eval.penalty {fda} | R Documentation |
A basis roughness penalty matrix is the matrix containing the possible inner products of pairs of basis functions. These inner products are typically defined in terms of the value of a derivative or of a linear differential operator applied to the basis function. The basis penalty matrix plays an important role in the computation of functions whose roughness is controlled by a roughness penalty.
eval.penalty(basisobj, Lfdobj=int2Lfd(0), rng=rangeval)
basisobj |
a basis object. |
Lfdobj |
either a nonnegative integer defining an order of a derivative or a linear differential operator. |
rng |
a vector of length 2 defining a restricted range. Optionally, the inner products can be computed over a range of argument values that lies within the interval covered by the basis function definition. |
The inner product can be computed exactly for many types of bases if $m$ is an integer. These include B-spline, fourier, exponential, monomial, polynomial and power bases. In other cases, and for noninteger operators, the inner products are computed by an iterative numerical integration method called Richard extrapolation using the trapezoidal rule.
The computation of the penalty matrix is a relatively
lengthy operation, and particular in S-PLUS code, so that
the calculations are made with a C++ function. The function
getbasispenaltyC
calls this function, but can
only do so for integer-valued linear differential operators.
If the penalty matrix must be evaluated repeatedly,
computation can be greatly speeded up by avoiding the use
of this function, and instead using quadrature points and
weights defined by Simpson's rule. These may be conveniently
computed using function quadset
, and stored along
with the associated basis function/derivative values as a
part of the basis object.
a square symmetric matrix whose order is equal
to the number of basis functions defined by
the basis function object basisobj
.
If Lfdobj
is $m$ or a linear
differential operator of order $m$, the rank
of the matrix should be at least approximately equal to
its order minus $m$.
getbasispenalty, eval.basis, quadset, getbasispenaltyC