Class 22: Calculating Smoothing Splines

Review from previous class

• We started to look at smoothing splines, and considered the foundations of their calculation.

Smoothing Splines

Lisp files for today's class

• splsm.lsp
• splsm.work
• utils.lsp
• splsm.so

Handout

This is a revised draft. It'll never be complete, but this is about all that I intend to add this year.

Bayesian Spin

Though it may be surprising, other arguments lead to the same time of estimating equations. These penalized least squares estimators can also be motivated from a "shrinkage perspective" (as in ridge regression) or by posing a prior distribution located at zero.

Smoothing and cross-validation

The trick in smoothing is to chose the smoothing parameter in the loss function. Once you have picked lambda, the rest of the smoothing is a linear problem (like a regression problem). This conditional linearity suggests a `kludge' for picking lambda from data: pretend the smoothing matrix is a projection matrix and use its diagonal as leverage values.

Computing a Spline

The code in the lisp files for today is pretty fast stuff, converted from the original Fortran code of deBoor. Makes it easy to do things like compute the smoothing matrix quickly. On the Sun, one builds a file like splsm.so using the commands

		       cc -c -xcg92 -xO3 splsm.c
               ld -G -o splsm.so splsm.o
        

Watch out for doing too much optimization; code starts to break here easily.

Next time