[Fis] probability versus power laws

Guy, and others --

Guy has a realist take on power laws. In effect, one could say that, in
any collection of the same kinds of events (say, earthquakes), it will be
the case that there will be far fewer large, powerful examples, and more
and more examples as one tallies weaker and weaker ones. In texts, one
finds some words used very infrequently and more words used more frequently
(Zipf's Law). This dichotomizing is the material substance of power laws.
These data can be plotted as magnitude (Xi) or frequency (Pi) by rank (i),
and will show a long tail into smaller (or rarer) instances. So, looking
at ensemble data, here we are focusing dichotomously, on the relative
amounts of large magnitude (or common) versus small magnitude (or rare)
values. This technique can be useful in comparing the slopes of log Xi by
log i plots. For example, Zipf found that all kinds of texts he
investigated in many languages all had the same slope. Only "schizophrenic
word salad" had a different slope.

How does this relate to frequency distributions?. If we take the same set
of magnitude data, we could compute the frequencies of the various
magnitudes (if they aren't already frequencies, as with words in texts),
and we could plot P ( Xi) by Xi, to get a frequency distribution graph.
Immediately we see the central tendency pop out as the mode of the curve
showing as a hump. Instead of privileging the extremes of the data as in
the power law plot based on an ordination imposed on the data from outside,
we are focusing on the middle of the data collection, which emerges from
the data itself. The technique becomes useful in statistics, where the mean
and variances of a sample can be compared with other samples.

In biology and some other fields the statistical mean is often supposed to
point at some canonical value for Xi, which gets distorted in individuals
to different degrees due to historical accidents. Whether that kind of
interpretation could work for, say, earthquakes, seems doubtful. I know of
no comparable attempt at canonical interpretation of the slope of log Xi /
log i power plots. Dichotomizing bigger versus smaller (size or presence),
there might be a hint in the fact that bigger systems are slower to change,
while smaller ones change more rapidly, delivering in the event more sizes
(or kinds) of small things, and fewer big ones. In earthquakes, then,
small slippages would be more common just because fast dynamics have more
frequent results (whatever THAT could mean!).

Several workers in the past (most elaborately, Belevitch, 1959, Ann. Soc.
Sci. Bruxelles 73:310-) have suggested that the power plot can be
interpreted as the cumulative probability distribution for a lognormal
probability density function, which he shows using a technique of
rearranging the data.

In any case, it seems that the two techniques are mutually exclusive ways
of focusing on a data set -- the ends, or the middle. As such, they are
techniques for probing the data -- tools for analysis. In and of
themselves they show little about the world, other than that not all
instances of a kind of thing are identical; rather they are tools for
comparing different data sets. A test case for the ontological perspective
might be, if we are looking at fluctuations at thermodynamic equilibrium,
we could either identify a canonical ensemble of them, showing them to be
random, or instead find them (a la Tsallis) to make up a power disribution.
I seem to be leaning toward the notion that this would depend upon one's
viewpoint on the data!

STAN

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Received on Wed Jun 30 21:00:00 2004