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In the Kalman Filter, is the one step look-ahead estimate generated before there is an observation, or after? i.e. if we have observations up until time t-1, do we use only this information to generate a prediction for time t?

Secondly, how do the one-step ahead predictions of the state differ from the one-step ahead predictions of the observation? My assumption is that the state predictions differ in that they are updated with the Kalman Gain, whereas the observation predictions are not.

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  • $\begingroup$ Before. First you predict than you use the observation to update your belief. Hope this video helps you: youtube.com/watch?v=Qa8YMP9dQYo $\endgroup$ – xboard Sep 18 '17 at 14:46
  • $\begingroup$ thanks, very helpful. I have a few more questions: (1) What statistical tests should we run to test the performance of the Kalman Filter? (2) Should I expect the difference between the forecast estimates and the observations to converge to zero over time? (3) Should I expect the difference between the posterior estimate and the observations to converge to zero? $\endgroup$ – user5211911 Sep 18 '17 at 17:08
  • $\begingroup$ Closely related: stats.stackexchange.com/questions/272736 $\endgroup$ – Juho Kokkala Sep 20 '17 at 5:52
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It's both. You have two steps: predict and smooth. The predict will use t-1 to predict t, then smooth will use t to update the state. This diagram from Wikipedia page on Kalman Flter summarizes the steps: enter image description here

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  • $\begingroup$ thanks, very helpful. I have a few more questions: (1) What statistical tests should we run to test the performance of the Kalman Filter? (2) Should I expect the difference between the forecast estimates and the observations to converge to zero over time? (3) Should I expect the difference between the posterior estimate and the observations to converge to zero? $\endgroup$ – user5211911 Sep 18 '17 at 17:06
  • $\begingroup$ 1. if it's a forecasting application then the main test is on forecast performance, look at MSFE for instance, look at whether the forecast errors correspond to model estimated errors 2. no, the measurement equation postulates that, your errors are stationary, they're not going anywhere. what does improve is your estimate of the state and parameters, which may improve your forecast, but errors will not disappear. 3. yes, in terms of the distribution, i.e. as you collect more observations your estimate of the distribution should improve. $\endgroup$ – Aksakal Sep 18 '17 at 17:28
  • $\begingroup$ thanks for the reply. In your reply to (2) you say that "what does improve is your estimate of the state and parameters, which may improve your forecast". When you speak of the estimate of the state, what do you mean? Are you referring to the posterior state estimate or the forecast? And when you say it may improve your forecast, how is this measured? Is it measured as the difference between your forecast and your posterior state estimate? $\endgroup$ – user5211911 Sep 18 '17 at 18:36
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    $\begingroup$ @Aksakal while this image is clearly licensed for such use (having been placed in the public domain) StackExchange's own policies ask you to credit the source. Something like "Image by Petteri Aimonen (placed in the public domain)" should be enough, though it would be good to link that to the wikimedia image source page which has license details. $\endgroup$ – Glen_b Sep 19 '17 at 3:36

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