Multifractal

Lesson

Introduction
Multifractal theory
Hierarchy of singularities
Multifractal and extremes
Universal multifractal model
- Universal multifractal model (UM)
- Comments on UM estimation
- UM and extremes
Exercises
Conclusion

Comments on UM estimation

The UM model provides also a parametric expression for function:

$c(\gamma-H)= \left \{ \begin{array}{l} C_1 \Big( \frac{\gamma}{C_1 \alpha^{'}} + \frac{1}{\alpha}\Big) \quad \alpha\neq1 \\ \quad \\ C_1 exp\Big( \frac{\gamma}{C_1 } - 1 \Big) \quad \alpha = 1 \end{array} \right.$

With the condition :

$\frac{1}{\alpha} + \frac{1}{\alpha^{'}} = 1$

The parameters are called a Universal parameters. They have both geometrical and physical meaning :

Is the co-dimension of mean singularity of the process. It measure the mean heterogeneity of the field. The phenomenon is homogenous if . More increases more the means singularity is scattered. It is unusual to observe a phenomenon which is greater than its mean, but this can happen with extreme way. For , where is the dimension of the support, the process degenerate.

represents the degree of multifractality and define the gap from the monfractality. Its value between 0 and 2. If , we can observe fractal process or a simple scale invariance. The case of corresponds to the maximum multifractality.

is the parameter which quantify the deviation from a conservative process. means that the process is conservative.

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