exponentiate.fd {fda}

Powers of a functional data ('fd') object

Description

Exponentiate a functional data object where feasible.

Usage

## S3 method for class 'fd'
e1 ^ e2
exponentiate.fd(e1, e2, tolint=.Machine$double.eps^0.75,
  basisobj=e1$basis,
  tolfd=sqrt(.Machine$double.eps)*
          sqrt(sum(e1$coefs^2)+.Machine$double.eps)^abs(e2),
  maxbasis=NULL, npoints=NULL)

Arguments

e1

object of class 'fd'.

e2

a numeric vector of length 1.

basisobj

reference basis

tolint

if abs(e2-round(e2))<tolint, we assume e2 is an integer. This simplifies the algorithm.

tolfd

the maximum error allowed in the difference between the direct computation eval.fd(e1)^e2 and the computed representation.

maxbasis

The maximum number of basis functions in growing referencebasis to achieve a fit within tolfd. Default = 2*nbasis12+1 where nbasis12 = nbasis of e1^floor(e2).

npoints

The number of points at which to compute eval.fd(e1)^e2 and the computed representation to evaluate the adequacy of the representation. Default = 2*maxbasis-1. For a max Fourier basis, this samples the highest frequency at all its extrema and zeros.

Details

If e1 has a B-spline basis, this uses the B-spline algorithm.

Otherwise it throws an error unless it finds one of the following special cases:

e2 = 0

Return an fd object with a constant basis that is everywhere 1

e2 is a positive integer to within tolint

Multiply e1 by itself e2 times

e2 is positive and e1 has a Fourier basis

e120 <- e1^floor(e2)

outBasis <- e120$basis

rng <- outBasis$rangeval

Time <- seq(rng[1], rng[2], npoints)

e1.2 <- predict(e1, Time)^e2

fd1.2 <- smooth.basis(Time, e1.2, outBasis)$

d1.2 <- (e1.2 - predict(fd1.2, Time))

if(all(abs(d1.2)<tolfd))return(fd1.2)

Else if(outBasis$nbasis<maxbasis) increase the size of outBasis and try again.

Else write a warning with the max(abs(d1.2)) and return fd1.2.

Value

A function data object approximating the desired 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

arithmetic.fd basisfd, basisfd.product

Examples

##
## sin^2
##

basis3 <- create.fourier.basis(nbasis=3)
oldpar <- par(no.readonly=TRUE)
plot(basis3)
# max = sqrt(2), so
# integral of the square of each basis function (from 0 to 1) is 1
integrate(function(x)sin(2*pi*x)^2, 0, 1) # = 0.5

# sin(theta)
fdsin <- fd(c(0,sqrt(0.5),0), basis3)
plot(fdsin)

fdsin2 <- fdsin^2

# check
fdsinsin <- fdsin*fdsin
# sin^2(pi*time) = 0.5*(1-cos(2*pi*theta) basic trig identity
plot(fdsinsin) # good


all.equal(fdsin2, fdsinsin)

par(oldpar)