Assume for simplicity that the problem is 1d, the transition we are studying is the very simple:

$ x_{t+1} = x_t + \epsilon $
$ z_{t} = x_t + u $

The measurement-update phase of the 1d Kalman filter looks like this:
$ k = \frac{p_{t-1}}{p_{t-1}+\sigma_u^2} $
$ \hat x_{t} = \hat x_{t-1} + k_d (z_t - \hat x_{t-1})$
$ p_t = p_{t-1} - p_{t-1} k $

What I don't understand is why $p$ (the estimated error variance) decreases at every measurement regardless of how bad the prediction is. It seems to me that when the prediction is off by a lot we should increase our uncertainty on $\hat x$ rather than decrease it.

  • $\begingroup$ An alternative way to write the last equation seems to be $p_t=(1-k)p_{t-1}=\frac{p_{t-1}\sigma^2_u}{p_{t-1}+\sigma^2_u}$, maybe that can help to see something (I don't know). $\endgroup$ – Richard Hardy Aug 10 '16 at 16:15

The Kalman filter assumes the data is noisy around the model you are using, say a noisy distribution around a parabolic curve. It assumes that the more data points you run through, the more accurate its estimation of the parabola's coefficients will become. Thus it slows down adaptation in order to converge on what it believes SHOULD be the correct coefficient estimations. Adaptation will slow down regardless of how bad the difference is between the filter's predictions and the measurements. If the data veers off course, and is no longer following the parabola's trajectory, the filter's estimates will get worse, unless the model is either reset, or a different model is employed.


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