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Suppose the time-series $X$ is hidden state Gaussian random walk and we observe $Y = X + e$, where $e$ is gaussian white noise independent of $X$.

The Kalman estimator of $X$ in this case has a steady state closed form solution and corresponds to an exponential moving average smoother with constant smoothing parameter. The optimal smoothing parameter looks like $\lambda = \frac{p}{1-p}$ where $p$ is a quadratic formula of signal to noise ratio between $e$ and $\Delta{x}$. See formula for closed form (by searching 'kalman solu‌​tion random walk noise').

If we have instead multiple independent observations $Y_1, Y_2,...$, i.e. independent $e_1, e_2,...$ is there a closed form solution for the optimal estimator of $X$? What would it look like?

If the $e_1, e_2$ were independent and identically distributed (same standard deviation), I can imagine the best estimator for $X$ is to simply average the estimators obtained treating the problem as univariate for each $Y$.

If the $e_1,e_2,...$ have different variance, then some $Y$ series should have low weightings in the overall estimator as their signal to noise ratio is poorer. Perhaps the closed form solution is linear with coefficients proportional to each $Y$ signal to noise ratio.

Is there a known closed form solution? Google seems not to be too helpful on this problem.

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  • $\begingroup$ formula for closed form: books.google.com/books?id=Kc6tnRHBwLcC&lpg=PA175&ots=I2VPQwXUGA&dq=kalman%20solution%20random%20walk%20noise&pg=PA175#v=onepage&q=kalman%20solution%20random%20walk%20noise&f=false $\endgroup$ – Matthew Pollard Feb 19 '13 at 6:55
  • $\begingroup$ First of all the constant smoothing parameter is achieved when the filter converges to the steady state. So in that respect exponential smoothing and kalman differ. Secondly I think the question requires clarification in the sense that when you mean Y1 and Y2 are they different time indicies or position in the vector Y for the same time? $\endgroup$ – Cagdas Ozgenc Nov 14 '13 at 12:51
  • $\begingroup$ Your observation noises $e_i$ are independent, but are your random walk $X$ increments independent? If they are, then you just have a bunch of 1-d independent filters and closed-form formula for 1-d case applies to each. $\endgroup$ – Kochede Dec 18 '13 at 2:04
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You haven't specified but if the observation noise is independent of the model noise then I think you should first combine the Y estimates with simple observation variance weighted mean as specified here:

http://en.wikipedia.org/wiki/Weighted_mean#Dealing_with_variance

after that apply kalman filter as if it is a univariate model. As a result you will have a lower observation noise hence a better estimate of x.

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