re: Cubic splines in package "mgcv"
I don't have access to Gu (2002) but clearly the function R(x,z) defined
on p126 of Simon Wood's book is piecewise quartic, not piecewise cubic.
Like Kunio Takezawa (below) I was puzzled by the word "cubic" on p126.
As Simon Wood writes, this basis is not actually used by mgcv when
specifying bs="cr".
Maybe the point is that at the knot, this continuous function has
continuous 1st and 2nd derivatives, but a discontinuous 3rd derivative, so
in that sense it behaves like a cubic spline.
Greg
#using the code from p127 of Wood:
#compare Wood Fig 3.4 (p125)
#if the function were piecewise cubic the 3rd derivative would be
piecewise constant
rk<-function(x,z)
{
((z-0.5)^2-1/12)*((x-0.5)^2-1/12)/4-((abs(x-z)-0.5)^4-(abs(x-z)-0.5)^2/2+7/240)/24
}
par(mfrow=c(2,2))
u<-seq(0,1,by=0.001)
plot(u,rk(u,5/6),main="function")
plot(u[-1],1e3*diff(rk(u,5/6),differences=1),main="1st derivative")
plot(u[-(1:2)],1e6*diff(rk(u,5/6),differences=2),main="2nd
derivative")
plot(u[-(1:3)],1e9*diff(rk(u,5/6),differences=3),main="3rd
derivative")
par(mfrow=c(1,1))
-------
From: Simon Wood <s.wood>
Date: Sun, 6 Jan 2008 16:59:35 +0000
On Wednesday 26 December 2007 04:14, Kunio takezawa
wrote:> R-users
> E-mail: r-help at r-project.org
> My understanding is that package "mgcv" is based on
> "Generalized Additive Models: An Introduction with R (by Simon N.
Wood)".
> On the page 126 of this book, eq(3.4) looks a quartic equation with respect
> to
> "x", not a cubic equation. I am wondering if all routines which
uses
> cubic splines in mgcv are based on this quartic equation.
--- No, `mgcv' does not use the basis given on page 126. See sections
4.1.2-4.1.8 of the same book for the bases used.
> In my humble opinion, the '^4' in the first term
> of the second line of this equation should be '^3'.
--- Perhaps take a look at section 2.3.3 of Gu (2002) "Smoothing Spline
ANOVA"
for a bit more detail on this/
>
> K. Takezawa
-- > Simon Wood, Mathematical Sciences, University of Bath, Bath, BA2 7AY UK
> +44 1225 386603 www.maths.bath.ac.uk/~sw283
