If I define a GP over observed values, $y$, of a sensor reading over time, $t$, as (for simplicity assuming discrete time series e.g lets say readings after every 5 mins) :
$y=f(t)+\epsilon$
where $t=[{1\dots N}$] is time and $\epsilon=\mathcal{N}(0,\sigma_{y}^2)$ is a zero mean gaussian noise.
Then a GP model for $y$ is:
$y(t)=GP\Big(0,K(t,t)+\sigma_{y}^2I\Big)$.
If $L$ is the cholesky decomposition of the above covariance matrix $K(t,t)$ (size NxN), and $S$ is a $Nx1$ vector computed as:
$S=L^T$$\backslash$$(L $$\backslash$$y)$
(A computationally optimised form of computing $S=K^{-1}y$ using cholesky decomposition $L$)
My question is, if $S$ and $L$ are already computed and i want to find out $S_h$ and $L_h$ for only the first $h$ inputs i.e $t=[1 \dots h]$, where $h < n$
I know that given $L$, $L_h$ can be computed by partitioning $L$ as:
$L_h=\begin{pmatrix}
L_{1,1} \dots L_{1,h}\\
\vdots \ddots \vdots\\
L_{h,1} \dots L_{h,h}
\end{pmatrix}$ i.e. a $h x h$ size square partition of L (top left to be exact).
However, would the following be true?: (my math knowledge is a little rusty)
$S_h=L_h^T \backslash (L_h \backslash y_h)$
implying that:
$Sh=S(1 \dots h$) i.e a sub vector of $S$
The reason i ask this is because I want these to deduce the following without computing anything additional.
$p(f_h|y_h;\theta)$
where:
$f_h = f[1 \dots h]$
&
$y_h=y[1 \dots h]$.
where the model is trained on $y[1 \dots n]$.
This is my first post i might not have described the problem clearly enough so please bear with me, i would really appreciate your help and will try to clarify any questions/problems you have in understanding the problem i tried to describe.
