Spline Toolbox

Smoothing

What if the data are noisy? For example, suppose that the given values are

• noisy = y + .1*(rand(size(x))-.5);
  

Then you might prefer to approximate instead. For example, you might try the cubic smoothing spline, obtained by the command

• scs = csaps(x,noisy);
  

and plotted by

• fnplt(scs), hold on, plot(x,noisy,'o')
  legend('smoothing spline','noisy data'), hold off
  

This produces a figure like this:

Figure 2-4: Cubic Smoothing Spline to Noisy Data

If you don't like the level of smoothing done by csaps(x,y), you can change it by specifying the smoothing parameter, p, as an optional third argument. Choose this number anywhere between 0 and 1. As p changes from 0 to 1, the smoothing spline changes, correspondingly, from one extreme, the least squares straight-line approximation to the data, to the other extreme, the "natural" cubic spline interpolant to the data. Since csaps returns the smoothing parameter actually used as an optional second output, you could now experiment, as follows:

• [scs,p] = csaps(x,noisy); fnplt(scs), hold on
  fnplt(csaps(x,noisy,p/2),'--')
  fnplt(csaps(x,noisy,(1+p)/2),':'), plot(x,noisy,'o')
  legend('smoothing spline','more smoothed','less smoothed',...
  'noisy data'), hold off
  

This produces the following picture.

Figure 2-5: Noisy Data More or Less Smoothed

At times, you might prefer simply to get the smoothest cubic spline sp that is within a specified tolerance tol of the given data in the sense that

• norm(noisy - fnval(sp,x))^2 <= tol
  

This spline is provided by the command

• sp = spaps(x,noisy,tol);
  

Knot Choices

Least Squares