View source for Talk:Kalman filter

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== Suggestion on notation between deterministic and stochastic variables ==

Not all variables are stochastic. An estimate is a deterministic value, but its estimator is often represented using the same notation and it is a stochastic variable. right? The input is usually seen as being deterministic, as well as the process and observation matrices (hence they can step out of the expected value). But what about the state vector? The state is well defined no (deterministic I mean)? But at the same time it is governed by a stochastic evolution in time (process noise). I'm confused.  <small>—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/77.54.101.212|77.54.101.212]] ([[User talk:77.54.101.212|talk]]) 14:52, 5 June 2008 (UTC)</small><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->

== Suggestion on new sub-topic for Kalman Filter ==

The only mention that KF gets in the area of econometrics is for the page to say it is used in teh reference section at the bottom. Can we make a new topic on the page, stating how it is used, and why. There is not too much in the literature about this, due to how secretive the work is by the companies that are using it (i guess) but there are some good papers (GROENEWOLD & FRASER).  The useage seems to center around the fact that CAPM Beta is unstable across time, and that use of KF can lead to improved stability - pretty important if you are taking your trading signals from Mod Beta. Any experts on this topic here?

== On the Unscented Kalman Filter ==

In the UKF, what is the significance of the term "unscented"? If not embarrassing, it should be included in the description...

Nice KF page overall - it gives enough on each point for more to investigate further, or if you just need the equations in a hurry.

I have some comments on UKF though.   Personally, I think it needs a separate page - if nothing else but for the number of different [[unscented transform|unscented transforms]] that are described by both Julier and Van der Merwe etc.

The other thing that isn't really clear is when calculating the cross-''covariance'' matrix.  (it isn't a cross-correlation matrix, sorry, I made that change to covariance there).  The problem is that often the state vector and the observation vector are not of the same length (e.g. GPS/INS integration), and therefore, over which indicies are you summing, and what are the corresponding weights to use - The Wc as calculated from the Sigma Points from the state, the Sigma points as calculated from the observation, some combination, or do you need to explicitly augment the state **and** observation noises into the augmented state vector - something that I didn;t think was absolutely nessessary (though most papers tend to do it that way anyway).  [[User:Damien d|Damien d]] 12:04, 30 April 2007 (UTC)

There seems to be an issue with the dimension of the sigma points and the state estimate. How can it be that weighted sum of the 
sigma points, whose dimension corresponds to the dimension of the augmented state, yields prediction/update of the state (which has a lower dimension)?  <small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/129.132.67.247|129.132.67.247]] ([[User talk:129.132.67.247|talk]]) 13:05, 27 April 2015 (UTC)</small><!-- Template:Unsigned IP --> <!--Autosigned by SineBot-->

== Excellent page ==

This is a quite lucid explanation of a (potentially) challenging topic/set of concepts.  As such is deserves recognition ... and inclusion, maybe, in the CD version of wikipedia? [[User:132.239.215.69|132.239.215.69]] 17:52, 9 August 2006 (UTC)

<br />
: Totally agree with you [[User:Pravincar|shampoo]] ([[User talk:Pravincar|talk]]) 12:25, 6 July 2008 (UTC)


Being a total noob, this page was the best resource I had found on the topic and I would have agreed with you until I found this: http://www.cs.unc.edu/~welch/kalman/maybeck.html . Perhaps a more general explanation/emphasis of /what/ the filter is doing (recursively refining a series of uncertain measurements over time based on a variance-weighted average) using graphs like the conditional probability densities (i.e. Fig 1.7 in that linked chapter) would be a good way to start this off. Explaining it in the one-dimensional case first, then introducing the matrix version would be good, too. One thing that really throws me off is the priority of the section "Underlying dynamic system model." I think one should be able to understand the Kalman filter before one understands Markov chains, etc (I sure don't). This may be really important to the theory underlying the filter or the precision of the mathematical definition, but it's confusing if you've just seen a reference to this topic and want to know what the heck is going on.

I don't really feel qualified to make huge changes to this page, but maybe I'll put up some graphs based on that Maybeck chapter if I get around to it and nobody objects.
[[User:Kronick|Kronick]] ([[User talk:Kronick|talk]]) 23:30, 23 January 2010 (UTC)

Many thanks for this exemplary explanation. The different perspectives and the comparison with HMM helped me to put it in the right context.  <span style="font-size: smaller;" class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/128.40.254.54|128.40.254.54]] ([[User talk:128.40.254.54|talk]]) 16:29, 21 June 2011 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->

Thanks! After trying to read from several sources (and fail), this intro cleared things up for me. [[Special:Contributions/2001:4643:EBFE:0:2C12:BB40:84B:4D31|2001:4643:EBFE:0:2C12:BB40:84B:4D31]] ([[User talk:2001:4643:EBFE:0:2C12:BB40:84B:4D31|talk]]) 00:21, 28 January 2016 (UTC)

== Typo ==

In the UKF section, there are four or five occurrences of sums from 1 to N to reconstruct the estimate and its covariance from the samples. These sums should run from 0 to 2L (see the original paper). I'm not really aware of wikipedia protocol etc. or I'd fix it myself. Best wishes, Nathaniel

: It seems like you are indeed right in that the summing indices are wrong (or at least unclear). I've attempted to correct the sums to run from 0 to 2L now. Please review my changes to make sure that they have been succesfully corrected.
: Please feel free to make your own edits in wikipedia in the future. It's easy, just read [[Wikipedia:How to edit a page]] to learn about wiki-formating. But it would be favourable if you register a username and log in prior to making edits, since that makes it easier to trace article history and users contributions. --[[User:Orderud|Fredrik Orderud]] 18:32, 25 May 2006 (UTC)

In the section titled "Underlying dynamic system model", it would appear there is a typo in the following line:  "Bk is the control-input model which is applied to the control vector uk".   I beleive this should be ".... control vector uk-1".   Dave E. --<small><span class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:64.132.58.18|64.132.58.18]] ([[User talk:64.132.58.18|talk]] • [[Special:Contributions/64.132.58.18|contribs]]) </span></small><!-- Template:Unsigned -->

: Fixed. --[[User:Drizzd|Drizzd]] ([[User talk:Drizzd|talk]]) 16:20, 2 July 2008 (UTC)


Why does the Wikipedia information about Kalman filters say : "Fk is the state transition model which is applied to the previous state xk−1;
Bk is the control-input model which is applied to the control vector uk;" ? Shouldn't the matrix F_k-1 (not F_k) be applied to the previous state (since the information about the matrix F at the previous time is F_k-1 (not F_k)? Similarly, based on previous state (ie. starting state) information, shouldn't the input matrix and the input itself have time index of "k-1" (instead if time index "k")? So is it a 'typo'? [[User:KorgBoy|KorgBoy]] ([[User talk:KorgBoy|talk]]) 20:33, 27 September 2017 (UTC)

== What does ''HMM'' mean? ==
In section ''Relationship to recursive Bayesian estimation'':
"Using these assumptions the probability distribution over all states of the HMM can be written simply as: ..."
Can somebody explain (and update in the article)? <small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/134.60.10.127|134.60.10.127]] ([[User talk:134.60.10.127|talk]]) 13:19, 18 November 2005‎ (UTC)</small><!-- Template:Unsigned IP -->

: ''HMM'' is [[Hidden Markov model]]. The Kalman filter model can be considered as a HMM, since it is both ''hidden'' and ''Markov''. It is hidden because the state ('''x'''-vector) is only indirectly observable through the observation model; and it is Markov because the current state ''only'' depends on the previous state and is therefore conditionally independent of any state before the previous state.  --[[User:Orderud|Fredrik Orderud]] 01:52, 20 November 2005 (UTC)

:  This is my first time contributing to a Wikipedia page, so I don't want to tread on any toes.  But I think it's a little confusing to say that the Kalman Filter is a type of HMM.  Both are latent variable models, for sure, in that the state variables are ''Hidden'' or ''latent''.  But the normal definition of the HMM is that the latent variables are ''discrete'', and the time update of the latent variables is governed by a transition probability matrix.  With the Kalman Filter, the latent variables are continuous, and the time update is giverned by the state matrix, which represents a linear dynamical system.  In principle, an HMM that modelled the Kalman filter to arbitrary accuracy could be calculated by discretizing the state variables, and then constructing a transition matrix that modelled the dynamics and the noise term.  However, in practice this would need a huge number of states for a Kalman filter where the dimension of state space is much greater than one due to the [[Curse of Dimensionality]].  Perhaps it would be more clear to give a more expanded discussion on the relationship between Kalman filters and the HMM.  I haven't yet figured how to put a reference into a Wiki discussion page, but a very good summary paper is ''A unifying review of Linear Gaussian Models'' by Roweis and Ghahramani (Neural Computation Vol 11 No 2, 1999).  Possibly a more accurate summary sentence would be to say that the Kalman filter is analogous to the Hidden Markov Model, where the state variables and distributions are continuous valued, rather than discrete. --[[User:Alan1507|Alan1507]] 09:07, 9 May 2006 (UTC)

I've edited the section on "Underlying Dynamical System" to attempt to clarify the relationship between Kalman Filters and HMMs.  However, I am still not happy with the section on ''Relationship to recursive Bayesian estimation'', particularly the sentence that states that the measurements are the ''observed states'' of the Hidden Markov Model.  The ''observations'' in a Hidden Markov Model  are used to ''infer'' the values of the Hidden States, which, by definition are not directly observed. So I do not understand this statement at all - the states of the Hidden Markov Model are ''hidden'', or ''latent variables'', and these are analogous to the system variables in the Kalman Filter.  But it's possible the author had some other meaning in mind - perhaps this could be clarified?  --[[User:Alan1507|Alan1507]] 20:42, 9 May 2006 (UTC)

: The ''Relationship to recursive Bayesian estimation'' section was written by [[User:Chrislloyd]] in February 2005, and has remained pretty much untouched since. The section is important, since it (attempts to) relates Kalman filtering into the bigger picture of sequential state estimation which it is a part of, but it could probably be formulated more clearly. Any help in improving the section is therefore greatly appreciated. --[[User:Orderud|Fredrik Orderud]] 00:23, 10 May 2006 (UTC)

I'll have a go when I've time - need to think it through carefully ;-) --[[User:Alan1507|Alan1507]] 07:51, 10 May 2006 (UTC)

== Relationship to recursive Bayesian estimation ==

[[User:Chrislloyd]], you added this section (which was then titled "Derivation") [http://en.wikipedia.org/w/index.php?title=Kalman_filter&diff=10129775&oldid=9918105 around February 2/10].  Now that we have a derivation section contributed by [[User:Orderud]], is it still necessary?  What does it add?  Thanks! &mdash; [[User:ciphergoth|ciphergoth]] 10:18, 2005 Apr 28 (UTC)

: I think the section is '''still importaint''', since it relates Kalman filtering to the "bigger picture" of recursive Bayesian estimation (which Kalman filtering is a part of). --[[User:Orderud|Fredrik Orderud]] 20:08, 28 Apr 2005 (UTC)

::In that case I think it needs substantial work to make its point clear, since I've tried very hard to understand it and come up with very little.  Is p(X) the probability density function of X?  It doesn't link to [[probability density function]] and that latter doesn't mention p(X) having that meaning.  How is PDF defined for vectors and joint distributions?  I think I can guess, but it's not discussed in [[probability density function]] making it a bit demanding to infer.  Even [[Lebesgue integration]] only defines integration from reals to reals, leaving one to infer how integration of functions such as p: (R x R) -> R is defined (though it seems straightforward to extend it to any real function whose domain is measurable).  What does "The probability distribution of updated" mean?  What is the denominator unimportant to?  What do the probability density functions given at the end mean?  How does it all tie together to say something cohesive and substantial?  &mdash; [[User:ciphergoth|ciphergoth]] 22:21, 2005 Apr 28 (UTC)

::: You're probably right in that it's poorly written (I haven't read until now myself), but it's still very importaint. The variable p(x) is, as you thought, the probability distribution of the state x. The Kalman filter replaces p(x) with a Gaussian distribution parametrized by a state estimate and a covariance. IEEE SignalProc. had a quite straightforward [http://www.ene.unb.br/~gaborges/disciplinas/pe/papers/arulampalam2002.pdf tutorial] in 2002, containing the derivation of Kalman filter from a recursive Bayesian estimator. It is absolutely worth a read. --[[User:Orderud|Fredrik Orderud]] 22:39, 28 Apr 2005 (UTC)

::::Thanks, that helps a lot! &mdash; [[User:ciphergoth|ciphergoth]] 23:05, 2005 Apr 28 (UTC)

::::At a glance it looks like that paper is the basis of this section.  I can follow the paper much better, since I can see what it's trying to get at.  Unfortunately, it doesnt AFAICT actually prove its assertions about the Kalman filter at all - it just states "if you do this, you get the correct conditional PDFs". If we're going to do the same, we should make it explicit that we're stating without proof that the Kalman filter gives the correct PDFs.  I think I can see how to do this.  (Also, it's a pity the equations are bitmaps rather than scalable fonts in the PDF of the paper!) &mdash; [[User:ciphergoth|ciphergoth]] 21:46, 2005 Apr 29 (UTC)

::::: This section is not there to prove the optimality of the Kalman filter. The "proof" section already does that. It's main intent is to demonstrate how recursive Bayestion estimation can be simplified into tractable linear equations with Gaussian PDFs when dealing with linear state-space models subject to Gaussian noise. The derivations are pretty standard, and found in many Kalman textbooks. Your can also find the paper on IEEE [http://ieeexplore.ieee.org/ Xplore] in much higher quality, but this requires an subscription. --[[User:Orderud|Fredrik Orderud]] 11:03, 30 Apr 2005 (UTC)

:::::: OK, but the paper makes it look as if our proof is insufficiently precise, because it talks about expected values, covariance and so forth without talking about what they're conditioned on.  Is it

:::::::<math>\textbf{P}_{k|k} = \textrm{cov}(\textbf{x}_k - \hat{\textbf{x}}_{k|k})</math>

::::::or 

:::::::<math>\textbf{P}_{k|k} = \textrm{cov}(\textbf{x}_k - \hat{\textbf{x}}_{k|k}|\textbf{z}_{1 \ldots k})</math>

::::::?  It feels as if there's big gaps in our proof that the Kalman filter is valid... &mdash; [[User:ciphergoth|ciphergoth]] 17:38, 2005 Apr 30 (UTC)

::::::: I'm pretty sure <math>\textbf{P}_{k|k} = \textrm{cov}(\textbf{x}_k - \hat{\textbf{x}}_{k|k}|\textbf{z}_{1 \ldots k})</math>, since the Kalman filter is a causal recursive estimator which incorporates the latest measurements available into its estimates. --[[User:Orderud|Fredrik Orderud]] 11:43, 1 May 2005 (UTC)

== Underlying dynamical system ==

I removed the reference to a Markov Chain, and replaced it with Probabilistic Graphical Model, as I think this introduces less confusion - although Markov Chains can be defined on continuous variables, it seems the most widely understood definition is as a Finite state machine, as, for example at http://www.nist.gov/dads/HTML/markovchain.html .  Hidden Markov Models and Kalman Filters are derived from the same Probabilistic Graphical Model.  When time permits, I might write a section illustrating the duality between HMM and the Kalman Filter.

: I disagree. 
: [[Probabilistic graphical model]] is a rarely used "nonsense" term, that does not say anything about the specific [[Bayesian network]] encountered in Kalman filtering. The process- & observation-models yields a Bayesian network on a special sequential/recursive form, consisting of 1st order [[Markov chains]] for state propagation and 0th order Markov chains for the measurements. This form is better known as a [[Hidden Markov Model]] with continous state.
: Please go change the [[Markov chain]] article first is you think that Markov chains are somewhat restricted to systems with discrete state. (no pun intended)
: Alternatively, you can use the term "Markov process" instead [http://mathworld.wolfram.com/MarkovProcess.html], which undoubtedly covers systems with continous state. --[[User:Orderud|Fredrik Orderud]] 19:15, 10 May 2006 (UTC)

::I agree with [[User:Orderud|Fredrik Orderud]] - just because the state modelled by a Kalman filter is continuous, doesn't mean it's not a Markov model.  I know that it's more usual to use Markov model to refer to things with discrete state but it's not the only application.  Please stop removing these references from the article! &mdash; [[User_talk:Ciphergoth/Please don't use this|ciphergoth]] 19:40, 10 May 2006 (UTC)



'''---Untrue that Kalman filter requires Gaussian Noise model---'''
I have developed Kalman filters for NASA and taught a graduate level course on the topic. I think the literature on the subject, including this Wiki article is horrible. For example, is untrue that the Kalman filter underlying model depends on Gaussian modeled noise. The derivation minimizes the generalized state variance (opaquely referred to as "covariance matrix") using the data and it's uncertainly. On the contrary, data does not need to be Gaussian distributed... its expected error needs to be known (or as is usual, guessed.)  <small>—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/70.139.190.35|70.139.190.35]] ([[User talk:70.139.190.35|talk]]) 17:12, 7 September 2008 (UTC)</small><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->
:I think the point is that the Kalman filter is an optimal linear filter if the noises are Gaussian (normally distributed). That is true (and very important point) that they do not however need to be Gaussian for Kalman filter to still work. I think the only required assumption for the data is that the noise average is zero (white noise). '''Assumptions section''' (with citations) that would state clearly what the Kalman filter usage requires would definitely make the article more understandable. I hope there is somebody with depth enough knowledge about the subject to add one! [[User:Yebbey|Yebbey]] ([[User talk:Yebbey|talk]]) 14:28, 3 December 2010 (UTC)

== Inferring backwards? ==

A Kalman model will use today's observation to estimate today's state.  What do you use when you want to use today's observation to improve your estimate of yesterday's state? &mdash; [[User_talk:Ciphergoth/Please don't use this|ciphergoth]] 09:59, 30 July 2006 (UTC)

:Well one could simply run the Kalman model in reverse, though if the dynamics aren't reversible you might have to modify your choice of state space and dynamics so you can invert the matrix that evolves the system forward in time  (I think this should always be doable by adding 'dummy' state variables).  Then you can just apply the Kalman model starting from the last time point and evolving it backwards in time.

:Now if you mean to ask something more involved, namely how can you use ''both'' past and future information together (using all the info together) to improve estimates the question is much harder.  One might try to use the Kalman method both in forward and reverse together, for instance instead of the real measurements use the output of the forward Kalman method as input into running it into reverse.  However, I think it very likely this sort of technique '''will not work''' but I don't really know.  In fact my motivation for answer this question was curiousity about whether this sort of use is possible. [[User:Logicnazi|Logicnazi]] 22:36, 10 August 2006 (UTC)

::There are definitely well-understood techniques that use both past and future information together.  I don't think they're based on the sort of Kalman variants you suggest.  I just don't know how to look for them because I don't know what they're called. &mdash; [[User_talk:Ciphergoth/Please don't use this|ciphergoth]] 10:49, 11 August 2006 (UTC)

:::Estimation using both past and future information is "interpolation".  There is a 1960 paper by Kalman at http://www.elo.utfsm.cl/~ipd481/Papers%20varios/kalman1960.pdf that has some references. [[User:Jrvz|Jrvz]] 20:56, 31 August 2006 (UTC)

:: "Smoothing" has been utillized extensively in actual applications, and software is currently in use at DoD test ranges based in the technique.  I see that smoothing is mentioned in the "examples" paragraph -- it might be enough to include references from the 70's and 80's that developed the algorithms involved.  The ones I am aware of came from Bierman, who used the "information filter" formulation for the forward pass, and then performed a backward pass for the smoothing.  I'll try to locate the references for possible inclusion.

:: It occurs to me that the other thing done back then by Bierman and others was to apply factorization techniques to reduce the ranges of numerical values being manipulated (since manipulation was of square roots of quantities instead of the quantities themselves).  It might be worthwhile to also include a brief discussion, and references, related to this.  paul.wilfong at ngc.com

== Peter Swerling ==

Peter Swerling was a radar engineer who is most famous for the "Swerling Models" and had many contributions to the field of electrical and electronic engineering. The fact that he discovered the Kalman filter (and published it) before Kalman is simply an interesting side note in his life. Why "Peter Swerling" gets re-directed to this page on the Kalman filter is beyond me. He should have a page of his own with a biography and so on.

== recursive link ==
The link
http://en.wikipedia.org/wiki/Peter_Swerling

entitled "Peter Swerling" leads back to this Kalman Filter Page!
[[User:Carrionluggage|Carrionluggage]] 21:42, 1 August 2006 (UTC)

== Data Fusion - Combining Information from Related Observations ==
I've been asked to design a Kalman Filter where we can observe several states of the process
(some of which have relationships) and to use the Kalman filter to combine related observations to get a better estimate of each.

Some texts I've been reading seem to indicate instead of making a prediction and measurement and using these to form the best estimate, two measurements are combined to form the best estimate (of one of the measurements).   In examples, the two measurements seem to be usually  a state and its derivative.   

I find all of this quite confusing - but if it's a technique used in Kalman filtering, perhaps it needs mentioning?

Let's say, for example, we can measure
<math> u,\ v,\ w,\ V_{tot} </math>
where <math>u,\ v\ and\ w</math> represent the speed in 3 dimensions and <math>V_{tot}</math> is the total velocity - i.e. <math>V_{tot}=\sqrt{(u^2+v^2+w^2)}</math>

I guess the rates of change of these variables are also observable.  How could the Kalman filter be used here (where functions to make predictions of the next step are unknown)...

Something else that would really be terrific would be an worked example of an ekf with a nonlinear system - many people seem to have difficulty understanding this (myself included, and I've been reading about them for over a year now!). 
--[[User:Ultimâ|Ultimâ]] 20:28, 16 September 2006 (UTC)


:The [[sensor fusion]] process in the 'update' section of the article can be used to blend together multiple sensors. The simulation is just another sensor. blending order doesn't matter. That's not clear in the article but the sensor fusion used here has properties identical to addition, (eventually I'll update the article to explain why)..


:'State' is anything that has memory (and is worth the trouble to model). Objects generally 'remember' their velocity (derivative of position) so where your text measures the position and velocity it's probably just making one measurement of the state vector.

     
:Unless an object remembers it's acceleration from step to step and you feel like modeling the acceleration, it should probably just be left in the noise model.
:[[User:Sukisuki|Sukisuki]] ([[User talk:Sukisuki|talk]]) 11:55, 8 January 2010 (UTC)

== Relationship between Digital and Kalman Filters ==

The Wikipedia article for Digital Filter has a reference to the Kalman Filter article.  Neither has a discussion about the relationship between them, and I know from experience that it would be very helpful to include a brief discussion about the relationship.  I have added such, and hope it is acceptable.

paul.wilong at ngc.com

== equation typo ==

I changed the (innovation (or residual) covariance) equation as it looked as if the R's and P's got switched.  If someone who is better versed in state space modeling could double check that this is a correct change I would appreciate it.  I'm still a newbie at this state space stuff.  Much appreciated.

--(Reply)--
Perhaps some prankster came along and switched the variables.  Equation 
:<math>\textbf{S}_{k} = \textbf{H}_{k}\textbf{P}_{k|k-1} \textbf{H}_{k}^{T}+\textbf{R}_{k} </math>
looks the same as it was last March. [[User:Ultimâ|Ultimâ]] 11:57, 14 November 2006 (UTC)

== Kalman filter implementation ==

Can be a good improvement to add an implementation part to the article. 
Can be usefull have some indication on the practical right algoritm to implement a robust, against ill conditioning, implementation of KF or EKF. Particularly, wich is the most indicated matrix decomposition to use for that. Thanks Mauro [[User:Oliverone|Oliverone]] 19:19, 3 April 2007 (UTC)oliverone.

== Unscented Kalman Filter Questions ==

I'm new to UKF.  I read the section on it, but I'm having a hard time understanding with the notation.  Take, for example, this step in the UKF prediction:

:<math>\chi_{k|k-1}^{i} = f(\chi_{k-1|k-1}^{i}) \quad i = 0..2L </math>

Here's where I'm stuck.  <math>f</math> is a function of <math>x</math>, the state vector.  Let's call its dimension <math>N</math>.  On the other hand, the sigma points, <math>\chi^i</math>, have dimension <math>2N</math> (if I understand everything correctly).  So I don't quite understand how to evaluate <math>f(\chi^i)</math>.  Is it really <math>f(\chi^i(1:N))</math> (to borrow MATLAB notation)?

The same thing applies when evaluating <math>h(\chi^i)</math> in the UKF update.


== Equation Fix Suggestion ==
I think unnecessary confusion is caused by using F in both the linear and non-linear Kalman filter equations.   'A' should be used for the first instance as the state transition matrix (this also means amending the diagram).  In the second instance, F is the Jacobian of the state transition matrix, the state matrix and the output matrix- ie. it should not be referred to as the state transition matrix.    --[[User:Ultimâ|Ultimâ]] ([[User talk:Ultimâ|talk]]) 10:57, 19 May 2008 (UTC)

Really, the whole page should be made to use the same notation as the [[linear-quadratic-Gaussian control]] article.  <small>—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Dave.rudolf|Dave.rudolf]] ([[User talk:Dave.rudolf|talk]] • [[Special:Contributions/Dave.rudolf|contribs]]) 18:14, 24 June 2008 (UTC)</small><!-- Template:Unsigned --> <!--Autosigned by SineBot-->

== Missing term in UKF equations? ==

I think that there is a missing term in this equation:-

<math>\textbf{P}_{z_{k}z_{k}} = \sum_{i=0}^{2L} W_{c}^{i}\ [\gamma_{k}^{i} - \hat{\textbf{z}}_{k}] [\gamma_{k}^{i} - \hat{\textbf{z}}_{k}]^{T} </math>

Comparing it with Box 3.1 in Julier's original paper ("A New Extension of the Kalman Filter to Nonlinear Systems"), there is a missing <math>\textbf{R}_k</math> which should be added on to this.  I think that it should read:-

<math>\textbf{P}_{z_{k}z_{k}} = \textbf{R}_k + \sum_{i=0}^{2L} W_{c}^{i}\ [\gamma_{k}^{i} - \hat{\textbf{z}}_{k}] [\gamma_{k}^{i} - \hat{\textbf{z}}_{k}]^{T} </math>

--[[Special:Contributions/155.198.108.162|155.198.108.162]] ([[User talk:155.198.108.162|talk]]) 06:51, 18 June 2008 (UTC)

==  I think the notion of dynamic system model equation is wrong ==

I think the formula 
<math>\textbf{x}_{k} = \textbf{F}_{k} \textbf{x}_{k-1} + \textbf{B}_{k}\textbf{u}_{k}  + \textbf{w}_{k}</math> is wrong. 

I have found many books talk about this equation.
such as <<kalman filtering, Theroy and practics using Matlab, second edition>> by G,S,Grewal. Another book is <<Optimal State Estimation>> by D.Simon. And the pdf tutorials in http://www.cs.unc.edu/~welch/media/pdf/kalman_intro.pdf

They all use the linear equation

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

So, I think it's necessory to change the equations to eliminate ambiguity.
Thank you --[[User:Ollydbg|Ollydbg]] ([[User talk:Ollydbg|talk]]) 02:51, 29 July 2008 (UTC)
::::I agree with you - which is why I'm here now. You're right (end of my comment here) [[User:KorgBoy|KorgBoy]] ([[User talk:KorgBoy|talk]]) 03:52, 29 September 2017 (UTC)


:A. Gelb's 'Applied Optimal Estimation' and others (e.g. Trawny's http://www-users.cs.umn.edu/~trawny/Publications/Quaternions_3D.pdf) use the formula

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

:where '''G''' is the noise matrix.

:They also make a clear distinction between continuous time matrices (which they call '''F''', '''L''' and '''G'''), and their discrete time counterparts, which they denote with Greek letters <math>\Phi</math>, <math>\Lambda</math> and <math>\Gamma</math>.

:Our '''F''' is their state transition matrix <math>\Phi</math> and is the solution of the differential equation

:<math>
d/dt \Phi(t, t_0) = \textbf{F}(t) \Phi(t, t_0),
</math>

:with <math>\Phi(t_0, t_0) = I</math>.

:Our '''B''' is their control matrix <math>\Lambda</math> and determined by

:<math>
\Lambda_k u_k = \int^{t_{k+1}}_{t_k} \Phi(t, t_k) \textbf{L}(t) \textbf{u}(t) dt
</math>

:Our '''G''' is their noise matrix <math>\Gamma</math> and is calculated as

:<math>
\Gamma_k w_k = \int^{t_{k+1}}_{t_k} \Phi(t, t_k) \textbf{G}(t) \textbf{w}(t) dt 
</math>
.

:The noise covariance matrix is calculated as

:<math>
\textbf{Q}_k = \int^{t_{k+1}}_{t_k} \textbf{F}(t_{k+1}, t) \textbf{G}(t) \textbf{Q}(t) \textbf{G}^T(t) \textbf{F}(t_{k+1}, t)^T dt
</math>
.

:It should also be noted, that when going from continuous time covariances <math>\sigma_c</math> to discrete time covariances <math>\sigma_d</math>, factors of <math>\sqrt {\Delta t}</math> have to be introduced. Trawny's paper has examples and cites ::R.O. Allen and D.H. Chang, "Performance Testing of the Systron Donner Quartz Gyro (...)", JPL, Tech. Rep. ENGINEERING MEMORANDUM EM #343-1297, 1993
:but I don't have access to this paper.  <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/188.174.16.15|188.174.16.15]] ([[User talk:188.174.16.15|talk]]) 11:42, 7 February 2010 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->


::I agree that '''x'''<sub>k</sub> is dependent on '''u'''<sub>k-1</sub> and '''w'''<sub>k-1</sub>, not on '''u'''<sub>k</sub> and '''w'''<sub>k</sub>. This is in accordance with what I was taught by Howard Rosenbrock, who worked closely with Kalman and with what I wrote in my own dissertation.
::PS - I took the liberty of indenting the previous editor's comments. [[User:Martinvl|Martinvl]] ([[User talk:Martinvl|talk]]) 09:04, 8 June 2011 (UTC)

:::[[File:Kalman_filter_model2.svg]]
:::Is this about how you would do it?  I previously made some comments against making this change, but I now think it is a good idea. I think it is more intuitive, and also more consistent with the current way the EKF is described. Does anyone think it would be a bad idea to change this? --[[User:Headlessplatter|Headlessplatter]] ([[User talk:Headlessplatter|talk]]) 21:28, 8 June 2011 (UTC)

==Error with diagram?==

[[Image:Kalman filter model.png|right|thumb|250px|Model underlying the Kalman filter. Circles are [[vector space|vectors]], squares are [[matrix (mathematics)|matrices]], and stars represent [[multivariate normal distribution|Gaussian noise]] with the associated [[covariance matrix]] at the lower right.]]

Does this diagram imply x(k) and x(k+1) are the same?  <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/144.189.100.25|144.189.100.25]] ([[User talk:144.189.100.25|talk]]) 21:50, 29 January 2009 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->

:The fact that it passes through F is enough to tell me that they are different. Ideally, x should be subscripted with k-1 and k respectively, however. --[[Special:Contributions/76.27.96.159|76.27.96.159]] ([[User talk:76.27.96.159|talk]]) 21:26, 27 February 2009 (UTC)

This diagram implies that F, B, and H are all hidden. In fact, these three functions must be known to the user because they are part of the update equations. This diagram needs to be redrawn to indicate that only x, R, v, Q, and w are hidden.--[[Special:Contributions/128.187.80.2|128.187.80.2]] ([[User talk:128.187.80.2|talk]]) 22:01, 25 February 2009 (UTC)

: Actually, R and Q should be visible too. They are specified by the user as parameters, although several techniques exist to estimate them. Only x, v, and w should really be shown in the hidden area. --[[Special:Contributions/128.187.80.2|128.187.80.2]] ([[User talk:128.187.80.2|talk]]) 23:28, 2 March 2009 (UTC)

can someone explain the differences between the model equation in the "Underlying dynamic system model" section and the prediction equation in "The Kalman filter" section. B[k] -> B[k-1] and u[k] -> u[k-1] and process noise term w[k] is dropped. Does the diagram need to be changed by moving B matrix and u vector to the k-1 region? Thanks.  <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/148.104.5.2|148.104.5.2]] ([[User talk:148.104.5.2|talk]]) 17:29, 31 March 2009 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->

[[File:Kalman filter model 2.svg|right|thumb|631px|Model underlying the Kalman filter. Squares represent [[matrix (mathematics)|matrices]], ellipses represent [[multivariate normal distribution|multivariate normal distributions]] (with the mean and covariance matrix enclosed), unenclosed values are [[vector space|vectors]].]]

Here's a candidate diagram that I made to replace that old one. My diagram expresses more information, it more accurately represents the Kalman filter, and I think it addresses the problems mentioned here. --[[User:Headlessplatter|Headlessplatter]] ([[User talk:Headlessplatter|talk]]) 18:44, 7 April 2009 (UTC)

: v and w should also be subscripted, because they represent different values in each time frame.--[[Special:Contributions/128.187.80.2|128.187.80.2]] ([[User talk:128.187.80.2|talk]]) 18:59, 7 April 2009 (UTC)

:: Okay, I fixed that. --[[User:Headlessplatter|Headlessplatter]] ([[User talk:Headlessplatter|talk]]) 17:17, 8 April 2009 (UTC)

: It's been nearly two months without comment, so I went ahead and updated that diagram. --[[User:Headlessplatter|Headlessplatter]] ([[User talk:Headlessplatter|talk]]) 16:59, 5 June 2009 (UTC)

There is another area in the diagram with which I am not happy - and which links up to earlier comments.  I think that the input u<sub>k-1</sub> and the martix '''B''' should feed into the oval containing P<sub>k</sub> and x<sub>k</sub>, not the oval containing P<sub>k-1</sub> and x<sub>k-1</sub>.  After all, this is part of the predictor section of the algorithm, not the corrector part. [[User:Martinvl|Martinvl]] ([[User talk:Martinvl|talk]]) 13:55, 4 March 2010 (UTC)

: This really seems to be a matter of deciding when "time" is incremented. If I understand your complaint, you would like time to advance after u fees into B, but before B influences P and x. The easiest way to implement this change would be to slide the shaded time-sections to the right a little bit. Unfortunately, the formula in the "Predict" section seems to agree with the diagram. That is, x_k is a function of u_k, not u_{k-1}. I am also somewhat confident that the scholarly literature on the subject agree with these formulas. So, I think the convention is consistent with the way the diagram shows the advancement of time, and there is too much momentum behind this convention to change it now. (It's like electricity flowing from - to +. It's too established to fix.)--[[User:Headlessplatter|Headlessplatter]] ([[User talk:Headlessplatter|talk]]) 22:28, 16 December 2010 (UTC)

[[File:Basic concept of Kalman filtering.svg|thumb|400px]]
I added another diagram which tries to give a higher level view on the subject, explain the different variables etc. -- [[User:Petteri Aimonen|Petteri Aimonen]] ([[User talk:Petteri Aimonen|talk]]) 18:20, 25 November 2011 (UTC)

==Rejection of data==
I had the delusion that the original Kalman filter (Kalman (1960)) included a method for culling (or rejecting, or abandoning, or filtering) outliers. Where did I get this from? [[User:Albmont|Albmont]] ([[User talk:Albmont|talk]]) 21:44, 10 February 2009 (UTC)

:The original Kalman filter does culling in an analog sort of way, in that an input that tends to be an outlier tends to get a high covariance and a low weight so the value ends up impacting the results by a small amount. But to do this in an "on vs. off" sort of way by "using vs. omitting" an input from the Kalman filter needs to be done by logic that is outside the Kalman filter. A large spike, pulse, or step in a measurement is not handled well by the analog-like response of the original Kalman filter, so this would need to be handled by separate logic.
:In Wikipedia, the articles [[Data validation]] and [[Bounds checking]] are not oriented to data validation of real-time measurements or signals. The article [[Robust statistics]] describes methods of dealing with outliers, some of which could be adapted to real-time measurements or signals. But I couldn't find a Wikipedia article that addressed the issue of culling (or rejecting, or abandoning, or filtering) outliers in real-time measurements or signals. Based on my experience with real-time data acquisition systems, I included below a list of methods to flag data as questionable or invalid. Using a Kalman filter with robust handling of outliers would fall in the category of "Propagation through calculated values (complex)". Simple smoothing (e.g., [[Low-pass_filter#Algorithmic_implementation|exponential filter]]) would fall in the category of "Propagation through calculated values (simple)".
:While I was working at a nuclear power plant, a co-worker tested the use of a Kalman filter to calculate data, in this case combine the 4 redundant sensors measuring the power output of the reactor into a single value. The goal was to compare the Kalman filter with simpler forms of [[sensor fusion]] (e.g., taking the maximum valid value). The results were that most of the time the Kalman filter produced improved outputs, but that there were occasional anomalous situations that disturbed the output of the Kalman filter and made it worse than simpler forms of sensor fusion. I think the problem in this application was exactly the same as your question, what is the proper way to deal with outliers in data. The outliers need to be flagged as questionable or invalid data, and the calculation handle these flagged values appropriately.

:These methods can be used with real-time data acquisition systems to flag data as questionable or invalid:
** Sensor range error - the measuring and/or data acquisition system reported a value that is out of range high or low, which can be used to flag data as invalid
** No value - the measuring and/or data acquisition system provided no value within the expected timeframe, which can be used to flag data as questionable or invalid depending on how stale the data is
** Change in value - a measurement or signal whose value changed by an amount that is not reasonable (high or low) since the previous value can be used to flag data as questionable or invalid
** Statistical validation - as a byproduct of curve fitting to smooth data and calculate rates of change, the goodness of fit can be used to flag data as questionable or invalid
** Redundant sensors - two or more sensors measuring the same value can be compared in order to flag data as questionable or invalid
** Redundant data paths - the same sensor that gets to the system by more than one path can be compared in order to flag data as questionable or invalid
** Cross-validation of sensors - two or more sensors that are not measuring the same thing, but can be related by a calculation, can be compared in order to flag data as questionable or invalid
** Propagation through calculated values (simple) - questionable or invalid data used as the input to a calculation is used to flag the results of the calculation as questionable or invalid; if any input is questionable then the output is flagged as questionable, and if any input is invalid then the output is flagged as invalid (invalid overrides questionable)
** Propagation through calculated values (complex) - a calculation or algorithm uses a set of tests, logic, and voting to use or omit values; the tests and logic looks at the questionable and invalid flags of the input data and the intermediate results in the calculation to determine the questionable or invalid flags of the outputs

:[[User:Obankston|Obankston]] ([[User talk:Obankston|talk]]) 20:11, 19 February 2010 (UTC)

== Example is inconsistent? ==

The example suggests that the B matrix is to be ignored: "There are no controls on the truck, so we ignore Bk and uk"
However, a couple of lines down we see:

From [[Newton's laws of motion]] we conclude that

:<math>\textbf{x}_{k} = \textbf{F} \textbf{x}_{k-1} + \textbf{B}a_{k}</math>

and

:<math>\textbf{B} = \begin{bmatrix} \frac{\Delta t^{2}}{2} \\ \Delta t \end{bmatrix} </math>

I find this very confusing. Since the <math>a_{k}</math> is used, it suggests that the acceleration is meassured, and used to update the predicted state of the truck. Maybe the description of what is going on needs to be updated?<span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/195.84.34.4|195.84.34.4]] ([[User talk:195.84.34.4|talk]]) 09:35, 16 December 2009 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->

: I agree that statement does not make any sense. Other than that, the example seems fine though. --[[User:Drizzd|Drizzd]] ([[User talk:Drizzd|talk]]) 19:14, 18 December 2009 (UTC)
@above: I would much rather use uk and Xk|k-1 to keep with the same notation.  <small><span class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Milte001|Milte001]] ([[User talk:Milte001|talk]] • [[Special:Contributions/Milte001|contribs]]) 10:58, 1 April 2011 (UTC)</span></small><!-- Template:Unsigned --> <!--Autosigned by SineBot-->

Also, in the example the matrix H is given as a 1x2 matrix. That doesn't make sense. In order to be used in the residual covariance equation Sk = HkPk|k-1Hk^T+Rk, wouldn't it need to be 2x2 like P? Also, doesn't that make R from the example need to be a 2x2 matrix as well? --[[User:Madprog|Madprog]] ([[User talk:Madprog|talk]]) 18:35, 5 January 2010 (UTC)

@Madprog: Yes, it does: <math>S_{1x1} = H_{1x2} P_{2x2} H^T_{2x1} + R_{1x1}</math> works perfectly.

The older version, which had '''G''' instead of '''B''' was (half) right and should be reinstated, but only after the noise matrix '''G''' has been properly introduced (see 'notion of dynamic system' above).

But due to the confusion of continuous and discrete time variables there is still a problem, because the formula for the covariance '''Q''' turns out to be wrong.

To do it properly, start with the continuous system (I use lowercase letters for the continuous time matrices instead of uppercase letters as in the sources cited above to avoid confusion):

The dynamics of position x, velocity <math>v = \dot{x}</math> and acceleration <math>a = \ddot{x}</math> of the truck are described by the linear differential equation

:<math>\begin{bmatrix} \dot{x} \\ \ddot{x} \end{bmatrix} =
\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} x \\ \dot{x} \end{bmatrix}
+ \begin{bmatrix} 0 \\ 1 \end{bmatrix} w_a         
</math>

where the dot denotes the derivative with respect to time,

<math>
\textbf{f} = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}
</math>

is the system matrix,

<math>
\textbf{g} = \begin{bmatrix} 0 \\ 1 \end{bmatrix}
</math>

is the noise matrix and <math>w_a</math> is a random acceleration with characteristics <math>\textrm{E}[w_a] = 0</math> and <math>\textrm{E}[w_a(t+\tau)w_a(t)] = \sigma^2_{a_c}\delta(\tau)</math>.

Integration of the homogenous differential equation yields the state transition matrix

<math>
\Phi(t_1, t_0) = \Phi(\Delta t=t_1 - t_0) = \begin{bmatrix} 1 & \Delta t \\ 0 & 1 \end{bmatrix}
</math>

and, assuming <math>w_a(t) = w_k = \textrm{const}</math> during the integration interval,

<math>
\Gamma_k w_k = \int_{t_k}^{t_{k+1}} \Phi(t_{k+1}, \tau) \textbf{g}(\tau) w_a(\tau) d\tau
= \begin{bmatrix} \frac{\Delta t^{2}}{2} \\ \Delta t \end{bmatrix} w_k 
</math>

is the discrete time inhomogenous driving force.

Together, this comprises the discrete time system model

<math>
\textbf{x}_{k+1} = \Phi_k \textbf{x}_k + \Gamma_k w_k
</math>

The discrete time noise covariance matrix is calculated as

<math>
\textbf{Q}_{k+1} = \int_{t_k}^{t_{k+1}} \Phi(t_{k+1}, \tau) \textbf{g}(\tau)
\textbf{Q}(\tau) \textbf{g}^T(t) \Phi^T(t_{k+1}, \tau) d\tau
</math>.

Assuming <math> Q(t) = \sigma^2_{a_c,k} = \textrm{const} </math> during the integration interval, we get

<math>
\textbf{Q}_{k+1} = \sigma^2_{a_c} \int_{t_k}^{t_{k+1}} \begin{bmatrix}t \\ 1 \end{bmatrix}
\begin{bmatrix}t & 1 \end{bmatrix} dt

= \sigma^2_{a_c} \int_{t_k}^{t_{k+1}} \begin{bmatrix}\frac{t^2}{2} & t \\ t & 1 \end{bmatrix} dt

= \sigma^2_{a_c} \begin{bmatrix}\frac{{\Delta t}^3}{6} & \frac{{\Delta t}^2}{2} \\ \frac{{\Delta t}^2}{2} & \Delta t \end{bmatrix} 
</math>
.

With <math>\sigma^2_v = \sigma^2_{a_c} \Delta t</math> this resolves to

<math>
\textbf{Q}_{k+1} = \sigma^2_v \begin{bmatrix}\frac{{\Delta t}^2}{6} & \frac{\Delta t}{2} \\ \frac{\Delta t}{2} & 1\end{bmatrix}
</math>

where <math>\textbf{Q}_{11, k+1}</math> is the position error and <math>\textbf{Q}_{22, k+1}</math> is the velocity error.

At each time step, a noisy measurement of the true position of the truck is made. Let us suppose the measurement noise ''v<sub>k</sub>'' is also normally distributed, with mean 0 and standard deviation ''σ''<sub>z</sub>.

:<math>\textbf{z}_{k} = \textbf{H x}_{k} + \textbf{v}_{k}</math>

where 

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

and

:<math>\textbf{R} = \textrm{E}[\textbf{v}_k \textbf{v}_k^{\text{T}}] = \begin{bmatrix} \sigma_z^2 \end{bmatrix} </math>  <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/188.174.16.15|188.174.16.15]] ([[User talk:188.174.16.15|talk]]) 15:26, 7 February 2010 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->

Your statement of the problem differs from the one in the article: you have a constantly-changing acceleration, accelerations at any two different times being uncorrelated, whereas the article has acceleration that's constant on each of the discrete time intervals (hence nonzero correlation when tau < delta-t). I think Q in the article is correct, given how the problem is stated there. I've fixed the B-versus-G confusion in the article and left Q as it was. [[User:Gareth McCaughan|Gareth McCaughan]] ([[User talk:Gareth McCaughan|talk]]) 14:06, 1 March 2010 (UTC)

This argument seems wrong to me: the derivation of the discrete time model '''does''' assume a constant acceleration during the integration interval. When you consider the discrete system, you can and do not make statements about the autocorrelation during the integration interval.  But even if we assume for a moment, that there is a different function at work, the formula in the article still does not work out, because if we set

<math>
\textbf{Q}_{article} = \begin{bmatrix}\frac{{\Delta t}^4}{4} & \frac{{\Delta t}^3}{2} \\ \frac{{\Delta t}^3}{2} & {\Delta t}^2\end{bmatrix}
= \int_{t_k}^{t_{k+1}} \begin{bmatrix}t \\ 1 \end{bmatrix} Q(t)
\begin{bmatrix}t & 1 \end{bmatrix} dt
</math>

with some unknown <math>Q(t)</math>, and take the derivative on both sides, it follows, that there exists no <math>Q(t)</math>, which can solve this matrix equation.

(Former [[User talk:188.174.16.15|talk]] with a different IP)

All that means is that the discrete model is not a discretization of a well-behaved continuous-time system. Specifically, the process noise is not happening in continuous time. Instead, it affects the acceleration which will then apply (constant, no further noise) over each time interval. So Q(t) consists of delta-function spikes and it makes no sense to take its derivative. Yes, this is a very artificial setup, but it has to be if it's going to give an opportunity to explain the discrete-time Kalman filter using a real-world-looking example. [[User:Gareth McCaughan|Gareth McCaughan]] ([[User talk:Gareth McCaughan|talk]]) 11:29, 9 March 2010 (UTC)

This argument makes no sense to me. Q(t) is never differentiated, only integrals involving Q(t):

Looking at the (2, 2) component of the matrix equation, <math>Q_{article, 2,2} = \Delta t^2 = \int_0^{\Delta t} Q(t) dt </math>. This integral is well defined, however nasty Q(t) is, and it follows, that <math>\frac{d}{d \Delta t} \Delta t^2 = 2 \Delta t = \frac{d}{d \Delta t} \int_0^{\Delta t} Q(t) dt = Q(\Delta t)</math>, a perfectly well-behaved function (ideed, it looks like the covariance of a random walk). With Q(t) now known, the other components of the matrix equation can be checked:
<math>\int_0^{\Delta t} t Q(t) dt = \frac{2}{3} \Delta t^3 \ne \frac{\Delta t^3}{2}</math> and 
<math>\int_0^{\Delta t} t^2 Q(t) dt = \frac{2}{4} \Delta t^4 \ne \frac{\Delta t^4}{4}</math>. Ooops!

First: you're right that my comment about differentiating Q(t) was nonsense. Sorry about that. However, about the main point at issue, I still think you're simply wrong. You are assuming that <math>Q_{discrete}=\int Q_{continuous} dt</math>, which assumes that the (co)variances are additive, which is true only when they are independent, which is not true for the system specified in the example because the process noise is forced to be constant on our time intervals of length <math>\Delta t</math>. In other words, the system described in the article is of the right form for the application of a discrete-time Kalman filter but is *not* of the right form for the application of a continuous-time Kalman filter, because the process noise isn't white. That would be a problem if the example purported to show how the continuous-time Kalman filter works, but it doesn't; it's an example of a discrete-time Kalman filter, and it seems to me to serve that purpose just fine. [[User:Gareth McCaughan|Gareth McCaughan]] ([[User talk:Gareth McCaughan|talk]]) 21:25, 21 March 2010 (UTC)

Huh? The formulas above aren't about a continuous-time Kalman filter. It is about a dynamic system, which is initially described as a linear differential equation system, excited by Gaussian random noise. Using time-honored methods, the discrete time system dynamics is determined, as well as a Covariance matrix. Everything works out perfectly, with one exception: the results for <math>\textbf{Q}_k</math> cannot be reconciled. Now, as nobody has shown an error in the deduction itself, I have to ask:

Where does <math>\textbf{Q} = \textbf{G} \textbf{G}^T \sigma_a^2</math> come from, and are you sure that this formula is correct?

[Note: the two paragraphs above were unsigned; they were posted from 188.174.72.117; I assume it's the same person as before.]

<math>\textbf{Q} = \textbf{G} \textbf{G}^T \sigma_a^2</math> is the formula for the covariance matrix of a multivariate Gaussian obtained by taking a standard Gaussian distribution (in this case <math>a_k</math>) and applying a linear transformation (in this case G) to it.

As you rightly say, there doesn't appear to be an error in the derivation in the article. As I've pointed out, however, there is an error in your derivation: it assumes that <math>Q_{discrete}=\int Q_{continuous} dt</math>, which is not true in this case because the <math>Q_{continuous}(t)</math> at different t are correlated: it's the acceleration, not merely the distribution of the acceleration, that's constant on each time interval. [[User:Gareth McCaughan|Gareth McCaughan]] ([[User talk:Gareth McCaughan|talk]]) 23:06, 23 March 2010 (UTC)

:Assuming that the speed is fluctuating continuously throughout the interval delta-t, I believe the denominator in the leading element of the Q matrix should be 3, not 4.
:The reason for this is that, (suggested text), "Given the precise speed at the start of the interval, the variance of the displacement is delta-t^2 x the variance of the AVERAGE speed over the interval, and the variance of the average speed over the interval is 1/3 the variance of the speed at the end of the interval". (It can be verified that the Q matrix grows consistently in this fashion with multiples of delta-t in the absence of measurement, (i.e. setting K=0)).
:This version of Q is full rank, so the comment about its rank becomes redundant.
:Mike Hodgson. [[Special:Contributions/2A00:23C6:4A00:7D01:4DF8:5D9:4B51:B993|2A00:23C6:4A00:7D01:4DF8:5D9:4B51:B993]] ([[User talk:2A00:23C6:4A00:7D01:4DF8:5D9:4B51:B993|talk]]) 21:26, 17 November 2024 (UTC)

After some deliberation I think I now understand enough of the matter to be able to reconcile the differences.

:Executive summary: The integral formulas above are correct and valid for arbitrary values of <math>\Delta t</math>, but need some clarification (the introduction of <math>\sigma^2_v</math> was a bad idea). The expression <math>\textbf{G} \textbf{G}^T \sigma_a^2</math> is an approximation, which is valid only for "small" <math>\Delta t</math>. The discrepancies arise, when the approximation approaches the limit of its validity.

For the explanation, let's reiterate the relevant formulas and the results for the example system and write <math>\Gamma_k Q_k \Gamma_k^T</math> instead of <math>\textbf{Q}_{k+1}</math>, as I should have from the beginning:

:<math>
\Gamma_k w_k = \int_{t_k}^{t_{k+1}} \Phi(t_{k+1}, \tau) \textbf{g}(\tau) w_a(\tau) d\tau
= \begin{bmatrix} \frac{\Delta t^{2}}{2} \\ \Delta t \end{bmatrix} w_k 
</math>

:<math>
\Gamma_k Q_k \Gamma_k^T = \int_{t_k}^{t_{k+1}} \Phi(t_{k+1}, \tau) \textbf{g}(\tau)
\textbf{Q}(\tau) \textbf{g}^T(t) \Phi^T(t_{k+1}, \tau) d\tau
= \sigma^2_{a_c} \begin{bmatrix}\frac{{\Delta t}^3}{6} & \frac{{\Delta t}^2}{2} \\ \frac{{\Delta t}^2}{2} & \Delta t \end{bmatrix} 
</math>

<math>\sigma^2_{a_c}</math> is a spectral density. The corresponding discrete time covariance <math>\sigma^2_{a_d}</math>is defined via

:<math>\sigma^2_{a_d} \Delta t = \int_0^{\Delta t} \sigma^2_{a_c} \delta(0) d\tau = \sigma^2_{a_c}</math>

thus

:<math>
\Gamma_k Q_k \Gamma_k^T = \sigma^2_{a_d} \begin{bmatrix}\frac{{\Delta t}^4}{6} & \frac{{\Delta t}^3}{2} \\ \frac{{\Delta t}^3}{2} & {\Delta t}^2\end{bmatrix} 
</math>

which is equal to the result of the article, as long as terms of <math>{\textbf{O}(\Delta t}^4)</math> can be neglected.

Where do the integrals come from? To answer that, the coarse-grained time scale with period <math>\Delta t</math>, on which the discrete Kalman-Filter operates, is overlayed by a fine-grained timescale with period <math>\Delta t_{tiny}</math>. With each tick on the tiny timescale our system is excited by a randomly distributed acceleration, which propagates according the system equation until the next tick of the coarse-grained timescale.
 
<math>\Gamma_k w_k</math> is the sum of all those tiny accelerations, and can be treated, as if there was only one random acceleration event <math>w_k</math> at each tick of the coarse-grained timescale, provided it is multiplied with a suitable matrix <math>\Gamma_k</math>. But in general, quoting from A. Gelb's book, ''only the product <math>\Gamma_k w_k</math> is uniquely defined, not the individual terms <math>\Gamma_k</math> and <math>w_k</math>''. This is even more relevant for <math>\Gamma_k Q_k \Gamma_k^T</math>, which should be treated as some fancy glyph, not a matrix expression.

The right-hand side of <math>\Gamma_k Q_k \Gamma_k^T</math> is derived as the expectation of an uncorrelated random sequence <math>\Gamma_k w_k</math>:

:<math>E[(\Gamma_k w_k)(\Gamma_l w_l)] = \Gamma_k Q_k \Gamma_k^T \delta_{kl}</math>

via the integral definition of <math>\Gamma_k w_k</math> (for details, see chapter 3.6 of Gelb's book).

The const-ness of <math>\sigma^2_{a_c}</math> during the integration interval <math>\Delta t</math> means, that the distribution of the individual random accelerations on the tiny timescale does not change during the integration.

The term <math>\sigma^2_{a_d} \Delta t</math>, on the other hand, could maybe be construed as the covariance of a random walk (which is what you get, if you integrate random Gaussians) while sweeping across the integration interval, but that's for somebody else to figure out.

Anyway, thanks for the learning experience! Peter Pöschl ([[Special:Contributions/188.174.83.40|188.174.83.40]] ([[User talk:188.174.83.40|talk]]) 10:48, 2 April 2010 (UTC)) and other anonymous comments above.

:Assuming that the speed is fluctuating continuously throughout the interval delta-t, I believe the denominator in the leading element of the Q matrix should be 3, not 4.
:The reason for this is that, (suggested text), "Given the precise speed at the start of the interval, the variance of the displacement is delta-t^2 x the variance of the AVERAGE speed over the interval, and the variance of the average speed over the interval is 1/3 the variance of the speed at the end of the interval".  (It can be verified that the Q matrix grows consistently in this fashion with multiples of delta-t in the absence of measurement, (i.e. setting K=0)).
:This version of Q is full rank, so the comment about its rank becomes redundant.
:Mike Hodgson. [[Special:Contributions/2A00:23C6:4A00:7D01:4DF8:5D9:4B51:B993|2A00:23C6:4A00:7D01:4DF8:5D9:4B51:B993]] ([[User talk:2A00:23C6:4A00:7D01:4DF8:5D9:4B51:B993|talk]]) 21:23, 17 November 2024 (UTC)

== "Technical" template ==

This article is listed in [[:Category:Articles needing expert attention from January 2010]] and [[:Category:Wikipedia articles that are too technical]]. I created a more accessible lead and shuffled some sections and paragraphs so that the information is approximately sorted in order of ascending technical content. Is the article ready to remove the <nowiki>{{technical|date=January 2010}}</nowiki> template? [[User:Obankston|Obankston]] ([[User talk:Obankston|talk]]) 23:50, 16 February 2010 (UTC)

: The Too Technical template was removed in [http://en.wikipedia.org/w/index.php?title=Kalman_filter&action=historysubmit&diff=349607636&oldid=349560166]. [[User:Obankston|Obankston]] ([[User talk:Obankston|talk]]) 02:44, 17 March 2010 (UTC)

== Are the UKF function dimensions correct? ==

It looks to me like there's a problem with the dimensions of the functions in the UKF discussion.

Under the EKF section, the argument to the non-linear function, f, is a vector of dimension n.  In the UKF discussion, the argument to the function f is passed a vector of larger dimension, because the vector is augmented.  What do I misunderstand?

[[Special:Contributions/204.15.2.146|204.15.2.146]] ([[User talk:204.15.2.146|talk]]) 23:10, 15 March 2010 (UTC)

== Unit mismatch in the [[Kalman filter#Kalman gain derivation|Kalman gain derivation]] section ==

How can it be meaningful to speak of minimizing the trace of the [[covariance matrix]]? The diagonals in the covariance matrix will normally have mixed units, i.e., m<sup>2</sup> and (m/s)<sup>2</sup> for a simple constant velocity two-state Kalman filter (the Kalman analogue of the [[alpha beta filter]]). And adding those give no meaning from a physical unit point-of-view. Thus, one would have to consider a normalization of the state vector with the physical units, e.g., [m] and [m/s] for this trace minimization to be meaningful. 

But that brings me to another doubt, I can normalize lenghts with [[nm]] or [[light year]]s which will yield covariance matrices with very different numerical values in the diagonal depending on the normalization unit. As a matter of fact, it seems unclear for me, why it is the difference between the true and filtered target state vectors, which should be minimized and not the difference normalized with some typical scale of the problem in each dimension of the target state vector. 

I was thinking that it should be possible to make a [[maximum likelihood estimate]] of the Kalman gain matrix instead, but I just cannot figure out quite how to do that. In the ML-view, the optimal Kalman gain should drop out as a matrix weight in a [[weighted mean]] between [[multivariate normal distribution|multivariate normally distributed]] random vectors, which gives the maximum likelihood matrix weighted addition between a new and the ''a priori'' covariance mapped to the observation space via the observation matrix. 

The multivariates have this nice unit-less trace term in the [[log-likelihood]]

:<math>\operatorname{Tr}\left[(\mathbf{x}-\boldsymbol{\mu})^T\boldsymbol{\Sigma}^{-1}(\mathbf{x}-\boldsymbol{\mu})\right]</math>

Which seems more concise to work with for me.

I would love to see such a ML derivation instead in that section, as I think it could be "neater". If only I could figure out how to do it the MLE way, I would do it myself......

--[[User:Slaunger|Slaunger]] ([[User talk:Slaunger|talk]]) 07:45, 29 June 2010 (UTC)

Your concerns are answered in chapter 4 of A. Gelb's book: They start with a weighted scalar sum of the diagonal elements of the covariance matrix, so the cost function to be minimized is

:<math>\mathbf{J} = E[\tilde{x_k}^T \mathbf{S} \tilde{x_k}]</math>

where <math>\mathbf{S}</math> is an arbitrary positive semidefinite matrix. The important fact is, that the minimum of <math>\mathbf{J}</math> is ''independent'' of <math>\mathbf{S}</math>, therefore <math>\mathbf{S}</math> can be ''chosen'' to be the unit matrix, which results in  <math>\mathbf{J} = Tr[\mathbf{P_k}]</math>.

(Peter Pöschl) [[Special:Contributions/188.174.71.81|188.174.71.81]] ([[User talk:188.174.71.81|talk]]) 21:29, 5 July 2010 (UTC)

:Hi Peter, thank you for that explanation, which sheds some light on my question. I suppose '''S''' is the inverse covariance matrix (the precision matrix) in the expression from Gelb? Still, replacing a matrix with different physical units in the individual elements with a unitless identity matrix is really ugly IMO. I have done some internal training of collegues in the Kalman filter recently where we went through the derivation as it is stated here, and two commented that the derivation of the optimal Kalman gain did not make sense due to the mixed physical units - i.e., they are left with a feeling of a dummy derivation whose sole purpose is to arrive at the correct result without being a stringent derivation. I suppose though that I could use the argument still and replace with a diagonal matrix, which just have values of one times time natural unit of the element in the matrix, such that the cost function is without physicial dimensions. --[[User:Slaunger|Slaunger]] ([[User talk:Slaunger|talk]]) 07:58, 16 July 2010 (UTC)

== Kalman gain derivation not obvious ==

I am especially confused as to how 
:<math>\textbf{x}_{k} - \hat{\textbf{x}}_{k|k}</math>
is expanded to the monster below it. Some step-by-step seems in order here.  <small><span class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Dave.rudolf|Dave.rudolf]] ([[User talk:Dave.rudolf|talk]] • [[Special:Contributions/Dave.rudolf|contribs]]) 00:53, 12 January 2011 (UTC)</span></small><!-- Template:Unsigned --> <!--Autosigned by SineBot-->

== Erroneous Modified Bryson-Frazier formula ? ==

I think the formula
:<math>\tilde{\lambda}_k =  -\textbf{H}_k^T \textbf{S}_k^{-1} \textbf{z}_k + \textbf{C}_k^T \hat{\lambda}_k</math>
really should be
:<math>\tilde{\lambda}_k =  -\textbf{H}_k^T \textbf{S}_k^{-1} (\textbf{z}_k -\textbf{C}_k  \hat{\textbf{x}}_{k|k-1})  + \textbf{C}_k^T \hat{\lambda}_k</math>
This fits better with the reference, and seems to give the correct numerical results for me...  <span style="font-size: smaller;" class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/89.236.1.222|89.236.1.222]] ([[User talk:89.236.1.222|talk]]) 00:02, 22 June 2011 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->

== Hybrid Kalman filter ==

The section [[Kalman filter#Hybrid Kalman filter|Hybrid Kalman filter]] has broken math code. Does anyone know how to fix it? [[User:Chris857|Chris857]] ([[User talk:Chris857|talk]]) 17:51, 5 October 2011 (UTC)

== Intro sentence ==

:''Its purpose is to use measurements observed over time, containing noise (random variations) and other inaccuracies, and produce values that tend to be closer to the true values of the measurements and their associated calculated values.''

I'm sure a lot of collective work went into this sentence, but I'm afraid it is still not very clear. The second half sentence has too many "values". What is a "true value of a measurement", and what "associated calculated values" are we talking about? It seems the author wants to avoid talking about estimating the system's state, which is fine at this early stage, but "state of the system" and "calculated values associated to measurements" are not the same thing, and the latter isn't any simpler than the former. Further, the sentence states that the produced values are "closer to" something, raising the question: closer than what?

Let's start brainstorming. How about:
:''Its purpose is to use a series of measurements observed over time, containing noise (random variations) and other inaccuracies, and produce estimates that tend to be closer to the true unknown values than those that would be based on a single measurement alone.''

[[User:AxelBoldt|AxelBoldt]] ([[User talk:AxelBoldt|talk]]) 03:23, 29 December 2011 (UTC)


:: Your suggestion is better than the original, though both are ok. It would be good to get a comment from someone *not* familiar with the subject.
:: [[User:Petteri Aimonen|Petteri Aimonen]] ([[User talk:Petteri Aimonen|talk]]) 04:35, 29 December 2011 (UTC)

:::Ok, I'll put it in; if somebody complains we can always switch back. Cheers, [[User:AxelBoldt|AxelBoldt]] ([[User talk:AxelBoldt|talk]]) 21:03, 2 January 2012 (UTC)

I've rewritten the introduction in an attempt to clarify and unify it.  After [http://en.wikipedia.org/w/index.php?title=Kalman_filter&action=historysubmit&diff=470038749&oldid=469815028 this edit] it read like two introductions mashed together.  However, I'm a student just learning about the filter and a newcomer to this page, so it would be great if someone with more experience could review it.  The introduction also should be significantly shorter - perhaps some of the explanatory paragraphs should be merged into the main article.  [[User:StevenBell|StevenBell]] ([[User talk:StevenBell|talk]]) 18:23, 20 February 2012 (UTC)

== Example Animation ==

[[File:Kalman filter animation, 1d, frame 018.svg|thumb|400px|link=http://upload.wikimedia.org/wikipedia/commons/5/5e/Kalman_filter_animation%2C_1d.gif|Click for animation]]

I made this animation. Does anyone have some constructive criticism? Maybe it should go in the [[Kalman_filter#Example_application| example application section]]? Is this the right way to link a custom thumbnail to an animation? Thanks for the feedback.
--[[User:Suki907|Suki907]] ([[User talk:Suki907|talk]]) 22:44, 4 March 2012 (UTC)

:Nice animation. It could be useful to write the corresponding model in the page next to the animation. Custom thumbnail is not possible, unless such a feature has been added just recently.
:--[[User:Petteri Aimonen|Petteri Aimonen]] ([[User talk:Petteri Aimonen|talk]]) 08:11, 5 March 2012 (UTC)

::That should be easy, it's just a few 2x2 matrices, I'll also put the parameters into the comments page for the file, with the next version. --[[User:Suki907|Suki907]] ([[User talk:Suki907|talk]]) 14:13, 11 March 2012 (UTC)

:Great work! But: You have spelled measurement wrong. And also, i would add the system function to the pic, just in case. And, it would be much easier to understand, if the scales were fix. I've watched it several times now and I'm still not sure if it's right or if the star that represents the actual system state jumps nonsensically. --[[Special:Contributions/92.206.63.2|92.206.63.2]] ([[User talk:92.206.63.2|talk]]) 23:38, 5 March 2012 (UTC)

::Ok. So in the updated version I fixed the spelling, and stopped shearing the axes with the update step (you're not the first person to find that this makes it less clear). 

:: The reason I have the star jumping is because of the process noise. The kalman filter models the system update as a (linear transform) + (gaussian process noise). In this example the system I'm filtering meets the expectations of the kalman filter -> the system is stepped forward with the shear, then gaussian process noise is added.

:: The same noise covariance is also added to the filter state (in a well calibrate filter, the noise value used in the filter should match the noise in the actual system). 

:: So on the "add noise" frame you see two red elipses, one adding the noise to the filter, the other adding the same noise to the system state (that's when the star gets re-sampled). I'm prety sure it's correct, it just might not be the best way to show it.

::Thanks. --[[User:Suki907|Suki907]] ([[User talk:Suki907|talk]]) 14:13, 11 March 2012 (UTC)

Fixed spelling of Title above. Михал Орела 17:55, 13 March 2012 (UTC)

== References for Applications ==

The applications section is unconvincing (yes, I know the Kalman filter is important) because there are no references.  Further, the applications link to other wiki pages that often do not mention the Kalman filter.  This makes the present page look a little bit hyped up. [[User:Jfgrcar|Jfgrcar]] ([[User talk:Jfgrcar|talk]]) 06:46, 20 April 2012 (UTC)

== x or x dot ? ==

In some of the sections,   the state update equation is
x = Fx + Bu + w
and in other sections it is
x' = Fx + Bu + w                   where x' is x(dot) ,  presumably the time derivative.

Can these both be right ?

: I can see two variations, <math>\hat{\mathbf{x}}</math> ("x hat") and <math>\dot{\mathbf{x}}</math> ("x dot"). The hat barely underlines the fact that <math>\hat{\mathbf{x}}</math> is an estimate. It is quite common to leave the hat off, though of course it would be good to have it used everywhere in the article or not at all

: On the other hand, x dot means time derivative like you guessed. It is only used in the continuous-time case, and this difference has a logical reason: in discrete case, we update at one moment <math>x_2 = F_d x_1</math>, in continuous case the x is changing continuously with the instantaneous derivative <math>F_c x</math>. The F matrix for these cases is different, and <math>F_d</math> can be found as a solution to the differential equation <math>\dot{x} = F_c x</math>. The solution is usually approximated with [[Taylor series]] as <math>F_d = I + T F_c + (T F_c)^2 / 2 + ...</math> (if I recall the equation correctly), where T is the timestep.

: Usually continuous time Kalman filters are not very nice computationally, but they are more closely related to the actual physics and can be easily converted to discrete version for calculations.

: -- [[User:Petteri Aimonen|Petteri Aimonen]] ([[User talk:Petteri Aimonen|talk]]) 15:31, 4 June 2012 (UTC)

::OK,  so if you have your previous estimate   x(k-1)   and then you calculate your estimated rate of change  x(dot),   then you calculate your a priori estimate of the current position  x(k) which would be something like  xhat(k) =  x(k-1) + ( delta_t x x(dot ) )      and then you would adjust this with all the other matrix stuff.    But I don't see this equation,  or anything like this,   anywhere.    I just see,  calculate  x(dot),   and then you got your xhat(k) from no where.    Is this bit just assumed to occur, because it is obvious ??  Or is it something completely different? [[User:Eregli bob|Eregli bob]] ([[User talk:Eregli bob|talk]]) 06:26, 13 January 2013 (UTC)

== Organization and rating ==

I recently read through this whole article top to bottom. There's good information here. I've found some problems and I don't believe this article quite merits a B rating. --[[User:Kvng|Kvng]] ([[User talk:Kvng|talk]]) 13:02, 1 July 2012 (UTC)
* There is unhelpful repetition of material
* Some of the special topics are not well integrated into the article
* Some of the math needs supporting copy

== Original research in Kalman’s original filter derivation ==

The whole section "Kalman’s original filter derivation" appears to be an original research of user Hikenstuff aka Mr. J. W. Bell. Significant portions of the referenced publication (single author, only published last month!) were copied directly to the Wikipedia article. I'd suggest removing this section altogether, and add it back only if the results are verified in future by other researchers.  <span style="font-size: smaller;" class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/46.7.118.35|46.7.118.35]] ([[User talk:46.7.118.35|talk]]) 14:57, 30 November 2012 (UTC)</span><!-- Template:Unsigned IP --> <!--Autosigned by SineBot-->

The research articles cited in that section are self-published, they are not reviewed nor do they seem serious. They are all from the same author. There is no mention of the "MFOE" anywhere, except in Mr. Bell's (self-published) articles. Given the relevance of the article, I think this section should be removed as soon as possible.
[[Special:Contributions/190.19.236.140|190.19.236.140]] ([[User talk:190.19.236.140|talk]]) 23:41, 2 December 2012 (UTC)

I concur with the previous criticisms.  I have published original research in first-rate journals on topics related to Kalman filtering.  Delete this section ASAP.  At best, it represents the personal opinion of Mr. Bell alone and, at worst, the conclusions are simply incorrect.  Must we provide a proof of the latter before deleting this section?  [[User:Butala|Butala]] ([[User talk:Butala|talk]]) 23:43, 14 December 2012 (UTC)

I just deleted the section.  Original research does not belong in the article, especially if the claims are dubious at best.  [[User:Butala|Butala]] ([[User talk:Butala|talk]]) 23:43, 14 December 2012 (UTC)

== Hats over variables ==

Why does x deserve a hat and P not?  <span style="font-size: smaller;" class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/192.193.116.137|192.193.116.137]] ([[User talk:192.193.116.137|talk]]) 12:11, 20 May 2013 (UTC)</span><!-- Template:Unsigned IP --> <!--Autosigned by SineBot-->

:There are still inconsistencies in hat notation the "Example application, technical" setion vs the "Details" section.  It might be useful to explain (or reference) what the hat notation means.[[User:Chris2crawford|Chris2crawford]] ([[User talk:Chris2crawford|talk]]) 01:24, 5 November 2023 (UTC)

== Notations ==
Notations should be consistent. [[User:Kkddkkdd|Kkddkkdd]] ([[User talk:Kkddkkdd|talk]]) 15:39, 15 June 2013 (UTC)

== information filter ==
I think the formulas of information filter should be improved to allow non-invertible ''Q''. 

My suggestions are following:
:<math> M_{k+1} \triangleq {F_{k+1}}^{-\textrm{T}} Y_{k|k} {F_{k+1}}^{-1} </math> <br>
:<math> C_k \triangleq P_{k|k} \, {F_{k+1}}^\textrm{T} {P_{k+1|k}}^{-1} </math> <br>
:<math> {C_k}^\textrm{T} = \left( \textrm{I} + M_{k+1} Q_{k+1} \right)^{-1} {F_{k+1}}^{-\textrm{T}} </math> <br>
:<math> Y_{k+1|k} =  {C_k}^\textrm{T} Y_{k|k} {F_{k+1}}^{-1} = \left( \textrm{I} + M_{k+1} Q_{k+1} \right)^{-1} M_{k+1} </math> <br>
:<math> \hat{\boldsymbol{y}}_{k+1|k} = {C_k}^\textrm{T} \hat{\boldsymbol{y}}_{k|k} + Y_{k+1|k} \boldsymbol{u}_k </math>

<math>C_k</math>in the section of RTS smoother should be also corrected.
[[User:Kkddkkdd|Kkddkkdd]] ([[User talk:Kkddkkdd|talk]]) 09:43, 14 July 2013 (UTC)

If we can use <math>G_k</math>, my other suggestions for <math>Y_{k+1|k}  </math> are following:

:<math>A_{k} \triangleq \left( {Q_{k+1}}^{-1} + {G_{k+1}}^\textrm{T}  M_{k+1} G_{k+1}  \right)^{-1} {G_{k+1}}^\textrm{T}  M_{k+1} </math>
:<math>C_{k} = {F_{k+1}}^{-1} \left( \mathrm{I} -  G_{k+1} A_k \right)  </math>
:<math>Y_{k+1|k} =  {C_k}^\textrm{T} Y_{k|k} {F_{k+1}}^{-1}  = {C_k}^\textrm{T} Y_{k|k} C_{k} + 
{A_k}^\textrm{T}  {Q_{k+1}}^{-1} A_{k} </math>
[[User:Kkddkkdd|Kkddkkdd]] ([[User talk:Kkddkkdd|talk]]) 14:45, 21 July 2013 (UTC)

== "Invariants" ==
I think the subtitle of "Invariants" should be "Unbiasedness." [[User:Kkddkkdd|Kkddkkdd]] ([[User talk:Kkddkkdd|talk]]) 11:42, 20 July 2013 (UTC)

== Noise Matrix ==
The noise Matrix <math>\textbf{G}_{k}</math>, which is used in the section of "Example application, technical", should be introduced in the section of "Underlying dynamic system model". And the dynamic model should be:
:<math> \textbf{x}_{k} = \textbf{F}_{k} \textbf{x}_{k-1} + \textbf{B}_{k} \textbf{u}_{k} + \textbf{G}_{k} \textbf{w}_{k} </math>
[[User:Kkddkkdd|Kkddkkdd]] ([[User talk:Kkddkkdd|talk]]) 12:21, 21 July 2013 (UTC)

== Asymptotic Properties ==
Could anyone add some asymptotic properties of the filter?
Mainly in relation to the causal Wiener Filter and how its MSE is related to Q and R?
[[User:Ramkakarala|Ramkakarala]] ([[User talk:Ramkakarala|talk]]) 20:43, 7 May 2019 (UTC) I've added a part about the asymptotic gain, with a reference, but it would be good to add more about the MSE.

== Misleading comparison of types of smoothers? ==

In the section on minimum variance smoothers, there's this:

''Taking the causal part of this minimum-variance smoother yields''

<math>\hat{\textbf{y}}_{k\mid k} = \textbf{z}_{k} - \textbf{R}_{k} \textbf{S}_k^{-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.''

The problem with this statement is that it seems to imply that the derivations give different answers. For any system of data, the minimum variance solution is the conditional mean. For a linear system with Gaussian data, the conditional mean can be determined as a linear combination of the data, so that the minimum variance estimator is in fact a linear estimator. For a linear system with non-Gaussian data, in general the minimum variance estimator is nonlinear, but one can of course choose to look for the minimum variance ''linear'' estimator. In the case of linear systems with Gaussian data, these coincide of course.

The point is that the RTS smoother can be derived as the optimal smoother for a linear system with Gaussian data, or as the optimum linear smoother for a linear system with non-Gaussian data, and the results will be the same, if the second-order statistics of the two systems are the same. The same could be said of the "minimum variance smoother". So the sentence ″Note that the Rauch–Tung–Striebel smoother derivation assumes that the underlying distributions are Gaussian, whereas the minimum-variance solutions do not.″ is confusing here, because it seems to imply a distinction that is in fact not that important. [[User:CessnaCutlass|CessnaCutlass]] ([[User talk:CessnaCutlass|talk]]) 17:12, 23 September 2014 (UTC)


== Error in Marginal log likelihood? ==

The recursive equation for the marginal log likelihood seems to assume that the observations are univariate.  When observations are d-dimensional, I believe the equation should change from

<math>\ell^{(k)} = \ell^{(k-1)} - \frac{1}{2} \left(\tilde{\mathbf{y}}_k^T \mathbf{S}^{-1}_k \tilde{\mathbf{y}}_k + \log \left|\mathbf{S}_k\right| + \log 2\pi \right)</math>

to 

<math>\ell^{(k)} = \ell^{(k-1)} - \frac{1}{2} \left(\tilde{\mathbf{y}}_k^T \mathbf{S}^{-1}_k \tilde{\mathbf{y}}_k + \log \left|\mathbf{S}_k\right| + d \log 2\pi \right)</math>

Can someone confirm?
[[User:Ksimek|Ksimek]] ([[User talk:Ksimek|talk]]) 01:14, 24 March 2015 (UTC)


Yes, you are right, I changed it. --[[User:Lipkenszaal|Lipkenszaal]] ([[User talk:Lipkenszaal|talk]]) 15:56, 16 April 2015 (UTC)

== Rauch–Tung–Striebel ==

In the article, <math>F_k</math> acts on <math>x_{k-1}</math>. It looks like there is a mismatch with literature using  <math>F_k</math> acting on <math>x_{k}</math>. Writing  <math>F_k = F_{k,k-1}</math> for clarity,
the article gives for RTS gain <math> \textbf{C}_k = \textbf{P}_{k\mid k} \textbf{F}_{k, k-1}^\mathrm{T} \textbf{P}_{k+1\mid k}^{-1} </math>.
In the Springer Handbook of Speech Processing, 2008 they give <math> \textbf{C}_k = \textbf{P}_{k\mid k} \textbf{F}_{k+1,k}^\mathrm{T} \textbf{P}_{k+1\mid k}^{-1} </math>. --[[User:Lipkenszaal|Lipkenszaal]] ([[User talk:Lipkenszaal|talk]]) 18:40, 20 April 2015 (UTC)

== Unscented Kalman Filter ==

This section only cites on of the papers from the authors who propose the UKF.  In fact and it has been shown analytically the second order performance of the UKF is somewhere between the first order and second order extended Kalman filter.  The problem is rather easy.  Imagine a second order system.  Then the second order taylor-series approximation is exactly the system.  Then a simple calculation of the first order and the UKF show to have less accurate values for the covariance.  This section parrots the promotional material that Julier et.al. pass rather than the subsequent research.  Also there is some controversy because the UKF is one kind of Linear Regression Kalman Filter with a particular way to pick sample points.  This section ignores all these problems.  I will post the paper about the second order characteristics when I can find it.  It came out in Feb. 2012 IEEE.  --[[User:Zoratao|Zoratao]] ([[User talk:Zoratao|talk]]) 19:53, 2 May 2015 (UTC)

== Noise Generality ==

The description of the system model for the Kalman filter on this page states that the process and measurement noise are assumed to be normally disributed, but this is not true: normally distributed noise is not a neccessary condition for derivation of the Kalman Filter. The Kalman filter is still the best ''linear'' estimator for the system described for '''all''' zero-mean finite-variance noise processes, even if they are not normally distributed. For normally distributed noise processes, the Kalman filter is the optimal estimator because the normal distribution is closed under linearity.  <!-- Template:Unsigned IP --><small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/173.61.14.195|173.61.14.195]] ([[User talk:173.61.14.195#top|talk]]) 14:32, 14 February 2017 (UTC)</small> <!--Autosigned by SineBot-->

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== Problems with explanation of unscented Kalman filter ==

The description of this currently says:
"The estimated state and covariance are augmented with the mean and covariance of the process noise.
<math>{\displaystyle {\begin{aligned}\mathbf {x} _{k-1\mid k-1}^{a}&=\left[{\hat {\mathbf {x} }}_{k-1\mid k-1}^{\mathrm {T} }\quad E\left[\mathbf {w} _{k}^{\mathrm {T} }\right]\right]^{\mathrm {T} }\\\mathbf {P} _{k-1\mid k-1}^{a}&={\begin{bmatrix}\mathbf {P} _{k-1\mid k-1}&&0\\0&&\mathbf {Q} _{k}\end{bmatrix}}\end{aligned}}} </math>

A set of 2L + 1 sigma points is derived from the augmented state and covariance where L is the dimension of the augmented state.
<math>{\displaystyle {\begin{aligned}\chi _{k-1\mid k-1}^{0}&=\mathbf {x} _{k-1\mid k-1}^{a}\\\chi _{k-1\mid k-1}^{i}&=\mathbf {x} _{k-1\mid k-1}^{a}+\left({\sqrt {(L+\lambda )\mathbf {P} _{k-1\mid k-1}^{a}}}\right)_{i},\qquad i=1,\ldots ,L\\\chi _{k-1\mid k-1}^{i}&=\mathbf {x} _{k-1\mid k-1}^{a}-\left({\sqrt {(L+\lambda )\mathbf {P} _{k-1\mid k-1}^{a}}}\right)_{i-L},\qquad i=L+1,\ldots ,2L\end{aligned}}} 
</math>

where <math>{\displaystyle \left({\sqrt {(L+\lambda )\mathbf {P} _{k-1\mid k-1}^{a}}}\right)_{i}} 

</math> is the ith column of the matrix square root of
<math>{\displaystyle (L+\lambda )\mathbf {P} _{k-1\mid k-1}^{a}} 
</math>

using the definition: square root <math>{\displaystyle \mathbf {A} }</math>  of matrix <math>{\displaystyle \mathbf {B} }</math>  satisfies <math>{\displaystyle \mathbf {B} \triangleq \mathbf {A} \mathbf {A} ^{\mathrm {T} }.\,} 
</math>
The matrix square root should be calculated using numerically efficient and stable methods such as the Cholesky decomposition.
The sigma points are propagated through the transition function f.
<math>{\displaystyle \chi _{k\mid k-1}^{i}=f\left(\chi _{k-1\mid k-1}^{i}\right)\quad i=0,\dots ,2L} 
</math>

There seem to be two problems with this
(1)  If your state variable is vector X, with some dimension N,  and you augment X by adding additional elements which are the expected values of the process noise distribution, Wk.   But isn't the expected value of the noise distribution, zero ?
(2)  The non-linear process function f(),  is a function of the N-dimensional state vector X,  which is supposed to have an output of the projected N dimensional state vector,  X or X hat or whatever.   So how is this function supposed to process the augmented states and sigma points,  which don't have dimension N ?
[[User:Lathamibird|Lathamibird]] ([[User talk:Lathamibird|talk]]) 10:03, 15 May 2017 (UTC)


:I agree. Without any straight-forward example problems that sets up each and every step associated with an unscented Kalman filter problem, all these cryptic formulae for unscented Kalman filters (as well as any other Kalman filter) are next to useless (pointless) and a waste of time. [[User:KorgBoy|KorgBoy]] ([[User talk:KorgBoy|talk]]) 03:10, 5 October 2017 (UTC)

:I believe that I have now fixed this issue by rewriting the section. Thank you for making this comment giving me the push to do it. [[User:Svendtt|Svendtt]] ([[User talk:Svendtt|talk]]) 12:16, 7 March 2019 (UTC) {{Resolved}}

== Kalman gain expression ==

The Wikipedia page has an expression for the Kalman gain. Some sources have the [HPH + R]^-1 term as an inverse matrix, while other sources have it has a reciprocal term. It would be helpful to discuss this matrix - is it an inverse matrix, or is it a reciprocal? [[User:KorgBoy|KorgBoy]] ([[User talk:KorgBoy|talk]]) 11:37, 7 July 2017 (UTC)

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

[update] The ^-1 is an inverse matrix operator [[User:KorgBoy|KorgBoy]] ([[User talk:KorgBoy|talk]]) 04:02, 29 September 2017 (UTC)

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== Brass tacks ==

For my money, I wish the lead were closer in substance to this passage, from the article text below: 

<blockquote>
The Kalman filter is an efficient recursive filter that estimates the internal state of a linear dynamic system from a series of 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, and is an important topic in control theory and control systems 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 in control theory.

In most applications, the internal state is much larger (more 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. 
</blockquote>

Note that I broke that up into single-sentence paragraphs which are more quickly parsed by the eye, because on the talk page (unlike article space) there's no need to conserve paper. &mdash; [[user:MaxEnt|MaxEnt]] 23:10, 1 November 2018 (UTC)

:I went back and added "In statistics and control theory," to the front of the lead, mainly because the topic of [[filter (signal processing)]] would fill a small library all by itself, and you don't want find yourself in the wrong library building right from the get go. &mdash; [[user:MaxEnt|MaxEnt]] 23:17, 1 November 2018 (UTC)

== Possible error in intuitive expression update step ==

Equation that might be faulty (Section 6.2):

:<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{H}_k \hat{\mathbf{x}}_{k\mid k-1} + \mathbf{v}_k)</math>

I think there might be an error in the second term of the equation where <math>\mathbf{z}_k</math> is substituted by <math>\mathbf{H}_k \hat{\mathbf{x}}_{k\mid k-1} + \mathbf{v}_k</math>. 

Should in this substitution the ''a posteriori'' state estimate <math> \hat{\mathbf{x}}_{k\mid k-1}</math> not be replaced by <math> \mathbf{x}_{k}</math>?

Groetjes,
[[User:KalmeMan69|KalmeMan69]] ([[User talk:KalmeMan69|talk]]) 21:29, 1 November 2021 (UTC)

:Looks like someone put z back in, which makes more sense to me.  And took the hat off xk on the LHS.  But I think xk|k-1 on the RHS is correct, at least from the above statement xk = xk|k-1 + Kk yk.[[User:Chris2crawford|Chris2crawford]] ([[User talk:Chris2crawford|talk]]) 00:40, 5 November 2023 (UTC)

== Eq. typo? Measurement post-fit residual ==

The right hand side of the eq. _Measurement post-fit residual_, looks the same as the eq. for the innovation. I think the post-fit residual should involve

<math>\hat{\mathbf{x}}_{k\mid k}</math>


[[Special:Contributions/181.114.120.18|181.114.120.18]] ([[User talk:181.114.120.18|talk]]) 22:07, 2 April 2022 (UTC)

== A more intuitive way to express the updated state estimate... ==

Looking at the section that says:
{{Blockquote
|A more intuitive way to express the updated state estimate (<math>\hat{\mathbf{x}}_{k\mid k}</math>) is:

<math>\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 Kalman gain (<math>\mathbf{K}_k</math>), a matrix that takes values from <math>0</math> (high error in the sensor) to <math>I</math> (low error).
}}

But, in the extreme of low sensor error, shouldn't <math>\mathbf{K}_k</math> become the inverse of <math>\mathbf{H}_k</math>?  <!-- Template:Unsigned --><small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Helicopter34234|Helicopter34234]] ([[User talk:Helicopter34234#top|talk]] • [[Special:Contributions/Helicopter34234|contribs]]) 19:13, 19 January 2023 (UTC)</small> <!--Autosigned by SineBot-->

:I agree, unless H is singular.  I edited the article to treat KH as t, which has the limits of 0 or I; however now, the statement that KH is the 'Kalman gain' might be incorrect.  Also, there is still the problem that b=H^{-1}z might not exist, in which case the limit would be a projection, not identity (see below).
:The statement Hx = (I-HK)Hx+(HK)z, using HK (which actually has limits of 0 and I) instead of KH avoids these problems, but equation is supposed to be for x, not Hx.  I guess the analogy is imperfect because we're working in two different spaces without a 1-1 correspondence.  
:It is interesting, though, that the Kalman filter doesn't need H to be invertible. If R=0, then K is the right inverse of H (HK=I and thus KH is a projection). [[User:Chris2crawford|Chris2crawford]] ([[User talk:Chris2crawford|talk]]) 23:22, 4 November 2023 (UTC)

== "The algorithm is recursive" ==

I think the sentence "The algorithm is recursive" from the lede is at best confusing, and arguably just wrong. The algorithm is perhaps *defined* recursively but it's *computed* with simple iteration. In fact, when you convert a recursive technique to an iterative one, you often have to explicitly factor out the state, and that's exactly the state vector at the core of the Kalman filter. My preference would be to simply delete it, but I notice it's been there for quite a few years so wanted to see if there were any objections first. [[User:Quietbritishjim|Quietbritishjim]] ([[User talk:Quietbritishjim|talk]]) 23:11, 26 January 2025 (UTC)

:Specifically, it was changed in [https://en.m.wikipedia.org/w/index.php?title=Kalman_filter&diff=prev&oldid=672721435 this revision in 2015] from "Because of the algorithm's recursive nature, it can run in [[real-time Control System|real time]], using only the present input measurements and the previously calculated state and its uncertainty matrix". Despite the anonymous editor's edit comment of "Incorrect causal reasoning removed (other information preserved).", it seems to me that the original version was correct and the newer one is not. [[User:Quietbritishjim|Quietbritishjim]] ([[User talk:Quietbritishjim|talk]]) 23:21, 26 January 2025 (UTC)
::The algorithm is recursive in the sense that the solution of differential equations is recursive. The notation (using subscripts) is directly applicable to a solution which finds the motion of a body, for example. Motion need not be physical, it can denote change in an abstract state. -- [[User:Ancheta Wis|Ancheta Wis]] [[User talk:Ancheta Wis| &nbsp; (talk]] [[Special:Contributions/Ancheta Wis| &#124; contribs)]] 08:33, 27 January 2025 (UTC)

== Page rewrite is required, reads like a list ==

should be organised for quick indexing, I'd recommend seven topics on the landing page max goes with c. known seven item short term memory limitation [[Special:Contributions/2A01:B340:84:E0B5:E1C6:4003:EAD:1E4B|2A01:B340:84:E0B5:E1C6:4003:EAD:1E4B]] ([[User talk:2A01:B340:84:E0B5:E1C6:4003:EAD:1E4B|talk]]) 23:26, 26 February 2025 (UTC)

==External links==
:Some things just grow during incremental edits and sometimes get out of hand. The "External links" section, one of the optional appendices, has grown to 17 entries. Three seems to be an acceptable number, and of course, everyone has their favorite to try to add for a fourth. [[WP:consensus|Consensus]] needs to determine this. 
:However, none is needed for article promotion.
:Some links may be included in [[WP:ELNO]], [[WP:NOTREPOSITORY]], [[WP:ELCITE]], [[WP:ELDEAD]], or [[WP:NOTGUIDE]]. Others, listed below:  
*[[WP:ELPOINTS|ELpoints #3)]] states: {{tq|Links in the "External links" section should be kept to a minimum. A lack of external links or a small number of external links is not a reason to add external links.}}
*[[WP:LINKFARM|LINKFARM]] states: {{tq|There is nothing wrong with adding one or more useful content-relevant links to the external links section of an article; however, excessive lists can dwarf articles and detract from the purpose of Wikipedia. On articles about topics with many fansites, for example, including a link to one major fansite may be appropriate.}}
*[[WP:ELMIN|ELMIN]]: {{tq|Minimize the number of links}}. -- 
*[[WP:ELCITE|ELCITE]]: {{tq|'''Do not use {{tl|cite web}}''' or other citation templates in the External links section. Citation templates are permitted in the Further reading section.}} 
:External links '''This page in a nutshell''': {{tq|External links in an article can be helpful to the reader, but they should be kept minimal, meritable, and directly relevant to the article. With rare exceptions, external links should not be used in the body of an article.}}
:Second paragraph, {{tq|acceptable external links include those that contain further research that is accurate and on-topic, information that could not be added to the article for reasons such as copyright or amount of detail, or other meaningful, relevant content that is not suitable for inclusion in an article for reasons unrelated to its accuracy.}}
**'''Please also note''':
*[[WP:ELBURDEN]]: {{tq|'''Disputed links should be excluded by default''' unless and until there is a consensus to include them}}.
:'''Moved links''':
* [http://www.eng.tau.ac.il/~liptser/lectures1/lect6.pdf ''Kalman–Bucy Filter''], a derivation of the Kalman–Bucy Filter
* {{YouTube|d0D3VwBh5UQ|MIT Video Lecture on the Kalman filter}}
* [https://github.com/piercus/kalman-filter Kalman filter in Javascript]. Open source Kalman filter library for node.js and the web browser.
* [http://www.cs.unc.edu/~tracker/media/pdf/SIGGRAPH2001_CoursePack_08.pdf An Introduction to the Kalman Filter] {{Webarchive|url=https://web.archive.org/web/20210224114008/http://www.cs.unc.edu/~tracker/media/pdf/SIGGRAPH2001_CoursePack_08.pdf |date=2021-02-24 }}, SIGGRAPH 2001 Course, Greg Welch and Gary Bishop
* [http://www.cs.unc.edu/~welch/kalman/ Kalman Filter] webpage, with many links
*[https://thekalmanfilter.com/kalman-filter-explained-simply/ Kalman Filter Explained Simply], Step-by-Step Tutorial of the Kalman Filter with Equations
* {{Cite journal|url=http://www.siam.org/pdf/news/362.pdf|title=Kalman filters used in Weather models|journal=SIAM News|volume=36|issue=8|date=October 2003|access-date=2007-01-27|archive-url=https://web.archive.org/web/20110517044040/http://www.siam.org/pdf/news/362.pdf|archive-date=2011-05-17}}
* {{cite journal |doi=10.1021/ie034308l|title=Critical Evaluation of Extended Kalman Filtering and Moving-Horizon Estimation|journal=Industrial & Engineering Chemistry Research|volume=44|issue=8|page=2451|year=2005|last1=Haseltine|first1=Eric L.|last2=Rawlings|first2=James B.}}
* [https://web.archive.org/web/20160304042652/http://www.cnel.ufl.edu/~weifeng/publication.htm The Kalman Filter in Reproducing Kernel Hilbert Spaces] A comprehensive introduction.
* [http://www.mathfinance.cn/kalman-filter-finance-revisited/ Matlab code to estimate Cox–Ingersoll–Ross interest rate model with Kalman Filter] {{Webarchive|url=https://web.archive.org/web/20140209065941/http://www.mathfinance.cn/kalman-filter-finance-revisited/ |date=2014-02-09 }}: Corresponds to the paper "estimating and testing exponential-affine term structure models by kalman filter" published by Review of Quantitative Finance and Accounting in 1999.
* [https://kalman-filter.com/ kalman-filter.com]. Insights into the use of Kalman Filters in different domains.
* {{cite journal |doi=10.3390/s110808164 |pmid=22164069|pmc=3231703|pages=1251–1259|title=FPGA-Based Multimodal Embedded Sensor System Integrating Low- and Mid-Level Vision|journal=Sensors|volume=11|issue=12|year=2011|last1=Botella|first1=Guillermo|last2=Martín h.|first2=José Antonio|last3=Santos|first3=Matilde|last4=Meyer-Baese|first4=Uwe|bibcode=2011Senso..11.8164B|doi-access=free}}
* [http://blog.sciencenet.cn/home.php?mod=space&uid=1565&do=blog&id=851754 Explaining Filtering (Estimation) in One Hour, Ten Minutes, One Minute, and One Sentence]  by Yu-Chi Ho
*Simo Särkkä (2013). "Bayesian Filtering and Smoothing". ''Cambridge University Press.'' Full text available on author's webpage https://users.aalto.fi/~ssarkka/.  --  [[User:Otr500|Otr500]] ([[User talk:Otr500|talk]]) 05:36, 30 May 2025 (UTC)