Multifractal

Lesson

Introduction
Multifractal theory
Hierarchy of singularities
- Statistical deception of multifractal processes
- Co-dimenion function c(g)
- Statistical moment function k(q)
- K(q) proprieties
- General proprieties of multifractal
- Conservative and non-conservative quantities
Multifractal and extremes
Universal multifractal model
Exercises
Conclusion

Statistical moment function k(q)

As mentioned, the statistical properties of a random variable can be described by the distribution function but also by the moments of different orders. Both representation are linked by the Mellin transformation:

$E(X^q) = \int \limits_{0}^{\infty} x^q p(x) dx$

Where is the moment order and is the probability density function. This can also be applied to a multifractal process.

For a multifractal variable, the statistical moment varies with according to the power law:

$E(\varepsilon_ {\lambda}^{q}) = \lambda^{k(q)}$

Left : represents the theoretical behavior of the statistical moment; right : shape of the statistical moment function

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