Some comments on the correlation dimension of $1/f^α$ noise

Details Some comments on the correlation dimension of $1/f^α$ noise Some comments on the correlation dimension of $1/f^α$ noise

1993-02-10

arxiv.org/abs/comp-gas/9302001v1

Nonlinear Sciences

Cellular Automata and Lattice Gases

CYCLER Paper 93feb005 Several PostScript files, compress'ed tar'ed uuencode'ed

Scientific paper

It has recently been observed that a stochastic (infinite degree of freedom) time series with a $1/f^\alpha$ power spectrum can exhibit a finite correlation dimension, even for arbitrarily large data sets. [A.R. Osborne and A.~Provenzale, {\sl Physica D} {\bf 35}, 357 (1989).] I will discuss the relevance of this observation to the practical estimation of dimension from a time series, and in particular I will argue that a good dimension algorithm need not be trapped by this anomalous fractal scaling. Further, I will analytically treat the case of gaussian \onefas noise, with explicit high and low frequency cutoffs, and derive the scaling of the correlation integral $C(N,r)$ in various regimes of the $(N,r)$ plane. Appears in: {\sl Phys. Lett. A} {\bf 155} (1991) 480--493.

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