Fitting Tensor Product P-splines, Penalty Parameters

I am working with the mgcv package in r and I am fitting tensor product P-splines.

test1<-function(x,z,sx=0.3,sz=0.4){ x<-x*20
(pi**sx*sz)*(1.2*exp(-(x-0.2)^2/sx^2-(z-0.3)^2/sz^2)+
0.8*exp(-(x-0.7)^2/sx^2-(z-0.8)^2/sz^2))
}
n<-500

x<-runif(n)/20;z<-runif(n);
xs<-seq(0,1,length=30)/20;zs<-seq(0,1,length=30)
pr<-data.frame(x=rep(xs,30),z=rep(zs,rep(30,30)))
truth<-matrix(test1(pr$x,pr$z),30,30)
f <- test1(x,z)
y <- f + rnorm(n)*0.2

b1<-gam(y~t2(x,z,bs=c("ps","ps")),drop.intercept=T)

I think that is the code for how to fit it with two variables and no intercepts.

My question is how do I find the P-spline smoothing parameters. I am under the assumption there should be two, one for each variable, and possibly one for the interaction between the two P-splines, but this should be the product of the first two. I could be wrong about this.

Anyway reading the documentation this should give me my smoothing parameters:

b1\$sp

# t2(x,z)rr       t2(x,z)nr        t2(x,z)rn

# 4.280221e-04     3.884871e+06     1.386065e+06

However, these are not products of eachother and I don't know how they relate to the x and z fitted p-spline.

The reason for me thinking this way is due to equation five in this paper.

Any help would be much appreciated.

edit

I have found these definitions for rr, nr, and rn

## label "rr" indicates interaction (range space times range space)

## label "nr" (null space for x0 times range space for x1) is main effect for x1.

## label "rn" is main effect for x0

x0 is x in our case and x1 is z.

Then according to my understanding nr*rn=rr should be the case, I wonder if my interpretation of equation 5 in the paper is wrong?

• Looking at "Straightforward intermediate rank tensor product smoothing in mixed models" page 348 bottom of left col, I read "if the smoothing parameter for the interaction penalty → ∞, the smooth tends to the additive model" which seems to imply that the smoothing parameter of the interaction term is not a product of the other two (otherwise they would have to go to infinity together). However, I am not sure this is referring to the t2 basis. Jun 30, 2017 at 12:51
• @MatteoFasiolo Thanks for the references. They should be helpful. I am including a link to the paper here. Jul 5, 2017 at 13:29