create.fourier.basis: Create a Fourier Basis

In fda: Functional Data Analysis

Create a Fourier Basis

Description

Create an Fourier basis object defining a set of Fourier functions with specified period.

Usage

create.fourier.basis(rangeval=c(0, 1), nbasis=3,
              period=diff(rangeval), dropind=NULL, quadvals=NULL,
              values=NULL, basisvalues=NULL, names=NULL,
              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	positive odd integer: If an even number is specified, it is rounded up to the nearest odd integer to preserve the pairing of sine and cosine functions. An even number of basis functions only makes sense when there are always only an even number of observations at equally spaced points; that case can be accommodated using dropind = nbasis-1 (because the bases are const, sin, cos, ...).
period	the width of any interval over which the Fourier functions repeat themselves or are periodic.
dropind	an optional vector of integers specifiying basis functions to be dropped.
quadvals	an optional 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	an optional 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	an optional 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 sublist corresponds to a specific set of argument values, and must have at least two components: a vector of argument values and a matrix of the values the basis functions evaluated at the arguments in the first component. Third and subsequent components, if present, contain matrices of values their derivatives. 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 simple stem used to construct such a vector. If nbasis = 3, names defaults to c('const', 'cos', 'sin'). If nbasis > 3, names defaults to c('const', outer(c('cos', 'sin'), 1:((nbasis-1)/2), paste, sep=”)). If names = NA, no names are used.
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 is to create CanadianWeather plots using list("axesIntervals")

Details

Functional data objects are constructed by specifying a set of basis functions and a set of coefficients defining a linear combination of these basis functions. The Fourier basis is a system that is usually used for periodic functions. It has the advantages of very fast computation and great flexibility. If the data are considered to be nonperiod, the Fourier basis is usually preferred. The first Fourier basis function is the constant function. The remainder are sine and cosine pairs with integer multiples of the base period. The number of basis functions generated is always odd.

Value

a basis object with the type fourier.

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.monomial.basis, create.polygonal.basis, create.power.basis

Examples

# Create a minimal Fourier basis for annual data
#  using 3 basis functions
yearbasis3 <- create.fourier.basis(c(0,365),
                    axes=list("axesIntervals") )
#  plot the basis
oldpar <- par(no.readonly=TRUE)
plot(yearbasis3)

# Identify the months with letters
plot(yearbasis3, axes=list('axesIntervals', labels=monthLetters))

# The same labels as part of the basis object
yearbasis3. <- create.fourier.basis(c(0,365),
       axes=list("axesIntervals", labels=monthLetters) )
plot(yearbasis3.)

# set up the Fourier basis for the monthly temperature data,
#  using 9 basis functions with period 12 months.
monthbasis <- create.fourier.basis(c(0,12), 9, 12.0)

#  plot the basis
plot(monthbasis)

# Create a false Fourier basis using 1 basis function.
falseFourierBasis <- create.fourier.basis(nbasis=1)
#  plot the basis:  constant
plot(falseFourierBasis)
par(oldpar)

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