Introduction to complex systems

Lesson

Introduction
Complex system
Environmental complex system
Water cycle
Atmospheric motion
Hydrodynamic
Ecology
Karst
Exercises
- Quiz
- Quiz: Vapor pressure calculation
- Quiz: Precipitation
- Quiz: Precipitation measurement
- Quiz: Probability distribution function (PDF)
To go further
Conclusion

Probability distribution function (PDF)

The tail of a PDF is linked to its kurtosis. This kurtosis gives the concentration of values around the central value of the law and thus the concentration for extreme values, which means, far from the average (mean). In this exercise we will compare the tail of empirical probability distribution and a normal distribution. Using

data.odt

Question

Using the data set, calculate:

The mean value

The variance

The standard deviation

Hint

$\text{Lets X a random variable}:\\ \quad \\ \text{The mean is}:\\ E[X] = \mu \\ \quad \\ \text{The variance is }: \\ \sigma^2 = \frac{\sum(X-\mu)^2}{N}, \text{Where} \quad N \quad \text{is the number scores} \quad \\ \quad \\ \text{The standar deviation}: \\ std = \sqrt{\sigma^2}$

Question

Using the given data set , compare the empirical PDF and Normal distribution

Hint

The empirical pdf is calculate using Weibull distribution

Solution

Python script

@author: yacine.mezemate
"""
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.mlab as mlab
#Import data
data = np.genfromtxt("data.txt", delimiter= '')
data = np.diff(data) # use the difference
# Statistics
mu = np.mean(data) # mean
Var = np.var(data) # variance
std = np.sqrt(Var) # standard deviation
# Calculate and plot the Empirical PDF using Weibull distribution
neg = np.sort(data[data<0]) # sort negative values
pos = np.sort(data[data>0]) # sort positive values
P_neg = np.arange(len(neg), dtype = np.double)/len(neg) # probability of negative values
P_pos = np.arange(len(pos), dtype = np.double)/len(pos) # probability of positive values
plt.plot(neg, np.log(P_neg), marker='+', label ="Empirical Estimation") # plot PDF
# Calculate and plot Normal PDF
mn = np.min(data) # min value
mx = np.max(data) # max value
x = np.linspace(mn, 0, 100) # bins number
Gauss=np.log(mlab.normpdf(x,mu,std)) # Normal distribution
plt.plot(x,Gauss, label="Normal distribution") # plot PDF
# Plot informations
plt.legend(loc='down left')
plt.title("Probability distribution", fontsize = 18)
plt.ylabel("$\log(Pr(\Delta v))$", fontsize=14)
plt.xlabel("$\Delta v$", fontsize=14)
plt.show()

@author: yacine.mezemate
"""
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.mlab as mlab
#Import data
data = np.genfromtxt("data.txt", delimiter= '')
data = np.diff(data) # use the difference
# Statistics
mu = np.mean(data) # mean
Var = np.var(data) # variance
std = np.sqrt(Var) # standard deviation
# Calculate and plot the Empirical PDF using Weibull distribution
neg = np.sort(data[data<0]) # sort negative values
pos = np.sort(data[data>0]) # sort positive values
P_neg = np.arange(len(neg), dtype = np.double)/len(neg) # probability of negative values
P_pos = np.arange(len(pos), dtype = np.double)/len(pos) # probability of positive values
plt.plot(neg, np.log(P_neg), marker='+', label ="Empirical Estimation") # plot PDF
# Calculate and plot Normal PDF
mn = np.min(data) # min value
mx = np.max(data) # max value
x = np.linspace(mn, 0, 100) # bins number
Gauss=np.log(mlab.normpdf(x,mu,std)) # Normal distribution
plt.plot(x,Gauss, label="Normal distribution") # plot PDF
# Plot informations
plt.legend(loc='down left')
plt.title("Probability distribution", fontsize = 18)
plt.ylabel("$\log(Pr(\Delta v))$", fontsize=14)
plt.xlabel("$\Delta v$", fontsize=14)
plt.show()

Comparison between empirical probability distribution and a normal distribution for geophysical field

The plot is logarithmic so as to emphasis the heavy tail of the distribution. The plot shows that the normal distribution does not fit the empirical one. In complex system such as in geophysics, extreme values can not be detected using a Gaussian distribution.

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