Self-similar Process - Self-similar Stochastic Processes Modeled By Tweedie Distributions

Self-similar Stochastic Processes Modeled By Tweedie Distributions

Leland et al have provided a mathematical formalism to describe self-similar stochastic 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 (Taylor's law), with p=2-d.

Tweedie distributions are a special case of exponential dispersion models, a class of models used to describe error distributions for the generalized linear model.

These Tweedie distributions are characterized by an inherent scale invariance and thus for any random variable Y that obeys a Tweedie distribution, the variance var(Y) relates to the mean E(Y) by the power law,

where a and p are positive constants. The exponent p for the variance to mean power law associated with certain self-similar stochastic processes ranges between 1 and 2 and thus may be modeled in part by a Tweedie compound Poisson–gamma distribution.

The additive form of the Tweedie compound Poisson-gamma model has the cumulant generating function (CGF),

,

where

,

is the cumulant function, α is the Tweedie exponent

,

s is the generating function variable, θ is the canonical parameter and λ is the index parameter.

The first and second derivatives of the CGF, with s=0, yields the mean and variance, respectively. One can thus confirm that for the additive models the variance relates to the mean by the power law,

.

Whereas this Tweedie compound Poisson-gamm CGF will represent the probability density function for certain self-similar stochastic processes, it does not return information regarding the long range correlations inherent to the sequence Y.

Nonetheless the Tweedie distributions provide a means understand the possible origins of self-similar stochastic processes for reason of their role as foci for for a central limit-like convergence effect known as the Tweedie convergence theorem. In nontechnical terms this theorem tells us that any exponential dispersion model that asymptotically manifests a variance-to-mean power law is required to have a variance function that comes within the domain of attraction of a Tweedie model.

The Tweedie convergence theorem can be used to explain the origin of the variance to mean power law, 1/f noise and multifractality, features associated with self-similar processes.