Kees Spoor (tu-delft/nl)

06-20-1994, 02:49 AM

Dear biomechanists,

In addition to the summary of responses to Ellen C. Ross I'd like to supply

some information on the c0pyright of the GCVSPL software package.

When the late Herman Woltring handed over a reprint on his GCV software to me,

he added the following note about the copyright and usage.

(Beginning of quotation)

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

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 much greater than 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.

(End of quotation)

The software package contains parts made by dr M.F. Hutchinson of Canberra

(Australia) and by Dr F.I. Utreras of Santiago (Chile), as was stated in the

abovementioned reference.

I did not check if the explanation of the test-programme is still up-to-date.

Kees Spoor, Delft University of Technology

In addition to the summary of responses to Ellen C. Ross I'd like to supply

some information on the c0pyright of the GCVSPL software package.

When the late Herman Woltring handed over a reprint on his GCV software to me,

he added the following note about the copyright and usage.

(Beginning of quotation)

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

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 much greater than 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.

(End of quotation)

The software package contains parts made by dr M.F. Hutchinson of Canberra

(Australia) and by Dr F.I. Utreras of Santiago (Chile), as was stated in the

abovementioned reference.

I did not check if the explanation of the test-programme is still up-to-date.

Kees Spoor, Delft University of Technology