powerpen {fda}

Power Penalty Matrix

Description

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

Usage

powerpen(basisobj, Lfdobj=int2Lfd(2))

Arguments

basisobj

a power basis object.

Lfdobj

either a nonnegative integer or a linear differential operator object.

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, and this is the default. To apply this roughness penalty, the matrix of inner products produced by this function is necessary.

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.

References

Ramsay, James O., Hooker, Giles, and Graves, Spencer (2009), Functional data analysis with R and Matlab, Springer, New York.

Ramsay, James O., and Silverman, Bernard W. (2005), Functional Data Analysis, 2nd ed., Springer, New York.

Ramsay, James O., and Silverman, Bernard W. (2002), Applied Functional Data Analysis, Springer, New York.

See Also

create.power.basis, powerbasis

Examples

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