fda (version 2.4.0)

create.power.basis: Create a Power Basis Object

Description

The basis system is a set of powers of argument $x$. That is, a basis function would be x^exponent, where exponent is a vector containing a set of powers or exponents. The power basis would normally only be used for positive values of x, since the power of a negative number is only defined for nonnegative integers, and the exponents here can be any real numbers.

Usage

create.power.basis(rangeval=c(0, 1), nbasis=NULL, exponents=NULL,
            dropind=NULL, quadvals=NULL, values=NULL,
            basisvalues=NULL, names='power', axes=NULL)

Arguments

rangeval

a vector of length 2 with the first element being the lower limit of the range of argument values, and the second the upper limit. Of course the lower limit must be less than the upper limit.

nbasis

the number of basis functions = length(exponents). Default = if(is.null(exponents)) 2 else length(exponents).

exponents

a numeric vector of length nbasis containing the powers of x in the basis.

dropind

a vector of integers specifiying the basis functions to be dropped, if any. For example, if it is required that a function be zero at the left boundary, this is achieved by dropping the first basis function, the only one that is nonzero at th

quadvals

a matrix with two columns and a number of rows equal to the number of quadrature points for numerical evaluation of the penalty integral. The first column of quadvals contains the quadrature points, and the second column the quad

values

a list of matrices with one row for each row of quadvals and one column for each basis function. The elements of the list correspond to the basis functions and their derivatives evaluated at the quadrature points contained in the

basisvalues

A list of lists, allocated by code such as vector("list",1). This field is designed to avoid evaluation of a basis system repeatedly at a set of argument values. Each list within the vector corresponds to a specific set of argument values, a

names

either a character vector of the same length as the number of basis functions or a simple stem used to construct such a vector.

For power bases, this defaults to paste(power', 0:(nbasis-1), sep='').

axes

an optional list used by selected plot functions to create custom axes. If this axes argument is not NULL, functions plot.basisfd, plot.fd, plot.fdSmooth

Value

• a basis object of type power.

Details

The power basis differs from the monomial and polynomial bases in two ways. First, the powers may be nonintegers. Secondly, they may be negative. Consequently, a power basis is usually used with arguments that only take positive values, although a zero value can be tolerated if none of the powers are negative.

See Also

basisfd, create.basis, create.bspline.basis, create.constant.basis, create.exponential.basis, create.fourier.basis, create.monomial.basis, create.polygonal.basis, create.polynomial.basis

Examples

#  Create a power basis over the interval [1e-7,1]
#  with powers or exponents -1, -0.5, 0, 0.5 and 1
basisobj <- create.power.basis(c(1e-7,1), 5, seq(-1,1,0.5))
#  plot the basis
plot(basisobj)