Inequality Measures

Introduction

Is important understand the uses of inequality Measures to stuy the widespread of a disease over a territory, the public

Gini index

Entropy

Idea

Is a concept shanon offert a perspective… comes from information theory…

\(p_{i}\) the probability of occurnce of the \(i\) event, and \(h(p_{i})\) is the value of of get the information that \(i\) has been occur before than other one known, not have value knonw with anticipation that occur a likyely event that everyone wait.

if we have N events or in a

\begin{equation} E = \sum p_{i} h(p_{i}) \end{equation}

The entropy reach the maximun value with all the envets have the same probability.

# Bag model

We can think to \(p_{i}\) as the probability of drawm from a bag with ‘pesos’ one ‘peso’ what is the probability of the ‘peso’ belong to the \(i\) person or unity.

\begin{equation} p_{i} = s_{i} = \frac{ x_{i}}{N \bar{y}} \end{equation}

the theil index will be the difference between the maximun theil possible and the real.

\begin{equation} T = \sum \frac{1}{N} h(\frac{1}{N}) - \sum s_{i}h(s_{i}) \end{equation}

if we assmume that \(h()\) is \(-ln()\) we can take adwhile of that some properties remember that a decresing function is good, and some logarithmic properties…

applying to the equations we have:

\begin{equation} T = \sum \frac{1}{N} -ln(\frac{1}{N}) - \sum s_{i} -ln(s_{i}) \end{equation}

changing the value function we have:

\begin{equation} -ln(\frac{1}{N}) - \sum (\frac{x_{i}}{N \bar{y}})- ln(\frac{x_{i}}{N\bar{y}}) \end{equation}

applying logartimic properties we have:

\begin{equation} T = -ln(\frac{1}{N}) - \sum \frac{x_{i}}{N\bar{y}} - ln(\frac{x_{i}}{\bar{y}}) - ln(\frac{1}{N}) \end{equation}

therefore:

\begin{equation} T = \frac{1}{N} \sum \frac{x_{i}}{\bar{y}}ln(\frac{x_{i}}{\bar{y}}) \end{equation}

#Restricted to \((0-1)\) theil index

Another point of view in SEN:

remeber that \(\sum_{i}^{n} x_{i} = 1\). \(h(x) = ln(1/x) \) while \(x\) be more improbably more value have \(x = \sum x_{i} log( \frac{1}{x_{i}} )\) therefore with \(x_{i} = \frac{1}{n}\) then \(H(x) = log(n)\), remeber that we can write \(log(n) = \sum x_{i} log(n)\), thus

\begin{equation} T = \sum x_{i}log(n) - \sum x_{i}log(\frac{1}{x_{i}}) \end{equation}

We can acotate Theil index:

\begin{equation} \frac{T}{log(n)} = \frac{log(n) - H(x)}{log(n)} \end{equation}

if \(H(x)\) is evenely distributed then \(H(x) = log(n)\) if one person concentrate all then \(H(x) = 0\). then wen a person concentrate all income \(1*log(1/1) =0\).

therefore our index will be \(1\) to perfect inequality and zero by uniform distribution.

code:

def theil(array):
  Y = sum(array)
  N = len(array)
  relative = [y/Y for y in array]
  Theil = sum([y * math.log(y * N) for y in relative])/math.log(N)
  return Theil

Theil in pandas Theil pandas

# Discrete gini

\begin{equation} G = \frac{0.5 - B}{0.5} \end{equation}

Now we can see that

\begin{equation} G = 1 - 2B \end{equation}

\begin{equation} B = \sum \frac{(y_{i-1} + y_{i})}{2}*(N_{i} - N_{i-1}) \end{equation}

the gini could be calculated with:

\begin{equation} G = 1 - \sum (y_{i-1} + y_{i}) * (N_{i} - N_{i-1}) \end{equation}

code:

def gini(array):
    def cumsum(array):
        output = [0]
        counter = 0
        total = sum(array)
        for e in array:
        counter = counter + e
        output.append(counter/total)
        return output
    array.sort()
    n = [i/len(array) for i in range(0,len(array)+1)]
    acum_array = cumsum(array)
    area = 0
    for i in range(1,len(acum_array)):
        area = area +  (acum_array[i] + acum_array[i-1]) * (n[i] - n[i-1])
    return 1  - area

codemaster

Theil index

Theil index was defined by the economist Theil, and is based in information theory….

This lectuer is very important to constrait the gini index to a value.

The Theil index will be defined as:

the equations are important \(h(X)=\sum_{i=1}^{N}x\)

\[H(x) = \sum_{i=1}^{N}\]

This is a excelent renderization, however never is enough..

Hamming distance

Hamming is a important distance

Sometimes we need establish..

K-means

The ideas of similarity also been linked to distance, in a simple way, the way in which measure the distance is a fucking problem, given there are a lot of fucking distances…

Inequality Measures

Introduction

Gini index

Entropy

Idea

codemaster

Theil index

Hamming distance

K-means

Euclidean

Manhattan

Another distance

Kullback - Leiber or KL divergence