Measures of Statistical Complexity: Why?


D. P. Feldman and J. P. Crutchfield. Measures of Statistical Complexity: Why?, Physics Letters A 238: (4-5), 244-252, (1998).


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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