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Suppose my state-space model is: $$x_{t+k} = Ax_{t}+\eta$$ $$z_{t} = Bx_{t}+\epsilon$$ ,where $\eta\sim N(0,\Sigma)$ and $\epsilon\sim N(0,\Phi)$.

Since I want a multi-step model instead of the usual one step in Kalman filter, where $k=1$, I trained $\Theta=\{A,B,\Sigma,\Phi\}$ on $S_0=\{z_0,z_k,z_{2k},...\}$,where $k>1$. However by then I waste other data, therefore I also train $\Theta$ on $S_1=\{{z_1,z_{k+1},z_{2k+1}\}},...,S_{k-1}=\{z_{k-1},z_{2k-1},...\}$,and obtained $\Theta_0,...,\Theta_{k-1}$.

Now I end up with $k$ state space models all describing the $k$-step hidden process of $x_t$. How am I going about combining them together ?

I have looked up the literature but not much of them describe how to train a multi-step Kalman Filter.

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    $\begingroup$ If $\Sigma $ I'd a covariance matrix you wrote your model down wrong. The name of your normal random error vector needs a different name than the parameters of its distribution $\endgroup$ – Taylor Jun 14 '16 at 3:48
  • $\begingroup$ I also don't really get a clear idea of what you're trying to do $\endgroup$ – Taylor Jun 14 '16 at 3:49
  • $\begingroup$ You are right. Updated $\endgroup$ – kchpchan Jun 15 '16 at 19:27
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    $\begingroup$ Yes, I really don't get what you're trying to achieve. There are two problems you seem to be confusing: 1) system identification to find your model parameters, and 2) Kalman prediction ($k$ step look ahead). Can you please explain a bit more about your application? $\endgroup$ – Peter K. Jun 15 '16 at 22:26

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