create.bspline.basis: Create a B-spline Basis

Description

Functional data objects are constructed by specifying a set of basis functions and a set of coefficients defining a linear combination of these basis functions. The B-spline basis is used for non-periodic functions. B-spline basis functions are polynomial segments jointed end-to-end at at argument values called knots, breaks or join points. The segments have specifiable smoothness across these breaks. B-splne basis functions have the advantages of very fast computation and great flexibility. A polygonal basis generated by create.polygonal.basis is essentially a B-spline basis of order 2, degree 1. Monomial and polynomial bases can be obtained as linear transformations of certain B-spline bases.

Usage

create.bspline.basis(rangeval=NULL, nbasis=NULL, norder=4,
      breaks=NULL, dropind=NULL, quadvals=NULL, values=NULL,
      basisvalues=NULL, names="bspl", axes=NULL)

Arguments

rangeval

a numeric vector of length 2 defining the interval over which the functional data object can be evaluated; default value is if(is.null(breaks)) 0:1 else range(breaks).

If length(rangeval) == 1 and rangeval

nbasis

an integer variable specifying the number of basis functions. Default value NULL.

This 'nbasis' argument is ignored if 'breaks' is supplied, in which case nbasis = nbreaks + norder - 2, where nbreaks = length(breaks).

norder

an integer specifying the order of b-splines, which is one higher than their degree. The default of 4 gives cubic splines.

breaks

a vector specifying the break points defining the b-spline. Also called knots, these are a strictly increasing sequence of junction points between piecewise polynomial segments. They must satisfy breaks[1] = rangeval[1] and

dropind

a vector of integers specifiying the basis functions to be dropped, if any. For example, if it is required that a function be zero at the left boundary, this is achieved by dropping the first basis function, the only one that is nonzero a

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 quad

values

a list containing the basis functions and their derivatives evaluated at the quadrature points contained in the first column of quadvals.

basisvalues

a vector of lists, allocated by code such as vector("list",1). This argument 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

names

either a character vector of the same length as the number of basis functions or a single character string to which norder, "." and 1:nbasis are appended as

paste(names, norder, ".",
    1:nbasis, sep="")

. F

axes

an optional list used by selected plot functions to create custom axes. If this axes argument is not NULL, functions plot.basisfd, plot.fd, plot.fdSmooth

Value

a basis object of the type bspline

Details

Spline functions are constructed by joining polynomials end-to-end at argument values called break points or knots. First, the interval is subdivided into a set of adjoining intervals separated the knots. Then a polynomial of order $m$ and degree $m-1$ is defined for each interval. In order to make the resulting piece-wise polynomial smooth, two adjoining polynomials are constrained to have their values and all their derivatives up to order $m-2$ match at the point where they join.

Consider as an illustration the very common case where the order is 4 for all polynomials, so that degree of each polynomials is 3. That is, the polynomials are cubic. Then at each break point or knot, the values of adjacent polynomials must match, and so also for their first and second derivatives. Only their third derivatives will differ at the point of junction.

The number of degrees of freedom of a cubic spline function of this nature is calculated as follows. First, for the first interval, there are four degrees of freedom. Then, for each additional interval, the polynomial over that interval now has only one degree of freedom because of the requirement for matching values and derivatives. This means that the number of degrees of freedom is the number of interior knots (that is, not counting the lower and upper limits) plus the order of the polynomials:

nbasis = norder + length(breaks) - 2

The consistency of the values of nbasis, norder and breaks is checked, and an error message results if this equation is not satisfied.

B-splines are a set of special spline functions that can be used to construct any such piece-wise polynomial by computing the appropriate linear combination. They derive their computational convience from the fact that any B-spline basis function is nonzero over at most m adjacent intervals. The number of basis functions is given by the rule above for the number of degrees of freedom.

The number of intervals controls the flexibility of the spline; the more knots, the more flexible the resulting spline will be. But the position of the knots also plays a role. Where do we position the knots? There is room for judgment here, but two considerations must be kept in mind: (1) you usually want at least one argument value between two adjacent knots, and (2) there should be more knots where the curve needs to have sharp curvatures such as a sharp peak or valley or an abrupt change of level, but only a few knots are required where the curve is changing very slowly.

This function automatically includes norder replicates of the end points rangeval. By contrast, the analogous functions splineDesign and spline.des in the splines package do NOT automatically replicate the end points. To compare answers, the end knots must be replicated manually when using splineDesign or spline.des.

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.

Examples

Run this code

##
## The simplest basis currently available with this function:
##
bspl1.1 <- create.bspline.basis(norder=1)
plot(bspl1.1)
# 1 basis function, order 1 = degree 0 = step function:

# should be the same as above:
b1.1 <- create.bspline.basis(0:1, nbasis=1, norder=1, breaks=0:1)
stopifnot(
all.equal(bspl1.1, b1.1)
)

bspl2.2 <- create.bspline.basis(norder=2)
plot(bspl2.2)

bspl3.3 <- create.bspline.basis(norder=3)
plot(bspl3.3)

bspl4.4 <- create.bspline.basis()
plot(bspl4.4)

bspl1.2 <- create.bspline.basis(norder=1, breaks=c(0,.5, 1))
plot(bspl1.2)
# 2 bases, order 1 = degree 0 = step functions:
# (1) constant 1 between 0 and 0.5 and 0 otherwise
# (2) constant 1 between 0.5 and 1 and 0 otherwise.

bspl2.3 <- create.bspline.basis(norder=2, breaks=c(0,.5, 1))
plot(bspl2.3)
# 3 bases:  order 2 = degree 1 = linear
# (1) line from (0,1) down to (0.5, 0), 0 after
# (2) line from (0,0) up to (0.5, 1), then down to (1,0)
# (3) 0 to (0.5, 0) then up to (1,1).

bspl3.4 <- create.bspline.basis(norder=3, breaks=c(0,.5, 1))
plot(bspl3.4)
# 4 bases:  order 3 = degree 2 = parabolas.
# (1) (x-.5)^2 from 0 to .5, 0 after
# (2) 2*(x-1)^2 from .5 to 1, and a parabola
#     from (0,0 to (.5, .5) to match
# (3 & 4) = complements to (2 & 1).

bSpl4. <- create.bspline.basis(c(-1,1))
plot(bSpl4.)
# Same as bSpl4.23 but over (-1,1) rather than (0,1).

# set up the b-spline basis for the lip data, using 23 basis functions,
#   order 4 (cubic), and equally spaced knots.
#  There will be 23 - 4 = 19 interior knots at 0.05, ..., 0.95
lipbasis <- create.bspline.basis(c(0,1), 23)
plot(lipbasis)

bSpl.growth <- create.bspline.basis(growth$age)
# cubic spline (order 4)

bSpl.growth6 <- create.bspline.basis(growth$age,norder=6)
# quintic spline (order 6)

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