I have 50 or so training examples involving a set of 200 or so real numbers (x1,x2,...,x200) (normalized to a 0 mean and std dev 1), and a single output real (y) in the range 0.0..1.0. I want to fit a linear model as follows:
y = w0 + w1 * x1 + w2 * x2 + ... + w200 * x200
So I need to calculate (w0,w1,w2,...,w200) based on the training examples. By what formula or algorithm should I calculate these weights?
There is a whole area of machine learning studying all different aglorithms to do that. You probably can't visualize your data because it lives in $\mathbb{R}^{200}.$
I guess you should try least squares. Let $X : \mathbb{R}^{50\times 201}$ be your feature matrix augmented with column of $1$'s (to account for $w_0$). $X$ has $50$ rows; one for each training example. Each row looks like: $$(1, x_1, x_2, \ldots, x_{200}).$$ Let $y : \mathbb{R}^{50\times 1}$ be your output vector.
The linear least squares will solve for $w$ in $Xw = y,$ where $w : \mathbb{R}^{201\times 1}$ are the weights you're looking for.
The naive solution is solving the following linear system for $w$: $$ X^{T}Xw = X^{T}y, $$ but better numerical methods exist.