Multifractal

Lesson

Introduction
Multifractal theory
Hierarchy of singularities
Multifractal and extremes
Universal multifractal model
Exercises
- Quiz: Probability Distribution Multiple Scaling
- Quiz: Hierarchy of singularities
Conclusion

Probability Distribution Multiple Scaling

During this lecture we have seen that it is of interest to estimate function to characterize the studied field. Then, it is important to calculate directly. In this exercise we will introduce the Probability Distribution Multiple Scaling technique which allows us to estimation the co-dimension function.

$C(\gamma) = - \frac{log (Pr(\varepsilon_\lambda \geq \lambda^{\gamma}))}{log(\lambda)}$

Question

Using the given data:

Write a script to estimate the singularities for each resolution

Write a script which calculates the probability that the field exceed the values

According the above equation write a script to calculate the function

data.zip

Hint

We choose in this exercise the Scilab programming language (free).

Solution

// Main
clear;
clc;
exec("Upscaling.sce");      // UpScale
exec("Probability.sce");    //ProbaSup
exec("Cgamma.sce");         //PDMS
exec("Gamma.sce");          // GammaValues
flux = fscanfMat("data.txt")
sample_res = 10;
[gamas]=GammaValues(flux,sample_res)
rng =[1:sample_res];
lambdas = 2^sample_res./(2.^(0:sample_res));
for j= 1:length(gamas)
    [c_gammas(j),y1,x1] = PDMS(flux,gamas(j),lambdas,rng);
end
plot(gamas,c_gammas)

// Main
clear;
clc;

exec("Upscaling.sce");      // UpScale
exec("Probability.sce");    //ProbaSup
exec("Cgamma.sce");         //PDMS
exec("Gamma.sce");          // GammaValues

flux = fscanfMat("data.txt")

sample_res = 10;

[gamas]=GammaValues(flux,sample_res)

rng =[1:sample_res];

lambdas = 2^sample_res./(2.^(0:sample_res));

for j= 1:length(gamas)
    [c_gammas(j),y1,x1] = PDMS(flux,gamas(j),lambdas,rng);

end
plot(gamas,c_gammas)

function [gamas]=GammaValues(data,sample_res)
    
    data = data(1:2^sample_res);
    
    for i = 1:sample_res
        new_data = UpScale(data,i);
        s = log(new_data./mean(new_data))./log(length(new_data));
        gamma_max_lambdas = max(s);
        gammas_max_lam(i) = gamma_max_lambdas;
    end
    
    gamma_max = min(gammas_max_lam);
    gamma_min = min(log(data./mean(data))./log(length(data)));
    gamas = linspace(0,gamma_max,1000);
    
endfunction

function [gamas]=GammaValues(data,sample_res)
    
    data = data(1:2^sample_res);
    
    for i = 1:sample_res
        new_data = UpScale(data,i);
        s = log(new_data./mean(new_data))./log(length(new_data));
        gamma_max_lambdas = max(s);
        gammas_max_lam(i) = gamma_max_lambdas;
    end
    
    gamma_max = min(gammas_max_lam);
    gamma_min = min(log(data./mean(data))./log(length(data)));
    gamas = linspace(0,gamma_max,1000);
    
endfunction

function [data_res_bas] = UpScale(data,iterations)
    
    data_res_bas=[];
    defactor = 2;
    step = defactor^(iterations-1);
    newlen=floor(length(data)/step);
    
    data_res_bas=zeros(1,newlen);
    
    for i = 1:newlen
        data_res_bas(i) = sum(data((i-1)*step+1:i*step))./step; 
    end
    
endfunction

function [data_res_bas] = UpScale(data,iterations)
    
    data_res_bas=[];
    defactor = 2;
    step = defactor^(iterations-1);
    newlen=floor(length(data)/step);
    
    data_res_bas=zeros(1,newlen);
    
    for i = 1:newlen
        data_res_bas(i) = sum(data((i-1)*step+1:i*step))./step; 
    end
    
endfunction

function [p]=ProbaSup(data,threshold)
    nb_sup=0;
    nb_tot=length(data);
    for i=1:length(data)
        if data(i)>=threshold
            nb_sup=nb_sup+1;
        end
    end
    p=nb_sup/nb_tot;
    
endfunction

function [p]=ProbaSup(data,threshold)
    nb_sup=0;
    nb_tot=length(data);
    for i=1:length(data)
        if data(i)>=threshold
            nb_sup=nb_sup+1;
        end
    end
    p=nb_sup/nb_tot;
    
endfunction

function [c_gamma,y1,x1] = PDMS(data,gama,lambdas,rng)
    
    data = data(1:2^length(rng));
    
    for i=1:length(rng)
        if i==1
            data_lambda=data./mean(data);
        else
            data_lambda=UpScale(data,i);
            data_lambda = data_lambda./mean(data_lambda);
        end
        proba(:,i) = ProbaSup(data_lambda,(lambdas(i))^gama);
    end
    
    l1=1; l2=length(rng);    
    x1=log(lambdas(l1:l2)); 
    y1=log(proba(:,l1:l2));   
    [pente]=reglin(x1,y1,1);
    c_gamma = -pente(1);
    
endfunction

function [c_gamma,y1,x1] = PDMS(data,gama,lambdas,rng)
    
    data = data(1:2^length(rng));
    
    for i=1:length(rng)
        if i==1
            data_lambda=data./mean(data);
        else
            data_lambda=UpScale(data,i);
            data_lambda = data_lambda./mean(data_lambda);
        end
        proba(:,i) = ProbaSup(data_lambda,(lambdas(i))^gama);
    end
    
    l1=1; l2=length(rng);    
    x1=log(lambdas(l1:l2)); 
    y1=log(proba(:,l1:l2));   
    [pente]=reglin(x1,y1,1);
    c_gamma = -pente(1);
    
endfunction

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