create.basis {fda} | R Documentation |
Functional data analysis proceeds by selecting a finite basis set and
fitting data to it. The current fda
package supports fitting
via least squares penalized with lambda times the integral over the
(finite) support of the basis set of the squared deviations from a
linear differential operator.
The most commonly used basis in fda
is probably B-splines. For
periodic phenomena, Fourier bases are quite useful. A constant basis
is provided to facilitation arithmetic with functional data objects.
To restrict attention to solutions of certain differential equations,
it may be useful to use a corresponding basis set such as exponential,
monomial, polynomial, or power basis sets.
Monomial and polynomial bases are similar. As noted in the table
below, create.monomial.basis
has an argument exponents
absent from create.polynomial.basis
, which has an argument
ctr
absent from create.monomial.basis
.
Power bases support the use of negative and fractional powers, while monomial bases are restricted only to nonnegative integer exponents.
The polygonal basis is essentialy a B-spline of order 2, degree 1.
The following summarizes arguments used by some or all of the current
create.basis
functions:
For bspline
bases, this can be inferred from
range(breaks). For polygonal
bases, this can be inferred
from range(argvals). In all other cases, this defaults to 0:1.
This is not used for two of the create.basis
functions:
For constant
this is 1, so there is no need to specify it.
For polygonal
bases, it is length(argvals), and again there
is no need to specify it.
For bspline
bases, if nbasis
is not specified, it
defaults to (length(breaks) + norder - 2) if breaks
is
provided. Otherwise, nbasis
defaults to 20 for
bspline
bases.
For exponential
bases, if nbasis
is not specified,
it defaults to length(ratevec) if ratevec
is provided.
Otherwise, in fda_2.0.2
, ratevec
defaults to 1,
which makes nbasis
= 1; in fda_2.0.4
,
ratevec
will default to 0:1, so nbasis
will then
default to 2.
For monomial
and power
bases, if nbasis
is
not specified, it defaults to length(exponents) if
exponents
is provided. Otherwise, nbasis
defaults
to 2 for monomial
and power
bases. (Temporary
exception: In fda_2.0.2
, the default nbasis
for
power
bases is 1. This will be increased to 2 in
fda_2.0.4
.)
For polynomial
bases, nbasis
defaults to 2.
In addition to rangeval
and nbasis
, all but
constant
bases have one or two parameters unique to that
basis type or shared with one other:
norder
= the order of the spline, which is one
more than the degree of the polynomials used. This defaults to
4, which gives cubic splines.
Argument breaks
= the locations of the break or join
points; also called knots
. This defaults to
seq(rangeval[1], rangeval[2], nbasis-norder+2).
argvals
= the locations of the break or join
points; also called knots
. This defaults to
seq(rangeval[1], rangeval[2], nbasis).
period
defaults to diff(rangeval).
ratevec
. In fda_2.0.2
, this defaulted to
1. In fda_2.0.3
, it will default to 0:1.
exponents
. Default = 0:(nbasis-1). For
monomial
bases, exponents
must be distinct
nonnegative integers. For power
bases, they must be
distinct real numbers.
ctr
must be a single number used to shift the
argument prior to computing its powers. Default =
mean(rangeval).
Beginning with fda_2.1.0
, the last 6 arguments for all the
create.basis
functions will be as follows; some but not all
are available in the previous versions of fda
:
quadvals
contains
the quadrature points, and the second column the quadrature
weights. A minimum of 5 values are required for each inter-knot
interval, and that is often enough. For Simpson's rule, these
points are equally spaced, and the weights are proportional to
1, 4, 2, 4, ..., 2, 4, 1.
quadvals
and one column for each basis function. The elements of the
list correspond to the basis functions and their derivatives
evaluated at the quadrature points contained in the first column
of quadvals
.
basisvalues
with code such as
the following:
basisobj$basisvalues <- vector("list",1)
basisobj$basisvalues[[1]] <- list(args=evalargs, values=basismat)
For bspline
bases, this defaults to paste('bspl', norder,
'.', 1:nbreaks, sep='').
For other bases, there are crudely similar defaults.
plot
functions to
create custom axes
. If this axes
argument is not
NULL, functions plot.basisfd
, plot.fd
,
plot.fdSmooth
plotfit.fd
, plotfit.fdSmooth
,
and plot.Lfd
will create axes via
do.call(x$axes[[1]], x$axes[-1])
. The primary example of
this is to create CanadianWeather
plots using
list("axesIntervals")
J. O. Ramsay and Spencer Graves
Ramsay, James O., and Silverman, Bernard W. (2006), Functional Data Analysis, 2nd ed., Springer, New York.
Ramsay, James O., and Silverman, Bernard W. (2002), Applied Functional Data Analysis, Springer, New York.
create.bspline.basis
create.constant.basis
create.exponential.basis
create.fourier.basis
create.monomial.basis
create.polygonal.basis
create.polynomial.basis
create.power.basis