Multifractal

Lesson

Introduction
Multifractal theory
Hierarchy of singularities
Multifractal and extremes
- Extremes and multifractal
- Divergence of statistical moment
- Sample's size effect
Universal multifractal model
Exercises
Conclusion

Extremes and multifractal

We will show in the following sections, that the empirical function and could differ from the theoretical forms presented in the following figure, even if the studied field present the scale invariance propriety.

(a) : Theoretical shape of K(q) function; (b): Theoretical shape of c(g) function

It is possible that the and functions which are theoretically convex become linear beyond some specific threshold. This behavior known as the multifractal phase transition, has involvement in the analysis of extreme values. Two kinds of multifractal phase transition are possible which are caused by two different phenomena : divergence of the statistical moment and the size of samples.

Comparison of theoretical and empirical estimation of K(q) function

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