[R] Fitting generalized additive models with constraints?

[Simon Wood]

On Tuesday 05 September 2006 20:32, David Reiss wrote:
> >> I am trying to fit a GAM for a simple model, a simple model, y ~ s(x0) +
> >> s(x1) ; with a constraint that the fitted smooth functions s(x0) and
> >> s(x1) have to each always be >0.
> >>
> >> From the library documentation and a search of the R-site and R-help
> >> archives I have not been able to decipher whether the following is
> >> possible using this, or other GAM libraries, or whether I will have to
> >> try to "roll my own". I see from the mgcv docs that GAMs need to be
> >> constrained such that the smooth functions have zero mean. Is there a
> >> way around this?
> >>
> >> Is such a constraint possible?
> >
> > It is possible to estimate a GAM subject to this constraint, but be aware
> > that the mean levels of your component smooths are not identifiable, so
> > there is an unavoidable abitrariness in the estimate....
> >
> > You have to have some sort of constraint on the smooths in a GAM to
> > ensure identifiability, and a convenient way to set the model up is to
> > write it as e.g.
> >
> > E(y) = a + f0(x0) + f1(x1)
> >
> > where `a' is the intercept and f0 and f1 are smooth functions which sum
> > to zero over their respective covariate values. In this parameterization
> > your constraint implies that
> >
> > a + f0(x0) + f1(x1)  > 0
> >
> > for all x0, x1. If this constraint is met then you can find constants b
> > and c such that b+c=a such that f0(x0)+b>0 and f1(x1)+c>0 for all x0,x1.
> > i.e. you redefine f0 as f0+b and f1 as f1+c, and you have a fitted model
> > meeting the constraints.
> >
> > To fit the GAM subject to the constraints you can use mgcv:::pcls...
> > ?pcls has some examples, but it does involve moderately low level
> > programming. It's hard to impose the constraint exactly, so the usual
> > approach would be to impose the constraint over a fairly fine grid of x0,
> > x1 values. Also, you'll need to figure out how to select smoothing
> > parameters. For many problems it suffices to estimate smoothing
> > parameters on the unconstrained fit, and then use these to fit subject to
> > constraints, but it depends on the problem....
> >
> > Hope that's some use.
> >
> > Simon
>
> Hi Simon,
> thanks very much for the advice. I will try to parse your response and
> the pcls docs and see if I can get this to work. In the meantime, I
> found a paper that tries to achieve a similar thing with the same
> constraints as I am working with, using quadprog:
> http://www.esajournals.org/esaonline/?request=get-abstract&issn=0012-9658&v
>olume=083&issue=08&page=2256 -David

- The additional wrinkle in gradient matching is that you have the equality 
constraints that the functions pass through zero at zero (a zero population 
can't produce offspring or corpses! ), which you don't have do you? If you do 
then it removes the ambiguity in the model, and makes everything a bit 
easier.
best,
Simon

-- 
> Simon Wood, Mathematical Sciences, University of Bath, Bath, BA2 7AY UK
> +44 1225 386603  www.maths.bath.ac.uk/~sw283