AmpPhaseDecomp.Rd

\name{AmpPhaseDecomp}
\alias{AmpPhaseDecomp}
\title{
  Decomposition for Amplitude and Phase Variation
}
\description{
  Registration is the process of aligning peaks, valleys and other
  features in a sample of curves.  Once the registration has taken
  place, this function computes two mean squared error measures, one for
  amplitude variation, and the other for phase variation.  It also
  computes a squared multiple correlation index of the amount of
  variation in the unregistered functions is due to  phase.
}
\usage{
AmpPhaseDecomp(xfd, yfd, hfd, rng=xrng)
}
\arguments{
  \item{xfd}{
    a functional data object containing the unregistered curves.
  }
  \item{yfd}{
    a functional data object containing the registered curves.
  }
  \item{hfd}{
    a functional data object containing the strictly monotone warping
    functions $h(t)$.  This is typically returned by the functions
    \code{landmarkreg} and \code{register.fd}.
  }
  \item{rng}{
    a vector of length 2 specifying a range of values over which the
    decomposition is to be computed.  Both values must be within the
    range of the functional data objects in the argument.  By default
    the whole range of the functional data objects is used.
  }
}
\details{
  The decomposition can yield negative values for \code{MS.phas} if the
  registration does not improve the alignment of the curves, or if used
  to compare two registration processes based on different principles,
  such as  is the case for functions \code{landmarkreg} and
  \code{register.fd}.
}
\value{
  a named list with the following components:

  \item{MS.amp}{
    the mean squared error for amplitude variation.
  }
  \item{MS.phas}{
    the mean squared error for phase variation.
  }
  \item{RSQR}{
    the squared correlation measure of the  proportion of the total
    variation that is due to phase variation.
  }
  \item{C}{
    a constant required for the decomposition.  Its value is one if the
    derivatives the warping functions are independent of the squared
    registered functions.
  }
}
\references{
  Ramsay, James O., Hooker, Giles, and Graves, Spencer (2009),
  \emph{Functional data analysis with R and Matlab}, Springer, New
  York.
  
  Ramsay, James O., and Silverman, Bernard W. (2005), \emph{Functional
    Data Analysis, 2nd ed.}, Springer, New York.
  
  Ramsay, James O., and Silverman, Bernard W. (2002), \emph{Applied
    Functional Data Analysis}, Springer, New York.
}
\seealso{
  \code{\link{landmarkreg}},
  \code{\link{register.fd}},
  \code{\link{smooth.morph}}
}
\examples{
#See the analysis for the growth data in the examples.
}
% docclass is function
\keyword{smooth}
\keyword{models}