Tweedie Distributions - Tweedie Convergence and 1/f Noise

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.