I have a problem that is similar to the state space model in Kalman filter but the observation matrix $G_t$ of $$y_t=G_tx_t +w_t,$$ is random. The elements of $G_t$ are i.i.d. random variables with a given distribution. Is it possible to use the Kalman filter to this problem? If it has already been used please provide a reference.

Above $w_t$ is noise vector of Gaussian distribution with known covariance matrix, $x_t$ is the state of the system at time $t$ and $y_t$ is the observation at time $t$. The state tarnsision is given by $x_{t+1}=F_tx_t+B_tu_t+v_t$ where $F_t$ is a known matrix and $v_t$ is Gaussian noise with known covariance, $B_t$ is also a known matrix and $u_t$ is Gaussian distributed random input.

Thanks a lot.

  • $\begingroup$ @JasonR Thanks. Do you know if people have done it? It seems so natural in many problems. $\endgroup$
    – triomphe
    Commented Oct 29, 2013 at 15:16

1 Answer 1


Very interesting question!

I do not know of any reason why you can't do this, but I have had a hard time finding an example. I finally found this paper, which does an analysis of what happens when the measurements are randomly lost due to transport delays / errors in communication systems.

The main problem that they report is that the error covariance matrix iteration is now stochastic rather than deterministic (as in the usual Kalman filter formulation).

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