Lfd {fda} | R Documentation |
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
nderiv |
a nonnegative integer specifying the order $m$ of the highest order derivative in the operator |
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 $m-1$. |
Details
To check that an object is of this class, use functions is.Lfd
or 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 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
$Lx(t) = b_0(t) x(t) + b_1(t)Dx(t) + ... + b_{m-1}(t) D^{m-1} x(t) + D^mx(t)$
There are print
, summary
, and plot
methods for
objects of class Lfd
.
Value
a linear differential operator object
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
int2Lfd
,
vec2Lfd
,
fdPar
,
pda.fd
plot.Lfd
Examples
# Set up the harmonic acceleration operator
dayrange <- c(0,365)
Lbasis <- create.constant.basis(dayrange,
axes=list("axesIntervals"))
Lcoef <- matrix(c(0,(2*pi/365)^2,0),1,3)
bfdobj <- fd(Lcoef,Lbasis)
bwtlist <- fd2list(bfdobj)
harmaccelLfd <- Lfd(3, bwtlist)