MEMO: GCVSPL software package
(C) COPYRIGHT 1985, 1986: H.J. Woltring
Philips Medical Systems Division, Eindhoven
University of Nijmegen (The Netherlands)
DATE: 1986-05-12
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NB: This software is copyrighted, and may be copied for excercise,
study and use without authorization from the copyright owner(s), in
compliance with paragraph 16b of the Dutch Copyright Act of 1912
("Auteurswet 1912"). Within the constraints of this legislation, all
forms of academic and research-oriented excercise, study, and use are
allowed, including any necessary modifications. Copying and use as
object for commercial exploitation are not allowed without permission
of the copyright owners, including those upon whose work the package
is based.
A full description of the package is provided in:
H.J. Woltring (1986), A FORTRAN package for generalized, cross-valida-
tory spline smoothing and differentiation. Advances in Engineering
Software 8(2):104-113 (U.K.).
For large datasets (N >> 0) and negligible boundary artefacts, the
behaviour of a natural spline approximates that of a periodic spline.
For the latter case, the frequency characteristic in the equidistantly
sampled, uniformly weighted case is that of a double, phase-symmetric
Butterworth filter, with transfer function H(w) = [1 + (w/wo)^2M]^-1,
where w is the frequency, wo = (p*T)^(-0.5/M) the filter's cut-off
frequency, p the smoothing parameter, T the sampling interval, and 2M
the order of the spline. If T is expressed in seconds, the frequen-
cies are expressed in radians/second.
It has been found empirically, that the effective number of estimated
spline parameters Np is related to the Butterworth cut-off frequency
wo as Np ~ M/2 + KM * wo * N * T, where Np ranges between M and N, and
where KM is the integral over x from 0 to infinity of (1 + x^2M)^-1
divided by PI. For large M, KM approaches 1/PI from above; values for
small M are: K1 = 1/2, K2 = 1/V8, K3 = 1/3. This relation has also
been found to apply for uniformly weighted data which are sampled
slightly anequidistantly, with T taken as the average sampling inter-
val. For large Np, the relation with wo * N * T becomes nonlinear.
A simple test-programme GCV is provided with the package. The test
data (a simple parabolic time signal) are not entirely appropriate for
smoothing and differentiation by means of natural splines: for low-
order splines, boundary artefacts prevail, while low-frequency noise
and the low-frequency signal are confounded for high-order splines.
Smoothing via the optimization criteria in the GCVSPL package assumes
a sufficiently large signal-to-noise ratio and number of measurements,
a sufficiently strong m-th signal derivative, and wide-band, uncorre-
lated noise; these conditions are not fully met by the given test
data. Perusal of GCV for different spline orders, smoothing modes,
and numbers of observations will allow the user to acquaint himself
with the GCVSPL package.