Gini index

The Gini index, named for Italian statistician Corrado Gini (1884-1965), was developed about 1912.

It is used as a measure of income inequality. The value ranges from 0.0 to 1.0.

0.0 represents perfect equality
1.0 represents perfect inequality.

Interestingly, Gina was developing his ideas at the same time as Pareto in Italy. And, in 1922, Benito Mussolini took over Italy.

The Pareto chart is named after Vilfredo Pareto, 1848-1923, an Italian economist.

Pareto made the observation that 20% of the population owned 80% of the wealth.

This idea has been called Pareto's principle.

A Pareto chart is a combination chart that shows an individual count or frequency using a column bar chart, ordered in decreasing order from left to right, and a cumulative distribution using a line chart.

Another Italian statistician, Bruno de Finetti (1902-1985) did important work in the field of "infinite measures" (Dirichlet distribution, stick breaking, etc.) that forms the theoretical basis of some aspects of LDA (Latent Dirichlet Allocation) as a generative Bayesian method.

A common way to get a Dirichlet process for "infinite measures" is via a simple stick breaking algorithm.

Another way is the Polya urn scheme.

Break a stick randomly in half. Use one part. Then break the other part in half. Use one part. Continue until the desired number of values are obtained.

Back to the Italian statistician Gini.

Here is a graphical depiction of the Gini coefficient from the Wikipedia page on the "Gini coefficient".

The Gini index can be represented graphically with the Lorenz curve.

Formula is from the Wikipedia page on the Gini Coefficient.

Here is the Python code.

import matplotlib.pyplot as plt
import rsPlot

import numpy as np

# distribution data - wealth, etc.
dataList1 = [0, 1, 5, 10, 100]
dataArray1 = np.array(dataList1)

def giniIndexDo1(dataArray1):
	n1 = dataArray1.size
	c1 = 2.0 / n1
	indexArray1 = np.arange(1, n1+ 1)
	sum1 = (indexArray1 * dataArray1).sum()
	total1 = dataArray1.sum()
	c2 = (n1+ 1) / n1
	return c1 * (sum1 / total1) - c2


def lorentzDo1(dataArray1):
	sum1 = dataArray1.sum()
	sumArray1 = dataArray1.cumsum() / sum1
	return sumArray1

plot1 = rsPlot.plot("dt53-1")
curveArray1 = lorentzDo1(dataArray1)
curveLen1 = len(curveArray1)
giniIndex1 = giniIndexDo1(dataArray1)
print("Gini coefficient and Lorentz curve:")
print("")
print("Data: {0:s}".format(str(dataList1)))
print("")
print("Gini index: {0:1.2f}".format(giniIndex1))
print("")
print("Curve: {0:s}".format(str(curveArray1)))
thick1 = 6
plt.plot(np.linspace(0.0, 1.0, curveLen1), curveArray1,linewidth=thick1)
plt.plot([0,1], [0,1],linewidth=thick1)

plot1.save()

Here is the output of the Python code.

(module rsPlot imported)
Gini coefficient and Lorentz curve:

Data: [0, 1, 5, 10, 100]

Gini index: 0.72

Curve: [0.         0.00862069 0.05172414 0.13793103 1.        ]
	./dt53-1-01.png : 557 x 414 , 24,275 bytes