Generate an approximation to the Demmler-Reinsch orthonormal bases for
smoothing splines, using orthogonal polynomials. basis.gen
generates a basis for a single x, and pseudo.bases
generates a list of bases for each column of the matrix x.
A vector of values for basis.gen, or a matrix for pseudo.bases
df
The degrees of freedom of the smoothing spline.
thresh
If the next eigenvector improves the approximation by less than
threshold, a truncated bases is returned. For pseudo.bases this can be a single
value or a vector of values, which are recycled sequentially for each
column of x
degree
The nominal number of basis elements. The basis returned has no more
than degree columns. For pseudo.bases this can be a single
value or a vector of values, which are recycled sequentially for each
column of x
parms
A parameter set. If included in the call, these are used to define the
basis. This is used for prediction.
parallel
For pseudo.bases, allows for parallel bases
computation in multiple cores.
...
other arguments for basis.gen can be passed through pseudo.bases
Details
basis.gen starts with a basis of orthogonal polynomials of total
degree. These are each smoothed using a smoothing spline, which
allows for a one-step approximation to the Demmler-Reinsch basis for a
smoothing spline of rank equal to the degree. See the reference for
details. The function also
approximates the appropriate diagonal penalty matrix for this basis, so
that the a approximate smoothing spline (generalized ridge regression)
has the target df.
Value
An orthonormal basis is returned (a list for pseudo.bases).
This has an attribute parms, which has elements
coefsCoefficients needed to generate the orthogonal
polynomials
rotateTransformation matrix for transforming the polynomial
basis
dpenalty values for the diagonal penalty
dfdf used
degreenumber of columns
Author(s)
Alexandra Chouldechova and Trevor Hastie
Maintainer: Trevor Hastie hastie@stanford.edu
References
T. Hastie Pseudosplines. (1996) JRSSB 58(2), 379-396.
Chouldechova, A. and Hastie, T. (2015) Generalized Additive Model Selection