null.space.dimension: Dimension of the space of un-penalized functions.
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. This routine 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, 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
$M=(m+d+1)!/(d!(m-d)!)$
which is the value returned.
References
Wahba (1990) spline models of observational data. SIAM.