I am interested doing the consistency check of the Kalman filter. There are several measures like Posterior Cramer Rao Lower Bound (PCRLB), NIS (Normalised Innovation Squared), NEES(Normalized Estimation Error Squared) autocorrelation, and MSE (Mean Square Error). I had following doubts.
In PCRLB equation, how can I initialize the J matrix i.e. J0 because its a recursive equation which requires an initialization. In a paper Posterior Cramer–Rao Bounds for Discrete-Time Nonlinear, Filtering eq.(29) gives an idea on how to initialize. But I could not understand how they initialized the J0 matrix.
If I do not know the true states and the true covariance matrices (P, Q and R) because I do not do any simulation, is it still possible to calculate PCRLB because one thing is clear for me is that NEES (Normalized Estimation Error Squared) and MSE (Mean Square Error) can not be done for real data. I can never get the exact true states and covariances.
NIS and autocorrelation can be done for real data. Can Monte Carlo runs be done for real data as well because I need to change R and Q at each run, and the rest will be same.
Any help in this context will be highly appreciated.