[R] [Rd] Formulas in gam function of mgcv package

[Simon Wood]

This will not work...
> 2) y~s(x1, .... ,x36)
Estimating a 36 dimensional functions reasonably well would require a 
tremendous quantity of data, but in any case the 36 dimensional TPS smoothnes 
measure will involve such high order derivatives that it will no longer be 
practically useful: in fact you will not have enough data to estimate the 
unpenalized coefficients of the smoother (and if you did R would run out of 
memory first).  

In such a high dimensional situation, I think that GAMs are really only useful 
if you have some prior knowledge of which variables are likely to interact 
(and it's not too many of them). If there's no prior information saying 
roughly what sort of smooth additive structure might be useful then, I'm not 
sure that GAMs are the right way to go, and some sort of machine learning 
approach might be better.

Then again, the real problem with 
y~s(x1, .... ,x36)
is that the data just won't contain enough information to estimate s, if all 
you can say is that s is smooth, but this also means that it's very unlikely 
that you really need to estimate s(x1, .... ,x36) in order to predict well. 
In that case, starting from 
y ~ s(x1) + .... + s(x36)
and building the model up might result in something that does a reasonable 
predictive job. 

On the subject of tensor product smoothing vs isotropic smoothing. Isotropic 
smooths are really only reasonable if you think  that the smooth should 
display approximately the same amount of wiggliness in all directions. If 
this is not the case then tensor product smoothing is a better bet. Centering 
and scaling alone is not enough to ensure that isotropy is reasonable 
(although in particular cases it may help, of course).

best,
Simon



> I am trying to build a predictive model. Since the the variables are
> centred and scaled, I think I need an isotropic smooth. I am also
> interested in having the interactions between the variables included, that
> is not a purely additive model.
>
> It is not clear to me when should I give preference to tensor smooths,
> possibly because I have not understood well how they work.
>
> I am reading Wood(2003) as recommended and I have also read rather
> extensively Simon N. Wood. Generalized Additive Models: An Introduction,
> 2006, but still I am stuck. Any additional suggestion or reading
> recommendation would be greatly appreciated.
>
> I have also some difficulties in understanding the values you have chosen
> for k in the first example (why 60?).
>
> Thanks
>
> Best,
>
> On Monday 24 August 2009 17:33:55 Gavin Simpson wrote:
> > [Note R-Devel is the wrong list for such questions. R-Help is where this
> > should have been directed - redirected there now]
> >
> > On Mon, 2009-08-24 at 17:02 +0100, Corrado wrote:
> > > Dear R-experts,
> > >
> > > I have a question on the formulas used in the gam function of the mgcv
> > > package.
> > >
> > > I am trying to understand the relationships between:
> > >
> > > y~s(x1)+s(x2)+s(x3)+s(x4)
> > >
> > > and
> > >
> > > y~s(x1,x2,x3,x4)
> > >
> > > Does the latter contain the former? what about the smoothers of all
> > > interaction terms?
> >
> > I'm not 100% certain how this scales to smooths of more than 2
> > variables, but Sections 4.10.2 and 5.2.2 of Simon Wood's book GAM: An
> > Introduction with R (2006, Chapman Hall/CRC) discuss this for smooths of
> > 2 variables.
> >
> > Strictly y ~ s(x1) + s(x2) is not nested in y ~ s(x1, x2) as the bases
> > used to produce the smoothers in the two models may not be the same in
> > both models. One option to ensure nestedness is to fit the more
> > complicated model as something like this:
> >
> > ## if simpler model were: y ~ s(x1, k=20) + s(x2, k = 20)
> > y ~ s(x1, k=20) + s(x2, k = 20) + s(x1, x2, k = 60)
> >                                   ^^^^^^^^^^^^^^^^^
> > where the last term (^^^ above) has the same k as used in s(x1, x2)
> >
> > Note that these are isotropic smooths; are x1 and x2 measured in the
> > same units etc.? Tensor product smooths may be more appropriate if not,
> > and if we specify the bases when fitting models s(x1) + s(x2) *is*
> > strictly nested in te(x1, x2), eg.
> >
> > y ~ s(x1, bs = "cr", k = 10) + s(x2, bs = "cr", k = 10)
> >
> > is strictly nested within
> >
> > y ~ te(x1, x2, k = 10)
> > ## is the same as y ~ te(x1, x2, bs = "cr", k = 10)
> >
> > [Note that bs = "cr" is the default basis in te() smooths, hence we
> > don't need to specify it, and k = 10 refers to each individual smooth in
> > the te().]
> >
> > HTH
> >
> > G
> >
> > > I have (tried to) read the manual pages of gam, formula.gam,
> > > smooth.terms, linear.functional.terms but could not understand
> > > properly.
> > >
> > > Regards

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