Kalman Filter Covariance 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
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.