pointwise multiplication method for basisfd class
# S3 method for basisfd
*(basisobj1, basisobj2)
objects of class basisfd
TIMES for (two basis objects sets up a basis suitable for expanding the pointwise product of two functional data objects with these respective bases. In the absence of a true product basis system in this code, the rules followed are inevitably a compromise: (1) if both bases are B-splines, the norder is the sum of the two orders - 1, and the breaks are the union of the two knot sequences, each knot multiplicity being the maximum of the multiplicities of the value in the two break sequences. That is, no knot in the product knot sequence will have a multiplicity greater than the multiplicities of this value in the two knot sequences. The rationale this rule is that order of differentiability of the product at eachy value will be controlled by whichever knot sequence has the greater multiplicity. In the case where one of the splines is order 1, or a step function, the problem is dealt with by replacing the original knot values by multiple values at that location to give a discontinuous derivative. (2) if both bases are Fourier bases, AND the periods are the the same, the product is a Fourier basis with number of basis functions the sum of the two numbers of basis fns. (3) if only one of the bases is B-spline, the product basis is B-spline with the same knot sequence and order two higher. (4) in all other cases, the product is a B-spline basis with number of basis functions equal to the sum of the two numbers of bases and equally spaced knots.
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.
basisfd
f1 <- create.fourier.basis()
f1.2 <- f1*f1
stopifnot(
all.equal(f1.2, create.fourier.basis(nbasis=5))
)
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