dssmoother: Evaluate the smoothing matrix, the radial basis matrix, the polynomial matrix and their associated coefficients

Description

The function evaluates the smoothing matrix H, the matrices Q and S and their associated coefficients c and s. This function is not intended to be used directly.

Usage

dssmoother(X,Y=NULL,lambda,m,s)

Arguments

Matrix of explanatory variables, size n,p.

Vector of response variable. If null, only the smoothing matrix is returned.

lambda

The smoothness coefficient lambda for thin plate splines of order m.

The order of derivatives for the penalty (for thin plate splines it is the order). This integer m must verify 2m+2s/d>1, where d is the number of explanatory variables.

The power of weighting function. For thin plate splines s is equal to 0. This real must be strictly smaller than d/2 (where d is the number of explanatory variables) and must verify 2m+2s/d. To get pseudo-cubic splines, choose m=2 and s=(d-1)/2 (See Duchon, 1977).

Value

H, and two matrices denoted Sgu (for null space) and Qgu.

Details

see the reference for detailed explanation of Q (the semi kernel or radial basis) and S (the polynomial null space).

References

Duchon, J. (1977) Splines minimizing rotation-invariant semi-norms in Solobev spaces. in W. Shemp and K. Zeller (eds) Construction theory of functions of several variables, 85-100, Springer, Berlin.