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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
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.
URI:	http://wrap.warwick.ac.uk/id/eprint/588