Consider $f(x, y) = a + bx + cy + dxy$. Given a set of n data points where the $i$’th data point is given by : $(x_{i} , y_{i} , z_{i})$, find the system of linear equations to solve for $a, b, c, d$ such that
(A) $\frac{1}{n}\sum_{i=1}^{n} (f(x_{i},y_{i}) -z_{i})^2$ is minimized. Solution Given as:
(B) MSE=$\frac{1}{n}\sum_{i=1}^{n}(a+bx_{i}+cy_{i}+dx_{i}y_{i}-z_{i})^2$
$\frac{∂}{∂a}MSE(a,b,c,d)=\frac{1}{n}\sum_{i=1}^{n}(1)(a+bx_{i}+cy_{i}+dx_{i}y_{i}-z_{i})$
$\frac{∂}{∂a}MSE(a,b,c,d)=\frac{1}{n}\sum_{i=1}^{n}(x)(a+bx_{i}+cy_{i}+dx_{i}y_{i}-z_{i})$
$\frac{∂}{∂a}MSE(a,b,c,d)=\frac{1}{n}\sum_{i=1}^{n}(y)(a+bx_{i}+cy_{i}+dx_{i}y_{i}-z_{i})$
$\frac{∂}{∂a}MSE(a,b,c,d)=\frac{1}{n}\sum_{i=1}^{n}(xy)(a+bx_{i}+cy_{i}+dx_{i}y_{i}-z_{i})$
Can someone please explain a bit what is happening here ? Completely lost, if correctly remember correctly,it uses least squares method? How can we get step B from step A? Thank you advance.
provide a pic for avoid typos
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