The function evaluates the matrix Q and S related to the explanatory variables \(X\) at any points. This function is not intended to be used directly.
dsSx(X,Xetoile,m=2,s=0)Returns a list containing two matrices denoted Sgu (for null
space) and Qgu
Matrix of explanatory variables, size n,p.
Matrix of new observations with the same number of variables as \(X\), size m,p.
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).
Pierre-Andre Cornillon, Nicolas Hengartner and Eric Matzner-Lober
see the reference for detailed explanation of Q (the semi kernel) and S (the polynomial null space).
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.
C. Gu (2002) Smoothing spline anova models. New York: Springer-Verlag.
ibr