From: Gary Knott <knott@civilized.com>
  To  : rasmb@bbri.harvard.edu
  Date: Mon, 10 Feb 97 12:10:34 -0500

non-parametric regression

Dear RASMBers,

Re the recent discussion of spline functions,
the general situation is to  solve the problem
of finding an appropriate
model for data, without much regard for the
functional form of that model.
What 'appropriate' is depends on the
motives of the investigator.  Often
we want a smooth curve so we can differentiate
it and use the derivative.

In statistics, solving this problem is
called doing non-parametric regression ('regression'
because Galton named straight-line curve-fitting
with this term, and  'non-parametric' because
we don't insist on a particular functional form.)

There are a number of ways to do non-parametric
regression: weighted moving means, moving medians
(when outliers are an issue), moving polynomial
regression (called the LOESS method in the linear
case), kernel estimators, and smoothing splines,
among others.

We often think of a C^1 or C^2 spline as a
piecewise-cubic curve that interpolates the
data points, so the adjective 'smoothing'
is used to indicate that interpolation is
not demanded, nor even desirable.  (Bezier
splines and B-splines, as commonly presented
are smoothing splines of a sort, but they
are more akin to kernel estimators.)

Two useful smoothing splines are the
least-squares smoothing spline mentioned
by Dr. Lechner, and the so-called Optimal
smoothing spline which minimized a mixed
functional consisting of a least-squares
term and an integrated  squared second-derivative
term.  The latter is particularly robust and
smooth enough to be used for differentiation.

A large family of smoothing methods, (recently
expanded to include optimal smoothing splines)
are provided in the MLAB (Modeling Laboratory)
software (www.civilized.com), and  they can
be found in a few other places as well,
notably S-PLUS. (www.mathsoft.com)   My
biased opinion (as the principle author of
MLAB) is that MLAB has the advantage of being
well-suited to data-analysis in chemistry,
including, in particular, ultracentrifuge modeling.

I hope this is useful. ---
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