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Scaling detection in time series: diffusion entropy analysis
The methods currently used to determine the scaling exponent of a complex
dynamic process described by a time series are based on the numerical
evaluation of variance. This means that all of them can be safely applied only
to the case where ordinary statistical properties hold true even if strange
kinetics are involved. We illustrate a method of statistical analysis based on
the Shannon entropy of the diffusion process generated by the time series,
called Diffusion Entropy Analysis (DEA). We adopt artificial Gauss and L\'{e}vy
time series, as prototypes of ordinary and anomalus statistics, respectively,
and we analyse them with the DEA and four ordinary methods of analysis, some of
which are very popular. We show that the DEA determines the correct scaling
exponent even when the statistical properties, as well as the dynamic
properties, are anomalous. The other four methods produce correct results in
the Gauss case but fail to detect the correct scaling in the case of L\'{e}vy
statistics.Comment: 21 pages,10 figures, 1 tabl
Turning a coin over instead of tossing it
Given a sequence of numbers in , consider the following
experiment. First, we flip a fair coin and then, at step , we turn the coin
over to the other side with probability , . What can we say about
the distribution of the empirical frequency of heads as ?
We show that a number of phase transitions take place as the turning gets
slower (i.e. is getting smaller), leading first to the breakdown of the
Central Limit Theorem and then to that of the Law of Large Numbers. It turns
out that the critical regime is . Among the scaling limits,
we obtain Uniform, Gaussian, Semicircle and Arcsine laws
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