I'm reading a book about Polynomial curve fitting and I'm trying to make some of the examples in python, but I'm stuck at the calculus part.
We have a set of $N$ training points ($x$ represents observations on $x$ axis with their corresponding observations of the values of $t$.
$ \mathbf x = (x_1,x_2,..x_N)^T$
$ \mathbf t = (t_1,t_2,..t_N)^T$
I'm going to fit the data using a polynomial function of the form
$y(x,\mathbf W) = w_0 + w_1x+w_2x^2+...+w_Mx^M = \sum_{j=0}^M w_jx^j $
The coefficients $w_0,...w_M$ are denoted by the vector $\mathbf W$.
To determine the values of the coefficients I'm going to minimize the error function: $ E(\mathbf W) = \frac{1}{2}\sum_{n=1}^N (y(x_n,\mathbf W)-t_n)^2 $
and to do that I have to calculate the gradient of $E(\mathbf W)$ and "take little steps on the opposite direction"
the problem is that I don't know how to calculate the partial derivative of a vector function that also includes a summatory, i.e: $\frac{\partial}{\mathbf W} E(\mathbf W)$
Any help would be appreciated.
Thank you.