[a, xc]=train_thin_plate_spline(x,y)

function [a, xc]=train_thin_plate_spline(x,y)
%[a, xc]=train_thin_plate_spline(x,y)
%Trains a thin plate spline Radial Basis Function Network.
%Inputs:
% x - network input formatted as N_dimension rows and N_points columns
% y - desired network output. Row vector with N_points columns
%Outputs:
% a - basis function weights.  
% x_c - basis function centres formatted as N_dimension rows and N_centres
%   columns.  This will be equal to x

%
%Copyright (c) 2009, Travis Wiens
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xc=x;%basis function centres
N=size(x,2);%number of points
r=zeros(N);%basis function radii
for i=1:N
    for j=1:N
        r(i,j)=sqrt(sum((xc(:,i)-x(:,j)).^2));
    end
end

E=r.^2.*log(r);%apply thin plate spline RBF
E(find(r==0))=0;%avoid -inf

a=y/E;%use mrdivide to solve system of equations. For large systems it may 
%make more sense to use an online method such as recursive least squares
%(RLS)