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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.

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  • $\begingroup$ @JasonR Thanks. Do you know if people have done it? It seems so natural in many problems. $\endgroup$ – triomphe Oct 29 '13 at 15:16
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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).

enter image description here

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