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I've recently started playing with the Kalman filter for a simple 2D (x,y,dx,dy) tracking toy problem. But I seem to have some misunderstanding on what I can expect from the filter. I'm interested in plotting the uncertainty ellipse from the corrected covariance matrix but noticed a few observations:

  • The covariance decreases to a steady state regardless of how much error I introduce into the measurement.

  • The variance for x and y are exactly the same even though I introduce more measurement errors in y.

Staring at the maths for a bit it seems that this is how the vanilla Kalman Filter works. What I'm expecting is the opposite of the two points mentioned above. In the end I want to plot an uncertainty ellipse that reflects the error I'm observing. Is this possible at all? Do I have to do some post processing on the covariance matrix?

ANSWER:

It occurred to me what I needed was a Kalman Filter that has the ability to adapt its covariance. I found this paper that details a few different methods to do this.

Evaluating the performances of adaptive Kalman filter methods in GPS/INS integration by Ali Almagbile, Jinling Wang, and Weidong Ding, 2010

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  • $\begingroup$ If you've found an answer, please post it as an answer and mark it as accepted. Thanks! $\endgroup$
    – Peter K. ♦
    May 12, 2015 at 7:53
  • $\begingroup$ The link doesn't work. Could you write the title of the paper? $\endgroup$
    – Mark
    Feb 23, 2021 at 5:39

2 Answers 2

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The covariance matrix of a Kalman filter is a function of the $ Q $ and $ R $ matrices of the model.
If you use a model where $ R $ and $ Q $ are time invariant or known in prior then the calculation of the covariance matrix $ P $ can be done offline and isn't a function of the measurements.

In some cases, advanced implementations of Kalman Filter estimate the covariances $ R $ and $ Q $ on line according to some data gathered in the process of calculating the result.

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  • $\begingroup$ I'm just learning about the Unscented Kalman Filter (UKF). It seems like the estimation of the predicted state covariance matrix needs to be done online. I thought the UKF would yield different estimates for the variances of x and y in some cases (e.g. dynamics moving only in the y direction). Should the UKF also yield identical estimates for variances for x and y? $\endgroup$
    – user3731622
    Dec 6, 2016 at 18:48
  • $\begingroup$ Without knowing the model I can't say much. But if it is simple movement on 2D, why would you use UKF? UKF should be used for Non Linear Model. For instance measureing $ r $ and $ \theta $ instead of $ x $ and $ y $, $\endgroup$
    – Royi
    Dec 6, 2016 at 18:57
  • $\begingroup$ One reason to use the UKF might be to implement a single KF that supports both linear and non linear dynamic models. Let's just assume a simple constant acceleration dynamic model. $\endgroup$
    – user3731622
    Dec 6, 2016 at 20:11
  • $\begingroup$ Would you open a new question with the whole model? Unless you show the non linear part of the model there is no point in UKF. $\endgroup$
    – Royi
    Dec 7, 2016 at 6:03
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The covariance decreases to a steady state regardless of how much error I introduce into the measurement.

Yes, as @Drazick notes, if the $Q$ and $R$ matrices are time invariant, then the $P$ matrix will converge to a steady state that does not depend on the data (measurements).

The variance for x and y are exactly the same even though I introduce more measurement errors in y.

When you did that, did you change the $R$ matrix to take account of this extra error in one component over the other? (I'm assuming you just used $x$ and $y$ as the measurements). If the $R$ matrix was chosen to be $\sigma I$, then you will not see any difference between $x$ and $y$ state variances.

Even the Wikipedia page on Kalman filtering mentions what you can do if you need to estimate $Q$ and $R$.

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