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My understanding is that one of the attractive properties of the B-spline basis is that the individual basis functions have local support, i.e. it will be >0 on the interval between $d+2$ adjacent knots (letting $d$ be the degree of the spline basis), but zero everywhere else. However, the p-spines used in mgcv do not appear to have this property.

library(MASS)
library(mgcv)
gamfit <- gam(accel ~ s(times, bs="ps"), data=mcycle)
mc_ord <- mcycle[order(mcycle$times), ]
X <- predict(gamfit, mc_ord, type='lpmatrix')
matplot(y=X, x=mc_ord$times, type='l')

After extracting the spline basis projection using predict(gamfit, mc_ord, type='lpmatrix') and plotting the columns against the original unprojected variable, I'm seeing that the splines used in mgcv (using bs="ps") do not appear to be non-negative.

Simon Wood is a super smart dude, so I assume I am misinformed/confused/ignorant somehow. Can someone help me become less misinformed/confused/ignorant?

My main questions:

  1. After some googling, I'm finding the many different types of splines confusing. Is there a standard nomenclature? What do we call this type of spline / B-spline variant?
  2. What is the advantage of using this type of spline basis function versus one that is non-zero?
  3. Is there a way to require mgcv to use splines with local support?
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  • $\begingroup$ Can anyone help me with this? I'm not finding much information out there (or perhaps just having trouble figuring out the right search terms) $\endgroup$ Oct 13, 2020 at 3:06

1 Answer 1

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It's because of the sum to zero constraints. If you only need single smooths in your model there are a couple of options...

library(mgcv)
## set up p-spline smoother without constraint
sm <- smoothCon(s(times,bs="ps",k=20),data=mcycle)[[1]]
X <- sm$X ## no constraint => compact support
S <- sm$S[[1]]
accel <- mcycle$accel
m <- gam(accel~X-1,paraPen=list(X=list(S)))
plot(mcycle)
lines(mcycle$times,fitted(m))

## or fool gam into not using a constraint... 
n <- nrow(mcycle)
## dum is basically a constant, but just variable enough for gam to 
## not impose a constraint... 
dum <- rep(1,n) + (runif(n)-.5) * 1e-8
m <- gam(accel~s(times,bs="ps",k=20,by=dum)-1,data=mcycle)
plot(m,scheme=1,residuals=T,pch=19,cex=.5,by.resids=T)
model.matrix(m)

If you need multiple smooths then you can just clone the p-spline smooth constructors and set constraints that maintain compact support - the problem is that they tend to lead to very wide CI's as you get confounding via the intercept...

Simon

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  • $\begingroup$ Thank you! I ended up creating the splines with compact support "by hand," but I'm intrigued by the solution using the dummy constraint variable! $\endgroup$ Oct 14, 2020 at 17:47

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