Measures of Statistical Complexity: Why?
Citation
D. P. Feldman and J. P. Crutchfield.
Measures of Statistical Complexity: Why?,
Physics Letters A 238: (4-5), 244-252, (1998).
Abstract
We review several statistical complexity measures proposed over
the last decade and a half as general indicators of structure
or correlation. Recently, Lopez-Ruiz, Mancini, and Calbet
[Phys. Lett. A 209 (1995) 321] introduced another measure of
statistical complexity C_LMC that, like others, satisfies
the "boundary conditions" of vanishing in the extreme ordered and
disordered limits. We examine some properties of C_LMC
and find that it is neither an intensive nor an extensive
thermodynamic variable. It depends nonlinearly on system size and
vanishes exponentially in the thermodynamic limit for all
one-dimensional finite-range spin systems. We propose a simple
alteration of C_LMC that renders it extensive. However,
this remedy results in a quantity that is a trivial function of
the entropy density and hence of no use as a measure of structure
or memory. We conclude by suggesting that a useful "statistical
complexity" must not only obey the ordered-random boundary
conditions of vanishing, it must also be defined in a setting
that gives a clear interpretation to what structures are quantified.
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