Dear Biomch-L readers,

Further to the various postings on averaging time series in reply to John Scholz
at the University of Delaware (USA), I might suggest the use of a non-commercial
software package which can be retrieved (subscribers only) from Biomch-L's list-
server by means of the request


to LISTSERV@HEARN.EARN (or .BITNET). The word BIOMCH-L is optional here, but
it speeds up the search process in LISTSERV@HEARN's database. If the request
should fail because of, e.g., internetwork gateway constraints, you might also
send the request

send gcvspl from gcv

to NETLIB@RESEARCH.ATT.COM (.COM is the commercial component on Internet). Note
that the request should be in lower case for the latter fileserver lest the data
are returned in all caps. For further details on the mathematical, public file-
server Netlib, located at A.T.&T./Bell Research Labs in Murray Hill/NY-USA, send
the request 'send index' in a separate line.

GCVSPL is a general spline package which allows optimal estimation of smoothing
factors; in John Scholtz' example, optimal processing of (non)equidistantly
spaced data is feasible, and the amount of smoothing may either be estimated
from the data using a procedure known as cross-validation, or imposed based on
prior experience with the amount of smoothing required. Published reference
material is quoted in the package, and simultaneous processing of multiple
datasets is possible provided that the same independent variables and weight
factors are used; however, additional weight factors can be used per data

While it may be useful to linearly interpolate raw data as suggested in a pre-
vious reply, I think that uncorrelated noise as largely caused by measurement
shortcomings should be removed prior to any further signal processing. The
GCVSPL package might be used to accomodate this; admittedly, this will require
some more programming than with an elegant subroutine like Ton van den Bogert's
linear interpolator. Furthermore, the package allows optimal derivative estim-
ation from noisy position data for cases where direct accelerometric measure-
ments (as in John Wann's case) are not available.

GCVSPL's main disadvantage is that the data are assumed to be stationary; thus,
transients like heel impact may result in damped peak estimates and in "Gibb's
phenomena" vibrations t h r o u g h o u t the record, especially in higher
derivative estimates. However, the effect may be countered to some extent by
judiciously chosen weight factors.

In short, averaging of (transformed) time sequences is not an easy task, and
there is scope for improvements in this field.

Herman J. Woltring