussian distributions. Kalman filtering has been used successfully in [[Sensor fusion|multi-sensor fusion]],<ref>{{Cite journal|last1=Li|first1=Wangyan|last2=Wang|first2=Zidong|last3=Wei|first3=Guoliang|last4=Ma|first4=Lifeng|last5=Hu|first5=Jun|last6=Ding|first6=Derui|date=2015|title=A Survey on Multisensor Fusion and Consensus Filtering for Sensor Networks|journal=Discrete Dynamics in Nature and Society|language=en|volume=2015|pages=1–12|doi=10.1155/2015/683701|issn=1026-0226|doi-access=free}}</ref> and distributed [[Sensor Networks|sensor networks]] to develop distributed or [[Consensus (computer science)|consensus]] Kalman filtering.<ref>{{Cite journal|last1=Li|first1=Wangyan|last2=Wang|first2=Zidong|last3=Ho|first3=Daniel W. C.|last4=Wei|first4=Guoliang|date=2019|title=On Boundedness of Error Covariances for Kalman Consensus Filtering Problems|journal=IEEE Transactions on Automatic Control|volume=65|issue=6|pages=2654–2661|doi=10.1109/TAC.2019.2942826|s2cid=204196474|issn=0018-9286}}</ref>

== History ==
The filtering method is named for Hungarian [[émigré]] [[Rudolf E. Kálmán]], although [[Thorvald Nicolai Thiele]]<ref>{{cite journal |last1=Lauritzen |first1=S.&nbsp;L. |date=December 1981 |title=Time series analysis in 1880. A discussion of contributions made by T.N.&nbsp;Thiele |journal=International Statistical Review |volume=49 |number=3 |pages=319–331 |doi=10.2307/1402616 |jstor=1402616 |quote=He derives a recursive procedure for estimating the regression component and predicting the Brownian motion. The procedure is now known as Kalman filtering.}}</ref><ref>{{cite book |last=Lauritzen |first=S.&nbsp;L. |date=2002 |title=Thiele: Pioneer in Statistics |url=https://books.google.com/books?id=irugmNUwuG4C&q=kalman |location=New York |publisher=[[Oxford University Press]] |page=41 |isbn=978-0-19-850972-1 |author-link=Steffen Lauritzen |quote=He solves the problem of estimating the regression coefficients and predicting the values of the Brownian motion by the method of least squares and gives an elegant recursive procedure for carrying out the calculations. The procedure is nowadays known as ''Kalman filtering''.}}</ref> and [[Peter Swerling]] developed a similar algorithm earlier. Richard S. Bucy of the [[Johns Hopkins Applied Physics Laboratory]] contributed to the theory, causing it to be known sometimes as Kalman–Bucy filtering. Kalman was inspired to derive the Kalman filter by applying state variables to the [[Wiener filter|Wiener filtering problem]].<ref>{{Cite book |last1=Grewal |first1=Mohinder S. |title=Kalman filtering: theory and practice using MATLAB |last2=Andrews |first2=Angus P. |date=2015 |publisher=Wiley |isbn=978-1-118-98498-7 |edition=4th |location=Hoboken, New Jersey |pages=16–18 |chapter=1}}</ref>
[[Stanley F. Schmidt]] is generally credited with developing the first implementation of a Kalman filter. He realized that the filter could be divided into two distinct parts, with one part for time periods between sensor outputs and another part for incorporating measurements.<ref>{{Cite web |url=http://ieeecss.org/CSM/library/2010/june10/11-HistoricalPerspectives.pdf |title=Mohinder S. Grewal and Angus P. Andrews |access-date=2015-04-23 |archive-date=2016-03-07 |archive-url=https://web.archive.org/web/20160307142746/http://ieeecss.org/CSM/library/2010/june10/11-HistoricalPerspectives.pdf  }}</ref> It was during a visit by Kálmán to the [[NASA Ames Research Center]] that Schmidt saw the applicability of Kálmán's ideas to the nonlinear problem of trajectory estimation for the [[Project Apollo|Apollo program]] resulting in its incorporation in the [[Apollo Guidance Computer|Apollo navigation computer]].<ref>{{cite report
|title=A Linearized Error Analysis Of Onboard Primary Navigation Systems For The Apollo Lunar Module, NASA TN D-4027
|date=August 1967
|publisher= National Aeronautics and Space Administration
|type=NASA Technical Note 
|author=Jerrold H. Suddath
|author2=Robert H. Kidd
|author3=Arnold G. Reinhold
|url=https://ntrs.nasa.gov/api/citations/19670025568/downloads/19670025568.pdf}}</ref>{{rp|16}}

This digital filter is sometimes termed the ''Stratonovich–Kalman–Bucy filter'' because it is a special case of a more general, nonlinear filter developed by the [[Soviet]] [[mathematician]] [[Ruslan Stratonovich]].<ref>Stratonovich, R. L. (1959). ''Optimum nonlinear systems which bring about a separation of a signal with constant parameters from noise''. Radiofizika, 2:6, pp.&nbsp;892–901.</ref><ref>Stratonovich, R. L. (1959). ''On the theory of optimal non-linear filtering of random functions''. Theory of Probability and Its Applications, 4, pp.&nbsp;223–225.</ref><ref>Stratonovich, R. L. (1960) ''Application of the Markov processes theory to optimal filtering''. Radio Engineering and Electronic Physics, 5:11, pp.&nbsp;1–19.</ref><ref>Stratonovich, R. L. (1960). ''Conditional Markov Processes''. Theory of Probability and Its Applications, 5, pp.&nbsp;156–178.</ref> In fact, some of the special case linear filter's equations appeared in papers by Stratonovich that were published before the summer of 1961, when Kalman met with Stratonovich during a conference in Moscow.<ref name="stepanov_on_kalman">{{cite journal |last1=Stepanov |first1=O. A. |title=Kalman filtering: Past and present. An outlook from Russia. (On the occasion of the 80th birthday of Rudolf Emil Kalman) |journal=Gyroscopy and Navigation |date=15 May 2011 |volume=2 |issue=2 |page=105 |doi=10.1134/S2075108711020076 |bibcode=2011GyNav...2...99S |s2cid=53120402 }}</ref>

This Kalman filtering was first described and developed partially in technical papers by Swerling (1958), Kalman (1960) and Kalman and Bucy (1961).

{{blockquote|The Apollo computer used 2k of magnetic core RAM and 36k wire rope [...]. The CPU was built from ICs [...]. Clock speed was under 100 kHz [...]. The fact that the MIT engineers were able to pack such good software (one of the very first applications of the Kalman filter) into such a tiny computer is truly remarkable.|Interview with Jack Crenshaw, by Matthew Reed, TRS-80.org (2009) [http://www.trs-80.org/interview-jack-crenshaw/] }}

Kalman filters have been vital in the implementation of the navigation systems of [[U.S. Navy]] nuclear [[ballistic missile submarine]]s, and in the guidance and navigation systems of cruise missiles such as the U.S. Navy's [[Tomahawk (missile family)|Tomahawk missile]] and the [[U.S. Air Force]]'s [[AGM-86 ALCM|Air Launched Cruise Missile]]. They are also used in the guidance and navigation systems of [[reusable launch vehicle]]s and the [[Attitude dynamics and control|attitude control]] and navigation systems of spacecraft which dock at the [[International Space Station]].<ref>{{Cite book|doi=10.2514/6.2003-5445|chapter=GPS/INS Kalman Filter Design for Spacecraft Operating in the Proximity of International Space Station|title=AIAA Guidance, Navigation, and Control Conference and Exhibit|year=2003|last1=Gaylor|first1=David|last2=Lightsey|first2=E. Glenn|isbn=978-1-62410-090-1}}</ref>

== Overview of the calculation ==
Kalman filtering uses a system's dynamic model (e.g., physical laws of motion), known control inputs to that system, and multiple sequential measurements (such as from sensors) to form an estimate of the system's varying quantities (its [[State space (controls)|state]]) that is better than the estimate obtained by using only one measurement alone. As such, it is a common [[sensor fusion]] and [[data fusion]] algorithm.

Noisy sensor data, approximations in the equations that describe the system evolution, and external factors that are not accounted for, all limit how well it is possible to determine the system's state. The Kalman filter deals effectively with the uncertainty due to noisy sensor data and, to some extent, with random external factors. The Kalman filter produces an estimate of the state of the system as an average of the system's predicted state and of the new measurement using a [[Weighted mean|weighted average]]. The purpose of the weights is that values with better (i.e., smaller) estimated uncertainty are "trusted" more. The weights are calculated from the [[covariance]], a measure of the estimated uncertainty of the prediction of the system's state. The result of the weighted average is a new state estimate that lies between the predicted and measured state, and has a better estimated uncertainty than either alone. This process is repeated at every time step, with the new estimate and its covariance informing the prediction used in the following iteration. This means that Kalman filter works [[recursive filter|recursively]] and requires only the last "best guess", rather than the entire history, of a system's state to calculate a new state.

The measurements' certainty-grading and current-state estimate are important considerations. It is common to discuss the filter's response in terms of the Kalman filter's ''[[gain (electronics)|gain]]''. The Kalman gain is the weight given to the measurements and current-state estimate, and can be "tuned" to achieve a particular performance. With a high gain, the filter places more weight on the most recent measurements, and thus conforms to them more responsively. With a low gain, the filter conforms to the model predictions more closely. At the extremes, a high gain (close to one) will result in a more jumpy estimated trajectory, while a low gain (close to zero) will smooth out noise but decrease the responsiveness.

When performing the actual calculations for the filter (as discussed below), the state estimate and covariances are coded into [[Matrix (mathematics)|matrices]] because of the multiple dimensions involved in a single set of calculations. This allows for a representation of linear relationships between different state variables (such as position, velocity, and acceleration) in any of the transition models or covariances.

== Example application ==
As an example application, consider the problem of determining the precise location of a truck. The truck can be equipped with a [[GPS]] unit that provides an estimate of the position within a few meters. The GPS estimate is likely to be noisy; readings 'jump around' rapidly, though remaining within a few meters of the real position. In addition, since the truck is expected to follow the laws of physics, its position can also be estimated by integrating its velocity over time, determined by keeping track of wheel revolutions and the angle of the steering wheel. This is a technique known as [[dead reckoning]]. Typically, the dead reckoning will provide a very smooth estimate of the truck's position, but it will [[Drift (telecommunication)|drift]] over time as small errors accumulate.

For this example, the Kalman filter can be thought of as operating in two distinct phases: predict and update. In the prediction phase, the truck's old position will be modified according to the physical [[Newton's laws of motion|laws of motion]] (the dynamic or "state transition" model). Not only will a new position estimate be calculated, but also a new covariance will be calculated as well. Perhaps the covariance is proportional to the speed of the truck because we are more uncertain about the accuracy of the dead reckoning position estimate at high speeds but very certain about the position estimate at low speeds. Next, in the update phase, a measurement of the truck's position is taken from the GPS unit. Along with this measurement comes some amount of uncertainty, and its covariance relative to that of the prediction from the previous phase determines how much the new measurement will affect the updated prediction. Ideally, as the dead reckoning estimates tend to drift away from the real position, the GPS measurement should pull the position estimate back toward the real position but not disturb it to the point of becoming noisy and rapidly jumping.

== 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.

== Underlying dynamic system model ==

Kalman filtering is based on [[linear dynamic system]]s discretized in the time domain. They are modeled on a [[Markov chain]] built on [[linear operator]]s perturbed by errors that may include [[normal distribution|Gaussian]] [[noise (physics)|noise]]. The [[state space (controls)|state]] of the target system refers to the ground truth (yet hidden) system configuration of interest, which is represented as a [[vector space|vector]] of [[real number]]s. At each [[discrete time]] increment, a linear operator is applied to the state to generate the new state, with some noise mixed in, and optionally some information from the controls on the system if they are known. Then, another linear operator mixed with more noise generates the measurable outputs (i.e., observation) from the true ("hidden") state. The Kalman filter may be regarded as analogous to the hidden Markov model, with the difference that the hidden state variables have values in a continuous space as opposed to a discrete state space as for the hidden Markov model. There is a strong analogy between the equations of a Kalman Filter and those of the hidden Markov model. A review of this and other models is given in Roweis and [[Zoubin Ghahramani|Ghahramani]] (1999)<ref name="roweisghahramani1999">{{cite journal|doi=10.1162/089976699300016674|pmid= 9950734|year= 1999|last1= Roweis|first1= S|title= A unifying review of linear gaussian models|journal= Neural Computation|volume= 11|issue= 2|pages= 305–45|last2= Ghahramani|first2= Z|s2cid= 2590898|url= https://authors.library.caltech.edu/13697/1/ROWnc99.pdf}}</ref> and Hamilton (1994), Chapter 13.<ref name="hamilton">Hamilton, J. (1994), ''Time Series Analysis'', Princeton University Press. Chapter 13, 'The Kalman Filter'</ref>

In order to use the Kalman filter to estimate the internal state of a process given only a sequence of noisy observations, one must model the process in accordance with the following framework. This means specifying the matrices, for each time-step ''<math>k</math>'', following:
* <math>\mathbf{F}_k</math>, the state-transition model;
* <math>\mathbf{H}_k</math>, the observation model;
* <math>\mathbf{Q}_k</math>, the [[covariance]] of the process noise;
* <math>\mathbf{R}_k</math>, the [[covariance]] of the observation noise;
* and sometimes <math>\mathbf{B}_k</math>, the control-input model as described below; if <math>\mathbf{B}_k</math> is included, then there is also
* <math>\mathbf{u}_k</math>, the control vector, representing the controlling input into control-input model.

As seen below, it is common in many applications that the matrices <math>\mathbf{F}</math>, <math>\mathbf{H}</math>, <math>\mathbf{Q}</math>, <math>\mathbf{R}</math>, and <math>\mathbf{B}</math> are constant across time, in which case their <math>k</math> index may be dropped.

[[File:Kalman filter model 2.svg|class=skin-invert-image|center|thumb|631px|Model underlying the Kalman filter. Squares represent matrices. Ellipses represent [[multivariate normal distribution]]s (with the mean and covariance matrix enclosed). Unenclosed values are [[vector space|vectors]].  For the simple case, the various matrices are constant with time, and thus the subscripts are not used, but Kalman filtering allows any of them to change each time step.]]

The Kalman filter model assumes the true state at time <math>k</math> is evolved from the state at <math>k-1</math> according to

:<math> \mathbf{x}_k = \mathbf{F}_k \mathbf{x}_{k-1} +\mathbf{B}_k \mathbf{u}_{k} + \mathbf{w}_k </math>

where
* <math>\mathbf{F}_k</math> is the state transition model which is applied to the previous state '''x'''<sub>''k''−1</sub>;
* <math>\mathbf{B}_k</math> is the control-input model which is applied to the control vector <math>\mathbf{u}_k</math>;
* <math>\mathbf{w}_k</math> is the process noise, which is assumed to be drawn from a zero mean [[multivariate normal distribution]], <math>\mathcal{N}</math>, with [[covariance matrix|covariance]], <math>\mathbf{Q}_k</math>: <math>\mathbf{w}_k \sim \mathcal{N}\left(0, \mathbf{Q}_k\right) </math>. 

If <math>\mathbf{Q}</math> is independent of time, one may, following Roweis and Ghahramani,<ref name="roweisghahramani1999" />{{Rp|page=307}} write <math>\mathbf{w}_\bull</math> instead of <math>\mathbf{w}_k</math> to emphasize that the noise has no explicit knowledge of time.

At time <math>k</math> an observation (or measurement) <math>\mathbf{z}_k</math>of the true state <math>\mathbf{x}_k</math> is made according to

:<math>\mathbf{z}_k = \mathbf{H}_k \mathbf{x}_k + \mathbf{v}_k</math>

where
* <math>\mathbf{H}_k</math> is the observation model, which maps the true state space into the observed space and
* <math>\mathbf{v}_k</math> is the observation noise, which is assumed to be zero mean Gaussian [[white noise]] with covariance <math>\mathbf{R}_k</math>: <math>\mathbf{v}_k \sim \mathcal{N}\left(0, \mathbf{R}_k\right) </math>.

Analogously to the situation for <math>\mathbf{w}_k</math>, one may write <math>\mathbf{v}_\bull</math> instead of <math>\mathbf{v}_k</math>  if <math>\mathbf{R}</math> is independent of time.

The initial state, and the noise vectors at each step <math>\{ \mathbf{x}_0, \mathbf{w}_1, \dots, \mathbf{w}_k, \mathbf{v}_1, \dots, \mathbf{v}_k  \} </math> are all assumed to be mutually [[statistical independence|independent]].

Many real-time dynamic systems do not exactly conform to this model. In fact, unmodeled dynamics can seriously degrade the filter performance, even when it was supposed to work with unknown stochastic signals as inputs. The reason for this is that the effect of unmodeled dynamics depends on the input, and, therefore, can bring the estimation algorithm to instability (it diverges). On the other hand, independent white noise signals will not make the algorithm diverge. The problem of distinguishing between measurement noise and unmodeled dynamics is a difficult one and is treated as a problem of control theory using [[robust control]].<ref name="ishihara06">{{cite journal |doi=10.1109/TAC.2006.878741|title=Robust Kalman Filter for Descriptor Systems|journal=IEEE Transactions on Automatic Control|volume=51|issue=8|page=1354|year=2006|last1=Ishihara|first1=J.Y.|last2=Terra|first2=M.H.|last3=Campos|first3=J.C.T.|bibcode=2006ITAC...51.1354I |s2cid=12741796}}</ref><ref name="terra14">{{cite journal |doi=10.1109/TAC.2014.2309282|title=Optimal Robust Linear Quadratic Regulator for Systems Subject to Uncertainties|journal=IEEE Transactions on Automatic Control|volume=59|issue=9|pages=2586–2591|year=2014|last1=Terra|first1=Marco H.|last2=Cerri|first2=Joao P.|last3=Ishihara|first3=Joao Y.|bibcode=2014ITAC...59.2586T |s2cid=8810105}}</ref>

== Details ==
The Kalman filter is a [[infinite impulse response|recursive]] estimator. This means that only the estimated state from the previous time step and the current measurement are needed to compute the estimate for the current state. In contrast to batch estimation techniques, no history of observations and/or estimates is required. In what follows, the notation <math>\hat{\mathbf{x}}_{n\mid m}</math> represents the estimate of <math>\mathbf{x}</math> at time ''n'' given observations up to and including at time {{nowrap|''m'' ≤ ''n''}}.

The state of the filter is represented by two variables:
* <math>\hat\mathbf{x}_{k\mid k}</math>, the ''[[a posteriori]]'' state estimate mean at time ''k'' given observations up to and including at time ''k'';
* <math>\mathbf{P}_{k\mid k}</math>, the ''a posteriori'' estimate covariance matrix (a measure of the estimated [[accuracy and precision|accuracy]] of the state estimate).

The algorithm structure of the Kalman filter resembles that of [[Alpha beta filter]]. The Kalman filter can be written as a single equation; however, it is most often conceptualized as two distinct phases: "Predict" and "Update". The predict phase uses the state estimate from the previous timestep to produce an estimate of the state at the current timestep. This predicted state estimate is also known as the ''a priori'' state estimate because, although it is an estimate of the state at the current timestep, it does not include observation information from the current timestep. In the update phase, the [[Innovation (signal processing)|innovation]] (the pre-fit residual), i.e. the difference between the current ''a priori'' prediction and the current observation information, is multiplied by the optimal Kalman gain and combined with the previous state estimate to refine the state estimate. This improved estimate based on the current observation is termed the ''a posteriori'' state estimate.

Typically, the two phases alternate, with the prediction advancing the state until the next scheduled observation, and the update incorporating the observation.  However, this is not necessary; if an observation is unavailable for some reason, the update may be skipped and multiple prediction procedures performed.  Likewise, if multiple independent observations are available at the same time, multiple update procedures may be performed (typically with different observation matrices '''H'''<sub>''k''</sub>).<ref>{{cite journal|last1=Kelly|first1=Alonzo|title=A 3D state space formulation of a navigation Kalman filter for autonomous vehicles|journal=DTIC Document|date=1994|page=13|url=http://apps.dtic.mil/dtic/tr/fulltext/u2/a282853.pdf|archive-url=https://web.archive.org/web/20141230004557/http://www.dtic.mil/dtic/tr/fulltext/u2/a282853.pdf|url-status=live|archive-date=December 30, 2014}} [http://www.frc.ri.cmu.edu/~alonzo/pubs/reports/kalman_V2.pdf 2006 Corrected Version] {{Webarchive|url=https://web.archive.org/web/20170110204109/http://www.frc.ri.cmu.edu/~alonzo/pubs/reports/kalman_V2.pdf |date=2017-01-10 }}</ref><ref>{{cite web|last1=Reid|first1=Ian|last2=Term|first2=Hilary|title=Estimation II|url=http://www.robots.ox.ac.uk/~ian/Teaching/Estimation/LectureNotes2.pdf|website=www.robots.ox.ac.uk|publisher=Oxford University|access-date=6 August 2014}}</ref>

=== Predict ===
{|
|-
| style="width:180pt;" | Predicted (''a priori'') state estimate
| <math>\hat{\mathbf{x}}_{k\mid k-1} = \mathbf{F}_k \hat\mathbf{x}_{k-1\mid k-1} + \mathbf{B}_k \mathbf{u}_{k}</math>
|-
| Predicted (''a priori'') estimate covariance
| <math>{\mathbf{P}}_{k\mid k-1} = \mathbf{F}_k \mathbf{P}_{k-1 \mid k-1} \mathbf{F}_k^\textsf{T} + \mathbf{Q}_{k}</math>
|}

=== Update ===
{|
|-
| style="width:180pt;" | [[Innovation (signal processing)|Innovation]] or measurement pre-fit residual
| <math>\tilde{\mathbf{y}}_k = \mathbf{z}_k - \mathbf{H}_k\hat{\mathbf{x}}_{k\mid k-1}</math>
|-
| Innovation (or pre-fit residual) covariance
| <math>\mathbf{S}_k = \mathbf{H}_k {\mathbf{P}}_{k\mid k-1} \mathbf{H}_k^\textsf{T} + \mathbf{R}_k</math>
|-
| ''Optimal'' Kalman gain
| <math>\mathbf{K}_k = {\mathbf{P}}_{k\mid k-1}\mathbf{H}_k^\textsf{T} \mathbf{S}_k^{-1}</math>
|-
| Updated (''a posteriori'') state estimate
| <math>\hat{\mathbf{x}}_{k\mid k} = \hat{\mathbf{x}}_{k\mid k-1} + \mathbf{K}_k\tilde{\mathbf{y}}_k</math>
|-
| Updated (''a posteriori'') estimate covariance (usual form)
|   <math>\mathbf{P}_{k \mid k} = \left(\mathbf{I} - \mathbf{K}_k \mathbf{H}_k\right) \mathbf{P}_{k|k-1}</math> 
|-
| Updated (''a posteriori'') estimate covariance (Joseph form)
| <math>\mathbf{P}_{k \mid k} = \left(\mathbf{I} - \mathbf{K}_k \mathbf{H}_k\right) \mathbf{P}_{k \mid k - 1} \left(\mathbf{I} - \mathbf{K}_k \mathbf{H}_k\right)^\textsf{T} + \mathbf{K}_k \mathbf{R}_k \mathbf{K}_k^\textsf{T}</math> 
|-
| Measurement post-fit [[residuals (statistics)|residual]]
| <math>\tilde{\mathbf{y}}_{k\mid k} = \mathbf{z}_k - \mathbf{H}_k\hat{\mathbf{x}}_{k\mid k}</math>
|}

The second formula for the updated (''a posteriori'') estimate covariance above is known as the "Joseph form" that is often used in applications (numerically more stable than the more simple usual formulation). Proof of the formulae is found in the ''[[#Derivations|derivations]]'' section, where the formula valid for any '''K'''<sub>k</sub> is also shown.

A more intuitive way to express the updated state estimate (<math>\hat{\mathbf{x}}_{k\mid k}</math>) is:

:<math>\hat{\mathbf{x}}_{k\mid k} = (\mathbf{I} - \mathbf{K}_k \mathbf{H}_k) \hat{\mathbf{x}}_{k\mid k-1} + \mathbf{K}_k \mathbf{z}_k</math>

This expression reminds us of a linear interpolation, <math>x = (1-t)(a) + t(b)</math> for <math>t</math> between [0,1]. 
In our case:
* <math>t</math> is the matrix <math>\mathbf{K}_k \mathbf{H}_k</math> that takes values from <math>0</math> (high error in the sensor) to <math>I</math> or a projection (low error).
* <math>a</math> is the internal state <math>\hat{\mathbf{x}}_{k\mid k-1}</math> estimated from the model.
* <math>b</math> is the internal state <math>\mathbf{H}_k^{-1} \mathbf{z}_k</math> estimated from the measurement, assuming <math>\mathbf{H}_k</math> is nonsingular (which in many applications is not a reasonable assumption e.g. when the state dimension is greater than the observation dimension).
This expression also resembles the [[alpha beta filter]] update step.

=== Invariants ===
If the model is accurate, and the values for <math>\hat{\mathbf{x}}_{0\mid 0}</math> and <math>\mathbf{P}_{0\mid 0}</math> accurately reflect the distribution of the initial state values, then the following invariants are preserved:
:<math>\begin{align}
  \operatorname{E}[\mathbf{x}_k - \hat{\mathbf{x}}_{k\mid k}] &= \operatorname{E}[\mathbf{x}_k - \hat{\mathbf{x}}_{k\mid k-1}] = 0 \\
                       \operatorname{E}[\tilde{\mathbf{y}}_k] &= 0
\end{align}</math>

where <math>\operatorname{E}[\xi]</math> is the [[expected value]] of <math>\xi</math>. That is, all estimates have a mean error of zero.

Also:
:<math>\begin{align}
    \mathbf{P}_{k\mid k} &= \operatorname{cov}\left(\mathbf{x}_k - \hat{\mathbf{x}}_{k\mid k}\right) \\
  \mathbf{P}_{k\mid k-1} &= \operatorname{cov}\left(\mathbf{x}_k - \hat{\mathbf{x}}_{k\mid k-1}\right) \\
            \mathbf{S}_k &= \operatorname{cov}\left(\tilde{\mathbf{y}}_k\right)
\end{align}</math>

so covariance matrices accurately reflect the covariance of estimates.

=== Estimation of the noise covariances Q<sub>''k''</sub> and R<sub>''k''</sub> ===
Practical implementation of a Kalman Filter is often difficult due to the difficulty of getting a good estimate of the noise covariance matrices '''Q'''<sub>''k''</sub> and '''R'''<sub>''k''</sub>.  Extensive research has been done to estimate these covariances from data.  One practical method of doing this is the ''autocovariance least-squares (ALS)'' technique that uses the time-lagged [[autocovariance]]s of routine operating data to estimate the covariances.<ref>{{cite thesis |url=http://jbrwww.che.wisc.edu/theses/rajamani.pdf |last=Rajamani |first=Murali |type=PhD Thesis |title=Data-based Techniques to Improve State Estimation in Model Predictive Control |location=University of Wisconsin–Madison |date=October 2007 |access-date=2011-04-04 |archive-url=https://web.archive.org/web/20160304194938/http://jbrwww.che.wisc.edu/theses/rajamani.pdf |archive-date=2016-03-04  }}</ref><ref>{{cite journal |last1=Rajamani |first1=Murali R. |last2=Rawlings |first2=James B. |title=Estimation of the disturbance structure from data using semidefinite programming and optimal weighting |journal=Automatica |volume=45 |issue=1 |pages=142–148 |year=2009 |doi=10.1016/j.automatica.2008.05.032 |s2cid=5699674 }}</ref>  The [[GNU Octave]] and [[Matlab]] code used to calculate the noise covariance matrices using the ALS technique is available online using the [[GNU General Public License]].<ref>{{cite web |url=https://sites.engineering.ucsb.edu/~jbraw/software/als/index.html |title=Autocovariance Least-Squares Toolbox |publisher=Jbrwww.che.wisc.edu |access-date=2021-08-18 }}</ref> Field Kalman Filter (FKF), a Bayesian algorithm, which allows simultaneous estimation of the state, parameters and noise covariance has been proposed.<ref>{{cite conference |url= https://www.researchgate.net/publication/312029167|title= Field Kalman Filter and its approximation|last1= Bania|first1= P.|last2= Baranowski|first2=J.|publisher=IEEE |date=12 December 2016|pages= 2875–2880|location= Las Vegas, NV, USA|conference= IEEE 55th Conference on Decision and Control (CDC)}}</ref> The FKF algorithm has a recursive formulation, good observed convergence, and relatively low complexity, thus suggesting that the FKF algorithm may possibly be a worthwhile alternative to the Autocovariance Least-Squares methods. Another approach is the ''Optimized Kalman Filter'' (''OKF''), which considers the covariance matrices not as representatives of the noise, but rather, as parameters aimed to achieve the most accurate state estimation.<ref name=":0">{{Cite journal |last1=Greenberg |first1=Ido |last2=Yannay |first2=Netanel |last3=Mannor |first3=Shie |date=2023-12-15 |title=Optimization or Architecture: How to Hack Kalman Filtering |url=https://proceedings.neurips.cc/paper_files/paper/2023/hash/9dfcc83c01e94d02c751c47517855c9f-Abstract-Conference.html |journal=Advances in Neural Information Processing Systems |language=en |volume=36 |pages=50482–50505|arxiv=2310.00675 }}</ref> These two views coincide under the KF assumptions, but often contradict each other in real systems. Thus, OKF's state estimation is more robust to modeling inaccuracies.

=== Optimality and performance ===
The Kalman filter provides an optimal state estimation in cases where a) the model matches the real system perfectly, b) the entering noise is "white" (uncorrelated), and c) the covariances of the noise are known exactly. Correlated noise can also be treated using Kalman filters.<ref>{{Cite book|last1=Bar-Shalom|first1=Yaakov|title=Estimation with Applications to Tracking and Navigation|last2=Li|first2=X.-Rong|last3=Kirubarajan|first3=Thiagalingam|date=2001|publisher=John Wiley & Sons, Inc.|isbn=0-471-41655-X|location=New York, USA|pages=319 ff|doi=10.1002/0471221279}}</ref> 
Several methods for the noise covariance estimation have been proposed during past decades, including ALS, mentioned in the section above. More generally, if the model assumptions do not match the real system perfectly, then optimal state estimation is not necessarily obtained by setting '''Q'''<sub>''k''</sub> and '''R'''<sub>''k''</sub> to the covariances of the noise. Instead, in that case, the parameters '''Q'''<sub>''k''</sub> and '''R'''<sub>''k''</sub> may be set to explicitly optimize the state estimation,<ref name=":0" /> e.g., using standard [[supervised learning]].

After the covariances are set, it is useful to evaluate the performance of the filter; i.e., whether it is possible to improve the state estimation quality. If the Kalman filter works optimally, the innovation sequence (the output prediction error) is a white noise, therefore the whiteness property of the [[Innovation (signal processing)|innovations]] measures filter performance. Several different methods can be used for this purpose.<ref>Three optimality tests with numerical examples are described in  {{cite book|doi=10.3182/20120711-3-BE-2027.00011|chapter=Optimality Tests and Adaptive Kalman Filter|title=16th IFAC Symposium on System Identification|series=IFAC Proceedings Volumes|volume=45|issue=16|pages=1523–1528|year=2012|last1=Peter|first1=Matisko|isbn=978-3-902823-06-9}}</ref> If the noise terms are distributed in a non-Gaussian manner, methods for assessing performance of the filter estimate, which use probability inequalities or large-sample theory, are known in the literature.<ref>{{cite journal|doi=10.1016/0005-1098(95)00069-9|title=The Kantorovich inequality for error analysis of the Kalman filter with unknown noise distributions|journal=Automatica|volume=31|issue=10|pages=1513–1517|year=1995|last1=Spall|first1=James C.}}</ref><ref>{{cite journal|doi=10.1109/TAC.2003.821415|title=Use of the Kalman Filter for Inference in State-Space Models with Unknown Noise Distributions|journal=IEEE Transactions on Automatic Control|volume=49|pages=87–90|year=2004|last1=Maryak|first1=J.L.|last2=Spall|first2=J.C.|last3=Heydon|first3=B.D.|issue=1 |bibcode=2004ITAC...49...87M |s2cid=21143516}}</ref>

== Example application, technical ==
[[File:kalman.png|class=skin-invert-image|thumb|
{{legend-line|solid light-dark(black,white)|Truth}}
{{legend-line|solid green|Filtered process}}
{{legend-line|dotted red|Observations}}
]]
Consider a truck on frictionless, straight rails. Initially, the truck is stationary at position 0, but it is buffeted this way and that by random uncontrolled forces. We measure the position of the truck every Δ''t'' seconds, but these measurements are imprecise; we want to maintain a model of the truck's position and [[velocity]]. We show here how we derive the model from which we create our Kalman filter.

Since <math>\mathbf{F}, \mathbf{H}, \mathbf{R}, \mathbf{Q}</math> are constant, their time indices are dropped.

The position and velocity of the truck are described by the linear state space
:<math>\mathbf{x}_k = \begin{bmatrix}
    x \\
    \dot{x}
  \end{bmatrix}
</math>

where <math>\dot{x}</math> is the velocity, that is, the derivative of position with respect to time.

We assume that between the (''k''&nbsp;−&nbsp;1) and ''k'' timestep, uncontrolled forces cause a constant acceleration of ''a''<sub>''k''</sub> that is [[normal distribution|normally distributed]] with mean 0 and standard deviation ''σ''<sub>''a''</sub>. From [[Newton's laws of motion]] we conclude that
:<math>\mathbf{x}_k = \mathbf{F} \mathbf{x}_{k-1} + \mathbf{G} a_k</math>

(there is no <math>\mathbf{B}u</math> term since there are no known control inputs. Instead, ''a''<sub>''k''</sub> is the effect of an unknown input and <math>\mathbf{G}</math> applies that effect to the state vector) where
:<math>\begin{align}
  \mathbf{F} &= \begin{bmatrix}
    1 & \Delta t \\
    0 & 1
  \end{bmatrix} \\[4pt]
  \mathbf{G} &= \begin{bmatrix}
    \frac{1}{2}{\Delta t}^2 \\[6pt]
    \Delta t
  \end{bmatrix}
\end{align}</math>

so that
:<math>\mathbf{x}_k = \mathbf{F} \mathbf{x}_{k-1} + \mathbf{w}_k</math>

where
:<math>\begin{align}
  \mathbf{w}_k &\sim N(0, \mathbf{Q}) \\
    \mathbf{Q} &=
  \mathbf{G}\mathbf{G}^\textsf{T}\sigma_a^2 =
  \begin{bmatrix}
    \frac{1}{4}{\Delta t}^4 & \frac{1}{2}{\Delta t}^3 \\[6pt]
    \frac{1}{2}{\Delta t}^3 & {\Delta t}^2
  \end{bmatrix}\sigma_a^2.
\end{align}</math>

The matrix <math>\mathbf{Q}</math> is not full rank (it is of rank one if <math>\Delta t \neq 0</math>). Hence, the distribution <math>N(0, \mathbf{Q})</math> is not absolutely continuous and has [[Multivariate normal distribution#Degenerate case|no probability density function]]. Another way to express this, avoiding explicit degenerate distributions is given by
:<math>\mathbf{w}_k \sim \mathbf{G} \cdot N\left(0, \sigma_a^2\right). </math>

At each time phase, a noisy measurement of the true position of the truck is made. Let us suppose the measurement noise ''v''<sub>''k''</sub> is also distributed normally, with mean 0 and standard deviation ''σ''<sub>''z''</sub>.
:<math>\mathbf{z}_k = \mathbf{H x}_k + \mathbf{v}_k</math>

where
:<math>\mathbf{H} = \begin{bmatrix} 1 & 0 \end{bmatrix} </math>

and
:<math>
  \mathbf{R} = \mathrm{E}\left[\mathbf{v}_k \mathbf{v}_k^\textsf{T}\right]
             = \begin{bmatrix} \sigma_z^2 \end{bmatrix}
</math>

We know the initial starting state of the truck with perfect precision, so we initialize
:<math>\hat{\mathbf{x}}_{0 \mid 0} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} </math>

and to tell the filter that we know the exact position and velocity, we give it a zero covariance matrix:
:<math>\mathbf{P}_{0 \mid 0} = \begin{bmatrix}
    0 & 0 \\
    0 & 0
  \end{bmatrix}
</math>

If the initial position and velocity are not known perfectly, the covariance matrix should be initialized with suitable variances on its diagonal:
:<math>\mathbf{P}_{0 \mid 0} = \begin{bmatrix}
    \sigma_x^2 & 0 \\
    0          & \sigma_\dot{x}^2
  \end{bmatrix}
</math>

The filter will then prefer the information from the first measurements over the information already in the model.

== Asymptotic form ==

For simplicity, assume that the control input <math>\mathbf{u}_k=\mathbf{0}</math>. Then the Kalman filter may be written:

:<math>\hat{\mathbf{x}}_{k\mid k} = \mathbf{F}_k \hat{\mathbf{x}}_{k-1\mid k-1} + \mathbf{K}_k[\mathbf{z}_k - \mathbf{H}_k \mathbf{F}_k\hat{\mathbf{x}}_{k-1\mid k-1}].</math>

A similar equation holds if we include a non-zero control input. Gain matrices <math>\mathbf{K}_k</math> and covariance matrices <math>\mathbf{P}_{k\mid k}</math> evolve independently of the measurements <math>\mathbf{z}_k</math>. From above, the four equations needed for updating the matrices are as follows:

:<math>\begin{align}
  \mathbf{P}_{k\mid k-1} &= \mathbf{F}_k \mathbf{P}_{k-1\mid k-1} \mathbf{F}_k^\textsf{T} + \mathbf{Q}_k, \\
            \mathbf{S}_k &= \mathbf{H}_k \mathbf{P}_{k\mid k-1} \mathbf{H}_k^\textsf{T} + \mathbf{R}_k, \\
            \mathbf{K}_k &= \mathbf{P}_{k\mid k-1}\mathbf{H}_k^\textsf{T} \mathbf{S}_k^{-1}, \\
         \mathbf{P}_{k|k} &= \left(\mathbf{I} - \mathbf{K}_k \mathbf{H}_k\right) \mathbf{P}_{k|k-1}.
\end{align}</math>

Since these depend only on the model, and not the measurements, they may be computed offline. Convergence of the gain matrices <math>\mathbf{K}_k</math> to an asymptotic matrix <math>\mathbf{K}_\infty</math> applies for conditions established in Walrand and Dimakis.<ref name=walrandnotes>{{cite book|last1=Walrand|first1=Jean|last2=Dimakis|first2=Antonis|date=August 2006|title=Random processes in Systems -- Lecture Notes|url=https://people.eecs.berkeley.edu/~wlr/226F06/226a.pdf|pages=69–70|access-date=2019-05-07|archive-date=2019-05-07|archive-url=https://web.archive.org/web/20190507181356/https://people.eecs.berkeley.edu/~wlr/226F06/226a.pdf|url-status=dead}}</ref> If the <math>\mathbf{P}_{k\mid k}</math> series converges, then it converges exponentially to an asymptotic <math>\mathbf{P}_\infty</math>, assuming non-zero plant noise.<ref>Kalman, Rudolf Emil, T. S. Englar, and Richard S. Bucy. ''Fundamental study of adaptive control systems''. Clearinghouse, US Department of Commerce, 1962.</ref> Recent analysis has shown that the ''rate and nature'' of this convergence can involve multiple coequal modes, including oscillatory components, depending on the eigenstructure of the Jacobian of the above Riccati map evaluated at <math>\mathbf{P}_\infty</math>.<ref>{{Cite journal |last=Herbst |first=Daniel C. |date=2024 |title=Exponential Convergence Rate and Oscillatory Modes of the Asymptotic Kalman Filter Covariance |journal=IEEE Access |volume=12 |pages=188137–188153 |doi=10.1109/ACCESS.2024.3508578 |bibcode=2024IEEEA..12r8137H |issn=2169-3536}}</ref> For the moving truck example described above, with <math>\Delta t = 1</math> and <math>\sigma_a^2=\sigma_z^2 =\sigma_x^2= \sigma_\dot{x}^2=1</math>, simulation shows convergence in <math>10</math> iterations.

Using the asymptotic gain, and assuming <math>\mathbf{H}_k</math> and <math>\mathbf{F}_k</math> are independent of <math>k</math>, the Kalman filter becomes a [[linear time-invariant]] filter:

:<math>\hat{\mathbf{x}}_{k} = \mathbf{F} \hat{\mathbf{x}}_{k-1} + \mathbf{K}_\infty[\mathbf{z}_k - \mathbf{H}\mathbf{F} \hat{\mathbf{x}}_{k-1}].</math>

The asymptotic gain <math>\mathbf{K}_\infty</math>, if it exists, can be computed by first solving the following discrete [[Riccati equation]] for the asymptotic state covariance <math>\mathbf{P}_\infty</math>:<ref name=walrandnotes/>

:<math>\mathbf{P}_\infty = \mathbf{F}\left(\mathbf{P}_\infty - \mathbf{P}_\infty \mathbf{H}^\textsf{T} \left(\mathbf{H}\mathbf{P}_\infty\mathbf{H}^\textsf{T} + \mathbf{R}\right) ^{-1} \mathbf{H}\mathbf{P}_\infty\right) \mathbf{F}^\textsf{T} + \mathbf{Q}.</math>

The asymptotic gain is then computed as before.

:<math>\mathbf{K}_\infty = \mathbf{P}_\infty \mathbf{H}^\textsf{T} \left( \mathbf{R} + \mathbf{H} \mathbf{P}_\infty \mathbf{H}^\textsf{T} \right) ^{-1}.</math>

Additionally, a form of the asymptotic Kalman filter more commonly used in control theory is given by

:<math>{\displaystyle {\hat {\mathbf {x} }}_{k+1}=\mathbf {F}{\hat {\mathbf {x} }}_{k}+ \mathbf {B}\mathbf{u}_k + \mathbf \overline{K} _{\infty}[\mathbf {z} _{k}-\mathbf {H}{\hat {\mathbf {x} }}_{k}],}</math>

where

:<math>\overline{\mathbf{K}}_\infty = \mathbf{F}\mathbf{P}_\infty \mathbf{H}^\textsf{T} \left( \mathbf{R} + \mathbf{H} \mathbf{P}_\infty \mathbf{H}^\textsf{T} \right) ^{-1}.</math>

This leads to an estimator of the form

:<math>{\displaystyle {\hat {\mathbf {x} }}_{k+1}=(\mathbf {F}-\overline{\mathbf{K}} _{\infty}\mathbf {H}){\hat {\mathbf {x} }}_{k} + \mathbf {B}\mathbf{u}_k  +\mathbf \overline{K} _{\infty}\mathbf {z} _{k},}</math>

== Derivations ==
{{More citations needed section|date=December 2010}}
The Kalman filter can be derived as a [[generalized least squares]] method operating on previous data.<ref>Sant, Donald T. "Generalized least squares applied to time varying parameter models." Annals of Economic and Social Measurement, Volume 6, number 3. NBER, 1977. 301-314. [https://www.nber.org/system/files/chapters/c10557/c10557.pdf Online Pdf]</ref>

=== Deriving the ''posteriori'' estimate covariance matrix ===
Starting with our invariant on the error covariance '''P'''<sub>''k''&nbsp;|&nbsp;''k''</sub> as above
:<math>\mathbf{P}_{k \mid k} = \operatorname{cov}\left(\mathbf{x}_k - \hat{\mathbf{x}}_{k \mid k}\right)</math>

substitute in the definition of <math>\hat{\mathbf{x}}_{k \mid k}</math>
:<math>\mathbf{P}_{k \mid k} = \operatorname{cov}\left[\mathbf{x}_k - \left(\hat{\mathbf{x}}_{k \mid k - 1} + \mathbf{K}_k\tilde{\mathbf{y}}_k\right)\right]</math>

and substitute  <math>\tilde{\mathbf{y}}_k</math>
:<math>\mathbf{P}_{k \mid k} = \operatorname{cov}\left(\mathbf{x}_k - \left[\hat{\mathbf{x}}_{k \mid k - 1} + \mathbf{K}_k\left(\mathbf{z}_k - \mathbf{H}_k\hat{\mathbf{x}}_{k \mid k - 1}\right)\right]\right)</math>

and <math>\mathbf{z}_k</math>
:<math>\mathbf{P}_{k \mid k} = \operatorname{cov}\left(\mathbf{x}_k - \left[\hat{\mathbf{x}}_{k \mid k - 1} + \mathbf{K}_k\left(\mathbf{H}_k\mathbf{x}_k + \mathbf{v}_k - \mathbf{H}_k\hat{\mathbf{x}}_{k \mid k - 1}\right)\right]\right)</math>

and by collecting the error vectors we get
:<math>\mathbf{P}_{k \mid k} = \operatorname{cov}\left[\left(\mathbf{I} - \mathbf{K}_k \mathbf{H}_k\right)\left(\mathbf{x}_k - \hat{\mathbf{x}}_{k \mid k - 1}\right) - \mathbf{K}_k \mathbf{v}_k\right]</math>

Since the measurement error '''v'''<sub>''k''</sub> is uncorrelated with the other terms, this becomes
:<math>\mathbf{P}_{k \mid k} = \operatorname{cov}\left[\left(\mathbf{I} - \mathbf{K}_k \mathbf{H}_k\right)\left(\mathbf{x}_k - \hat{\mathbf{x}}_{k \mid k - 1}\right)\right] + \operatorname{cov}\left[\mathbf{K}_k \mathbf{v}_k\right]</math>

by the properties of [[covariance matrix|vector covariance]] this becomes
:<math>\mathbf{P}_{k \mid k} = \left(\mathbf{I} - \mathbf{K}_k \mathbf{H}_k\right)\operatorname{cov}\left(\mathbf{x}_k - \hat{\mathbf{x}}_{k \mid k - 1}\right)\left(\mathbf{I} - \mathbf{K}_k \mathbf{H}_k\right)^\textsf{T} + \mathbf{K}_k\operatorname{cov}\left(\mathbf{v}_k\right)\mathbf{K}_k^\textsf{T}</math>

which, using our invariant on '''P'''<sub>''k''&nbsp;|&nbsp;''k''−1</sub> and the definition of '''R'''<sub>''k''</sub> becomes
:<math>\mathbf{P}_{k \mid k} = \left(\mathbf{I} - \mathbf{K}_k \mathbf{H}_k\right) \mathbf{P}_{k \mid k - 1} \left(\mathbf{I} - \mathbf{K}_k \mathbf{H}_k\right)^\textsf{T} + \mathbf{K}_k \mathbf{R}_k \mathbf{K}_k^\textsf{T}</math>

This formula (sometimes known as the '''Joseph form''' of the covariance update equation) is valid for any value of '''K'''<sub>''k''</sub>. It turns out that if '''K'''<sub>''k''</sub> is the optimal Kalman gain, this can be simplified further as shown below.

=== Kalman gain derivation ===
The Kalman filter is a [[minimum mean-square error| minimum mean-square error (MMSE)]] estimator. The error in the ''a posteriori'' state estimation is
:<math>\mathbf{x}_k - \hat{\mathbf{x}}_{k \mid k}</math>

We seek to minimize the expected value of the square of the magnitude of this vector, <math>\operatorname{E}\left[\left\|\mathbf{x}_{k} - \hat{\mathbf{x}}_{k|k}\right\|^2\right]</math>. This is equivalent to minimizing the [[trace (matrix)|trace]] of the ''a posteriori'' estimate [[covariance matrix]] <math> \mathbf{P}_{k|k} </math>. By expanding out the terms in the equation above and collecting, we get:
:<math>\begin{align}
  \mathbf{P}_{k\mid k} & = \mathbf{P}_{k\mid k-1} - \mathbf{K}_k \mathbf{H}_k \mathbf{P}_{k\mid k-1} - \mathbf{P}_{k\mid k-1} \mathbf{H}_k^\textsf{T} \mathbf{K}_k^\textsf{T} + \mathbf{K}_k \left(\mathbf{H}_k \mathbf{P}_{k\mid k-1} \mathbf{H}_k^\textsf{T} + \mathbf{R}_k\right) \mathbf{K}_k^\textsf{T} \\[6pt]
                       &= \mathbf{P}_{k\mid k-1} - \mathbf{K}_k \mathbf{H}_k \mathbf{P}_{k\mid k-1} - \mathbf{P}_{k\mid k-1} \mathbf{H}_k^\textsf{T} \mathbf{K}_k^\textsf{T} + \mathbf{K}_k \mathbf{S}_k\mathbf{K}_k^\textsf{T}
\end{align}</math>

The trace is minimized when its [[matrix calculus|matrix derivative]] with respect to the gain matrix is zero. Using the [[matrix calculus#Identities|gradient matrix rules]] and the symmetry of the matrices involved we find that
:<math>\frac{\partial \; \operatorname{tr}(\mathbf{P}_{k\mid k})}{\partial \;\mathbf{K}_k} = -2 \left(\mathbf{H}_k \mathbf{P}_{k\mid k-1}\right)^\textsf{T} + 2 \mathbf{K}_k \mathbf{S}_k = 0.</math>

Solving this for '''K'''<sub>''k''</sub> yields the Kalman gain:
:<math>\begin{align}
  \mathbf{K}_k \mathbf{S}_k &= \left(\mathbf{H}_k \mathbf{P}_{k\mid k-1}\right)^\textsf{T} = \mathbf{P}_{k\mid k-1} \mathbf{H}_k^\textsf{T} \\
   \Rightarrow \mathbf{K}_k &= \mathbf{P}_{k\mid k-1} \mathbf{H}_k^\textsf{T} \mathbf{S}_k^{-1}
\end{align}</math>

This gain, which is known as the ''optimal Kalman gain'', is the one that yields MMSE estimates when used.

=== Simplification of the ''posteriori'' error covariance formula ===
The formula used to calculate the ''a posteriori'' error covariance can be simplified when the Kalman gain equals the optimal value derived above. Multiplying both sides of our Kalman gain formula on the right by '''S'''<sub>''k''</sub>'''K'''<sub>''k''</sub><sup>T</sup>, it follows that
:<math>\mathbf{K}_k \mathbf{S}_k \mathbf{K}_k^\textsf{T} = \mathbf{P}_{k\mid k-1} \mathbf{H}_k^\textsf{T} \mathbf{K}_k^\textsf{T}</math>

Referring back to our expanded formula for the ''a posteriori'' error covariance,
:<math> \mathbf{P}_{k\mid k} = \mathbf{P}_{k\mid k-1} - \mathbf{K}_k \mathbf{H}_k \mathbf{P}_{k\mid k-1}  - \mathbf{P}_{k\mid k-1} \mathbf{H}_k^\textsf{T} \mathbf{K}_k^\textsf{T} + \mathbf{K}_k \mathbf{S}_k \mathbf{K}_k^\textsf{T}</math>

we find the last two terms cancel out, giving
:<math> \mathbf{P}_{k\mid k} = \mathbf{P}_{k\mid k-1} - \mathbf{K}_k \mathbf{H}_k \mathbf{P}_{k\mid k-1} = (\mathbf{I} - \mathbf{K}_k \mathbf{H}_k) \mathbf{P}_{k\mid k-1}</math>

This formula is computationally cheaper and thus nearly always used in practice, but is only correct for the optimal gain. If arithmetic precision is unusually low causing problems with [[numerical stability]], or if a non-optimal Kalman gain is deliberately used, this simplification cannot be applied; the ''a posteriori'' error covariance formula as derived above (Joseph form) must be used.

== Sensitivity analysis ==
{{More citations needed section|date=December 2010}}

The Kalman filtering equations provide an estimate of the state <math>\hat{\mathbf{x}}_{k\mid k}</math> and its error covariance <math>\mathbf{P}_{k\mid k}</math> recursively. The estimate and its quality depend on the system parameters and the noise statistics fed as inputs to the estimator. This section analyzes the effect of uncertainties in the statistical inputs to the filter.<ref name=anderson>{{cite book|last1= Anderson|first1= Brian D. O.|last2=Moore|first2=John B.|year=1979 |title=Optimal Filtering |publisher=[[Prentice Hall]]|place=New York|pages=129–133|isbn= 978-0-13-638122-8}}</ref> In the absence of reliable statistics or the true values of noise covariance matrices <math>\mathbf{Q}_{k}</math> and <math>\mathbf{R}_k</math>, the expression
:<math>\mathbf{P}_{k\mid k} = \left(\mathbf{I} - \mathbf{K}_k\mathbf{H}_k\right)\mathbf{P}_{k\mid k-1}\left(\mathbf{I} - \mathbf{K}_k\mathbf{H}_k\right)^\textsf{T} + \mathbf{K}_k\mathbf{R}_k\mathbf{K}_k^\textsf{T}</math>

no longer provides the actual error covariance. In other words, <math>\mathbf{P}_{k \mid k} \neq E\left[\left(\mathbf{x}_k - \hat{\mathbf{x}}_{k\mid k}\right)\left(\mathbf{x}_k - \hat{\mathbf{x}}_{k \mid k}\right)^\textsf{T}\right]</math>.  In most real-time applications, the covariance matrices that are used in designing the Kalman filter are different from the actual (true) noise covariances matrices.{{citation needed|date=December 2010}} This sensitivity analysis describes the behavior of the estimation error covariance when the noise covariances as well as the system matrices <math>\mathbf{F}_k</math> and <math>\mathbf{H}_k</math> that are fed as inputs to the filter are incorrect. Thus, the sensitivity analysis describes the robustness (or sensitivity) of the estimator to misspecified statistical and parametric inputs to the estimator.

This discussion is limited to the error sensitivity analysis for the case of statistical uncertainties. Here the actual noise covariances are denoted by <math>\mathbf{Q}^a_k</math> and <math>\mathbf{R}^a_k</math> respectively, whereas the design values used in the estimator are <math>\mathbf{Q}_k</math> and <math>\mathbf{R}_k</math> respectively. The actual error covariance is denoted by <math>\mathbf{P}_{k \mid k}^a</math> and <math>\mathbf{P}_{k \mid k}</math> as computed by the Kalman filter is referred to as the Riccati variable. When <math>\mathbf{Q}_k \equiv \mathbf{Q}^a_k</math> and <math>\mathbf{R}_k \equiv \mathbf{R}^a_k</math>, this means that <math>\mathbf{P}_{k \mid k} = \mathbf{P}_{k \mid k}^a</math>. While computing the actual error covariance using <math>\mathbf{P}_{k \mid k}^a = E\left[\left(\mathbf{x}_k - \hat{\mathbf{x}}_{k \mid k}\right)\left(\mathbf{x}_k - \hat{\mathbf{x}}_{k \mid k}\right)^\textsf{T}\right] </math>, substituting for <math>\widehat{\mathbf{x}}_{k \mid k}</math> and using the fact that <math>E\left[\mathbf{w}_k\mathbf{w}_k^\textsf{T}\right] = \mathbf{Q}_k^a</math> and <math>E\left[\mathbf{v}_k \mathbf{v}_k^\textsf{T}\right] = \mathbf{R}_k^a</math>, results in the following recursive equations for <math>\mathbf{P}_{k \mid k}^a</math> :
:<math>\mathbf{P}_{k \mid k-1}^a = \mathbf{F}_k\mathbf{P}_{k-1 \mid k-1}^a \mathbf{F}_k^\textsf{T} + \mathbf{Q}_k^a </math>

and
:<math>\mathbf{P}_{k \mid k}^a = \left(\mathbf{I} - \mathbf{K}_k \mathbf{H}_k\right)\mathbf{P}_{k \mid k-1}^a \left(\mathbf{I} - \mathbf{K}_k \mathbf{H}_k\right)^\textsf{T} + \mathbf{K}_k \mathbf{R}_k^a \mathbf{K}_k^\textsf{T}</math>

While computing <math>\mathbf{P}_{k \mid k}</math>, by design the filter implicitly assumes that <math>E\left[\mathbf{w}_k \mathbf{w}_k^\textsf{T}\right] = \mathbf{Q}_k</math> and <math>E\left[\mathbf{v}_k \mathbf{v}_k^\textsf{T}\right] = \mathbf{R}_k</math>. The recursive expressions for <math>\mathbf{P}_{k \mid k}^a</math> and <math>\mathbf{P}_{k \mid k}</math> are identical except for the presence of <math>\mathbf{Q}_k^a</math> and <math>\mathbf{R}_k^a</math> in place of the design values <math>\mathbf{Q}_k</math> and <math>\mathbf{R}_k</math> respectively. Researches have been done to analyze Kalman filter system's robustness.<ref>Jingyang Lu.  [https://web.archive.org/web/20200224230739/https://ieeexplore.ieee.org/document/6916211, "False information injection attack on dynamic state estimation in multi-sensor systems"], Fusion 2014</ref>

== Factored form ==
One problem with the Kalman filter is its [[numerical stability]].  If the process noise covariance '''Q'''<sub>''k''</sub> is small, round-off error often causes a small positive eigenvalue of the state covariance matrix '''P''' to be computed as a negative number.  This renders the numerical representation of '''P''' [[Positive-semidefinite matrix|indefinite]], while its true form is [[Positive-definite matrix|positive-definite]].

Positive definite matrices have the property that they have a factorization into the product of a [[Non-singular matrix|non-singular]], [[lower-triangular matrix]] '''S'''  and its [[Matrix transpose|transpose]] :  '''P'''&nbsp;=&nbsp;'''S'''·'''S'''<sup>T</sup> . The factor '''S''' can be computed efficiently using the [[Cholesky factorization]] algorithm. This product form of the covariance matrix '''P''' is guaranteed to be symmetric, and for all 1 <= k <= n, the k-th diagonal element  '''P'''<sub>kk</sub> is equal to the square of the [[euclidean norm]] of the k-th row of '''S''', which is necessarily positive. An equivalent form, which avoids many of the [[square root]] operations involved in the [[Cholesky factorization]] algorithm, yet preserves the desirable numerical properties, is the U-D decomposition form, '''P'''&nbsp;=&nbsp;'''U'''·'''D'''·'''U'''<sup>T</sup>, where '''U''' is a [[unit triangular matrix]] (with unit diagonal), and '''D''' is a diagonal matrix.

Between the two, the U-D factorization uses the same amount of storage, and somewhat less computation, and is the most commonly used triangular factorization.  (Early literature on the relative efficiency is somewhat misleading, as it assumed that square roots were much more time-consuming than divisions,<ref name=thornton />{{rp|69}} while on 21st-century computers they are only slightly more expensive.)

Efficient algorithms for the Kalman prediction and update steps in the factored form were developed by G. J. Bierman and C. L. Thornton.<ref name=thornton>{{cite thesis|title=Triangular Covariance Factorizations for Kalman Filtering |url=https://ntrs.nasa.gov/citations/19770005172 |type=PhD |first=Catherine L. |last=Thornton |date=15 October 1976|id=NASA Technical Memorandum 33-798 |publisher=[[NASA]] }}</ref><ref name="bierman" />

The [[LDL decomposition|'''L'''·'''D'''·'''L'''<sup>T</sup> decomposition]] of the innovation covariance matrix '''S'''<sub>k</sub> is the basis for another type of numerically efficient and robust square root filter.<ref name=barshalom>{{cite book|last1=Bar-Shalom|first1= Yaakov|author-link1=Yaakov Bar-Shalom|last2=Li|first2=X. Rong|last3=Kirubarajan|first3=Thiagalingam |date=July 2001 |title=Estimation with Applications to Tracking and Navigation |publisher=[[John Wiley & Sons]]|place=New York|isbn =978-0-471-41655-5  |pages=308–317}}</ref> The algorithm starts with the LU decomposition as implemented in the Linear Algebra PACKage ([[LAPACK]]). These results are further factored into the '''L'''·'''D'''·'''L'''<sup>T</sup> structure with methods given by Golub and Van Loan (algorithm 4.1.2) for a symmetric nonsingular matrix.<ref name=golub>{{cite book|last1=Golub |first1=Gene H. |first2=Charles F. |last2=Van Loan |year=1996|title=Matrix Computations|publisher=[[Johns Hopkins University]]|edition=Third|page=139|isbn =978-0-8018-5414-9 |place =Baltimore, Maryland|series=Johns Hopkins Studies in the Mathematical Sciences}}</ref> Any singular covariance matrix is [[Pivot element|pivoted]] so that the first diagonal partition is [[Invertible matrix|nonsingular]] and [[Condition number|well-conditioned]]. The pivoting algorithm must retain any portion of the innovation covariance matrix directly corresponding to observed state-variables '''H'''<sub>k</sub>·'''x'''<sub>k|k-1</sub> that are associated with auxiliary observations in
'''y'''<sub>k</sub>. The '''l'''·'''d'''·'''l'''<sup>t</sup> square-root filter requires [[orthogonalization]] of the observation vector.<ref name="bierman" /><ref name="barshalom" /> This may be done with the inverse square-root of the covariance matrix for the auxiliary variables using Method 2 in Higham (2002, p.&nbsp;263).<ref name=higham>{{cite book|first=Nicholas J. |last=Higham|year=2002|title=Accuracy and Stability of Numerical Algorithms|edition=Second|isbn =978-0-89871-521-7|page=680|publisher=[[Society for Industrial and Applied Mathematics]]|place =Philadelphia, PA}}</ref>

== Parallel form ==
The Kalman filter is efficient for sequential data processing on [[central processing units]] (CPUs), but in its original form it is inefficient on parallel architectures such as [[graphics processing units]] (GPUs). It is however possible to express the filter-update routine in terms of an associative operator using the formulation in Särkkä and García-Fernández (2021).<ref name=parallel>{{cite journal | author = Särkkä, S.| author2 = Ángel F. García-Fernández | year = 2021 | title = Temporal Parallelization of Bayesian Smoothers | journal = IEEE Transactions on Automatic Control | volume = 66 | issue = 1 | pages = 299–306 | doi = 10.1109/TAC.2020.2976316 | arxiv = 1905.13002 | bibcode = 2021ITAC...66..299S | s2cid = 213695560 }}</ref> The filter solution can then be retrieved by the use of a [[prefix sum]] algorithm which can be efficiently implemented on GPU.<ref name=prefix-nvida>{{cite web |url=https://developer.nvidia.com/gpugems/gpugems3/part-vi-gpu-computing/chapter-39-parallel-prefix-sum-scan-cuda |title=Parallel Prefix Sum (Scan) with CUDA |author=<!--Not stated--> |website=developer.nvidia.com/ |access-date=2020-02-21 |quote=The scan operation is a simple and powerful parallel primitive with a broad range of applications. In this chapter we have explained an efficient implementation of scan using CUDA, which achieves a significant speedup compared to a sequential implementation on a fast CPU, and compared to a parallel implementation in OpenGL on the same GPU. Due to the increasing power of commodity parallel processors such as GPUs, we expect to see data-parallel algorithms such as scan to increase in importance over the coming years.}}</ref> This reduces the [[computational complexity]] from <math>O(N)</math> in the number of time steps to <math>O(\log(N))</math>.

== Relationship to recursive Bayesian estimation ==
The Kalman filter can be presented as one of the simplest [[dynamic Bayesian network]]s. The Kalman filter calculates estimates of the true values of states recursively over time using incoming measurements and a mathematical process model. Similarly, [[recursive Bayesian estimation]] calculates [[density estimation|estimates]] of an unknown [[probability density function]] (PDF) recursively over time using incoming measurements and a mathematical process model.<ref>{{cite journal|first1=C. Johan |last1=Masreliez|first2= R D|last2= Martin |year=1977|title=Robust Bayesian estimation for the linear model and robustifying the Kalman filter|journal= IEEE Transactions on Automatic Control|volume= 22|issue= 3|pages= 361–371|doi=10.1109/TAC.1977.1101538|author-link1= C. Johan Masreliez}}</ref>

In recursive Bayesian estimation, the true state is assumed to be an unobserved [[Markov process]], and the measurements are the observed states of a hidden Markov model (HMM).
[[File:HMM Kalman Filter Derivation.svg|class=skin-invert-image|center|hidden markov model]]

Because of the Markov assumption, the true state is conditionally independent of all earlier states given the immediately previous state.
:<math>p(\mathbf{x}_k\mid \mathbf{x}_0,\dots,\mathbf{x}_{k-1}) = p(\mathbf{x}_k\mid \mathbf{x}_{k-1})</math>

Similarly, the measurement at the ''k''-th timestep is dependent only upon the current state and is conditionally independent of all other states given the current state.
:<math>p(\mathbf{z}_k\mid\mathbf{x}_0,\dots,\mathbf{x}_k) = p(\mathbf{z}_k\mid \mathbf{x}_k )</math>

Using these assumptions the probability distribution over all states of the hidden Markov model can be written simply as:
:<math>p\left(\mathbf{x}_0, \dots, \mathbf{x}_k, \mathbf{z}_1, \dots, \mathbf{z}_k\right) = p\left(\mathbf{x}_0\right)\prod_{i=1}^k p\left(\mathbf{z}_i \mid \mathbf{x}_i\right)p\left(\mathbf{x}_i \mid \mathbf{x}_{i-1}\right)</math>

However, when a Kalman filter is used to estimate the state '''x''', the probability distribution of interest is that associated with the current states conditioned on the measurements up to the current timestep. This is achieved by marginalizing out the previous states and dividing by the probability of the measurement set.

This results in the ''predict'' and ''update'' phases of the Kalman filter written probabilistically. The probability distribution associated with the predicted state is the sum (integral) of the products of the probability distribution associated with the transition from the (''k''&nbsp;−&nbsp;1)-th timestep to the ''k''-th and the probability distribution associated with the previous state, over all possible <math>x_{k-1}</math>.

:<math>p\left(\mathbf{x}_k \mid \mathbf{Z}_{k-1}\right) = \int p\left(\mathbf{x}_k \mid \mathbf{x}_{k-1}\right) p\left(\mathbf{x}_{k-1} \mid \mathbf{Z}_{k-1}\right)\, d\mathbf{x}_{k-1}</math>

The measurement set up to time ''t'' is
:<math>\mathbf{Z}_t = \left\{\mathbf{z}_1, \dots, \mathbf{z}_t\right\}</math>

The probability distribution of the update is proportional to the product of the measurement likelihood and the predicted state.
:<math>p\left(\mathbf{x}_k \mid \mathbf{Z}_k\right) = \frac{p\left(\mathbf{z}_k \mid \mathbf{x}_k\right) p\left(\mathbf{x}_k \mid \mathbf{Z}_{k-1}\right)}{p\left(\mathbf{z}_k \mid \mathbf{Z}_{k-1}\right)}</math>

The denominator
:<math>p\left(\mathbf{z}_k \mid \mathbf{Z}_{k-1}\right) = \int p\left(\mathbf{z}_k \mid \mathbf{x}_k\right) p\left(\mathbf{x}_k \mid \mathbf{Z}_{k-1}\right)\, d\mathbf{x}_k</math>

is a normalization term.

The remaining probability density functions are
:<math>\begin{align}
      p\left(\mathbf{x}_k \mid \mathbf{x}_{k-1}\right) &= \mathcal{N}\left(\mathbf{F}_k\mathbf{x}_{k-1}, \mathbf{Q}_k\right) \\
          p\left(\mathbf{z}_k \mid \mathbf{x}_k\right) &= \mathcal{N}\left(\mathbf{H}_k\mathbf{x}_k, \mathbf{R}_k\right) \\
  p\left(\mathbf{x}_{k-1} \mid \mathbf{Z}_{k-1}\right) &= \mathcal{N}\left(\hat{\mathbf{x}}_{k-1}, \mathbf{P}_{k-1}\right)
\end{align}</math>

The PDF at the previous timestep is assumed inductively to be the estimated state and covariance. This is justified because, as an optimal estimator, the Kalman filter makes best use of the measurements, therefore the PDF for <math>\mathbf{x}_k</math>  given the measurements <math>\mathbf{Z}_k</math> is the Kalman filter estimate.

== Marginal likelihood ==
Related to the recursive Bayesian interpretation described above, the Kalman filter can be viewed as a [[generative model]], i.e., a process for ''generating'' a stream of random observations '''z''' = ('''z'''<sub>0</sub>, '''z'''<sub>1</sub>, '''z'''<sub>2</sub>, ...). Specifically, the process is

# Sample a hidden state <math>\mathbf{x}_0</math> from the Gaussian prior distribution <math>p\left(\mathbf{x}_0\right) = \mathcal{N}\left(\hat{\mathbf{x}}_{0 \mid 0}, \mathbf{P}_{0 \mid 0}\right)</math>.
# Sample an observation <math>\mathbf{z}_0</math> from the observation model <math>p\left(\mathbf{z}_0 \mid \mathbf{x}_0\right) = \mathcal{N}\left(\mathbf{H}_0\mathbf{x}_0, \mathbf{R}_0\right)</math>.
# For <math>k = 1, 2, 3, \ldots</math>, do
## Sample the next hidden state <math>\mathbf{x}_k</math> from the transition model <math>p\left(\mathbf{x}_k \mid \mathbf{x}_{k-1}\right) = \mathcal{N}\left(\mathbf{F}_k \mathbf{x}_{k-1} + \mathbf{B}_k\mathbf{u}_k, \mathbf{Q}_k\right).</math>
## Sample an observation <math>\mathbf{z}_k</math> from the observation model <math>p\left(\mathbf{z}_k \mid \mathbf{x}_k\right) = \mathcal{N}\left(\mathbf{H}_k\mathbf{x}_k, \mathbf{R}_k\right).</math>

This process has identical structure to the [[hidden Markov model]], except that the discrete state and observations are replaced with continuous variables sampled from Gaussian distributions.

In some applications, it is useful to compute the ''probability'' that a Kalman filter with a given set of parameters (prior distribution, transition and observation models, and control inputs) would generate a particular observed signal. This probability is known as the [[marginal likelihood]] because it integrates over ("marginalizes out") the values of the hidden state variables, so it can be computed using only the observed signal. The marginal likelihood can be useful to evaluate different parameter choices, or to compare the Kalman filter against other models using [[Bayesian model comparison]].

It is straightforward to compute the marginal likelihood as a side effect of the recursive filtering computation. By the [[Chain rule (probability)|chain rule]], the likelihood can be factored as the product of the probability of each observation given previous observations,
:<math>p(\mathbf{z}) = \prod_{k=0}^T p\left(\mathbf{z}_k \mid \mathbf{z}_{k-1}, \ldots, \mathbf{z}_0\right)</math>,

and because the Kalman filter describes a Markov process, all relevant information from previous observations is contained in the current state estimate <math>\hat{\mathbf{x}}_{k \mid k-1}, \mathbf{P}_{k \mid k-1}.</math> Thus the marginal likelihood is given by
:<math>\begin{align}
  p(\mathbf{z}) &= \prod_{k=0}^T \int p\left(\mathbf{z}_k \mid \mathbf{x}_k\right) p\left(\mathbf{x}_k \mid \mathbf{z}_{k-1}, \ldots,\mathbf{z}_0\right) d\mathbf{x}_k\\
                &= \prod_{k=0}^T \int \mathcal{N}\left(\mathbf{z}_k; \mathbf{H}_k\mathbf{x}_k, \mathbf{R}_k\right) \mathcal{N}\left(\mathbf{x}_k; \hat{\mathbf{x}}_{k \mid k-1}, \mathbf{P}_{k \mid k-1}\right) d\mathbf{x}_k\\
                &= \prod_{k=0}^T \mathcal{N}\left(\mathbf{z}_k;  \mathbf{H}_k\hat{\mathbf{x}}_{k \mid k-1}, \mathbf{R}_k + \mathbf{H}_k \mathbf{P}_{k \mid k-1} \mathbf{H}_k^\textsf{T}\right)\\
                &= \prod_{k=0}^T \mathcal{N}\left(\mathbf{z}_k;  \mathbf{H}_k\hat{\mathbf{x}}_{k \mid k-1}, \mathbf{S}_k\right),
\end{align}</math>

i.e., a product of Gaussian densities, each corresponding to the density of one observation '''z'''<sub>''k''</sub> under the current filtering distribution <math>\mathbf{H}_k\hat{\mathbf{x}}_{k \mid k-1}, \mathbf{S}_k</math>. This can easily be computed as a simple recursive update; however, to avoid [[Arithmetic underflow|numeric underflow]], in a practical implementation it is usually desirable to compute the ''log'' marginal likelihood <math>\ell = \log p(\mathbf{z})</math> instead. Adopting the convention <math>\ell^{(-1)} = 0</math>, this can be done via the recursive update rule
:<math>\ell^{(k)} = \ell^{(k-1)} - \frac{1}{2} \left(\tilde{\mathbf{y}}_k^\textsf{T} \mathbf{S}^{-1}_k \tilde{\mathbf{y}}_k + \log \left|\mathbf{S}_k\right| + d_y\log 2\pi \right),</math>
where <math>d_y</math> is the dimension of the measurement vector.<ref>{{Cite book|last=Lütkepohl|first= Helmut|title=Introduction to Multiple Time Series Analysis|publisher= Springer-Verlag Berlin |location=Heidelberg|year= 1991|page=435}}</ref>

An important application where such a (log) likelihood of the observations (given the filter parameters) is used is multi-target tracking. For example, consider an object tracking scenario where a stream of observations is the input, however, it is unknown how many objects are in the scene (or, the number of objects is known but is greater than one). For such a scenario, it can be unknown apriori which observations/measurements were generated by which object. A multiple hypothesis tracker (MHT) typically will form different track association hypotheses, where each hypothesis can be considered as a Kalman filter (for the linear Gaussian case) with a specific set of parameters associated with the hypothesized object. Thus, it is important to compute the likelihood of the observations for the different hypotheses under consideration, such that the most-likely one can be found.

== Information filter ==
{{More citations needed section|date=April 2016}}

In cases where the dimension of the observation vector '''y''' is bigger than the dimension of the state space vector '''x''', the information filter can avoid the inversion of a bigger matrix in the Kalman gain calculation at the price of inverting a smaller matrix in the prediction step, thus saving computing time. Additionally, the information filter allows for system information initialization according to <math>{I_{1|0} = P^{-1}_{1|0} = 0}</math>, which would not be possible for the regular Kalman filter.<ref>{{Cite book |last=Gustafsson |first=Fredrik |title=Statistical sensor fusion |date=2018 |publisher=Studentlitteratur |isbn=978-91-44-12724-8 |edition=Third |location=Lund |pages=160–162}}</ref> In the information filter, or inverse covariance filter, the estimated covariance and estimated state are replaced by the [[Fisher information matrix|information matrix]] and [[Fisher information|information]] vector respectively. These are defined as:
:<math>\begin{align}
        \mathbf{Y}_{k \mid k} &= \mathbf{P}_{k \mid k}^{-1} \\
  \hat{\mathbf{y}}_{k \mid k} &= \mathbf{P}_{k \mid k}^{-1}\hat{\mathbf{x}}_{k \mid k}
\end{align}</math>

Similarly the predicted covariance and state have equivalent information forms, defined as:
:<math>\begin{align}
        \mathbf{Y}_{k \mid k-1} &= \mathbf{P}_{k \mid k-1}^{-1} \\
  \hat{\mathbf{y}}_{k \mid k-1} &= \mathbf{P}_{k \mid k-1}^{-1}\hat{\mathbf{x}}_{k \mid k-1}
\end{align}</math>

and the measurement covariance and measurement vector, which are defined as:
:<math>\begin{align}
  \mathbf{I}_k &= \mathbf{H}_k^\textsf{T} \mathbf{R}_k^{-1} \mathbf{H}_k \\
  \mathbf{i}_k &= \mathbf{H}_k^\textsf{T} \mathbf{R}_k^{-1} \mathbf{z}_k
\end{align}</math>

The information update now becomes a trivial sum.<ref name=terejanu>{{cite web |title=Discrete Kalman Filter Tutorial |author=Gabriel T. Terejanu |date=2012-08-04 |access-date=2016-04-13 |url=https://cse.sc.edu/~terejanu/files/tutorialKF.pdf |archive-date=2020-08-17 |archive-url=https://web.archive.org/web/20200817230005/https://cse.sc.edu/~terejanu/files/tutorialKF.pdf |url-status=dead }}</ref>
:<math>\begin{align}
        \mathbf{Y}_{k \mid k} &= \mathbf{Y}_{k \mid k-1} + \mathbf{I}_k \\
  \hat{\mathbf{y}}_{k \mid k} &= \hat{\mathbf{y}}_{k \mid k-1} + \mathbf{i}_k
\end{align}</math>

The main advantage of the information filter is that ''N'' measurements can be filtered at each time step simply by summing their information matrices and vectors.
:<math>\begin{align}
        \mathbf{Y}_{k \mid k} &= \mathbf{Y}_{k \mid k-1} + \sum_{j=1}^N \mathbf{I}_{k,j} \\
  \hat{\mathbf{y}}_{k \mid k} &= \hat{\mathbf{y}}_{k \mid k-1} + \sum_{j=1}^N \mathbf{i}_{k,j}
\end{align}</math>

To predict the information filter the information matrix and vector can be converted back to their state space equivalents, or alternatively the information space prediction can be used.<ref name=terejanu />
:<math>\begin{align}
                 \mathbf{M}_k &=
    \left[\mathbf{F}_k^{-1}\right]^\textsf{T} \mathbf{Y}_{k-1 \mid k-1} \mathbf{F}_k^{-1} \\
                 \mathbf{C}_k &=
    \mathbf{M}_k \left[\mathbf{M}_k + \mathbf{Q}_k^{-1}\right]^{-1} \\
                   \mathbf{L}_k &=
    \mathbf{I} - \mathbf{C}_k \\
        \mathbf{Y}_{k \mid k-1} &=
    \mathbf{L}_k \mathbf{M}_k +
    \mathbf{C}_k \mathbf{Q}_k^{-1} \mathbf{C}_k^\textsf{T} \\
  \hat{\mathbf{y}}_{k \mid k-1} &=
    \mathbf{L}_k \left[\mathbf{F}_k^{-1}\right]^\textsf{T} \hat{\mathbf{y}}_{k-1 \mid k-1}
\end{align}</math>

== Fixed-lag smoother ==
{{More citations needed section|date=December 2010}}

The optimal fixed-lag smoother provides the optimal estimate of <math>\hat{\mathbf{x}}_{k-N \mid  k}</math> for a given fixed-lag <math>N</math> using the measurements from <math>\mathbf{z}_1</math> to <math>\mathbf{z}_k</math>.<ref>{{cite book|last1=Anderson|first1=Brian D. O.|last2=Moore|first2=John B.|title=Optimal Filtering|date=1979|publisher=Prentice Hall, Inc.|location=Englewood Cliffs, NJ|isbn=978-0-13-638122-8|pages=176–190}}</ref> It can be derived using the previous theory via an augmented state, and the main equation of the filter is the following:
:<math>
 \begin{bmatrix}
  \hat{\mathbf{x}}_{t \mid t} \\
  \hat{\mathbf{x}}_{t-1 \mid t} \\
  \vdots \\
  \hat{\mathbf{x}}_{t-N+1 \mid t} \\
 \end{bmatrix}
 =
 \begin{bmatrix}
  \mathbf{I} \\
  0 \\
  \vdots \\
  0 \\
 \end{bmatrix}
 \hat{\mathbf{x}}_{t \mid t-1}
 +
 \begin{bmatrix}
  0          & \ldots & 0 \\
  \mathbf{I} & 0      & \vdots \\
  \vdots     & \ddots & \vdots \\
  0          & \ldots & \mathbf{I} \\
 \end{bmatrix}
 \begin{bmatrix}
  \hat{\mathbf{x}}_{t-1 \mid t-1} \\
  \hat{\mathbf{x}}_{t-2 \mid t-1} \\
  \vdots \\
  \hat{\mathbf{x}}_{t-N+1 \mid t-1} \\
 \end{bmatrix}
 +
 \begin{bmatrix}
  \mathbf{K}^{(0)} \\
  \mathbf{K}^{(1)} \\
  \vdots \\
  \mathbf{K}^{(N-1)} \\
 \end{bmatrix}
 \mathbf{y}_{t \mid t-1}
</math>

where:
* <math> \hat{\mathbf{x}}_{t \mid t-1} </math> is estimated via a standard Kalman filter;
* <math> \mathbf{y}_{t \mid t-1} = \mathbf{z}_t - \mathbf{H}\hat{\mathbf{x}}_{t \mid t-1} </math> is the innovation produced considering the estimate of the standard Kalman filter;
* the various <math> \hat{\mathbf{x}}_{t-i \mid t} </math> with <math> i = 1, \ldots, N-1 </math> are new variables; i.e., they do not appear in the standard Kalman filter;
* the gains are computed via the following scheme:
*:<math>
  \mathbf{K}^{(i+1)} =
  \mathbf{P}^{(i)} \mathbf{H}^\textsf{T}
  \left[
    \mathbf{H} \mathbf{P} \mathbf{H}^\textsf{T} + \mathbf{R}
  \right]^{-1}
</math>
:and
::<math>
    \mathbf{P}^{(i)} =
    \mathbf{P}
    \left[
      \left(
        \mathbf{F} - \mathbf{K} \mathbf{H}
      \right)^\textsf{T}
    \right]^i
  </math>
:where <math> \mathbf{P} </math> and <math> \mathbf{K} </math> are the prediction error covariance and the gains of the standard Kalman filter (i.e., <math> \mathbf{P}_{t \mid t-1} </math>).

If the estimation error covariance is defined so that
:<math>
\mathbf{P}_i :=
    E \left[
      \left(
        \mathbf{x}_{t-i} - \hat{\mathbf{x}}_{t-i \mid t}
      \right)^{*}
      \left(
        \mathbf{x}_{t-i} - \hat{\mathbf{x}}_{t-i \mid t}
       \right)
       \mid
       z_1 \ldots z_t
    \right],
</math>

then we have that the improvement on the estimation of <math> \mathbf{x}_{t-i} </math> is given by:
:<math>
  \mathbf{P} - \mathbf{P}_i =
  \sum_{j = 0}^i
  \left[
    \mathbf{P}^{(j)} \mathbf{H}^\textsf{T}
    \left(
      \mathbf{H} \mathbf{P} \mathbf{H}^\textsf{T} + \mathbf{R}
    \right)^{-1}
    \mathbf{H} \left( \mathbf{P}^{(i)} \right)^\textsf{T}
  \right]
</math>

== Fixed-interval smoothers ==
The optimal fixed-interval smoother provides the optimal estimate of <math>\hat{\mathbf{x}}_{k \mid  n}</math> (<math>k < n</math>) using the measurements from a fixed interval <math>\mathbf{z}_1</math> to <math>\mathbf{z}_n</math>. This is also called "Kalman Smoothing". There are several smoothing algorithms in common use.

=== Rauch–Tung–Striebel ===
The Rauch–Tung–Striebel (RTS) smoother is an efficient two-pass algorithm for fixed interval smoothing.<ref>{{cite journal | last1 = Rauch | first1 =  H.E. | last2 = Tung | first2 = F. | last3 = Striebel | first3 = C. T. | title = Maximum likelihood estimates of linear dynamic systems | journal = AIAA Journal| volume = 3 | issue = 8 | pages = 1445–1450 | date=August 1965 | doi = 10.2514/3.3166 | bibcode = 1965AIAAJ...3.1445R }}</ref>

The forward pass is the same as the regular Kalman filter algorithm. These ''filtered'' a-priori and a-posteriori state estimates <math>\hat{\mathbf{x}}_{k \mid k-1}</math>, <math>\hat{\mathbf{x}}_{k \mid k}</math> and covariances <math>\mathbf{P}_{k \mid k-1}</math>, <math>\mathbf{P}_{k \mid k}</math> are saved for use in the backward pass (for [[retrodiction]]).

In the backward pass, we compute the ''smoothed'' state estimates <math>\hat{\mathbf{x}}_{k \mid n}</math> and covariances <math>\mathbf{P}_{k \mid n}</math>. We start at the last time step and proceed backward in time using the following recursive equations:
:<math>\begin{align}
  \hat{\mathbf{x}}_{k \mid n} &= \hat{\mathbf{x}}_{k \mid k} + \mathbf{C}_k \left(\hat{\mathbf{x}}_{k+1 \mid n} - \hat{\mathbf{x}}_{k+1 \mid k}\right) \\
        \mathbf{P}_{k \mid n} &= \mathbf{P}_{k \mid k} + \mathbf{C}_k \left(\mathbf{P}_{k+1 \mid n} - \mathbf{P}_{k+1 \mid k}\right) \mathbf{C}_k^\textsf{T}
\end{align}</math>

where
:<math>\mathbf{C}_k = \mathbf{P}_{k \mid k} \mathbf{F}_{k+1}^\textsf{T} \mathbf{P}_{k+1 \mid k}^{-1}.</math>

<math> \mathbf{x}_{k \mid k}</math> is the a-posteriori state estimate of timestep <math>k</math> and <math>\mathbf{x}_{k+1 \mid k}</math> is the a-priori state estimate of timestep <math>k + 1</math>. The same notation applies to the covariance.

=== Modified Bryson–Frazier smoother ===
An alternative to the RTS algorithm is the modified Bryson–Frazier (MBF) fixed interval smoother developed by Bierman.<ref name=bierman>{{cite journal | last = Bierman | first =  G.J. | title = Factorization Methods for Discrete Sequential Estimation | year = 1977 | bibcode = 1977fmds.book.....B | journal = Factorization Methods for Discrete Sequential Estimation }}</ref> This also uses a backward pass that processes data saved from the Kalman filter forward pass. The equations for the backward pass involve the recursive
computation of data which are used at each observation time to compute the smoothed state and covariance.

The recursive equations are
:<math>\begin{align}
    \tilde{\Lambda}_k &= \mathbf{H}_k^\textsf{T} \mathbf{S}_k^{-1} \mathbf{H}_k + \hat{\mathbf{C}}_k^\textsf{T} \hat{\Lambda}_k \hat{\mathbf{C}}_k \\
  \hat{\Lambda}_{k-1} &= \mathbf{F}_k^\textsf{T}\tilde{\Lambda}_k\mathbf{F}_k \\
      \hat{\Lambda}_n &= 0 \\
    \tilde{\lambda}_k &=  -\mathbf{H}_k^\textsf{T} \mathbf{S}_k^{-1} \mathbf{y}_k + \hat{\mathbf{C}}_k^\textsf{T} \hat{\lambda}_k \\
  \hat{\lambda}_{k-1} &= \mathbf{F}_k^\textsf{T}\tilde{\lambda}_k \\
      \hat{\lambda}_n &= 0
\end{align}</math>

where <math>\mathbf{S}_k</math> is the residual covariance and <math>\hat{\mathbf{C}}_k = \mathbf{I} - \mathbf{K}_k \mathbf{H}_k</math>. The smoothed state and covariance can then be found by substitution in the equations
:<math>\begin{align}
  \mathbf{P}_{k \mid n} &= \mathbf{P}_{k \mid k} - \mathbf{P}_{k \mid k}\hat{\Lambda}_k\mathbf{P}_{k \mid k} \\
  \mathbf{x}_{k \mid n} &= \mathbf{x}_{k \mid k} - \mathbf{P}_{k \mid k}\hat{\lambda}_k
\end{align}</math>

or
:<math>\begin{align}
  \mathbf{P}_{k \mid n} &= \mathbf{P}_{k \mid k-1} - \mathbf{P}_{k \mid k-1}\tilde{\Lambda}_k\mathbf{P}_{k \mid k-1} \\
  \mathbf{x}_{k \mid n} &= \mathbf{x}_{k \mid k-1} - \mathbf{P}_{k \mid k-1}\tilde{\lambda}_k.
\end{align}</math>

An important advantage of the MBF is that it does not require finding the inverse of the covariance matrix.  Bierman's derivation is based on the RTS smoother, which assumes that the underlying distributions are Gaussian. However, a derivation of the MBF based on the concept of the fixed point smoother, which does not require the Gaussian assumption, is given by Gibbs.<ref name=gibbs1>{{cite journal |last1=Gibbs |first1=Richard G. |date=February 2011 |title=Square Root Modified Bryson–Frazier Smoother |journal=IEEE Transactions on Automatic Control |volume=56 |issue=2 |pages=452–456 |doi=10.1109/TAC.2010.2089753 |bibcode=2011ITAC...56..452G }}</ref>

The MBF can also be used to perform consistency checks on the filter residuals and the difference between the value of a filter state after an update and the smoothed value of the state, that is <math>\mathbf{x}_{k \mid k} - \mathbf{x}_{k \mid n}</math>.<ref name=gibbs2>{{cite journal |last1=Gibbs |first1=Richard G. |date=2013 |title=New Kalman filter and smoother consistency tests |url=https://www.sciencedirect.com/science/article/pii/S0005109813003610 |journal=Automatica |volume=49 |issue=10 |pages=3141–3144 |doi=10.1016/j.automatica.2013.07.013 }}</ref>

=== Minimum-variance smoother ===
The minimum-variance smoother can attain the best-possible error performance, provided that the models are linear, their parameters and the noise statistics are known precisely.<ref>{{cite journal | last = Einicke | first =  G.A. | title = Optimal and Robust Noncausal Filter Formulations | journal =  IEEE Transactions on Signal Processing| volume = 54 | issue = 3 | pages = 1069–1077 | date=March 2006 | bibcode = 2006ITSP...54.1069E | doi = 10.1109/TSP.2005.863042 | s2cid = 15376718 }}</ref> This smoother is a time-varying state-space generalization of the optimal non-causal [[Wiener filter]].

The smoother calculations are done in two passes. The forward calculations involve a one-step-ahead predictor and are given by
:<math>\begin{align}
  \hat{\mathbf{x}}_{k+1 \mid k} &= (\mathbf{F}_k - \mathbf{K}_k\mathbf{H}_k)\hat{\mathbf{x}}_{k \mid k-1} + \mathbf{K}_k\mathbf{z}_k \\
                       \alpha_k &= -\mathbf{S}_k^{-\frac{1}{2}}\mathbf{H}_k\hat{\mathbf{x}}_{k \mid k-1} + \mathbf{S}_k^{-\frac{1}{2}}\mathbf{z}_k
\end{align}</math>

The above system is known as the inverse Wiener-Hopf factor. The backward recursion is the adjoint of the above forward system. The result of the backward pass <math>\beta_k</math>  may be calculated by operating the forward equations on the time-reversed <math>\alpha_k</math>  and time reversing the result. In the case of output estimation, the smoothed estimate is given by
:<math>\hat{\mathbf{y}}_{k \mid N} = \mathbf{z}_k - \mathbf{R}_k\beta_k</math>

Taking the causal part of this minimum-variance smoother yields
:<math>\hat{\mathbf{y}}_{k \mid k} = \mathbf{z}_k - \mathbf{R}_k \mathbf{S}_k^{-\frac{1}{2}} \alpha_k</math>

which is identical to the minimum-variance Kalman filter. The above solutions minimize the variance of the output estimation error. Note that the Rauch–Tung–Striebel smoother derivation assumes that the underlying distributions are Gaussian, whereas the minimum-variance solutions do not. Optimal smoothers for state estimation and input estimation can be constructed similarly.

A continuous-time version of the above smoother is described in.<ref>{{cite journal | last = Einicke | first =  G.A. | title = Asymptotic Optimality of the Minimum-Variance Fixed-Interval Smoother | journal =  IEEE Transactions on Signal Processing| volume = 55 | issue = 4 | pages = 1543–1547 | date=April 2007 | bibcode = 2007ITSP...55.1543E | doi = 10.1109/TSP.2006.889402 | s2cid = 16218530 }}</ref><ref>{{cite journal | last1 = Einicke | first1 =  G.A. | last2 = Ralston | first2 = J.C. | last3 = Hargrave | first3 = C.O. | last4 = Reid | first4 = D.C. | last5 = Hainsworth | first5 = D.W. | title = Longwall Mining Automation. An Application of Minimum-Variance Smoothing | journal = IEEE Control Systems Magazine | volume = 28 | issue = 6 |pages=28–37 |doi=10.1109/MCS.2008.929281 | date=December 2008 | bibcode = 2008ICSys..28f..28E | s2cid = 36072082 }}</ref>

[[Expectation–maximization algorithm]]s may be employed to calculate approximate [[maximum likelihood]] estimates of unknown state-space parameters within minimum-variance filters and smoothers. Often uncertainties remain within problem assumptions. A smoother that accommodates uncertainties can be designed by adding a positive definite term to the Riccati equation.<ref>{{cite journal | last = Einicke | first =  G.A. | title = Asymptotic Optimality of the Minimum-Variance Fixed-Interval Smoother | journal = IEEE Transactions on Automatic Control | volume = 54 | issue = 12 | pages = 2904–2908 | date=December 2009 | bibcode = 2007ITSP...55.1543E | doi = 10.1109/TSP.2006.889402 | s2cid = 16218530 }}</ref>

In cases where the models are nonlinear, step-wise linearizations may be within the minimum-variance filter and smoother recursions ([[extended Kalman filter]]ing).

== Frequency-weighted Kalman filters ==
Pioneering research on the perception of sounds at different frequencies was conducted by Fletcher and Munson in the 1930s. Their work led to a standard way of weighting measured sound levels within investigations of industrial noise and hearing loss. Frequency weightings have since been used within filter and controller designs to manage performance within bands of interest.

Typically, a frequency shaping function is used to weight the average power of the error spectral density in a specified frequency band. Let <math>\mathbf{y} - \hat{\mathbf{y}}</math> denote the output estimation error exhibited by a conventional Kalman filter. Also, let <math>\mathbf{W}</math> denote a causal frequency weighting transfer function. The optimum solution which minimizes the variance of <math>\mathbf{W}\left(\mathbf{y} - \hat{\mathbf{y}}\right)</math> arises by simply constructing <math>\mathbf{W}^{-1} \hat{\mathbf{y}}</math>.

The design of <math>\mathbf{W}</math> remains an open question. One way of proceeding is to identify a system which generates the estimation error and setting <math>\mathbf{W}</math> equal to the inverse of that system.<ref>{{cite journal | last = Einicke | first =  G.A. | title = Iterative Frequency-Weighted Filtering and Smoothing Procedures | journal = IEEE Signal Processing Letters | volume = 21 | issue = 12 | pages = 1467–1470 | date=December 2014 | doi = 10.1109/LSP.2014.2341641 |bibcode = 2014ISPL...21.1467E | s2cid = 13569109 }}</ref> This procedure may be iterated to obtain mean-square error improvement at the cost of increased filter order. The same technique can be applied to smoothers.

== Nonlinear filters ==
The basic Kalman filter is limited to a linear assumption. More complex systems, however, can be [[nonlinear filter|nonlinear]]. The nonlinearity can be associated either with the process model or with the observation model or with both.

The most common variants of Kalman filters for non-linear systems are the Extended Kalman Filter and Unscented Kalman filter. The suitability of which filter to use depends on the non-linearity indices of the process and observation model.<ref>{{Cite journal|last1=Biswas|first1=Sanat K.|last2=Qiao|first2=Li|last3=Dempster|first3=Andrew G.|date=2020-12-01|title=A quantified approach of predicting suitability of using the Unscented Kalman Filter in a non-linear application|url=http://www.sciencedirect.com/science/article/pii/S0005109820304398|journal=Automatica|language=en|volume=122|article-number=109241|doi=10.1016/j.automatica.2020.109241|s2cid=225028760|issn=0005-1098}}</ref>

=== Extended Kalman filter ===
{{Main|Extended Kalman filter}}

In the extended Kalman filter (EKF), the state transition and observation models need not be linear functions of the state but may instead be nonlinear functions. These functions are of [[Differentiable function|differentiable]] type.
:<math>\begin{align}
  \mathbf{x}_k &= f(\mathbf{x}_{k-1}, \mathbf{u}_k) + \mathbf{w}_k \\
  \mathbf{z}_k &= h(\mathbf{x}_k) + \mathbf{v}_k
\end{align}</math>

The function ''f'' can be used to compute the predicted state from the previous estimate and similarly the function ''h'' can be used to compute the predicted measurement from the predicted state. However, ''f'' and ''h'' cannot be applied to the covariance directly. Instead a matrix of partial derivatives (the [[Jacobian matrix|Jacobian]]) is computed.

At each timestep the Jacobian is evaluated with current predicted states. These matrices can be used in the Kalman filter equations. This process essentially linearizes the nonlinear function around the current estimate.

=== Unscented Kalman filter ===
When the state transition and observation models—that is, the predict and update functions <math>f</math> and <math>h</math>—are highly nonlinear, the extended Kalman filter can give particularly poor performance.<ref name="JU2004">{{cite journal
 | author1 = Julier, Simon J.
 | author2 = Uhlmann, Jeffrey K.
 | year = 2004
 | title = Unscented filtering and nonlinear estimation
 | journal = Proceedings of the IEEE
 | volume = 92
 | issue = 3
 | pages = 401–422
 | doi=10.1109/JPROC.2003.823141
 | bibcode = 2004IEEEP..92..401J
 | s2cid = 9614092
 }}</ref>
<ref name="JU97">{{cite book
 | author1 = Julier, Simon J.
 | author2 = Uhlmann, Jeffrey K.
 | year = 1997
 | title = Signal Processing, Sensor Fusion, and Target Recognition VI
 | volume = 3
 | pages = 182–193
 | chapter-url = http://www.cs.unc.edu/~welch/kalman/media/pdf/Julier1997_SPIE_KF.pdf
 | access-date = 2008-05-03
 | bibcode = 1997SPIE.3068..182J
 | doi=10.1117/12.280797
 | series=Proceedings of SPIE
 | citeseerx=10.1.1.5.2891
| chapter = New extension of the Kalman filter to nonlinear systems
 | s2cid = 7937456
 | editor1-last = Kadar
 | editor1-first = Ivan
 }}</ref> This is because the covariance is propagated through linearization of the underlying nonlinear model. The unscented Kalman filter (UKF)&nbsp;<ref name="JU2004" /> uses a deterministic sampling technique known as the [[unscented transform|unscented transformation (UT)]] to pick a minimal set of sample points (called sigma points) around the mean. The sigma points are then propagated through the nonlinear functions, from which a new mean and covariance estimate are formed. The resulting filter depends on how the transformed statistics of the UT are calculated and which set of sigma points are used. It should be remarked that it is always possible to construct new UKFs in a consistent way.<ref>{{Cite journal|last1=Menegaz|first1=H. M. T.|last2=Ishihara|first2=J. Y.|last3=Borges|first3=G. A.|last4=Vargas|first4=A. N.|date=October 2015|title=A Systematization of the Unscented Kalman Filter Theory|journal=IEEE Transactions on Automatic Control|volume=60|issue=10|pages=2583–2598|doi=10.1109/tac.2015.2404511|bibcode=2015ITAC...60.2583M |issn=0018-9286|hdl=20.500.11824/251|s2cid=12606055|hdl-access=free}}</ref>  For certain systems, the resulting UKF more accurately estimates the true mean and covariance.<ref name="GH2012">{{cite journal
 | author  = Gustafsson, Fredrik
 | author2 = Hendeby, Gustaf
 | year    = 2012
 | title   = Some Relations Between Extended and Unscented Kalman Filters
 | journal = IEEE Transactions on Signal Processing
 | volume  = 60
 | issue = 2
 | pages   = 545–555
 | doi     = 10.1109/tsp.2011.2172431
 | bibcode= 2012ITSP...60..545G
 | s2cid = 17876531
 | url = http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-75272
 }}</ref>  This can be verified with [[Monte Carlo sampling]] or [[Taylor series]] expansion of the posterior statistics.  In addition, this technique removes the requirement to explicitly calculate Jacobians, which for complex functions can be a difficult task in itself (i.e., requiring complicated derivatives if done analytically or being computationally costly if done numerically), if not impossible (if those functions are not differentiable).

==== Sigma points ====
For a [[Random variable|random]] vector <math>\mathbf{x}=(x_1, \dots, x_L)</math>, sigma points are any set of vectors
:<math> \{\mathbf{s}_0,\dots, \mathbf{s}_N \}=\bigl\{\begin{pmatrix} s_{0,1}& s_{0,2}&\ldots& s_{0,L} \end{pmatrix}, \dots, \begin{pmatrix} s_{N,1}& s_{N,2}&\ldots& s_{N,L} \end{pmatrix}\bigr\}</math>
attributed with

* first-order weights <math>W_0^a, \dots, W_N^a</math> that fulfill
# <math> \sum_{j=0}^N W_j^a=1 </math> 
# for all <math>i=1, \dots, L</math>: <math> E[x_i]=\sum_{j=0}^N W_j^a s_{j,i} </math>
* second-order weights <math>W_0^c, \dots, W_N^c</math> that fulfill 
# <math> \sum_{j=0}^N W_j^c=1 </math>  
# for all pairs <math> (i,l) \in \{1,\dots, L\}^2: E[x_ix_l]=\sum_{j=0}^N W_j^c s_{j,i}s_{j,l} </math>.

A simple choice of sigma points and weights for <math>\mathbf{x}_{k-1\mid k-1}</math> in the UKF algorithm is
:<math>\begin{align}
\mathbf{s}_0&=\hat \mathbf{x}_{k-1\mid k-1}\\
-1&<W_0^a=W_0^c<1\\
\mathbf{s}_j&=\hat \mathbf{x}_{k-1\mid k-1} + \sqrt{\frac{L}{1-W_0}} \mathbf{A}_j, \quad j=1, \dots, L\\
\mathbf{s}_{L+j}&=\hat \mathbf{x}_{k-1\mid k-1} - \sqrt{\frac{L}{1-W_0}} \mathbf{A}_j, \quad j=1, \dots, L\\
W_j^a&=W_j^c=\frac{1-W_0}{2L}, \quad j=1, \dots, 2L
\end{align}
</math>
where <math>\hat \mathbf{x}_{k-1\mid k-1}</math> is the mean estimate of <math>\mathbf{x}_{k-1\mid k-1}</math>. The vector <math>\mathbf{A}_j</math> is the ''j''th column of <math>\mathbf{A}</math> where <math>\mathbf{P}_{k-1\mid k-1}=\mathbf{AA}^\textsf{T}</math>. Typically, <math>\mathbf{A}</math> is obtained via [[Cholesky decomposition]] of <math>\mathbf{P}_{k-1\mid k-1}</math>. With some care the filter equations can be expressed in such a way that <math>\mathbf{A}</math> is evaluated directly without intermediate calculations of <math>\mathbf{P}_{k-1\mid k-1}</math>.  This is referred to as the ''square-root unscented Kalman filter''.<ref>{{cite book |last1=Van der Merwe |first1=R. |last2=Wan |first2=E.A. |title=2001 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No.01CH37221) |chapter=The square-root unscented Kalman filter for state and parameter-estimation |date=2001 |volume=6 |pages=3461–3464 |doi=10.1109/ICASSP.2001.940586|isbn=0-7803-7041-4 |s2cid=7290857 }}</ref>

The weight of the mean value, <math>W_0</math>, can be chosen arbitrarily.

Another popular parameterization (which generalizes the above) is 
:<math>\begin{align}
\mathbf{s}_0&=\hat \mathbf{x}_{k-1\mid k-1}\\
W_0^a&= \frac{\alpha^2\kappa-L}{\alpha^2\kappa}\\
W_0^c&= W_0^a + 1-\alpha^2+\beta \\
\mathbf{s}_j&=\hat \mathbf{x}_{k-1\mid k-1} + \alpha\sqrt{\kappa} \mathbf{A}_j, \quad j=1, \dots, L\\
\mathbf{s}_{L+j}&=\hat \mathbf{x}_{k-1\mid k-1} - \alpha\sqrt{\kappa} \mathbf{A}_j, \quad j=1, \dots, L\\
W_j^a&=W_j^c=\frac{1}{2\alpha^2\kappa}, \quad j=1, \dots, 2L.
\end{align}
</math>

<math>\alpha</math> and <math>\kappa</math> control the spread of the sigma points.  <math>\beta</math> is related to the distribution of <math>x</math>. Note that this is an overparameterization in the sense that any one of <math>\alpha</math>, <math>\beta</math> and <math>\kappa</math> can be chosen arbitrarily.

Appropriate values depend on the problem at hand, but a typical recommendation is <math>\alpha = 1</math>, <math>\beta = 0</math>, and <math>\kappa \approx 3L/2</math>.{{cn|date=January 2025}} If the true distribution of <math>x</math> is Gaussian, <math>\beta = 2</math> is optimal.<ref>{{Cite book |doi=10.1109/ASSPCC.2000.882463 |chapter-url=http://www.lara.unb.br/~gaborges/disciplinas/efe/papers/wan2000.pdf |chapter=The unscented Kalman filter for nonlinear estimation |title=Proceedings of the IEEE 2000 Adaptive Systems for Signal Processing, Communications, and Control Symposium (Cat. No.00EX373) |page=153 |year=2000 |last1=Wan |first1=E.A. |last2=Van Der Merwe |first2=R. |isbn=978-0-7803-5800-3 |citeseerx=10.1.1.361.9373 |s2cid=13992571 |access-date=2010-01-31 |archive-date=2012-03-03 |archive-url=https://web.archive.org/web/20120303020429/http://www.lara.unb.br/~gaborges/disciplinas/efe/papers/wan2000.pdf |url-status=dead }}</ref>

==== Predict ====
As with the EKF, the UKF prediction can be used independently from the UKF update, in combination with a linear (or indeed EKF) update, or vice versa.

Given estimates of the mean and covariance, <math> \hat\mathbf{x}_{k-1\mid k-1}</math> and <math>\mathbf{P}_{k-1\mid k-1}</math>, one obtains <math> N = 2L+1 </math> sigma points as described in the section above. The sigma points are propagated through the transition function ''f''.
:<math>\mathbf{x}_{j} = f\left(\mathbf{s}_{j}\right) \quad j = 0, \dots, 2L </math>.

The propagated sigma points are weighed to produce the predicted mean and covariance. 
:<math>\begin{align}
   \hat{\mathbf{x}}_{k \mid k-1} &= \sum_{j=0}^{2L} W_j^a \mathbf{x}_j \\
         \mathbf{P}_{k \mid k-1} &= \sum_{j=0}^{2L} W_j^c \left(\mathbf{x}_j - \hat{\mathbf{x}}_{k \mid k-1}\right)\left(\mathbf{x}_j - \hat{\mathbf{x}}_{k \mid k-1}\right)^\textsf{T}+\mathbf{Q}_k
\end{align}</math>
where <math>W_j^a</math> are the first-order weights of the original sigma points, and <math>W_j^c</math> are the second-order weights. The matrix <math> \mathbf{Q}_k </math> is the covariance of the transition noise, <math>\mathbf{w}_k</math>.

==== Update ====
Given prediction estimates <math>\hat{\mathbf{x}}_{k \mid k-1}</math> and <math>\mathbf{P}_{k \mid k-1}</math>, a new set of <math>N = 2L+1</math> sigma points <math>\mathbf{s}_0, \dots, \mathbf{s}_{2L}</math> with corresponding first-order weights <math> W_0^a,\dots W_{2L}^a</math> and second-order weights <math>W_0^c,\dots, W_{2L}^c</math> is calculated.<ref>{{cite journal |last1=Sarkka |first1=Simo |title=On Unscented Kalman Filtering for State Estimation of Continuous-Time Nonlinear Systems |journal=IEEE Transactions on Automatic Control |date=September 2007 |volume=52 |issue=9 |pages=1631–1641 |doi=10.1109/TAC.2007.904453 |bibcode=2007ITAC...52.1631S }}</ref> These sigma points are transformed through the measurement function <math>h</math>.
:<math> \mathbf{z}_j=h(\mathbf{s}_j), \,\, j=0,1, \dots, 2L </math>.

Then the empirical mean and covariance of the transformed points are calculated.
:<math>\begin{align}
  \hat{\mathbf{z}} &= \sum_{j=0}^{2L} W_j^a \mathbf{z}_j \\[6pt]
  \hat{\mathbf{S}}_k &= \sum_{j=0}^{2L} W_j^c (\mathbf{z}_j-\hat{\mathbf{z}})(\mathbf{z}_j-\hat{\mathbf{z}})^\textsf{T} + \mathbf{R_k}
\end{align}</math>
where <math>\mathbf{R}_k</math> is the covariance matrix of the observation noise, <math>\mathbf{v}_k</math>. Additionally, the cross covariance matrix is also needed
:<math>\begin{align}
  \mathbf{C_{xz}} &= \sum_{j=0}^{2L} W_j^c (\mathbf{x}_j-\hat\mathbf{x}_{k|k-1})(\mathbf{z}_j-\hat\mathbf{z})^\textsf{T}.
\end{align}</math>

The Kalman gain is
: <math>\begin{align}
  \mathbf{K}_k=\mathbf{C_{xz}}\hat{\mathbf{S}}_k^{-1}.
\end{align}</math>

The updated mean and covariance estimates are
:<math>
\begin{align}
\hat\mathbf{x}_{k\mid k}&=\hat\mathbf{x}_{k|k-1}+\mathbf{K}_k(\mathbf{z}_k-\hat\mathbf{z})\\
\mathbf{P}_{k\mid k}&=\mathbf{P}_{k\mid k-1}-\mathbf{K}_k\hat{\mathbf{S}}_k\mathbf{K}_k^\textsf{T}.
\end{align}
</math>

=== Discriminative Kalman filter ===
When the observation model <math>p(\mathbf{z}_k\mid\mathbf{x}_k)</math> is highly non-linear and/or non-Gaussian, it may prove advantageous to apply [[Bayes' rule]] and estimate
:<math>
p(\mathbf{z}_k\mid\mathbf{x}_k) \approx 
\frac{p(\mathbf{x}_k\mid\mathbf{z}_k)}{p(\mathbf{x}_k)}
</math>
where <math>p(\mathbf{x}_k\mid\mathbf{z}_k) \approx \mathcal{N}(g(\mathbf{z}_k),Q(\mathbf{z}_k))</math> for nonlinear functions <math>g,Q</math>. This replaces the generative specification of the standard Kalman filter with a [[discriminative model]] for the latent states given observations.

Under a [[Stationary process|stationary]] state model
:<math>
\begin{align}
p(\mathbf{x}_1) &= \mathcal{N}(0, \mathbf{T}), \\
p(\mathbf{x}_k\mid\mathbf{x}_{k-1}) &= 
\mathcal{N}(\mathbf{F}\mathbf{x}_{k-1}, \mathbf{C}),
\end{align}
</math>
where <math>\mathbf{T}= \mathbf{F}\mathbf{T}\mathbf{F}^\intercal + \mathbf{C}</math>, if 
:<math>
p(\mathbf{x}_k\mid\mathbf{z}_{1:k}) \approx \mathcal{N}(\hat{\mathbf{x}}_{k|k-1}, \mathbf{P}_{k|k-1}),
</math>
then given a new observation <math>\mathbf{z}_k</math>, it follows that<ref name="Bur20">{{cite journal |last1=Burkhart |first1=Michael C. |last2=Brandman |first2=David M. |last3=Franco |first3=Brian |last4=Hochberg |first4=Leigh |last5=Harrison |first5=Matthew T. |title=The Discriminative Kalman Filter for Bayesian Filtering with Nonlinear and Nongaussian Observation Models |journal=Neural Computation |date=2020 |volume=32 |issue=5 |pages=969–1017 |doi=10.1162/neco_a_01275 |pmid=32187000 |pmc=8259355 |s2cid=212748230 |access-date=26 March 2021 | url=https://direct.mit.edu/neco/article/32/5/969/95592/The-Discriminative-Kalman-Filter-for-Bayesian}}</ref>
:<math>
p(\mathbf{x}_{k+1}\mid\mathbf{z}_{1:k+1}) \approx \mathcal{N}(\hat{ \mathbf{x}}_{k+1|k}, \mathbf{P}_{k+1|k})
</math>
where
:<math>
\begin{align}
\mathbf{M}_{k+1} &= \mathbf{F}\mathbf{P}_{k|k-1}\mathbf{F}^\intercal + \mathbf{C}, \\
\mathbf{P}_{k+1|k} &= (\mathbf{M}_{k+1}^{-1} + Q(\mathbf{z}_k)^{-1} - \mathbf{T}^{-1})^{-1}, \\
\hat{\mathbf{x}}_{k+1|k} &= \mathbf{P}_{k+1|k} (\mathbf{M}_{k+1}^{-1}\mathbf{F}\hat{\mathbf{x}}_{k|k-1} + \mathbf{P}_{k+1|k}^{-1}g(\mathbf{z}_k) ).
\end{align}
</math>
Note that this approximation requires <math> Q(\mathbf{z}_k)^{-1} - \mathbf{T}^{-1} </math> to be positive-definite; in the case that it is not, 
:<math>
\mathbf{P}_{k+1|k} = (\mathbf{M}_{k+1}^{-1} + Q(\mathbf{z}_k)^{-1})^{-1}
</math>
is used instead. Such an approach proves particularly useful when the dimensionality of the observations is much greater than that of the latent states<ref name="Bur19">{{cite thesis |last1=Burkhart |first1=Michael C. |title=A Discriminative Approach to Bayesian Filtering with Applications to Human Neural Decoding |date=2019 |publisher=Brown University |location=Providence, RI, USA |doi=10.26300/nhfp-xv22 }}</ref> and can be used build filters that are particularly robust to nonstationarities in the observation model.<ref name="Bra18">{{cite journal |last1=Brandman |first1=David M. |last2=Burkhart |first2=Michael C. |last3=Kelemen |first3=Jessica |last4=Franco |first4=Brian |last5=Harrison |first5=Matthew T. |last6=Hochberg |first6=Leigh R. |title=Robust Closed-Loop Control of a Cursor in a Person with Tetraplegia using Gaussian Process Regression |journal=Neural Computation |date=2018 |volume=30 |issue=11 |pages=2986–3008 |doi=10.1162/neco_a_01129 |pmid=30216140 |pmc=6685768 |url=https://direct.mit.edu/neco/article/30/11/2986/8418/Robust-Closed-Loop-Control-of-a-Cursor-in-a-Person |access-date=26 March 2021}}</ref>

== Adaptive Kalman filter ==
Adaptive Kalman filters allow to adapt for process dynamics which are not modeled in the process model <math>\mathbf{F}(t)</math>, which happens for example in the context of a maneuvering target when a constant velocity (reduced order) Kalman filter is employed for tracking.<ref>{{Cite book|last1=Bar-Shalom|first1=Yaakov|title=Estimation with Applications to Tracking and Navigation|last2=Li|first2=X.-Rong|last3=Kirubarajan|first3=Thiagalingam|date=2001|publisher=John Wiley & Sons, Inc.|isbn=0-471-41655-X|location=New York, USA|pages=421 ff|doi=10.1002/0471221279}}</ref>

== Kalman–Bucy filter ==
Kalman–Bucy filtering (named for Richard Snowden Bucy) is a continuous time version of Kalman filtering.<ref>Bucy, R.S. and Joseph, P.D., ''Filtering for Stochastic Processes with Applications to Guidance,'' John Wiley & Sons, 1968; 2nd Edition, AMS Chelsea Publ., 2005. {{isbn|0-8218-3782-6}}</ref><ref>Jazwinski, Andrew H., ''Stochastic processes and filtering theory,'' Academic Press, New York, 1970. {{isbn|0-12-381550-9}}</ref>

It is based on the state space model
:<math>\begin{align}
  \frac{d}{dt}\mathbf{x}(t) &= \mathbf{F}(t)\mathbf{x}(t) + \mathbf{B}(t)\mathbf{u}(t) + \mathbf{w}(t) \\
              \mathbf{z}(t) &= \mathbf{H}(t) \mathbf{x}(t) + \mathbf{v}(t)
\end{align}</math>

where <math>\mathbf{Q}(t)</math> and <math>\mathbf{R}(t)</math> represent the intensities of the two white noise terms <math>\mathbf{w}(t)</math> and <math>\mathbf{v}(t)</math>, respectively.

The filter consists of two differential equations, one for the state estimate and one for the covariance:
:<math>\begin{align}
  \frac{d}{dt}\hat{\mathbf{x}}(t) &= \mathbf{F}(t)\hat{\mathbf{x}}(t) + \mathbf{B}(t)\mathbf{u}(t) + \mathbf{K}(t) \left(\mathbf{z}(t) - \mathbf{H}(t)\hat{\mathbf{x}}(t)\right) \\
        \frac{d}{dt}\mathbf{P}(t) &= \mathbf{F}(t)\mathbf{P}(t) + \mathbf{P}(t)\mathbf{F}^\textsf{T}(t) + \mathbf{Q}(t) - \mathbf{K}(t)\mathbf{R}(t)\mathbf{K}^\textsf{T}(t)
\end{align}</math>

where the Kalman gain is given by
:<math>\mathbf{K}(t) = \mathbf{P}(t)\mathbf{H}^\textsf{T}(t)\mathbf{R}^{-1}(t)</math>

Note that in this expression for <math>\mathbf{K}(t)</math> the covariance of the observation noise <math>\mathbf{R}(t)</math> represents at the same time the covariance of the prediction error (or ''innovation'') <math>\tilde{\mathbf{y}}(t) = \mathbf{z}(t) - \mathbf{H}(t)\hat{\mathbf{x}}(t)</math>; these covariances are equal only in the case of continuous time.<ref>{{cite journal|pages= 646–655|doi=10.1109/TAC.1968.1099025|title=An innovations approach to least-squares estimation--Part I: Linear filtering in additive white noise|journal=IEEE Transactions on Automatic Control|volume=13|issue=6|year=1968|last1=Kailath|first1=T.}}</ref>

The distinction between the prediction and update steps of discrete-time Kalman filtering does not exist in continuous time.

The second differential equation, for the covariance, is an example of a [[Riccati equation]]. Nonlinear generalizations to Kalman–Bucy filters include continuous time extended Kalman filter.

== Hybrid Kalman filter ==
Most physical systems are represented as continuous-time models while discrete-time measurements are made frequently for state estimation via a digital processor. Therefore, the system model and measurement model are given by
:<math>\begin{align}
  \dot{\mathbf{x}}(t) &= \mathbf{F}(t)\mathbf{x}(t) + \mathbf{B}(t)\mathbf{u}(t) + \mathbf{w}(t),
           &\mathbf{w}(t) &\sim N\left(\mathbf{0}, \mathbf{Q}(t)\right) \\
         \mathbf{z}_k &= \mathbf{H}_k\mathbf{x}_k + \mathbf{v}_k,
           &\mathbf{v}_k  &\sim N(\mathbf{0},\mathbf{R}_k)
\end{align}</math>

where
:<math>\mathbf{x}_k = \mathbf{x}(t_k)</math>.

=== Initialize ===
:<math>\hat{\mathbf{x}}_{0 \mid 0} = E\left[\mathbf{x}(t_0)\right], \mathbf{P}_{0 \mid 0} = \operatorname{Var}\left[\mathbf{x}\left(t_0\right)\right]</math>

=== Predict ===
:<math>\begin{align}
                  \dot{\hat{\mathbf{x}}}(t) &= \mathbf{F}(t) \hat{\mathbf{x}}(t) + \mathbf{B}(t) \mathbf{u}(t)
    \text{, with } \hat{\mathbf{x}}\left(t_{k-1}\right) = \hat{\mathbf{x}}_{k-1 \mid k-1} \\
  \Rightarrow \hat{\mathbf{x}}_{k \mid k-1} &= \hat{\mathbf{x}}\left(t_k\right) \\
          \dot{\mathbf{P}}(t) &= \mathbf{F}(t)\mathbf{P}(t) + \mathbf{P}(t)\mathbf{F}(t)^\textsf{T} + \mathbf{Q}(t)
    \text{, with } \mathbf{P}\left(t_{k-1}\right) = \mathbf{P}_{k-1 \mid k-1} \\
        \Rightarrow \mathbf{P}_{k \mid k-1} &= \mathbf{P}\left(t_k\right)
\end{align}</math>

The prediction equations are derived from those of continuous-time Kalman filter without update from measurements, i.e., <math> \mathbf{K}(t) = 0 </math>. The predicted state and covariance are calculated respectively by solving a set of differential equations with the initial value equal to the estimate at the previous step.

For the case of [[linear time invariant]] systems, the continuous time dynamics can be exactly [[discretized]] into a discrete time system using [[matrix exponential]]s.

=== Update ===
:<math>\begin{align}
                 \mathbf{K}_k &= \mathbf{P}_{k \mid k-1}\mathbf{H}_k^\textsf{T} \left(\mathbf{H}_k\mathbf{P}_{k \mid k-1}\mathbf{H}_k^\textsf{T} + \mathbf{R}_k\right)^{-1} \\
  \hat{\mathbf{x}}_{k \mid k} &= \hat{\mathbf{x}}_{k \mid k-1} + \mathbf{K}_k\left(\mathbf{z}_k - \mathbf{H}_k\hat{\mathbf{x}}_{k \mid k-1}\right) \\
        \mathbf{P}_{k \mid k} &= \left(\mathbf{I} - \mathbf{K}_k\mathbf{H}_k\right)\mathbf{P}_{k \mid k-1}
\end{align}</math>

The update equations are identical to those of the discrete-time Kalman filter.

== Variants for the recovery of sparse signals ==
The traditional Kalman filter has also been employed for the recovery of [[sparse signal|sparse]], possibly dynamic, signals from noisy observations. Recent works<ref>{{cite book|doi=10.1109/ICIP.2008.4711899|chapter=Kalman filtered Compressed Sensing|title=2008 15th IEEE International Conference on Image Processing|pages=893–896|year=2008|last1=Vaswani|first1=Namrata|author1-link= Namrata Vaswani |isbn=978-1-4244-1765-0|arxiv=0804.0819|s2cid=9282476}}</ref><ref>{{Cite journal|title=Methods for sparse signal recovery using Kalman filtering with embedded pseudo-measurement norms and quasi-norms|journal=IEEE Transactions on Signal Processing|volume=58|issue=4|pages=2405–2409|doi=10.1109/TSP.2009.2038959|year=2010|last1=Carmi|first1=Avishy|author2-link=Pini Gurfil|last2=Gurfil|first2=Pini|last3=Kanevsky|first3=Dimitri|bibcode=2010ITSP...58.2405C|s2cid=10569233}}</ref><ref>{{Cite journal|title=Dynamic Iterative Pursuit|journal=IEEE Transactions on Signal Processing|volume=60|issue=9|pages=4967–4972|doi=10.1109/TSP.2012.2203813|year=2012|last1=Zachariah|first1=Dave|last2=Chatterjee|first2=Saikat|last3=Jansson|first3=Magnus|arxiv=1206.2496|bibcode=2012ITSP...60.4967Z|s2cid=18467024}}</ref> utilize notions from the theory of [[compressed sensing]]/sampling, such as the restricted isometry property and related probabilistic recovery arguments, for sequentially estimating the sparse state in intrinsically low-dimensional systems.

== Relation to Gaussian processes ==
Since linear Gaussian state-space models lead to Gaussian processes, Kalman filters can be viewed as sequential solvers for [[Gaussian Processes|Gaussian process regression]].<ref>{{cite arXiv|last1=Särkkä|first1=Simo|last2=Hartikainen|first2=Jouni|last3=Svensson|first3=Lennart|last4=Sandblom|first4=Fredrik|date=2015-04-22|title=On the relation between Gaussian process quadratures and sigma-point methods|class=stat.ME|eprint=1504.05994}}</ref>

== Applications ==
{{Div col}}
* [[Attitude and heading reference systems]]
* [[Autopilot]]
* [[Electric battery]] [[state of charge]] (SoC) estimation<ref>{{cite journal|doi=10.1016/j.jpowsour.2007.04.011|title=A novel combined battery model for state-of-charge estimation in lead-acid batteries based on extended Kalman filter for hybrid electric vehicle applications|journal=Journal of Power Sources|volume=174|issue=1|pages=30–40|year=2007|last1=Vasebi|first1=Amir|last2=Partovibakhsh|first2=Maral|last3=Bathaee|first3=S. Mohammad Taghi|bibcode=2007JPS...174...30V}}</ref><ref>{{cite journal|doi=10.1016/j.enconman.2007.05.017|title=Predicting state of charge of lead-acid batteries for hybrid electric vehicles by extended Kalman filter|journal=Energy Conversion and Management|volume=49|pages=75–82|year=2008|last1=Vasebi|first1=A.|last2=Bathaee|first2=S.M.T.|last3=Partovibakhsh|first3=M.|issue=1 |bibcode=2008ECM....49...75V }}</ref>
* [[Brain–computer interfaces]]<ref name="Bur20" /><ref name="Bra18" /><ref name="Bur19" />
* Tracking and vertex fitting of [[charged particles]] in [[particle detector]]s<ref>{{cite journal |journal= Nuclear Instruments and Methods in Physics Research Section A |volume=262 |issue=2–3 |year=1987 |pages=444–450 |title=Application of Kalman filtering to track and vertex fitting |first=R. |last=Fruhwirth |bibcode=1987NIMPA.262..444F |doi=10.1016/0168-9002(87)90887-4 }}</ref>
* Tracking of objects in [[computer vision]]
* [[Dynamic positioning]] in shipping
* [[Economics]], in particular [[macroeconomics]], [[time series analysis]], and [[econometrics]]<ref>{{cite book |last=Harvey |first=Andrew C. |chapter=Applications of the Kalman filter in econometrics |title=Advances in Econometrics |editor-first=Truman |editor-last=Bewley |editor-link=Truman Bewley |location=New York |publisher=Cambridge University Press |year=1994 |isbn=978-0-521-46726-1 |pages=[https://archive.org/details/advancesineconom00iiecon/page/285 285f] |chapter-url=https://archive.org/details/advancesineconom00iiecon/page/285 }}</ref>
* [[Inertial guidance system]]
* [[Nuclear medicine]] – single photon emission computed tomography image restoration<ref>{{Cite journal|pages=102–109|doi=10.1109/42.276148|pmid=18218487|title=Two-dimensional restoration of single photon emission computed tomography images using the Kalman filter|journal=IEEE Transactions on Medical Imaging|volume=13|issue=1|year=1994|last1=Boulfelfel|first1=D.|last2=Rangayyan|first2=R.M.|last3=Hahn|first3=L.J.|last4=Kloiber|first4=R.|last5=Kuduvalli|first5=G.R.}}</ref>
* [[Orbit determination]]
* Power system state estimation
* [[Radar tracker]]
* [[Satellite navigation system]]s
* [[Seismology]]<ref>{{cite journal|bibcode=2008AGUFM.G43B..01B|title=Fusion of High-Rate GPS and Seismic Data: Applications to Early Warning Systems for Mitigation of Geological Hazards|journal=AGU Fall Meeting Abstracts|volume=43|pages=G43B–01|last1=Bock|first1=Y.|last2=Crowell|first2=B.|last3=Webb|first3=F.|last4=Kedar|first4=S.|last5=Clayton|first5=R.|last6=Miyahara|first6=B.|year=2008}}</ref>
* Sensorless control of AC motor [[variable-frequency drive]]s
* [[Simultaneous localization and mapping]]
* [[Speech enhancement]]
* [[Visual odometry]]
* [[Weather forecasting]]
* [[Navigation system]]
* [[3D modeling]]
* [[Structural health monitoring]]
* [[Human sensorimotor processing]]<ref>{{cite journal |journal= Neural Networks|volume=9 |issue=8 |pages=1265–1279 |title=Forward Models for Physiological Motor Control |last1=Wolpert |first1=D. M. |last2=Miall |first2=R. C. |pmid=12662535 |year=1996 |doi=10.1016/S0893-6080(96)00035-4}}</ref>
{{div col end}}

== See also ==
{{div col|}}
* [[Alpha beta filter]]
* [[Inverse-variance weighting]]
* [[Covariance intersection]]
* [[Data assimilation]]
* [[Ensemble Kalman filter]]
* [[Extended Kalman filter]]
* [[Fast Kalman filter]]
* [[Filtering problem (stochastic processes)]]
* [[Generalized filtering]]
* [[Invariant extended Kalman filter]]
* [[Kernel adaptive filter]]
* [[Masreliez's theorem]]
* [[Moving horizon estimation]]
* [[Particle filter]] estimator
* [[PID controller]]
* [[Predictor–corrector method]]
* [[Recursive least squares filter]]
* [[Schmidt–Kalman filter]]
* [[Separation principle]]
* [[Sliding mode control]]
* [[State-transition matrix]]
* [[Stochastic differential equation]]s
* [[Switching Kalman filter]]
{{div col end}}

== References ==
{{reflist}}

== Further reading ==
{{refbegin}}
* {{cite book | last1 = Bar-Shalom | first1 = Yaakov | author-link1 = Yaakov Bar-Shalom | last2 = Li | first2 = X. Rong | last3 = Kirubarajan | first3 = Thiagalingam | title = Estimation with Applications to Tracking and Navigation: Theory Algorithms and Software | publisher = Wiley | year = 2004 }}
* {{cite book | last = Bierman | first = G.J. | year = 1977 | title = Factorization Methods for Discrete Sequential Estimation | series = Mathematics in Science and Engineering | volume = 128 | isbn = 978-0-486-44981-4 | publisher = Dover Publications | location = Mineola, N.Y. }}
* {{cite book | last = Bozic | first = S.M. | year = 1994 | title = Digital and Kalman filtering | publisher = Butterworth–Heinemann }}
* {{cite book | last1 = Chui | first1 = Charles K. | last2 = Chen | first2 = Guanrong | year = 2009 | title = Kalman Filtering with Real-Time Applications | publisher = [[Springer Science+Business Media|Springer]] | location=New York | isbn = 978-3-540-87848-3 | volume = 17 | series = Springer Series in Information Sciences | edition = 4th | page=229 }}
* {{cite book | last = Gelb | first = A. | year = 1974 | title = Applied Optimal Estimation | publisher = MIT Press }}
* {{cite book | last = Harvey | first = A.C. | year = 1990 | title = Forecasting, Structural Time Series Models and the Kalman Filter | publisher = Cambridge University Press | isbn = 978-0-521-40573-7 | url = https://books.google.com/books?id=Kc6tnRHBwLcC }}
* {{cite book | last = Haykin | first = S. | year = 2002 | title = Adaptive Filter Theory | url = https://archive.org/details/adaptivefilterth0000hayk_v4l9 | publisher = Prentice Hall | isbn = 978-0-13-090126-2 }}
* {{cite book | last = Jazwinski | first = Andrew H. | year = 1970 | title = Stochastic Processes and Filtering | publisher = [[Academic Press]] | location=New York | series = Mathematics in Science and Engineering | page=376 | isbn=978-0-12-381550-7 }}
* {{cite book | last1 = Kailath | first1 = Thomas | author-link1=Thomas Kailath | last2 = Sayed | first2 = Ali H. | author-link2=Ali H. Sayed | last3 = Hassibi | first3 = Babak | author-link3=Babak Hassibi | title=Linear Estimation | publisher=Prentice–Hall | location=NJ | year=2000 | isbn=978-0-13-022464-4 }}
* {{cite journal | last = Kalman | first = R.E. | year = 1960 | title = A new approach to linear filtering and prediction problems | journal = Journal of Basic Engineering | volume = 82 | issue = 1 | pages = 35–45 | url = http://www.elo.utfsm.cl/~ipd481/Papers%20varios/kalman1960.pdf | access-date = 2008-05-03 | doi = 10.1115/1.3662552 | s2cid = 1242324 | archive-url = https://web.archive.org/web/20080529105724/http://www.elo.utfsm.cl/~ipd481/Papers%20varios/kalman1960.pdf | archive-date = 2008-05-29 }}
* {{cite journal | last1 = Kalman | first1 = R.E. | last2 = Bucy | first2 = R.S. | year = 1961 | title = New Results in Linear Filtering and Prediction Theory | journal = Journal of Basic Engineering | volume = 83 | pages = 95–108 | citeseerx = 10.1.1.361.6851 | doi = 10.1115/1.3658902 | s2cid = 8141345 }}
* {{cite book | last1 = Liu | first1 = W. | last2 = Principe | first2 = J.C. | last3 = Haykin | first3 = S. | year = 2010 | title = Kernel Adaptive Filtering: A Comprehensive Introduction | publisher = John Wiley }}
* {{cite book | last = Maybeck | first = Peter S. | year = 1979 | title = Stochastic Models, Estimation, and Control | publisher = [[Academic Press]] | location = New York | isbn = 978-0-12-480701-3 | volume = 141-1 | series = Mathematics in Science and Engineering | chapter = Chapter 1 | chapter-url = http://www.cs.unc.edu/~welch/kalman/media/pdf/maybeck_ch1.pdf }}
* {{cite book | last = Manolakis | first = D.G. | year = 1999 | title = Statistical and Adaptive signal processing | publisher = Artech House }}
* {{cite book | last = Simon | first = D. | year = 2006 | title = Optimal State Estimation: Kalman, H Infinity, and Nonlinear Approaches | publisher = Wiley-Interscience | url = http://academic.csuohio.edu/simond/estimation/ | access-date = 2006-07-05 | archive-date = 2010-12-30 | archive-url = https://web.archive.org/web/20101230101133/http://academic.csuohio.edu/simond/estimation/ | url-status = dead }}
* {{cite book | last = Sayed | first = Ali H. | author-link=Ali H. Sayed | title = Adaptive Filters | publisher = Wiley | location = NJ | year = 2008 | isbn = 978-0-470-25388-5 }}
* {{cite journal | last1 = Roweis | first1 = S. | last2 = Ghahramani | first2 = Z. | author-link2 = Zoubin Ghahramani | year = 1999 | title = A Unifying Review of Linear Gaussian Models | journal = Neural Computation | volume = 11 | issue = 2 | pages = 305–345 | doi = 10.1162/089976699300016674 | pmid = 9950734 | s2cid = 2590898 | url = https://authors.library.caltech.edu/13697/1/ROWnc99.pdf }}
* {{cite journal | last = Warwick | first = Kevin | author-link = Kevin Warwick | year = 1987 | title = Optimal observers for ARMA models | journal = International Journal of Control | volume = 46 | issue = 5 | pages = 1493–1503 | doi = 10.1080/00207178708933989 }}
{{refend}}

== External links ==

{{commons|Kalman filters}}
* [http://www.cs.unc.edu/~welch/kalman/kalmanPaper.html A New Approach to Linear Filtering and Prediction Problems], by R. E. Kalman, 1960
* [https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python Kalman and Bayesian Filters in Python]. Open source Kalman filtering textbook.
*[http://www.bzarg.com/p/how-a-kalman-filter-works-in-pictures/ How a Kalman filter works, in pictures]. Illuminates the Kalman filter with pictures and colors


{{control theory}}
{{satellite navigation systems}}
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[[Category:Control theory]]
[[Category:Linear filters]]
[[Category:Markov models]]
[[Category:Nonlinear filters]]
[[Category:Robot control]]
[[Category:Signal estimation]]
[[Category:Stochastic differential equations]]
[[Category:Hungarian inventions]]