create.basis {fda} | R Documentation |
Create Basis Set for Functional Data Analysis
Description
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.
Details
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 or power basis sets.
Power bases support the use of negative and fractional powers, while monomial bases are restricted only to nonnegative integer exponents.
The polygonal basis is essentially a B-spline of order 2, degree 1.
The following summarizes arguments used by some or all of the current
create.basis
functions:
- rangeval
-
a vector of length 2 giving the lower and upper limits of the range of permissible values for the function argument.
For
bspline
bases, this can be inferred from range(breaks). Forpolygonal
bases, this can be inferred from range(argvals). In all other cases, this defaults to 0:1. - nbasis
-
an integer giving the number of basis functions.
This is not used for two of the
create.basis
functions: Forconstant
this is 1, so there is no need to specify it. Forpolygonal
bases, it is length(argvals), and again there is no need to specify it.For
bspline
bases, ifnbasis
is not specified, it defaults to (length(breaks) + norder - 2) ifbreaks
is provided. Otherwise,nbasis
defaults to 20 forbspline
bases.For
exponential
bases, ifnbasis
is not specified, it defaults to length(ratevec) ifratevec
is provided. Otherwise, infda_2.0.2
,ratevec
defaults to 1, which makesnbasis
= 1; infda_2.0.4
,ratevec
will default to 0:1, sonbasis
will then default to 2.For
monomial
andpower
bases, ifnbasis
is not specified, it defaults to length(exponents) ifexponents
is provided. Otherwise,nbasis
defaults to 2 formonomial
andpower
bases. (Temporary exception: Infda_2.0.2
, the defaultnbasis
forpower
bases is 1. This will be increased to 2 infda_2.0.4
.)
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:
- bspline
-
Argument
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 calledknots
. This defaults to seq(rangeval[1], rangeval[2], nbasis-norder+2). - polygonal
-
Argument
argvals
= the locations of the break or join points; also calledknots
. This defaults to seq(rangeval[1], rangeval[2], nbasis). - fourier
-
Argument
period
defaults to diff(rangeval). - exponential
-
Argument
ratevec
. Infda_2.0.2
, this defaulted to 1. Infda_2.0.3
, it will default to 0:1. - monomial, power
-
Argument
exponents
. Default = 0:(nbasis-1). Formonomial
bases,exponents
must be distinct nonnegative integers. Forpower
bases, they must be distinct real numbers.
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
:
- dropind
-
a vector of integers specifiying the basis functions to be dropped, if any.
- quadvals
-
a matrix with two columns and a number of rows equal to the number of quadrature points for numerical evaluation of the penalty integral. The first column of
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. - values
-
a list of matrices with one row for each row of
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 ofquadvals
. - basisvalues
-
A list of lists, allocated by code such as vector("list",1). This field is designed to avoid evaluation of a basis system repeatedly at a set of argument values. Each list within the vector corresponds to a specific set of argument values, and must have at least two components, which may be tagged as you wish. 'The first component in an element of the list vector contains the argument values. The second component in an element of the list vector contains a matrix of values of the basis functions evaluated at the arguments in the first component. The third and subsequent components, if present, contain matrices of values their derivatives up to a maximum derivative order. Whenever function getbasismatrix is called, it checks the first list in each row to see, first, if the number of argument values corresponds to the size of the first dimension, and if this test succeeds, checks that all of the argument values match. This takes time, of course, but is much faster than re-evaluation of the basis system. Even this time can be avoided by direct retrieval of the desired array. For example, you might set up a vector of argument values called "evalargs" along with a matrix of basis function values for these argument values called "basismat". You might want too use tags like "args" and "values", respectively for these. You would then assign them to
basisvalues
with code such as the following:basisobj$basisvalues <- vector("list",1)
basisobj$basisvalues[[1]] <- list(args=evalargs, values=basismat)
- names
-
either a character vector of the same length as the number of basis functions or a simple stem used to construct such a vector.
For
bspline
bases, this defaults to paste('bspl', norder, '.', 1:nbreaks, sep=”).For other bases, there are crudely similar defaults.
- axes
-
an optional list used by selected
plot
functions to create customaxes
. If thisaxes
argument is not NULL, functionsplot.basisfd
,plot.fd
,plot.fdSmooth
plotfit.fd
,plotfit.fdSmooth
, andplot.Lfd
will create axes viado.call(x$axes[[1]], x$axes[-1])
. The primary example of this is to createCanadianWeather
plots usinglist("axesIntervals")
Author(s)
J. O. Ramsay and Spencer Graves
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
create.bspline.basis
create.constant.basis
create.exponential.basis
create.fourier.basis
create.monomial.basis
create.polygonal.basis
create.power.basis