srf: Generate design for penalized surface estimation.

Description

This function generates the design for a 2-D penalized spline using (almost) radial basis functions. Use this type of term to account for spatial variation. Smooth interactions between covariates are often better fitted via the interactions of lin and sm terms, because they allow a decomposition into (linear and smooth) marginal trends and (linear-linear, linear-smooth/"varying coefficients", and smooth-smooth) interactions. This decomposition usually makes no sense for spatial data.

Usage

srf(coords, K = min(50, sum(nd)/4), rankZ = 0.999,
    centerBase = TRUE, baseType = c("B", "thinPlate"),
    decomposition = c("ortho", "MM", "asIs"), tol = 1e-10)

Arguments

coords

a data.frame with two columns containing the coordinates

(approximate) number of basis functions in the original basis (defaults to 50). If baseType="B" you can specify a vector giving the number of marginal basis functions in each direction.

rankZ

how many eigenvectors to retain from the eigen decomposition: either a number > 3 or the proportion of the sum of eigenvalues the retained eigenvectors must represent at least. Defaults to .999.

centerBase

project the basis of the penalized part into the complement of the column space of the basis of the unpenalized part? defaults to TRUE

baseType

Defaults to "B", i.e. a tensor product basis based on marginal cubic B-splines with ridge penalty (i.e. penalizing deviations from the constant). Set to "thinPlate" if cubic thin plate splines are desired, see note below

decomposition

use a (truncated) spectral decomposition of the implied prior covariance of $f(x,y)$ for a low rank representation with orthogonal basis functions and i.i.d. coefficients ("ortho"), or use the mixed model reparameterization for non-

tol

count eigenvalues smaller than this as zero

Value

a design matrix for the 2-D spline.

Details

Note that srf() expects coords to be a data.frame within the larger data.frame supplied to spikeSlabGAM in its data argument, i.e. coords is considered a two-dimensional covariate.

If baseType is 'thinPlate', knot locations for the thin plate spline basis are chosen via a space-filling algorithm (i.e. medoids returned by clara) as suggested in Ruppert/Wand/Carroll, ch. 13.5. Since the thin plate penalty term penalizes deviations from a linear trend, it is recommended to add marginal linear trends and their interaction to the model if baseType="thinPlate" to improve the fit.

References

Ruppert, D., Wand, M.P., Carroll, R.J. (2003). Semiparametric Regression. Cambridge University Press