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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 ?

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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.

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