create.polygonal.basis: Create a Polygonal Basis

In fda: Functional Data Analysis

Create a Polygonal Basis

Description

A basis is set up for constructing polygonal lines, consisting of straight line segments that join together.

Usage

create.polygonal.basis(rangeval=NULL, argvals=NULL, dropind=NULL,
        quadvals=NULL, values=NULL, basisvalues=NULL, names='polygon',
        axes=NULL)

Arguments

rangeval	a numeric vector of length 2 defining the interval over which the functional data object can be evaluated; default value is if(is.null(argvals)) 0:1 else range(argvals). If length(rangeval) == 1 and rangeval <= 0, this is an error. Otherwise, if length(rangeval) == 1, rangeval is replaced by c(0,rangeval). If length(rangeval)>2 and argvals is not provided, this extra long rangeval argument is assigned to argvals, and then rangeval = range(argvale).
argvals	a strictly increasing vector of argument values at which line segments join to form a polygonal line.
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 These are proportional to 1, 4, 2, 4, ..., 2, 4, 1.
values	a list containing 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 is designed to avoid evaluation of a basis system repeatedly at a set of argument values. Each sublist corresponds to a specific set of argument values, and must have at least two components, which may be named as you wish. The first component of a sublist contains the argument values. The second component 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 tags 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 single character string to which 1:nbasis are appended as paste(names, 1:nbasis, sep=''. For example, if nbasis = 4, this defaults to c('polygon1', 'polygon2', 'polygon3', 'polygon4').
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 actual basis functions consist of triangles, each with its apex over an argument value. Note that in effect the polygonal basis is identical to a B-spline basis of order 2 and a knot or break value at each argument value. The range of the polygonal basis is set to the interval defined by the smallest and largest argument values.

Value

a basis object with the type polyg.

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.power.basis

Examples

#  Create a polygonal basis over the interval [0,1]
#  with break points at 0, 0.1, ..., 0.95, 1
(basisobj <- create.polygonal.basis(seq(0,1,0.1)))
#  plot the basis
oldpar <- par(no.readonly=TRUE)
plot(basisobj)
par(oldpar)

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