# Radial basis function network - G function?

I'm trying to understand Radial Basis Function Network. I have (don' know how to write proper formatter mathematical functions here..):

$x = [ -1.0000, -0.5000, 0,0.5000,1.0000]$

$y_i = f(x_i)$

$f(x) = \ \frac{1}{(1+x^2)}$

$y = [0.5000, 0.8000,1.0000, 0.8000,0.5000]$

xy = [ -1.0000    0.5000
-0.5000    0.8000
0    1.0000
0.5000    0.8000
1.0000    0.5000]


Then there is:

$Phi_j(x) = exp(-(|x - t_j|/4))$

$t_j = -1 + (j - 1)\ \times\ \frac{2}{m_1 - 1}$

$m_1 <= N$

And then G matrix is written like this:

$G = (Phi_j(x_i,t_j))$

This matrix is used to calculate weights:

$w = G^+y$

$G^+ = (G'G)^-1G'$

So it looks for me that G uses Phi function with two inputs, but Phi is written as one input function. Do I miss something here?

• please fix your latex -- this is indecipherable. – Sycorax Nov 6 '17 at 18:30

$$f(x) = \sum_{n}w_{n}G(\lVert x-x_{n} \rVert)$$
where $x_{n}$ are the training data points. The two parameters you are referring to are the centers ($t_{n}$ in your notation) and the point you want to evaluate the function at, $x$. There are actually many possible elections for the kernel function, $G$. Each choice corresponds to a different type of regularization. One you make a choice, solving the problem reduces to a linear algebraic equation.