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Extremum statistics: a framework for data analysis

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Chapman, Sandra C., Rowlands, G. (George) and Watkins, Nicholas W.. (2002) Extremum statistics: a framework for data analysis. Nonlinear Processes in Geophysics, Vol.9 . pp. 409-418. ISSN 1023-5809

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Official URL: http://www.nonlin-processes-geophys.net/9/409/2002...

Abstract

Recent work has suggested that in highly correlated
systems, such as sandpiles, turbulent fluids, ignited
trees in forest fires and magnetization in a ferromagnet close to a critical point, the probability distribution of a global quantity (i.e. total energy dissipation, magnetization and so forth) that has been normalized to the first two moments follows a specific non-Gaussian curve. This curve follows a form suggested by extremum statistics, which is specified by a single parameter a (a = 1 corresponds to the Fisher-Tippett Type I (“Gumbel”) distribution).
Here we present a framework for testing for extremal
statistics in a global observable. In any given system, we
wish to obtain a, in order to distinguish between the different Fisher-Tippett asymptotes, and to compare with the
above work. The normalizations of the extremal curves are
obtained as a function of a. We find that for realistic ranges of data, the various extremal distributions, when normalized to the first two moments, are difficult to distinguish. In addition, the convergence to the limiting extremal distributions for finite data sets is both slow and varies with the asymptote.
However, when the third moment is expressed as a function
of a, this is found to be a more sensitive method.

Item Type:

Journal Article

Subjects:

G Geography. Anthropology. Recreation > G Geography (General)
Q Science > QC Physics

Divisions:

Faculty of Science > Physics

Library of Congress Subject Headings (LCSH):

Geophysics -- Data processing, Statistics

Journal or Publication Title:

Nonlinear Processes in Geophysics

Publisher:

Copernicus GmbH

ISSN:

1023-5809

Official Date:

8 February 2002

Dates:

Date	Event
8 February 2002	Submitted

Volume:

Vol.9

Page Range:

pp. 409-418

Status:

Peer Reviewed

Access rights to Published version:

Open Access

References:

Aji, V., and Goldenfeld, N.: Fluctuations in finite critical and turbulent
systems, Phys. Rev. Lett., 84, 1007–1010, 2001.
Bak, P.: How nature works: the science of self-organised criticality,
Oxford University Press, Oxford, 1997.
Bhavsar, S. P. and Barrow, J. D.: First ranked galaxies in groups and
clusters, Mon. Not. Roy. Astron. Soc., 213, 857–869, 1985.
Bohr, T., Jensen, M. H., Paladin, G., and Vulpiani, A.: Dynamical
systems approach to turbulence, Cambridge University Press,
Cambridge, 1998.
Bouchaud, J. P. and Mezard, M.: Universality classes for extreme
value statistics, J. Phys. A, 30, 7997, 1997.
Bouchaud, J. P. and Potters, M.: Theory of financial risks: from statistical
physics to risk management, Cambridge University Press,
Cambridge, 2000.
Bramwell, S. T., Holdsworth, P. C. W., and Pinton, J. F.: Universality
of rare fluctuations in turbulence and critical phenomena,
Nature, 396, 552–554, 1998.
Bramwell, S. T., Christensen, K., Fortin, J. Y., et al.: Universal
fluctuations in correlated systems, Phys. Rev. Lett., 84, 3744–
3747, 2000.
Bramwell, S. T. et al.: Reply to comment on “universal fluctuations
in correlated systems”, Phys. Rev. Lett., 87, 18 8902, 2001.
Bury, K.: Statistical distributions in engineering, Cambridge University
Press, Cambridge, 1999.
Cardy, J. E.: Scaling and renormalization in statistical physics,
Cambridge University Press, Cambridge, 1996.
Chapman, S. C. and Watkins, N. W.: Avalanching and selforganised
criticality: a paradigm for geomagnetic activity?,
Space. Sci. Rev., 95, 293–307, 2001.
Fisher, R. A. and Tippett, L. H. C.: Limiting forms of the frequency
distribution of the largest and smallest members of a sample,
Proc. Camb. Phil. Soc., 24, 180–190, 1928.
Gradshteyn, I. S. and Ryzhik, I. M.: Table of integrals, series and
products, Academic Press, 1980.
Gumbel, E. J.: Statistics of extremes, Columbia university press,
New York, 1958.
Jenkinson, A. F.: The frequency distribution of the annual maximum
(or minimum) values of meteorological elements, Q. J.
Roy. Meteorological Soc., 81, 158–171, 1955.
Jensen, H. J.: Self-organised criticality: emergent complex behaviour
in physical and biological systems, Cambridge University
Press, Cambridge, 1998.
Labbe, R., Pinton, J. F., and Fauve, S.: Power fluctuations in turbulent
swirling flows, J. Phys. II, France, 6, 1099–1110, 1996.
Lui, A. T. Y., Chapman, S. C., Liou, K., Newell, P. T., Meng, C. I.,
Brittnacher, M., and Parks, G. K.: Is the dynamic magnetosphere
an avalanching system?, Geophys. Res. Lett., 27, 911–915, 2000.
Pinton, J. F., Holdsworth, P., and Labbe, R.: Power fluctuations in
a closed turbulent shear flow, Phys. Rev. E, 60, R2452–R2455,
1999.
Sornette, D.: Critical phenomena in the natural sciences – chaos,
fractals, selforganization and disorder: concepts and tools,
Springer, Berlin, 2000.
Uritsky, V. M., Klimas, A. J., and Vassiliadis, D.: J. Geophys. Res.,
submitted, 2001.
Zheng, B., and Trimper, S.: Comment on “universal fluctuations in
correlated systems”, Phys. Rev. Lett., 87, 18 8901, 2001.