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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?

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  • $\begingroup$ please fix your latex -- this is indecipherable. $\endgroup$ – Sycorax Nov 6 '17 at 18:30
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Actually the point is that in RBF you approximate the density of probability of the function (in the case of regression) through a linear combination of kernel functions of the form,

$$f(x) = \sum_{n}w_{n}G(\lVert x-x_{n} \rVert)$$

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.

If you are interested on the mathematical basis of RBF, you may take a look at this paper.

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You can view RBF as linear regression with a non linear transformation of the inputs. ( just as you could use x^2, x^3 etc). The G+ Matrix is just standard linear regression formula. So first step, fix the t's - this could be either by taking random x (x[r_1], x[r_2],...)samples from your inputs, or again by performing unsupervised learning on your inputs to find "prototypes". These ts are now fixed, and you calculate G by applying phi to all x[i's] and t[j]s. Finally you perform linear regression, namely the G(+)

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