TaylorSpline: Taylor representation of a B-Spline

Description

Convert B-Spline coefficients into a local Taylor series representation expanded about the midpoint between each pair of distinct knots.

Usage

TaylorSpline(object, ...)
## S3 method for class 'fd':
TaylorSpline(object, ...)
## S3 method for class 'fdPar':
TaylorSpline(object, ...)
## S3 method for class 'fdSmooth':
TaylorSpline(object, ...)
## S3 method for class 'dierckx':
TaylorSpline(object, ...)

Arguments

object

a spline object, e.g., of class 'dierckx'.

...

optional arguments

Value

a list with the following components:
knotsa numeric vector of knots(object, interior=FALSE)
midpointsmidpoints of intervals defined by unique(knots)
coefA matrix of dim = c(nKnots-1, norder) containing the coeffients of a polynomial in (x-midpoints[i]) for interval i, where nKnots = length(unique(knots)).
derivA matrix of dim = c(nKnots-1, norder) containing the derivatives of the spline evaluated at midpoints.

Details

1. Is object a spline object with a B-spline basis? If no, throw an error. 2. Find knots and midpoints. 3. Obtain coef(object). 4. Determine the number of dimensions of coef(object) and create empty coef and deriv arrays to match. Then fill the arrays.

Examples

Run this code

##
## The simplest b-spline basis:  order 1, degree 0, zero interior knots:
##       a single step function
##
library(DierckxSpline)
bspl1.1 <- create.bspline.basis(norder=1, breaks=0:1)
# ... jump to pi to check the code
fd.bspl1.1pi <- fd(pi, basisobj=bspl1.1)
bspl1.1pi <- TaylorSpline(fd.bspl1.1pi)
bsp1.1ref <- list(knots=0:1, midpoints=0.5,
  coef=array(pi, dim=rep(1, 3), dimnames=list(NULL, 'b0', 'reps 1')),
  deriv=array(pi, dim=rep(1, 3), dimnames=list(NULL, 'D0', 'reps 1')) )
class(bsp1.1ref) <- "Taylor"
stopifnot(all.equal(bspl1.1pi, bsp1.1ref))

##
## Cubic spline:  4  basis functions
##
bspl4 <- create.bspline.basis(nbasis=4)
plot(bspl4)
parab4.5 <- fd(c(3, -1, -1, 3)/3, bspl4)
# = 4*(x-.5)
TaylorSpline(parab4.5)

##
## A more realistic example
##
data(titanium)
#  Cubic spline with 5 interior knots (6 segments)
# do not run this example on CRAN to save test time
#if(!CRAN()){
#titan10 <- with(titanium, curfit.free.knot(x, y))
#(titan10T <- TaylorSpline(titan10) )
#}