Measures of Statistical Complexity: Why?

David P. Feldman and James P. Crutchfield

97-07-064

We review several statistical complexity measures proposed over the last decade and a half as general indicators of structure or correlation. Recently, Lòpez-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 bounary conditions of vanishing, it must also be defined in a setting that gives a clear interpretation to what structures are quantified.