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I'm trying to estimate the state of a Gaussian random walk with central tendency based on time series measurements with varying uncertainties. My random variable has the following form:

$ \frac{d x}{d t} \equiv F(t) - \alpha x $

Where F is a Gaussian random variable. I've noticed that this problem is analogous to the velocity of a bubble experiencing Brownian motion. (See for example, F. Reif, Fundamentals of Statistical and Thermal Physics, p. 565). As a result of the $ -\alpha x $ term, the position has a central tendency (i.e. the variance does not become infinity as time approaches infinity).

Now, like any good physicist, I know that I cannot exactly measure the value $x$. The best I can do is to measure it at time $ t_i $ within some uncertainty, $ \sigma_i $. Using a Kalman filter, I can estimate the value of $x$ from several measurements. Let's call that $ \hat x $. The approach is as follows. For each measurement, we compute:

$ \delta t = t_i - t_{i-1} $

$ P(t) = P(t_{i-1}) * e^{-\alpha\, \delta t} + \langle x^2 | \delta t \rangle $

$ K = {{P}\over{P + \sigma_i}} $

Our incoming est estimate of $x_i$:

$ \hat x_{i-} = \hat x_{i-1} \, e^{-\alpha\,\delta t} $

$ \hat x_i = \hat x_{i-} + K [x_{obs} - \hat x_{i-}] $

This works great for propagating our estimates forward in time. My question is: If I have measurements at times that span the time at which I want the best estimate, how do I compute an $\hat x(t) $ where $ t_i < t < t_{i+1} $?

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    $\begingroup$ I'm a bit confused, perhaps by the title. Are you wanting to interpolate between two time points or are you asking about something like smoothing (as opposed to filtering)? $\endgroup$ – Wayne Feb 24 '11 at 15:12
  • $\begingroup$ @Wayne I'd like to interpolate between the two measurement times, but include information about the future measurement -- not just the previous measurements. $\endgroup$ – Carl F. Feb 24 '11 at 19:34
  • $\begingroup$ just to clarify, by * you mean the usual product or convolution? $\endgroup$ – mpiktas Feb 25 '11 at 12:49
  • $\begingroup$ @mpiktas product. One of the hazards of copying from code. :) I've corrected the post for clarity. $\endgroup$ – Carl F. Feb 26 '11 at 2:39
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Seems to me like a Kalman smoothing problem, which essentially computes the mean value of the state given past, present and future observations, with a computational effort similar to two Kalman filter passes. You might want to check any of the good books in existence which deal with the Kalman smoother, among them Durbin-Koopman and Anderson-Moore.

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  • $\begingroup$ Thanks. I was hoping to see a worked example, but the right key words to search on was helpful. I think I've found a good description. Now to apply it. $\endgroup$ – Carl F. Feb 27 '11 at 2:19

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