create.monomial.basis: Create a Monomial Basis

In fda: Functional Data Analysis

Create a Monomial Basis

Description

Creates a set of basis functions consisting of powers of the argument.

Usage

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

Arguments

rangeval	a vector of length 2 containing the initial and final values of the interval over which the functional data object can be evaluated.
nbasis	the number of basis functions = length(exponents). Default = if(is.null(exponents)) 2 else length(exponents).
exponents	the nonnegative integer powers to be used. By default, these are 0, 1, 2, ..., (nbasis-1).
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 when rangeval[1] = 0, 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 monomial bases, this defaults to paste('monomial', 1:nbreaks, 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.

Value

a basis object with the type monom.

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, link{create.basis} create.bspline.basis, create.constant.basis, create.fourier.basis, create.exponential.basis, create.polygonal.basis, create.power.basis

Examples

##
## simplest example: one constant 'basis function'
##
m0 <- create.monomial.basis(nbasis=1)
plot(m0)

##
## Create a monomial basis over the interval [-1,1]
##  consisting of the first three powers of t
##
basisobj <- create.monomial.basis(c(-1,1), 5)
#  plot the basis
oldpar <- par(no.readonly=TRUE)
plot(basisobj)

##
## rangeval of class Date or POSIXct
##
# Date
invasion1 <- as.Date('1775-09-04')
invasion2 <- as.Date('1812-07-12')
earlyUS.Canada <- c(invasion1, invasion2)
BspInvade1 <- create.monomial.basis(earlyUS.Canada)

# POSIXct
AmRev.ct <- as.POSIXct1970(c('1776-07-04', '1789-04-30'))
BspRev1.ct <- create.monomial.basis(AmRev.ct)
par(oldpar)

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