[R] [Rd] Formulas in gam function of mgcv package
s.wood at bath.ac.uk
Wed Aug 26 10:56:13 CEST 2009
> > 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?
The first says that you want a model
E(y) = f_1(x_1) + f_2(x_2) + f_3(x_3) + f_4(x_4) (1)
where the f_j are smooth functions. The additive decomposition is quite a
strong assumption, since it assumes that the effect of x_j is not dependent
on x_k unless j=k. The second model is just
E(y) = f(x_1,x_2,x_3,x4) (2)
where f is a smooth function. This looks very general, but actually `s' terms
assume isotropic smoothness, which is also quite a strong assumption.
Now if I simply state that f and the f_j are `smooth functions', and leave it
at that, then (2) would of course contain (1), but to actually estimate the
models I need to state, mathematically, what I mean by `smooth'. Once I've
done that I've pretty much determined the function spaces in which f and the
f_j will lie, and in general (2) will no longer strictly contain (1). mgcv's
`s' terms use a thin plate spline measure of smoothness for multivariate
smooths, and this means that (1) will not be strictly nested within (2),
since e.g. a 4D thin plate spline can not generally represent exactly what
the sum of 4 1D splines can represent.
If you want to acheive exact nesting then using tensor product smooths with
will do the trick (because the function space for (4) is built up from the
function spaces used in (3)).
As to where all the 2 and 3 way interactions have gone in (4)... it's just
like ANOVA - if you put in a 4 way interaction then the lower order
interactions are not identifiable, unless you choose to add constraints to
make them so. `mgcv' will allow you add main effects and interactions, and
will handle the constraints automatically, but if this sort of functional
ANOVA is a major component of what you want to do, then it is probably worth
checking out the gss package and Chong Gu's book on smoothing spline ANOVA.
> Simon Wood, Mathematical Sciences, University of Bath, Bath, BA2 7AY UK
> +44 1225 386603 www.maths.bath.ac.uk/~sw283
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