fda (version 1.2.3)

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=length(exponents), exponents=1,
            dropind=NULL, quadvals=NULL, values=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 exponential functions.

exponents

a vector of length nbasis containing the powers.

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 a

quadvals

a matrix with two columns and a number of rows equal to the number of argument values used to approximate an integral using Simpson's rule. The first column contains these argument values. A minimum of 5 values are required for each inter-

values

a list containing the basis functions and their derivatives evaluated at the quadrature points contained in the first column of quadvals.

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.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)