fourierpen {fda}

Fourier Penalty Matrix

Description

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

Usage

fourierpen(basisobj, Lfdobj=int2Lfd(2))

Arguments

basisobj

a Fourier 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 of the basis functions (possibly after applying the linear differential operator to them) defining this function is necessary. This function just calls the roughness penalty evaluation function specific to the basis involved.

Value

a symmetric matrix of order equal to the number of basis functions defined by the Fourier basis object. Each element is the inner product of two Fourier 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

fourier, eval.penalty, getbasispenalty

Examples

#  set up a Fourier basis with 13 basis functions
#  and and period 1.0.
basisobj <- create.fourier.basis(c(0,1),13)
#  compute the 13 by 13 matrix of inner products
#  of second derivatives
penmat <- fourierpen(basisobj)