smooth.construct.sos.smooth.spec {mgcv} | R Documentation |
gam
can use isotropic smooths on the sphere, via terms like
s(la,lo,bs="sos",m=2,k=100)
. There must be exactly 2 arguments to such a smooth.
The first is taken to be latitude (in degrees) and the second longitude (in degrees).
m
(default 1) is an integer in the range 0 to 4 determining the order of the penalty used.
For m>0
, (m+2)/2
is the penalty order, so m=2
is the equivalent to the usual second
derivative penalty. m=0
signals to use the 2nd order spline on the sphere, computed by
Wendelberger's (1981) method. m = -1
results in a Duchon.spline
being
used (with m=2 and s=1/2).
k
(default 50) is the basis dimension.
## S3 method for class 'sos.smooth.spec' smooth.construct(object, data, knots) ## S3 method for class 'sos.smooth' Predict.matrix(object, data)
object |
a smooth specification object, usually generated by a term s(...,bs="sos",...) . |
data |
a list containing just the data (including any by variable) required by this term,
with names corresponding to object$term (and object$by ). The by variable
is the last element. |
knots |
a list containing any knots supplied for basis setup — in same order and with same names as data .
Can be NULL |
For m>0
, the smooths implemented here are based on the pseudosplines on the sphere of Wahba (1981)
(there is a correction of table 1 in 1982, but the correction has a misprint in the definition of A — the A
given in the 1981 paper is correct). For m=0
then a second order spline on the sphere is used which is the analogue of
a second order thin plate spline in 2D: the computation is based on Chapter 4 of Wendelberger, 1981.
Optimal low rank approximations are obtained using exactly the approach given in Wood (2003).
Note that the null space of the penalty is always the space of constant functions on the sphere, whatever the order of penalty.
An object of class "sos.smooth"
. In addition to the usual elements of a
smooth class documented under smooth.construct
, this object will contain:
Xu |
A matrix of the unique covariate combinations for this smooth (the basis is constructed by first stripping out duplicate locations). |
UZ |
The matrix mapping the t.p.r.s. parameters back to the parameters of a full thin plate spline. |
Simon N. Wood simon.wood@r-project.org, with help from Grace Wahba (m=0 case) and Jean Duchon (m = -1, case).
Wahba, G. (1981) Spline interpolation and smoothing on the sphere. SIAM J. Sci. Stat. Comput. 2(1):5-16
Wahba, G. (1982) Erratum. SIAM J. Sci. Stat. Comput. 3(3):385-386.
Wendelberger, J. (1981) PhD Thesis, University of Winsconsin.
Wood, S.N. (2003) Thin plate regression splines. J.R.Statist.Soc.B 65(1):95-114
set.seed(0) n <- 400 f <- function(la,lo) { ## a test function... sin(lo)*cos(la-.3) } ## generate with uniform density on sphere... lo <- runif(n)*2*pi-pi ## longitude la <- runif(3*n)*pi-pi/2 ind <- runif(3*n)<=cos(la) la <- la[ind]; la <- la[1:n] ff <- f(la,lo) y <- ff + rnorm(n)*.2 ## test data ## generate data for plotting truth... lam <- seq(-pi/2,pi/2,length=30) lom <- seq(-pi,pi,length=60) gr <- expand.grid(la=lam,lo=lom) fz <- f(gr$la,gr$lo) zm <- matrix(fz,30,60) require(mgcv) dat <- data.frame(la = la *180/pi,lo = lo *180/pi,y=y) method <- "GCV.Cp";m <- 3 ## fit spline on sphere model... bp <- gam(y~s(la,lo,bs="sos",k=60),data=dat) ## pure knot based alternative... ind <- sample(1:n,100) bk <- gam(y~s(la,lo,bs="sos",k=60),knots=list(la=dat$la[ind],lo=dat$lo[ind]),data=dat) b <- bp cor(fitted(b),ff) ## plot results and truth... pd <- data.frame(la=gr$la*180/pi,lo=gr$lo*180/pi) fv <- matrix(predict(b,pd),30,60) par(mfrow=c(2,1),mar=c(4,4,1,1)) contour(lom,lam,t(zm)) contour(lom,lam,t(fv))