I have implemented Kalman Filter for state estimation of AR(2) univariate model and wanted to plot the Kalman Gain. When implementing, I saw that Kalman gain for every sample is getting computed and is a matrix of dimension 2 rows by 1 column. After few samples, the Kalman Gain does not change and the value is


What does this imply? Can convergence be inferred from the plot of Kalman Gain values so computed? If so, then which value should I take out of the 2 rows?


The following graph shows the plot of KAlman GAin computed for a uni-dimensional non - linear dynamical system and EKF when doing state estimation.

kalman gain


1 Answer 1


If you have an AR(2) in state-space form, the state dimension will be 2, and the Kalman gain matrix tells you what portion of the innovation enters the state vector at each point of time. Since the state dimension is 2, the gain matrix has that many rows.

If the state-space model is time-invariant and the covariance matrix of the state converges to a constant matrix, so will the Kalman gain matrix. You should not choose any of the rows.

  • $\begingroup$ Thank you for your reply. But one thing is not clear which is if each row of the Kalman Gain represents the state, then while plotting should I plot the mean of the gain matrix, otherwise how will I get a single curve? In some cases, the Kalman Gain converges to zero or a constant. What should be the ideal value? Zero or constant? $\endgroup$
    – SKM
    Commented Mar 23, 2015 at 17:28
  • $\begingroup$ If an element of the gain matrix as you have it goes to zero, it would mean that the filter does no learn anymore from observations. I think you are misunderstanding the meaning of the gain matrix, what you probable mean to plot is the filtered or smoothed series. $\endgroup$
    – F. Tusell
    Commented Mar 23, 2015 at 17:36
  • $\begingroup$ Tussell: I have added a picture of a graph which shows convergence characteristic of Kalman GAin, when using EKF for state estimation of a non-linear dynamical system. Fir this case, there is only 1 state variable. In similar lines, I was wondering how to obtain convergence when the state is 2 dimensional, i.e., when I have 2 state variables. Could you provide some insights, please? Thank you once again. $\endgroup$
    – SKM
    Commented Mar 24, 2015 at 5:02
  • $\begingroup$ Not quite sure what you mean by "obtain convergence". For linear time invariant state space models (you mention EKF, so your's is not linear, apparently) it is typically the case that the covariance matrix of the state vector "settles down" after a number of iterations. The intution is that after a while the reduction of uncertainty provided by a new observation and the new uncertainty which enters the model via the state vector disturbances "cancel". When this happens, the covariance matrix of the state vector (and hence the gain matrix) converge to a constant matrix. $\endgroup$
    – F. Tusell
    Commented Mar 24, 2015 at 8:08

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