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Fitting Values at Scattered 2-D Sites with Thin-Plate Smoothing Splines

Tensor-product splines are good for gridded (bivariate and even multivariate) data. For work with scattered bivariate data, the toolbox provides the thin-plate smoothing spline. Suppose you have given data values y(j) at scattered data sites x(:,j), j=1:N, in the plane. To give a specific example,

n = 65; t = linspace(0,2*pi,n+1); 
x = [cos(t);sin(t)]; x(:,end) = [0;0];

provides 65 sites, namely 64 points equally spaced on the unit circle, plus the center of that circle. Here are corresponding data values, namely noisy values of the very nice function .

y = (x(1,:)+.5).^2 + (x(2,:)+.5).^2;
noisy = y + (rand(size(y))-.5)/3;

Then you can compute a reasonable approximation to these data by

st = tpaps(x,noisy);

and plot the resulting approximation along with the noisy data by

fnplt(st); hold on
hold off

and so produce the following picture:

Thin-Plate Smoothing Spline Approximation to Noisy Data

The three-dimensional plot contains a surface and a collection of black dots. The dots lie close to the surface in the shape of a circle.