I have a problem with a linear Kalman filter algorithm that gets as input some sensor measurements $z_i$ with known measurement error with standard deviation $\sigma_{i,{measured}}$ (assumed normally distributed) and gives as output the updated (a posteriori) state estimate of that measurement $x_i$ and the updated a posteriori covariance error of the estimate from which we get $\sigma_{i, {estimated}} $.
I am searching for statistical methods to assess the performance of the estimator algorithm. As a first approach, I am thinking of computing the difference of the measured to the estimated value ($|z_i-x_i|$)and check if the 66.66% of these differences-assuming that the errors of both vectors are normally distributed- lies between the sum of their uncertainties $\sigma_{i,{measured}}+\sigma_{i,{estimated}}$. Do you think it is a good approach to understand if the estimator is erroneous or not?
Is there any other idea of validating the performance of the Kalman filter? Searching in the literature I have found a lot of papers that compare the estimate to the true value but I do not know the true value of the model. I just want to infer from the measurements and the estimates along with their documented/predicted uncertainties the accuracy of the estimator. And if an error can be identified is there a way to separate the measurement model error(the error that is introduced from the multiplication $Hx$ ) from a process model error ?
Best Answer
There are methods to check on the performance of the filter in the absence of truth data. One method is to recompute the measurement residuals after the state update (the a posteriori residual). Once the filter covariance has stabilized, and with constant measurement noise variance (R), the a posteriori residuals should be zero-mean normally distributed. If they're not, something is wrong. It could be an unmodeled state, non-Gaussian state or measurement noise, or non-linearity. One can't really say what's wrong but it is at least a fault-detection mechanism.