Dimensional Analysis Via PI Versus Factor/Cluster/Scaling/PCA

From: Osher Doctorow (mdoctorow_at_comcast.net)
Date: 12/29/04 

Date: Wed, 29 Dec 2004 13:46:52 +0000 (UTC)

 From Osher Doctorow mdoctorow@comcast.net

COPYRIGHT NOTICE
Dimensional Analysis Via PI Versus Factor/Cluster/Scaling/PCA
Copyright By Owner Osher Doctorow Ph.D.
First Published 2004.

In my last thread (Probability Transform), I discussed briefly the
value of dimensional analysis which emerges from the equation

1) PT(xy) = x + y

It is important for mathematical probability-statistics as well as
the applied probability-statistics fields to include dimensional
analysis in their repertoire of methods alongside factor analysis,
cluster analysis, multidimensional scaling, PCA (Principal Compo-
nents Analysis), etc.

Roughly speaking, factor/cluster/scaling/PCA seek to obtain patterns
 from (random) data, with considerable variation between methods as
to what those patterns are - ranging all the way from rather simple
spatial or temporal concentrations of data to theoretically cohesive
relatively abstract patterns including latent traits. The Bayesians
have already plunged deep into factor analysis, as several recent
volumes attest.

Dimensional analysis, including dimensionless ratios, seems at first
glance purely theoretical and almost unrelated to (random) data, but
it has in common with the above methods (a) exploratory analysis
of problems and (b) derivation of patterns - in this case, equations
themselves derived from "almost nothing". The applied mathematician
or scientist tries to intuitively select exactly the right variables
for a new or relatively little explored problem area, and the
variables are algebraically processed so to speak by the Buckingham
PI Theorem, and if they've been selected correctly their functional
equation emerges perfectly! If something is slightly "off" in the
choice, an "almost perfect" equation emerges with one or two varia-
bles not completely specified as to their functional relationship
beyond a form such as f(v1, v2) where f is some function of variables
v1 and v2 which aren't completely specified.

If dimensional analysis is so useful, then why is it almost entirely
absent from mathematical probability-statistics courses? That is a
problem in course selection. The literature on dimensional analysis
continues to increase, although not at a spectacular rate resembling
superstring theory or experimental physics or applied regression.
Moreover, the literature is in surprising directions - not in the
usual psychology, social sciences including economics and management
fields so much, but in physics, astronomy/astrophysics, engineering,
and to a somewhat lesser extent biology (although economics is also
represented by some research). Finally, computerization isn't
really necessary for dimensional analysis at least for typical
problems, so the tendency to computerize everything may pass by
dimensional analysis.

I'll conclude by mentioning "Fine structure constants in n-dimension-
al physical spaces through dimensional analysis," by Fabricio
Casrejos, Jaime F. Villas da Rocha, and Roberto Moreira Xavier,
U. do Estado do Rio de Janeiro, arXiv:physics/0309040 v1 7 Sep 2003
and "Analisi dimensionale: due interessanti applicazioni" by Germano
D'Abramo (U. Roma) arXiv:physics/0306042 v1 5 Jun 2003, as well as
several other papers that can be accessed on arXiv under keywords
"dimensional analysis" including physics/0304077 (turbulence),
gr-qc/0212029 (harmonic oscillators, inflation), physics/0002022
(cosmology), physics/9811016 (cosmology).

Osher Doctorow

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