Is it possible that kalman-filter algorithm result in a singular variance matrix?

Question

Is it possible that in a Kalman Filter implementation the 'Predicted estimate covariance', P(k|k-1), (see the wiki here) could be a singular matrix? If not, is something is wrong with my code?

This is the state-space model

%    y{t}=Z{t} b{t} +  eps{t},                  eps{t} ~ N(0,H{t})
%              b{t} = Pi{t} b{t-1} + tao{t}     tao{t} ~ N(0,Q{t})
%                                               b{1} ~ N(b0,P0)
%                                               t=1,...,T

and this is the backward recursion as the main part of the kalman-filter algorithm:

for t=1:T

    v{t} = y{t} - Z{t} * b_tt_1{t};
    M{t} = P_tt_1{t} * Z{t}';
    F{t} = Z{t} * M{t} + H{t};

       F_{t}= inv(F{t});
       MF_{t}= M{t} * F_{t};

    b_tt{t}=b_tt_1{t} + MF_{t} * v{t};
    P_tt{t}=P_tt_1{t} - MF_{t} * M{t}';

    b_tt_1{t+1} = Pi{t} * b_tt{t};
    P_tt_1{t+1} = Pi{t} * P_tt{t} * Pi{t}' + Q{t};

end

This happened when I used actual data. To see where the problem might be, I wrote some code to generate random state-space models (I can provide the code, if it is needed).

When T is large, After some t0, P_tt_1{t0} is singular and the states (b{t0}) diverge.

Edit: I used "Joseph form" of the covariance update equation (see Wikipedia). It helped, but the result still diverges when the state-space model is Large (in sense of number of equations or states). I think this means the problem is related to numerical stability. Is there a way to solve this?

Answer 1

the problem is with inversion process of inv or pinv functions. the matrices are large, but they are also positive definite. so I used cholesky decomposition for inversion. the function is:

function A_=inversePD(A)
%A:positive definite matrix
M=size(A,1);
[R b] = chol(A);
if b~=0
    return
end
R_ = R \ eye(M);
A_ = R_ * R_';
end

Answer 2

The only place your matrix can become singular is in the line:

F_{t}= inv(F{t});

You could use the the pseudo-inverse `pinv' instead.

Or even better if you rewrite the lines:

F_{t}= inv(F{t});
MF_{t}= M{t} * F{t};

to

MF_{t}= M{t} / F{t};

Matlab will solve the linear equation: MF_{t} * F{t} = M{t} - which might have a solution even if F_{t} is singular - or if it is still singular solve via pseudo inverse.