(Co)variance interpretation in Kalman filter

Question

Let's say I have a device which uses Kalman filter to fuse sensor data and produce an optimal estimate of the system parameters. As it should, it also estimates parameter covariance matrices at each step. Its diagonal elements are used to derive current error estimates for output variables.

The system that the device measures is quickly changing. What is the error variance then exactly? To be able to measure (co)variances, there has to be some population and some samples should be drawn out of it. Then one can calculate sample mean and covariance. In this case I can't develop intuition of what is the population and how is it sampled to get the variance.

If I understand it correctly, getting a parameter error estimate (variance) of N units, I should expect that only 68% of such data points would "really" have an error within ±N units? If so, how do I transform the filter output to get an error estimate with 95% confidence?

Also, should I expect any particular distribution of this variance values? Like chi-square or something like that? Or does is only depend on the measurement conditions?

Answer 1

On the specfic question: Should expect that only 68% of such data points would "really" have an error within ±N units? If so, how do I transform the filter output to get an error estimate with 95% confidence?

Assume the data is normal (or, per an analysis of transformation, it is made closer to normal), now here is how estimate of sigma can be developed. Per Wikipedia, to quote:

Three standard deviations account for 99.7% of the sample population being studied, assuming the distribution is normal or bell-shaped (see the 68-95-99.7 rule, or the empirical rule, for more information).

Also, develop a sigma from, say, the perspective of a range control chart discussed here https://www.spcforexcel.com/spc-blog/how-sigma-estimated-control-chart.

Having arrived at an estimate of sigma, proceed to develop say a 95% confidence interval.

Personally, I would develop a process and simulation test for accuracy for a select 'n' and specified values of sigma and even consider employing possible non-normal parent distributions based on the source of the data. More generally, this is a difficult task, but with more specifics, one may be able to display an operational improvement.