α-model
As we have seen in the previous exercise, the β-model is used to characterize the occurrence of a given field. However, in order to a have a complete characterization it is better to have the occurrence and the intensity of this field. As it has shown in this lecture, the α-model allows us to curry out this can of analysis.
Question
1 - According to the given elements in this lecture, give the expression of the singularities after n steps.
2 - What can be expect if we tend
3 - Generate different field with different values of the parameter and comment the obtained results.
Solution
Different steps of field generation using α-model cascade with C = 0.2 and α = 1.1 
Different steps of field generation using α-model cascade with C = 0.8 and α = 1.7
1 - After n steps :
with
2 - When
we obtain the β-model.
1
# -*- coding: utf-8 -*-2
"""3
@author: yacine.mezemate4
"""5
import scipy as sp
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from numpy import*
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import matplotlib.pyplot as plt
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from mpl_toolkits import mplot3d
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######################## AlphaModel Function ##################################12
def AlphaModelCascade(alpha, C, Lambda, nsteps, dim):
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alphaPrim = 1/(1-(1/alpha))
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GammaPlus = C/alpha
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GammaMinus = C/alphaPrim
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if dim ==1:
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resolution = sp.power(Lambda,nsteps)
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data = sp.ones((resolution))
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for n in range(nsteps):
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LambdaMu = sp.power(Lambda,n+1)
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Mu = random.uniform(0,1,LambdaMu)
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Mu = Mu / sp.mean(Mu)
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RangeList = range(0, resolution+1, resolution/LambdaMu)
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for i in range(len(RangeList)-1):
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rangeI = range(RangeList[i], RangeList[i+1])
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if Mu[i] > sp.power(LambdaMu, GammaPlus):
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data[rangeI]= Mu[i]*data[rangeI]
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elif Mu[i] < sp.power(LambdaMu, GammaMinus):
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data[rangeI] = Mu[i]*data[rangeI]
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data = data
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# Plot43
if n % 2:
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plt.subplot(nsteps,1,n)
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plt.plot(data)
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plt.title("Step number:" + str(n))
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frame = plt.gca()
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frame.axes.get_xaxis().set_ticks([])
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frame.axes.get_yaxis().set_ticks([])
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if dim == 2:
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resolution = sp.power(Lambda,nsteps)
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data = sp.ones((resolution, resolution))
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for n in range(nsteps):
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LambdaMu = sp.power(Lambda,n+1)
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Mu = random.uniform(0,1,LambdaMu)
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Mu = Mu / sp.mean(Mu)
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RangeList = range(0, resolution+1, resolution/LambdaMu)
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for i in range(len(RangeList)-1):
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for j in range(len(RangeList)-1):
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rangeI = range(RangeList[i], RangeList[i+1])
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rangeJ = range(RangeList[j], RangeList[j+1])
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if Mu[i] > sp.power(LambdaMu, GammaPlus):
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data[rangeI,rangeJ] = Mu[i]*data[rangeI,rangeJ]
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elif Mu[i] < sp.power(LambdaMu, GammaMinus):
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data[rangeI,rangeJ] = Mu[i]*data[rangeI,rangeJ]
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return data
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###############################################################################80
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############################# Parameters ######################################83
alpha = 1.7
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C = 0.8
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nsteps = 10
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Lambda = 2
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dim = 1
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############################# Cascade ######################################### 90
dataSim = AlphaModelCascade(alpha, C, Lambda, nsteps, dim)
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