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[Hans Blom, 950830c]
In my long-time-ago mail [Hans Blom, 950613] in reply to (Bill Powers
(950607.0630 MDT)), who asked:
And for that matter, how are you going to model even a low-frequency
disturbance that doesn't follow any analytically-describable waveform
and that doesn't repeat in a predictable way?
I answered:
I'm assigning this as homework to myself, and I'll show you a demo when
I'm finished. The theory behind modelling a signal, including a disturb-
ance, is called time series analysis. Lots of exams to review now, so
not any time soon...
I haven't found the time to do a demo yet (when, o when?), but I found an
article where the most recent theory on this subject is nicely described,
and one (Kalman-like) method is fully developed. It is:
Methods of dynamic spectral analysis by self-exciting
autoregressive moving average models and their application to
analysing biosignals. B. Schack, E. Bareshova, G. Griesbach,
and H. Witte. Medical & Biological Engineering & Computing,
vol. 3, no. 3, May 1955.
Abstract: Dynamic methods in the spectral domain are necessary to ana-
lyse biological signals because of the frequently nonstationa-
ry character of the signals. The paper presents an adaptive
procedure of fitting time-dependent ARMA [autoregressive mov-
ing average] models [i.e. models with parameters that vary in
time] to nonstationary signals, which is suitable for on-line
calculations. The properties of the model parameter estimat-
ions are examined, and in the stationary case are compared
with the results of convergent estimation methods. On this
basis time-varying spectral parameters with high temporal and
spectral resolution are calculated, and the possibility of
their application is shown in EEG analysis and laser-Doppler-
flowmetry.
From the paper: "The basis for the evaluation and statistical analysis of
technical or biological signals is the mathematical methods of time
series analysis. ... The application of parametric methods ... offers the
advantage of an immense data reduction, because such analysis methods are
based on only a few parameters."
And that is, of course, what one wants in a model: describing the "world"
(the perceived signals) with only a few parameters, yet have an accurate
description.
"Often the signal is only composed of single stationary parts, or the
signal power varies, or there are changes of slow and fast waves."
And all that must be, and can be, modelled "with high temporal and
spectral resolution", i.e. with little error. "The [model] parameters
vary around their true value [but, I think, closely enough for accurate
control to be possible], but do not converge to them [in order to be able
to keep on learning]."
Greetings,
Hans