create.basis: 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.

Arguments

code

list("axesIntervals")

itemize

dropind

item

quadvals
values
basisvalues
names
axes

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, 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:

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). For polygonal 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: 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:

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 called knots. This defaults to seq(rangeval[1], rangeval[2], nbasis-norder+2). } polygonal{ Argument argvals = the locations of the break or join points; also called knots. This defaults to seq(rangeval[1], rangeval[2], nbasis). } fourier{ Argument period defaults to diff(rangeval). } exponential{ Argument ratevec. In fda_2.0.2, this defaulted to 1. In fda_2.0.3, it will default to 0:1. } monomial, power{ Argument exponents. Default = 0:(nbasis-1). For monomial bases, exponents must be distinct nonnegative integers. For power bases, they must be distinct real numbers. } polynomial{ Argument ctr must be a single number used to shift the argument prior to computing its powers. Default = mean(rangeval). }

References

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.