Tweedie Convergence and 1/f Noise
Pink noise, or 1/f noise, refers to a pattern of noise characterized by a power-law relationship between its intensities S(f) at different frequencies f,
- ,
where the dimensionless exponent γ∈ . It is found within a diverse number of natural processes. Many different explanations for 1/f noise exist, a widely-held hypothesis is based on Self-organized criticality where dynamical systems close to a critical point are thought to manifest scale-invariant spatial and/or temporal behavior.
In this subsection a mathematical connection between 1/f noise and the Tweedie variance-to-mean power law will be described. To begin, we first need to introduce self-similar processes: For the sequence of numbers
with mean
- ,
deviations
- ,
variance
- ,
and autocorrelation function
with lag k, if the autocorrelation of this sequence has the long range behavior
as k→∞ and where L(k) is a slowly varying function at large values of k, this sequence is called a self-similar process.
The method of expanding bins can be used to analyze self-similar processes. Consider a set of equal-sized non-overlapping bins that divides the original sequence of N elements into groups of m equal-sized segments (N/m is integer) so that new reproductive sequences, based on the mean values, can be defined:
- .
The variance determined from this sequence will scale as the bin size changes such that
if and only if the autocorrelation has the limiting form
- .
One can also construct a set of corresponding additive sequences
- ,
based on the expanding bins,
- .
Provided the autocorrelation function exhibits the same behavior, the additive sequences will obey the relationship
Since and are constants this relationship constitutes a variance-to-mean power law, with p=2-d.
The biconditional relationship above between the variance-to-mean power law and power law autocorrelation function, and the Wiener–Khinchin theorem imply that any sequence that exhibits a variance-to-mean power law by the method of expanding bins will also manifest 1/f noise, and vice versa. Moreover, the Tweedie convergence theorem, by virtue of its central limit-like effect of generating distributions that manifest variance-to-mean power functions, will also generate processes that manifest 1/f noise. The Tweedie convergence theorem thus allows provides an alternative explanation for the origin of 1/f noise, based its central limit-like effect.
Much as the central limit theorem requires certain kinds of random processes to have as a focus of their convergence the Gaussian distribution and thus express white noise, the Tweedie convergence theorem requires certain non-Gaussian processes to have as a focus of convergence the Tweedie distributions that express 1/f noise.
Read more about this topic: Tweedie Distributions
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