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 ?
• 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 – Taylor Jun 14 '16 at 3:48
• 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? – Peter K. Jun 15 '16 at 22:26