fda (version 6.2.0)

monfn: Evaluate the a monotone function

Description

Evaluate a monotone function defined as the indefinite integral of $exp(W(t))$ where $W$ is a function defined by a basis expansion. Function $W$ is the logarithm of the derivative of the monotone function.

Usage

monfn(argvals, Wfdobj, basislist=vector("list", JMAX), returnMatrix=FALSE)

Value

A numerical vector or matrix containing the values the warping function h.

Arguments

argvals

A numerical vector at which function and derivative are evaluated.

Wfdobj

A functional data object.

basislist

A list containing values of basis functions.

returnMatrix

logical: If TRUE, a two-dimensional is returned using a special class from the Matrix package.

Author

J. O. Ramsay

Details

This function evaluates a strictly monotone function of the form $$h(x) = [D^{-1} exp(Wfdobj)](x),$$ where \(D^{-1}\) means taking the indefinite integral. The interval over which the integration takes places is defined in the basis object in Wfdobj.

References

Ramsay, James O., Hooker, G. and Graves, S. (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

mongrad, landmarkreg, smooth.morph

Examples

oldpar <- par(no.readonly=TRUE)
## basically this example resembles part of landmarkreg.R that uses monfn.R to
## estimate the warping function.

## Specify the curve subject to be registered
n=21
tbreaks = seq(0, 2*pi, len=n)
xval <- sin(tbreaks)
rangeval <- range(tbreaks)

## Establish a B-spline basis for the curve
wbasis <- create.bspline.basis(rangeval=rangeval, breaks=tbreaks)
Wfd0   <- fd(matrix(0,wbasis$nbasis,1),wbasis)
WfdPar <- fdPar(Wfd0, 1, 1e-4)
fdObj  <- smooth.basis(tbreaks, xval, WfdPar)$fd

## Set the mean landmark times. Note that the objective of the warping
## function is to transform the curve such that the landmarks of the curve
## occur at the designated mean landmark times.

## Specify the mean landmark times: tbreak[8]=2.2 and tbreaks[13]=3.76
meanmarks <- c(rangeval[1], tbreaks[8], tbreaks[13], rangeval[2])
## Specify landmark locations of the curve: tbreaks[6] and tbreaks[16]
cmarks <- c(rangeval[1], tbreaks[6], tbreaks[16], rangeval[2])

## Establish a B-basis object for the warping function
Wfd <- smooth.morph(x=meanmarks, y=cmarks, ylim=rangeval, 
                    WfdPar=WfdPar)$Wfdobj

## Estimate the warping function
h = monfn(tbreaks, Wfd)

## scale using a linear equation h such that h(0)=0 and h(END)=END
b <- (rangeval[2]-rangeval[1])/ (h[n]-h[1])
a <- rangeval[1] - b*h[1]
h <- a + b*h
plot(tbreaks, h, xlab="Time", ylab="Transformed time", type="b")
par(oldpar)