https://github.com/cran/fda
Tip revision: 94c6ce860922b18293ed95dbc4b4260534bf5cc9 authored by J. O. Ramsay on 18 August 2008, 00:00:00 UTC
version 2.0.0
version 2.0.0
Tip revision: 94c6ce8
Lfd.Rd
\name{Lfd}
\alias{Lfd}
\title{
Define a Linear Differential Operator Object
}
\description{
A linear differential operator of order $m$ is defined,
usually to specify a roughness penalty.
}
\usage{
Lfd(nderiv=0, bwtlist=vector("list", 0))
}
\arguments{
\item{nderiv}{
a nonnegative integer specifying the order $m$ of the
highest order derivative in the operator
}
\item{bwtlist}{
a list of length $m$. Each member contains a
functional data object that acts as a weight function for a
derivative. The first member weights the function, the
second the first derivative, and so on up to order $m-1$.
}
}
\value{
a linear differential operator object
}
\details{
To check that an object is of this class, use functions
\code{is.Lfd} or \code{int2Lfd}.
Linear differential operator objects are often used to
define roughness penalties for smoothing towards a
"hypersmooth" function that is annihilated by the operator.
For example, the harmonic acceleration operator used in the
analysis of the Canadian daily weather data annihilates linear
combinations of $1, sin(2 pi t/365)$ and $cos(2 pi t/365)$,
and the larger the smoothing parameter, the closer the smooth
function will be to a function of this shape.
Function \code{pda.fd} estimates a linear differential
operator object that comes as close as possible to annihilating
a functional data object.
A linear differential operator of order $m$ is a
linear combination of the derivatives of a functional
data object up to order $m$. The derivatives of
orders 0, 1, ..., $m-1$ can each be multiplied
by a weight function $b(t)$ that may or may not vary with
argument $t$.
If the notation $D^j$ is taken to
mean "take the derivative of order $j$", then a linear
differental operator $L$ applied to function $x$
has the expression
$Lx(t) = b_0(t) x(t) + b_1(t)Dx(t) + ... + b_\{m-1\}(t) D^\{m-1\} x(t)
+ D^mx(t)$
}
\seealso{
\code{\link{int2Lfd}},
\code{\link{vec2Lfd}},
\code{\link{fdPar}},
\code{\link{pda.fd}}
}
\examples{
# Set up the harmonic acceleration operator
dayrange <- c(0,365)
Lbasis <- create.constant.basis(dayrange)
Lcoef <- matrix(c(0,(2*pi/365)^2,0),1,3)
bfdobj <- fd(Lcoef,Lbasis)
bwtlist <- fd2list(bfdobj)
harmaccelLfd <- Lfd(3, bwtlist)
}
% docclass is function
\keyword{smooth}