create.power.basis {fda}

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 that point.

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 quadrature weights. A minimum of 5 values are required for each inter-knot interval, and that is often enough. For Simpson's rule, these points are equally spaced, and the weights are proportional to 1, 4, 2, 4, ..., 2, 4, 1.

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 first column of quadvals.

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, and must have at least two components, which may be tagged as you wish. 'The first component in an element of the list vector contains the argument values. The second component in an element of the list vector contains a matrix of values of the basis functions evaluated at the arguments in the first component. The third and subsequent components, if present, contain matrices of values their derivatives up to a maximum derivative order. Whenever function getbasismatrix is called, it checks the first list in each row to see, first, if the number of argument values corresponds to the size of the first dimension, and if this test succeeds, checks that all of the argument values match. This takes time, of course, but is much faster than re-evaluation of the basis system. Even this time can be avoided by direct retrieval of the desired array. For example, you might set up a vector of argument values called "evalargs" along with a matrix of basis function values for these argument values called "basismat". You might want too use names like "args" and "values", respectively for these. You would then assign them to basisvalues with code such as the following:

basisobj$basisvalues <- vector("list",1)

basisobj$basisvalues[[1]] <- list(args=evalargs, values=basismat)

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 plotfit.fd, plotfit.fdSmooth, and plot.Lfd will create axes via x$axes[[1]] and x$axes[-1]. The primary example of this uses list("axesIntervals", ...), e.g., with Fourier bases to create CanadianWeather plots

Details

The power basis differs from the monomial basis 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.

Value

a basis object of type power.

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

basisfd, create.bspline.basis, create.constant.basis, create.exponential.basis, create.fourier.basis, create.monomial.basis, create.polygonal.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
oldpar <- par(no.readonly=TRUE)
plot(basisobj)
par(oldpar)