Talk:Kalman filter

message: 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.  —Preceding unsigned comment added by 77.54.101.212 (talk) 14:52, 5 June 2008 (UTC)
thread_title:  Suggestion on notation between deterministic and stochastic variables 

message: 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 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).  Damien d 12:04, 30 April 2007 (UTC)
thread_title:  On the Unscented Kalman Filter 

message: 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? 132.239.215.69 17:52, 9 August 2006 (UTC)
replies: text: Totally agree with you shampoo (talk) 12:25, 6 July 2008 (UTC)
thread_title:  Excellent page 

message: 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. --Fredrik Orderud 18:32, 25 May 2006 (UTC)
replies: text: Fixed. --Drizzd (talk) 16:20, 2 July 2008 (UTC)
thread_title:  Typo 

message: 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)? — Preceding unsigned comment added by 134.60.10.127 (talk) 13:19, 18 November 2005‎ (UTC)
: 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.  --Fredrik Orderud 01:52, 20 November 2005 (UTC)
replies: text: 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. --Alan1507 09:07, 9 May 2006 (UTC)

text: 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. --Fredrik Orderud 00:23, 10 May 2006 (UTC)
thread_title:  What does ''HMM'' mean? 

message: 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 , which undoubtedly covers systems with continous state. --Fredrik Orderud 19:15, 10 May 2006 (UTC)
replies: text: I agree with 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! — ciphergoth 19:40, 10 May 2006 (UTC)

text: 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! Yebbey (talk) 14:28, 3 December 2010 (UTC)
thread_title:  Underlying dynamical system 

message: The link
http://en.wikipedia.org/wiki/Peter_Swerling
entitled "Peter Swerling" leads back to this Kalman Filter Page!
Carrionluggage 21:42, 1 August 2006 (UTC)
thread_title:  recursive link 

message: 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 
:\textbf{S}_{k} = \textbf{H}_{k}\textbf{P}_{k|k-1} \textbf{H}_{k}^{T}+\textbf{R}_{k} 
looks the same as it was last March. Ultimâ 11:57, 14 November 2006 (UTC)
thread_title:  equation typo 

message: 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 Oliverone 19:19, 3 April 2007 (UTC)
thread_title:  Kalman filter implementation 

message: 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.    --Ultimâ (talk) 10:57, 19 May 2008 (UTC)
thread_title:  Equation Fix Suggestion 

message: I think that there is a missing term in this equation:-
\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} 
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 \textbf{R}_k which should be added on to this.  I think that it should read:-
\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} 
--155.198.108.162 (talk) 06:51, 18 June 2008 (UTC)
thread_title:  Missing term in UKF equations? 

message: Does this diagram imply x(k) and x(k+1) are the same?  —Preceding unsigned comment added by 144.189.100.25 (talk) 21:50, 29 January 2009 (UTC)
replies: text: 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. --76.27.96.159 (talk) 21:26, 27 February 2009 (UTC)

text: 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. --128.187.80.2 (talk) 23:28, 2 March 2009 (UTC)

text: v and w should also be subscripted, because they represent different values in each time frame.--128.187.80.2 (talk) 18:59, 7 April 2009 (UTC)
replies: text:  Okay, I fixed that. --Headlessplatter (talk) 17:17, 8 April 2009 (UTC)

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

text: 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.)--Headlessplatter (talk) 22:28, 16 December 2010 (UTC)
thread_title: Error with diagram?

message: 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  template? Obankston (talk) 23:50, 16 February 2010 (UTC)
replies: text: The Too Technical template was removed in . Obankston (talk) 02:44, 17 March 2010 (UTC)
thread_title:  "Technical" template 

message: 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?
204.15.2.146 (talk) 23:10, 15 March 2010 (UTC)
thread_title:  Are the UKF function dimensions correct? 

message: 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., m2 and (m/s)2 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 years 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 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
:\operatorname{Tr}\left[(\mathbf{x}-\boldsymbol{\mu})^T\boldsymbol{\Sigma}^{-1}(\mathbf{x}-\boldsymbol{\mu})\right]
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......
--Slaunger (talk) 07:45, 29 June 2010 (UTC)
replies: text: \mathbf{J} = E[\tilde{x_k}^T \mathbf{S} \tilde{x_k}]
where \mathbf{S} is an arbitrary positive semidefinite matrix. The important fact is, that the minimum of \mathbf{J} is independent of \mathbf{S}, therefore \mathbf{S} can be chosen to be the unit matrix, which results in  \mathbf{J} = Tr[\mathbf{P_k}].
(Peter Pöschl) 188.174.71.81 (talk) 21:29, 5 July 2010 (UTC)

text: 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. --Slaunger (talk) 07:58, 16 July 2010 (UTC)
thread_title:  Unit mismatch in the [[Kalman filter#Kalman gain derivation|Kalman gain derivation]] section 

message: I am especially confused as to how 
:\textbf{x}_{k} - \hat{\textbf{x}}_{k|k}
is expanded to the monster below it. Some step-by-step seems in order here.  — Preceding unsigned comment added by Dave.rudolf (talk • contribs) 00:53, 12 January 2011 (UTC)
thread_title:  Kalman gain derivation not obvious 

message: I think the formula
:\tilde{\lambda}_k =  -\textbf{H}_k^T \textbf{S}_k^{-1} \textbf{z}_k + \textbf{C}_k^T \hat{\lambda}_k
really should be
:\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
This fits better with the reference, and seems to give the correct numerical results for me...  — Preceding unsigned comment added by 89.236.1.222 (talk) 00:02, 22 June 2011 (UTC)
thread_title:  Erroneous Modified Bryson-Frazier formula ? 

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

message: 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. Jfgrcar (talk) 06:46, 20 April 2012 (UTC)
thread_title:  References for Applications 

message: 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, \hat{\mathbf{x}} ("x hat") and \dot{\mathbf{x}} ("x dot"). The hat barely underlines the fact that \hat{\mathbf{x}} 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 x_2 = F_d x_1, in continuous case the x is changing continuously with the instantaneous derivative F_c x. The F matrix for these cases is different, and F_d can be found as a solution to the differential equation \dot{x} = F_c x. The solution is usually approximated with Taylor series as F_d = I + T F_c + (T F_c)^2 / 2 + ... (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.
: -- Petteri Aimonen (talk) 15:31, 4 June 2012 (UTC)
replies: text: 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? Eregli bob (talk) 06:26, 13 January 2013 (UTC)
thread_title:  x or x dot ? 

message: 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. --Kvng (talk) 13:02, 1 July 2012 (UTC)
thread_title:  Organization and rating 

message: 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.  — Preceding unsigned comment added by 46.7.118.35 (talk) 14:57, 30 November 2012 (UTC)
thread_title:  Original research in Kalman’s original filter derivation 

message: Why does x deserve a hat and P not?  — Preceding unsigned comment added by 192.193.116.137 (talk) 12:11, 20 May 2013 (UTC)
replies: text: 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.Chris2crawford (talk) 01:24, 5 November 2023 (UTC)
thread_title:  Hats over variables 

message: Notations should be consistent. Kkddkkdd (talk) 15:39, 15 June 2013 (UTC)
thread_title:  Notations 

message: I think the subtitle of "Invariants" should be "Unbiasedness." Kkddkkdd (talk) 11:42, 20 July 2013 (UTC)
thread_title:  "Invariants" 

message: The noise Matrix \textbf{G}_{k}, 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:
: \textbf{x}_{k} = \textbf{F}_{k} \textbf{x}_{k-1} + \textbf{B}_{k} \textbf{u}_{k} + \textbf{G}_{k} \textbf{w}_{k} 
Kkddkkdd (talk) 12:21, 21 July 2013 (UTC)
thread_title:  Noise Matrix 

message: 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?
Ramkakarala (talk) 20:43, 7 May 2019 (UTC)
thread_title:  Asymptotic Properties 

message: In the section on minimum variance smoothers, there's this:
Taking the causal part of this minimum-variance smoother yields
\hat{\textbf{y}}_{k\mid k} = \textbf{z}_{k} - \textbf{R}_{k} \textbf{S}_k^{-1/2} \alpha_{k}   
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. CessnaCutlass (talk) 17:12, 23 September 2014 (UTC)
thread_title:  Misleading comparison of types of smoothers? 

message: 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
\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)
to 
\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)
Can someone confirm?
Ksimek (talk) 01:14, 24 March 2015 (UTC)
thread_title:  Error in Marginal log likelihood? 

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

message: 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.  --Zoratao (talk) 19:53, 2 May 2015 (UTC)
thread_title:  Unscented Kalman Filter 

message: 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.  — Preceding unsigned comment added by 173.61.14.195 (talk) 14:32, 14 February 2017 (UTC)
thread_title:  Noise Generality 

message: Hello fellow Wikipedians,
I have just modified one external link on Kalman filter. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
*Added archive https://web.archive.org/web/20080529105724/http://www.elo.utfsm.cl/~ipd481/Papers%20varios/kalman1960.pdf to http://www.elo.utfsm.cl/~ipd481/Papers%20varios/kalman1960.pdf
When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.
Cheers.—InternetArchiveBot (Report bug) 15:31, 1 May 2017 (UTC)
thread_title:  External links modified 

message: The description of this currently says:
"The estimated state and covariance are augmented with the mean and covariance of the process noise.
{\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}}} 
A set of 2L + 1 sigma points is derived from the augmented state and covariance where L is the dimension of the augmented state.
{\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}}} 
where {\displaystyle \left({\sqrt {(L+\lambda )\mathbf {P} _{k-1\mid k-1}^{a}}}\right)_{i}} 
 is the ith column of the matrix square root of
{\displaystyle (L+\lambda )\mathbf {P} _{k-1\mid k-1}^{a}} 
using the definition: square root {\displaystyle \mathbf {A} }  of matrix {\displaystyle \mathbf {B} }  satisfies {\displaystyle \mathbf {B} \triangleq \mathbf {A} \mathbf {A} ^{\mathrm {T} }.\,} 
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.
{\displaystyle \chi _{k\mid k-1}^{i}=f\left(\chi _{k-1\mid k-1}^{i}\right)\quad i=0,\dots ,2L} 
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 ?
Lathamibird (talk) 10:03, 15 May 2017 (UTC)
replies: text: 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. KorgBoy (talk) 03:10, 5 October 2017 (UTC)

text: 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. Svendtt (talk) 12:16, 7 March 2019 (UTC)
thread_title:  Problems with explanation of unscented Kalman filter 

message: 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? KorgBoy (talk) 11:37, 7 July 2017 (UTC)
thread_title:  Kalman gain expression 

message: Hello fellow Wikipedians,
I have just modified one external link on Kalman filter. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
*Added archive https://web.archive.org/web/20160304042652/http://www.cnel.ufl.edu/~weifeng/publication.htm to http://www.cnel.ufl.edu/~weifeng/publication.htm
When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.
Cheers.—InternetArchiveBot (Report bug) 12:02, 5 December 2017 (UTC)
thread_title:  External links modified 

message: For my money, I wish the lead were closer in substance to this passage, from the article text below: 
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. 
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. — MaxEnt 23:10, 1 November 2018 (UTC)
replies: text: 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. — MaxEnt 23:17, 1 November 2018 (UTC)
thread_title:  Brass tacks 

message: Equation that might be faulty (Section 6.2):
:\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)
I think there might be an error in the second term of the equation where \mathbf{z}_k is substituted by \mathbf{H}_k \hat{\mathbf{x}}_{k\mid k-1} + \mathbf{v}_k. 
Should in this substitution the a posteriori state estimate  \hat{\mathbf{x}}_{k\mid k-1} not be replaced by  \mathbf{x}_{k}?
Groetjes,
KalmeMan69 (talk) 21:29, 1 November 2021 (UTC)
replies: text: 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.Chris2crawford (talk) 00:40, 5 November 2023 (UTC)
thread_title:  Possible error in intuitive expression update step 

message: 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
\hat{\mathbf{x}}_{k\mid k}
181.114.120.18 (talk) 22:07, 2 April 2022 (UTC)
thread_title:  Eq. typo? Measurement post-fit residual