Numerical Instability Kalman Filter in MatLab

Question

I am trying to run a standard Kalman Filter algorithm to calculate likelihoods, but I keep getting a problema of a non positive definite variance matrix when calculating normal densities. I've researched a little and seen that there may be in fact some numerical instabitlity; tried some numerical ways to avoid a non-positive definite matrix, using both choleski decomposition and its variant LDL' decomposition. I am using MatLab. Does anyone suggest anything? Thanks.

Answer 1

post your code.

As a rule of thumb, if the model is not accurate and the regularization (ie the model noise matrix Q) is not sufficiently "large" an underfitting will occur and the covariance matrix of the estimator will be ill-conditioned. Try fine tuning your Q matrix.

Answer 2

I have encountered what might be the same problem before when I needed to run a Kalman filter for long periods but over time my covariance matrix would degenerate. It might just be a problem of losing symmetry due to numerical error. One simple way to enforce your covariance matrix (let's call it P ) to remain symmetric is to do:

P = (P + P')/2  # where P' is transpose(P)

right after estimating P .

Answer 3

The Kalman Filter implemented using the Joseph Form is known to be numerically unstable, as any old timer who once worked with single precision implementation of the filter can tell. This problem was discovered zillions of years ago and prompt a lot of research in implementing the filter in a stable manner. Probably the best well-known implementation is the UD, where the Covariance matrix is factorized as UDU' and the two factors are updated and propagated using special formulas (see Thoronton and Bierman). U is an upper diagonal matrix with "1" in its diagonal, and D is a diagonal matrix.