Self-dissimilarity as a high dimensional complexity measure
David Wolpert, Intelligent Systems Division, NASA Ames
tc.arc.nasa.gov/people/dhw
For many systems characterized as ``complex'' the
patterns exhibited on different scales differ markedly from one
another. For example the biomass distribution in a human body ``looks
very different'' depending on the scale at which one examines it.
Conversely, the patterns at different scales in ``simple'' systems
(e.g., gases, mountains, crystals) vary little from one scale to
another. Accordingly, the degrees of self-{\it dis}similarity between
the patterns of a system at various scales constitute a complexity
``signature'' of that system.
Here I present a novel quantification of
self-dissimilarity. This quantification can be measured for many kinds
of real-world data. This allows comparisons of the complexity
signatures of wholly different kinds of systems (e.g., systems
involving information density in a digital computer vs. species
densities in a rain-forest vs. capital density in an economy,
etc.). Moreover, in contrast to many other suggested complexity
measures, evaluating the self-dissimilarity of a system does not
require one to already have a model of the system.
These facts may allow self-dissimilarity signatures to be used
as the underlying observational variables of an eventual overarching
theory relating all complex systems. To illustrate self-dissimilarity
I present several numerical experiments. In particular, I show that
underlying structure of the logistic map is picked out by the
self-dissimilarity signature of time series' produced by that map.