I have a function which I'm trying to estimate:
$U(x, y) = \begin{bmatrix} a_0 + a_1x + a_2y \\ a_3 + a_4x + a_5y\end{bmatrix}$
I have an initial estimate of the parameters. I want to iterate n times and identify a $U_n$ such that $E(U) = \sum_{X}[I_{t}(X) - I_{t-1}(X - U(X)]^2$ is minimum.
Here, $X = \begin{bmatrix}x\\ y\end{bmatrix}$. You can assume that the function $I_{0,1,2...l}$ are known. Also, $t$ remains constant for this optimization.
I've been reading up online and came across the Newton-Gauss optimization method. It looks like this method would help me find $U$ and it'll be relatively easy to write code for this.
I've been trying to follow this video lecture: https://www.youtube.com/watch?v=22y9BPrgL_o - but the professor lost me after a certain point.
I'm not sure where to look for more information on this. Any suggestions/help? Please let me know if there's more information I can provide.
(My question for the video: why do you have multiple 'observations' with the same weight $w$? Why can't you have just one observation and move on to the next iteration?)
