Multifractal

UM and extremes

The UM model can be used to estimate the critical moment and introduced previously. The estimation is performed from the sampling space , the dimension quantify the extension of the probability space. Lets the realization or the available sampling. We observe observation with :

D_s = \frac{\log N_s}{\log \lambda}

The co-dimension of the maximum singularity is given by :

c(\gamma_s) = D + D_s

The singularities characterized by a co-dimension larger then can not be observed. The statistical moment can be deduced from the following equation:

q_s = \frac{d c(\gamma_s)}{ d \gamma_s}

In the multifractal framework we obtain :

q_s = \Big(\frac{D + D_s}{C_1} \Big)^{1/\alpha}

When the upper limit of the UM model is the divergence of the statistical moment, the divergence order is the solution of :

D = \frac{C_1}{\alpha-1} \frac{q_{D}^{\alpha} - q_D }{q_D -1}
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