mgcv (version 0.9-6)

null.space.dimension: The basis of the space of un-penalized functions for a t.p.r.s.

Description

The thin plate spline penalties give zero penalty to some functions. The space of these functions is spanned by a set of polynomial terms. null.space.dimension finds the dimension of this space, $M$, given the number of covariates that the smoother is a function of, $d$, and the order of the smoothing penalty, $m$. If $m$ does not satisfy $2m>d$ then the smallest possible dimension for the null space is found given $d$ and the requirement that the smooth should be visually smooth. null.space.basis.powers() produces an M by d array, the ith row of which gives the powers of each of the d covariates featuring in the ith basis term. null.space.basis.names() gives the equivalent information as a list of basis labels - the key feature of which is that the same term will always have the same label, irrespective of the order in which variables are supplied to the routine. For some models a smooth may be multiplied by a by variable: in such cases the basis labels can include this name as well, but note that the constant label will be returned as if the by variable didn't exist (to meet requirements of function gam.side.conditions).

Usage

null.space.dimension(d,m)
null.space.basis.powers(m,d)
null.space.basis.labels(names,m,by="NA")

Arguments

d

is a positive integer - the number of variables of which the t.p.s. is a function.

m

a non-negative integer giving the order of the penalty functional, or signalling that the default order should be used.

names

is an array of d variable names

by

the name of any by variable, which mutliplies all the terms in the penalty null space basis.

Value

• null.space.dimension returns an integer (array), the null space dimension $M$: this is the only one of these functions that accepts array arguments. null.space.basis.powers produces a 2-d array, p, say: each row corresponds to one of the M basis vectors, while each column corresponds to one covariate. So if x1, x2, x3, etc are the covariates, the ith basis vector is given by x1^p[i,1]*x2^p[i,2]*x3^p[i,3].... null.space.basis.labels() just produces uniques labels for the basis terms.

WARNING

This routine operates symbolically, not numerically: your model may still be numerically un-identifiable, and the routine can't tell if two variables with different names are really the same.

Details

Thin plate splines are only visually smooth if the order of the wiggliness penalty, $m$, satisfies $2m > d+1$. If $2m

$M=(m+d+1)!/(d!(m-d)!)$

which is the value returned.

References

Wood, S.N. (2003) Thin plate regression splines. J.R.Statist.Soc.B 65(1):95-114

http://www.stats.gla.ac.uk/~simon/

See Also

gam.side.conditions, gam

Examples

null.space.dimension(2,0)
null.space.basis.powers(2,2)
null.space.basis.labels(c("x","z"),m=2)
null.space.basis.labels(c("z","x"),m=2) # labels are invariant