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nearness and sequential order not absolute distances), symbolic dynamical processes which as either (1) or (2) can be analyzed with respect to its various entropies, algorithmic complexities and word content and syntax. A variety of techniques aimed at deciding between the relevance of one or another of these underlying assumptions (such as series and Fourier phase shuffling to destroy statistical autocorrelations and vectorial continuities but leaving the probability density distributions intact ) may at times help emphasize one or another of these orientations in the analyses (see Ott et al, 1994 for a collection of articles on this topic).

Nonconvergent Distributions and Power Law Scaling in Biologically Relevant Time Series

The statistical distribution with which most of us are familiar is the Gaussian which can be generated by summing and averaging a series of independent random events. The average behavior head/tails probabilities observed by one person flipping a fair coin for a very long time or by many people flipping similar coins for shorter times converges upon the invariant measure of 0.5. The variance, “second moment” in the distribution of a population of coin flipping sequences will be finite and computable. In a graph of this distribution, the tails will converge to the x axis in a Gaussian exponential manner. The longer or the more numerous the “sample” series of observations, the closer they will approximate the “ergodic” invariant measures representing the true “central moments” of the behavior of this “population” of fair flipping coins. Since the coins are not changing their relevant characteristics over the time of observation, we say that the series is not time dependent but instead is “stationary.” Computation of correlations over increasing lags to determine how much and for how many flips the sequences continue to resemble themselves yield an exponential decay with a single characteristic correlation length. This reflects the existence of a finite variance from which its amplitude is derived and serves as the single characteristic temporal scale of the random process.

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