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What is the appropriate way to implement KF if your sensor confidence is time and observation dependent?

Ex, you asses the quality of camera tracking by the percentage of features correctly matched by a homography or the variance of a GPS output over a short time window.

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Just look at the Kalman equations:

Kalman update equations

Whereas normally, $Q_t$ and $R_t$ are constants (do not depend on $t$), your measurement noise covariance ($Q_t$) will be time varying.

The only real upshot is that the Kalman gain $K_t$ and the state covariance $P_t$ won't converge to constants $K_\infty$ and $P_\infty$ as $t \rightarrow \infty$.

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    $\begingroup$ While you're wrapping your head around that, consider that it applies to all of the input matrices in the Kalman -- so extending the Kalman filter to deal with a fully time-varying linear system is arithmetically trivial (if not, perhaps, trivial to know in advance how well it'll perform). $\endgroup$
    – TimWescott
    Oct 22 at 15:04

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