Power Penalty Matrix

Power Penalty Matrix

DESCRIPTION:

Computes the matrix defining the roughness penalty for functions expressed in terms of a power basis.

USAGE:

powerpen(basisobj, Lfdobj=2)

REQUIRED ARGUMENTS:

basisobj
A power basis object.

OPTIONAL ARGUMENTS:

Lfdobj
Either a nonnegative integer or a linear differential operator object.

VALUE:

A symmetric matrix of order equal to the number of basis functions defined by the power basis object. Each element is the inner product of two power basis functions after applying the derivative or linear differential operator defined by Lfdobj.

DETAILS:

A roughness penalty for a function x(t) is defined by integrating the square of either the derivative of x(t) or, more generally, the result of applying a linear differential operator L to it. The most common roughness penalty is the integral of the square of the second derivative D2x(t), and this is the default. To apply this roughness penalty, the matrix of inner products produced by this function is necessary.

EXAMPLES:

#  set up an power basis with 3 basis functions.
#  the powers are 0, 1, and 2.
basis <- create.power.basis(c(0,1),3,c(0,1,2))
#  compute the 3 by 3 matrix of inner products of second derivatives
penmat <- powerpen(basis)