eval.penalty: Evaluate a Basis Penalty Matrix

In fda: Functional Data Analysis

Evaluate a Basis Penalty Matrix

Description

A basis roughness penalty matrix is the matrix containing the possible inner products of pairs of basis functions. These inner products are typically defined in terms of the value of a derivative or of a linear differential operator applied to the basis function. The basis penalty matrix plays an important role in the computation of functions whose roughness is controlled by a roughness penalty.

Usage

eval.penalty(basisobj, Lfdobj=int2Lfd(0), rng=rangeval)

Arguments

basisobj	a basis object.
Lfdobj	either a nonnegative integer defining an order of a derivative or a linear differential operator.
rng	a vector of length 2 defining a restricted range. Optionally, the inner products can be computed over a range of argument values that lies within the interval covered by the basis function definition.

Details

The inner product can be computed exactly for many types of bases if $m$ is an integer. These include B-spline, fourier, exponential, monomial, polynomial and power bases. In other cases, and for noninteger operators, the inner products are computed by an iterative numerical integration method called Richard extrapolation using the trapezoidal rule.

If the penalty matrix must be evaluated repeatedly, computation can be greatly speeded up by avoiding the use of this function, and instead using quadrature points and weights defined by Simpson's rule.

Value

a square symmetric matrix whose order is equal to the number of basis functions defined by the basis function object basisobj . If Lfdobj is $m$ or a linear differential operator of order $m$, the rank of the matrix should be at least approximately equal to its order minus $m$.

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

getbasispenalty, eval.basis,

fda documentation built on Sept. 30, 2024, 9:19 a.m.