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Quantifying emergence in terms of persistent mutual information

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Ball, Robin, Diakonova, Marina and MacKay, R. S. (Robert Sinclair). (2010) Quantifying emergence in terms of persistent mutual information. Advances in Complex Systems, Vol.13 (No.3). pp. 327-338. ISSN 0219-5259

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Official URL: http://dx.doi.org/10.1142/S021952591000258X

Abstract

We define Persistent Mutual Information (PMI) as the Mutual (Shannon) Information between the past history of a system and its evolution significantly later in the future. This quantifies how much past observations enable long-term prediction, which we propose as the primary signature of (Strong) Emergent Behavior. The key feature of our definition of PMI is the omission of an interval of "present" time, so that the mutual information between close times is excluded: this renders PMI robust to superposed noise or chaotic behavior or graininess of data, distinguishing it from a range of established Complexity Measures. For the logistic map, we compare predicted with measured long-time PMI data. We show that measured PMI data captures not just the period doubling cascade but also the associated cascade of banded chaos, without confusion by the overlayer of chaotic decoration. We find that the standard map has apparently infinite PMI, but with well-defined fractal scaling which we can interpret in terms of the relative information codimension. Whilst our main focus is in terms of PMI over time, we can also apply the idea to PMI across space in spatially-extended systems as a generalization of the notion of ordered phases.

Item Type:

Journal Article

Subjects:

Q Science > QA Mathematics
Q Science > QC Physics

Divisions:

Faculty of Science > Mathematics
Faculty of Science > Physics

Library of Congress Subject Headings (LCSH):

Dynamics, Mathematical physics, Self-Organizing Systems

Journal or Publication Title:

Advances in Complex Systems

Publisher:

World Scientific Publishing

ISSN:

0219-5259

Official Date:

June 2010

Dates:

Date	Event
June 2010	Published

Volume:

Vol.13

Number:

No.3

Number of Pages:

Page Range:

pp. 327-338

Identification Number:

10.1142/S021952591000258X

Status:

Peer Reviewed

Publication Status:

Published

Access rights to Published version:

Open Access

Funder:

Engineering and Physical Sciences Research Council (EPSRC), University of Warwick. Complexity Science Doctoral Training Centre

Grant number:

EP/E501311/1

References:

Bar-Yam, Y., A mathematical theory of strong emergence using multiscale variety,
Complexity 9 (2004) 15–24.
Bialek, W., Nemenman, I., and Tishby, N., Predictability, complexity, and learning,
Neural Comput. 13 (2001) 2409–2463.
Chalmers, D. J., Strong and Weak Emergence, Vol. The Re-Emergence of Emergence.
(Oxford University Press., 2002).
Chirikov, B. V. and Shepelyansky, D. L., Asymptotic statistics of poincar´e recurrences
in hamiltonian systems with divided phase space, Phys. Rev. Lett. 82 (1999) 528–531.
Collet, J.-P., P. Eckmann, Iterated Maps on the Interval as Dynamical System, Modern
Birkh¨auser Classics (Birkh¨auser, 1980).
Coullet, P. and Tresser, C., Iterations d’endomorphismes et groupe de renormalisation.,
J. Phys. Colloque C 539 (1978) C5 – 25.
Crutchfield, J. and Packard, N., Symbolic Dynamics of Noisy Chaos, Physica D 7
(1983) 201–223.
Crutchfield, J. and Young, K., Inferring Statistical Complexity, Phys. Rev. Lett. 63
(1989) 105–108.
Ellison, C. J., Mahoney, J. R., and Crutchfield, J. P., Prediction, Retrodiction, and
the Amount of Information Stored in the Present, J. Stat. Phys. 136 (2009) 1005–
1034.
Feigenbaum, M. J., Quantitative universality for a class of nonlinear transfor mations.,
J. Stat. Phys. 19 (1978) 25 – 52.
Feldman, D. P., McTague, C. S., and Crutchfield, J. P., The organization of intrinsic
computation: Complexity-entropy diagrams and the diversity of natural information
processing, CHAOS 18 (2008).
Grassberger, P., Toward a quantitative theory of self-generated complexity, International
Journal of Theoretical Physics 25 (1986) 907–938.
Grassberger, P., On symbolic dynamics of one-humped maps of the interval, Z. Naturforschung
43 a (1988) 671 – 680.
Greene, J. M., A method for determining a stochastic transition, Journal of Mathematical
Physics 20 (1979) 1183–1201.
Grossman, S. and Thomae, S., Z. Naturforschung 32a (1977) 1353.
Kraskov, A., Stogbauer, H., and Grassberger, P., Estimating mutual information,
Phys. Rev. E 69 (2004).
Lindgren, K. and Nordahl, M. G., Complexity measures and cellular automata, Complex
Syst. 2 (1988) 409–440.
MacKay, R. and Percival, I., Converse KAM - Theory and Practice, Commun. in
Math. Phys. 98 (1985) 469–512.
MacKay, R. S., Nonlinearity in complexity science, Nonlinearity 21 (2008) T273–
T281.
Misiurewicz, M., Absolutely Continuous Measures for Certain Maps of an Interval,
Publications Mathematiques (1981) 17–51.
Shalizi, C., Shalizi, K., and Haslinger, R., Quantifying self-organization with optimal
predictors, Phys. Rev. Lett. 93 (2004).
Zambella, D. and Grassberger, P., Complexity of forcasting in a class of simple models,
Complex Systems 2 (1988) 269–303.