mgcv (version 1.8-31)

null.space.dimension: The basis of the space of un-penalized functions for a TPRS

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.

Usage

null.space.dimension(d,m)

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.

Value

An integer (array), the null space dimension $M$.

Details

Thin plate splines are only visually smooth if the order of the wiggliness penalty, $m$, satisfies $2m>d+1$. If $2m<d+1$ then this routine finds the smallest $m$ giving visual smoothness for the given $d$, otherwise the supplied $m$ is used. The null space dimension is given by:

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

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.maths.bris.ac.uk/~sw15190/

See Also

tprs

Examples

# NOT RUN {
require(mgcv)
null.space.dimension(2,0)
# }