Smoothing spline (spaps)

Hi, I've got a question concerning the behaviour of the 'spaps' function (smoothing spline in curve fitting toolbox). It works as expected if the input vector contains large numbers, but does not seem to work for small numbers. An example:

This behaves as I would expect it:

x = 1:20;
y = 10 + 2*randn(20,1);
tol = ones(20,1);

sp = spaps(x,y,tol); % smoothing spline

subplot(2,1,1)
errorbar(x,y,tol,'LineStyle','none');
hold on
xi = 1:0.1:20;
plot(xi,fnval(sp,xi),'r');
title('spaps(x,y,tol)')
hold off

Now the same as above, but the y and tol values multiplied by 0.01. I would expect the same curve as above (also multiplied by 0.01), but instead the smoothing parameter is 0 resulting in a straight line fit.

scale = 0.01;
y = y*scale;
tol = tol*scale;

sp = spaps(x,y,tol);

subplot(2,1,2)
errorbar(x,y,tol,'LineStyle','none');
hold on
xi = 1:0.1:20;
plot(xi,fnval(sp,xi),'r');
title('spaps(x,0.01*y,0.01*tol)')
hold off

Why is 'spaps' not scaleable?

Answer 1

[Andrew Newell]

Interesting question. I don't know anything about the algorithm, but the documentation says that the distance being minimized is a sum of squares, so I thought I'd try

tol = tol*scale^2;

It works, producing a fit that lies on top of the first fit when divided by scale.