Is there an R library that estimates a multivariate natural cubic spline (or similar) function?

Question

note: originally posted on Cross Validated (stats SE) on 07-26-2011, with no correct answers to date.

Background

I have a model,

Answer 1

I'm not sure if this is precisely what you are looking for, but you could try Tps() in the R package fields. It's meant for doing thin-plate splines interpolations (2D equivalent of cubic splines) for spatial data, but will take up to four covariates, although it will expect them to be euclidean x,y,z + time, so you need to be clear that you are selecting the correct options for your particular case. If you want to interpolate, set the smoothing parameter lambda to zero. You might also try the function polymars() in the R package polspline.