# Fitting multivariate, natural cubic spline

note: with no correct answers after a month, I have reposted to SO

Background

I have a model, $f$, where $Y=f(\textbf{X})$

$\textbf{X}$ is an $n \times m$ matrix of samples from $m$ parameters and $Y$ is the $n \times 1$ vector of model outputs.

$f$ is computationally intensive, so I would like to approximate $f$ using a multivariate cubic spline through $(X,Y)$ points, so that I can evaluate $Y$ at a larger number of points.

Question

Is there an R function that will calculate an arbitrary relationship between X and Y?

Specifically, I am looking for a multivariate version of the splinefun function, which generates a spline function for the univariate case.

e.g. this is how splinefun works for the univariate case

x <- 1:10
y <- runif(10)
foo <- splinefun(x,y)
foo(1:10) #returns y, as example
all(y == foo(1:10))
## TRUE


What I have tried

I have reviewed the mda package, and it seems that the following should work:

library(mda)
x   <- data.frame(a = 1:10, b = 1:10/2, c = 1:10*2)
y   <- runif(10)
foo <- mars(x,y)
predict(foo, x) #all the same value
all(y == predict(foo,x))
## FALSE


but I could not find any way to implement a cubic-spline in mars

update since offering the bounty, I changed the title – If there is no R function, I would accept, in order of preference: an R function that outputs a gaussian process function, or another multivariate interpolating function that passes through the design points, preferably in R, else Matlab.

This paper presented at UseR! 2009 seems to address a similar problem

http://www.r-project.org/conferences/useR-2009/slides/Roustant+Ginsbourger+Deville.pdf

It suggests the DiceKriging package http://cran.r-project.org/web/packages/DiceKriging/index.html

In particular, check the functions km and predict.

Here is a an example of three dimensional interpolation. It looks to be straightforward to generalise.

x <- c(0, 0.4, 0.6, 0.8, 1)
y <- c(0, 0.2, 0.3, 0.4, 0.5)
z <- c(0, 0.3, 0.4, 0.6, 0.8)

model <- function(param){
2*param[1] + 3*param[2] +4*param[3]
}

model.in <- expand.grid(x,y,z)
names(model.in) <- c('x','y','z')

model.out <- apply(model.in, 1, model)

# fit a kriging model
m.1 <- km(design=model.in, response=model.out, covtype="matern5_2")

# estimate a response
interp <- predict(m.1, newdata=data.frame(x=0.5, y=0.5, z=0.5), type="UK",    se.compute=FALSE)
# check against model output
interp$mean # [1] 4.498902 model(c(0.5,0.5,0.5)) # [1] 4.5 # check we get back what we put in interp <- predict(m.1, newdata=model.in, type="UK", se.compute=FALSE) all.equal(model.out, interp$mean)
# TRUE