×

zbMATH — the first resource for mathematics

Measures of statistical complexity: why? (English) Zbl 1026.82505
Summary: We review several statistical complexity measures proposed over the last decade and a half as general indicators of structure or correlation. Recently, López-Ruiz, Mancini, and Calbet (1995) introduced another measure of statistical complexity \(C_{\text{LMC}}\) that, like others, satisfies the “boundary conditions” of vanishing in the extreme ordered and disordered limits. We examine some properties of \(C_{\text{LMC}}\) and find that it is neither an intensive nor an extensive thermodynamic variable and that it vanishes exponentially in the thermodynamic limit for all one-dimensional finite-range spin systems. We propose a simple alteration of \(C_{\text{LMC}}\) 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.

MSC:
82B03 Foundations of equilibrium statistical mechanics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Shannon, C.E; Weaver, W, ()
[2] Cover, T.M; Thomas, J.A, ()
[3] (Russian) Math. Rev. 21, No. 2035a.
[4] Sinai, Ja.G, Dokl. akad. nauk. SSSR, 124, 768, (1959)
[5] Huberman, B.A; Hogg, T, Physica D, 22, 376, (1986)
[6] Grassberger, P, Int. J. theor. phys., 25, 907, (1986)
[7] Crutchfield, J.P; Young, K, Phys. rev. lett., 63, 105, (1989)
[8] Crutchfield, J.P; Packard, N.H, Physica D, 7, 201, (1983)
[9] Szépfalusy, P; Györgyi, G, Phys. rev. A, 33, 2852, (1986)
[10] Wolfram, S, Physica D, 10, 1, (1984)
[11] Shaw, R, ()
[12] Bennett, C.H, Found. phys., 16, 585, (1986)
[13] Lindgren, K; Norhdal, M.G, Complex systems, 2, 409, (1988)
[14] Li, W, Complex systems, 5, 381, (1991)
[15] Crutchfield, J.P, Physica D, 75, 11, (1994)
[16] Wackerbauer, B; Witt, A; Atmanspacher, H; Kurths, J; Scheingraber, H, Chaos, solitons fractals, 4, 133, (1994)
[17] Gell-Mann, M; Lloyd, S, Complexity, 2, 44, (1996)
[18] Kolmogorov, A.N, Prob. info. trans., 1, 1, (1965)
[19] Chaitin, G, J. ACM, 13, 145, (1966)
[20] Li, M; Vitanyi, P.M.B, ()
[21] Hopcroft, J.E; Ullman, J.D, ()
[22] Crutchfield, J.P; Hanson, J.E, Physica D, 69, 279, (1993)
[23] Crutchfield, J.P; Feldman, D.P, Phys. rev. E, 55, R1239, (1997)
[24] Upper, D.R, Theory and algorithms for hidden Markov models and generalized hidden Markov models, ()
[25] Pagels, H; Lloyd, S, Ann. phys., 188, 186, (1988)
[26] Lòpez-Ruiz, R; Mancini, H.L; Calbet, X, Phys. lett. A, 209, 321, (1995)
[27] Anteneodo, C; Plastino, A.R, Phys. lett. A, 223, 348, (1996)
[28] Bennett, C.H, (), 137
[29] Rissanen, J, ()
[30] Bronson, R, ()
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.