FRACTAL STATISTICAL ANALYSIS

Adriano V. Patac, Jr., Roberto N. Padua

Abstract


This paper provides the statistical foundations for fractal analysis of real-life data. It begins by developing a fractal probability distribution based on a power-law formulation and proceeds by estimating the parameters through (a) maximum likelihood estimation method, (b) exponential parameter estimation, and (c) regression approach. Data analysis based on a fractal distribution hypothesis is heavily guided by the fact that for a random variable X with fractal distribution f(x;θ,λ), the random variable y=log(x/θ) has an exponential distribution with rate parameter β = λ-1. A new Q-Q plot is introduced for assessing the fractality of observations. Likewise, tests of hypotheses about the fractal dimension λ are introduced based on the pivotal statistic y=log(x/θ). Test statistics are constructed whose null distributions are shown to be chi-square, for testing H0: λ=λ0 or beta distributions, for testing H0: λ1=λ2.

Keywords


fractal statistical dimension, fractal random variable, power-law distribution

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References


Albert, J.S. & Reis, R.E., eds. (2011). Historical biogeography of neotropical freshwater fishes. Berkeley: University of California Press.388pp.

Graybill, F. A. (1976). Theory and application of the linear model. Duxbury, North Scituate, Massachusetts.

Humphries, N.E., Queiroz, N., Dyer, J.R., Pade, N.G., Musyl, M.K., Schaefer, K.M., Fuller, D.W., Brunnschweiler, J.M., Doyle, T.K., Houghton, J.D., Hays, G.C., Jones, C.S., Noble, L.R., Wearmouth, V.J., Southall, E.J., Sims, D.W. (2010). Environmental context explains Lévy and Brownian movement patterns of marine predators. Nature, 465 (7301), 1066– 1069. doi:10.1038/nature09116

Klaus, A., Yu, S. & Plenz, D. (2011). Statistical analyses support power law distributions found in neuronal avalanches. PLoS ONE, 6(5), e19779. doi:10.1371/ journal.pone.0019779

Mandelbrot, B.B. (1983). The fractal geometry of nature. W.H. Freeman. San Francisco. California.

Neukum, G. & Ivanov, B.A. (1994). Crater size distributions and impact probabilities on Earth from lunar, terrestrial-planet, and asteroid cratering data. In T. Gehrels (ed.), Hazards Due to Comets and Asteroids, University of Arizona Press, Tucson, Arizona, 359–416.

Newman, M.E. J. (2005). Power laws, pareto distributions and Zipf’s law”. Contemporary Physics, 46(5): 323–351.

Nydick, S.W. (2012). A different(ial) way matrix derivatives again. University of Minnesota. Retrieve from http://www.tc.umn.edu/~nydic001/docs/unpubs/ Magnus_Matrix_Differentials_Presentation.pdf on Dec. 5, 2014.

Palmer, M.W. (1988). Fractal geometry: a tool for describing spatial patterns of plant communities. Vegetatio, 75 (1), 91–102. doi:10.1007/BF00044631


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