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?
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.