inequality: Inequality Measures
In genieclust: Fast and Robust Hierarchical Clustering with Noise Points Detection

inequality

R Documentation

Inequality Measures

Description

gini_index() gives the normalised Gini index, bonferroni_index() implements the Bonferroni index, and devergottini_index() implements the De Vergottini index.

Usage

gini_index(x)

bonferroni_index(x)

devergottini_index(x)

Arguments

`x`	numeric vector of non-negative values

Details

These indices can be used to quantify the "inequality" of a numeric sample. They can be conceived as normalised measures of data dispersion. For constant vectors (perfect equity), the indices yield values of 0. Vectors with all elements but one equal to 0 (perfect inequality), are assigned scores of 1. They follow the Pigou-Dalton principle (are Schur-convex): setting x_i = x_i - h and x_j = x_j + h with h > 0 and x_i - h \geq x_j + h (taking from the "rich" and giving to the "poor") decreases the inequality

These indices have applications in economics, amongst others. The Genie clustering algorithm uses the Gini index as a measure of the inequality of cluster sizes.

The normalised Gini index is given by:

G(x_1,\dots,x_n) = \frac{ \sum_{i=1}^{n} (n-2i+1) x_{\sigma(n-i+1)} }{ (n-1) \sum_{i=1}^n x_i },

The normalised Bonferroni index is given by:

B(x_1,\dots,x_n) = \frac{ \sum_{i=1}^{n} (n-\sum_{j=1}^i \frac{n}{n-j+1}) x_{\sigma(n-i+1)} }{ (n-1) \sum_{i=1}^n x_i }.

The normalised De Vergottini index is given by:

V(x_1,\dots,x_n) = \frac{1}{\sum_{i=2}^n \frac{1}{i}} \left( \frac{ \sum_{i=1}^n \left( \sum_{j=i}^{n} \frac{1}{j}\right) x_{\sigma(n-i+1)} }{\sum_{i=1}^{n} x_i} - 1 \right).

Here, \sigma is an ordering permutation of (x_1,\dots,x_n).

Time complexity: O(n) for sorted (increasingly) data. Otherwise, the vector will be sorted.

Value

The value of the inequality index, a number in [0, 1].

Author(s)

Marek Gagolewski and other contributors

References

Bonferroni C., Elementi di Statistica Generale, Libreria Seber, Firenze, 1930.

Gagolewski M., Bartoszuk M., Cena A., Genie: A new, fast, and outlier-resistant hierarchical clustering algorithm, Information Sciences 363, 2016, pp. 8-23. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/j.ins.2016.05.003")}

Gini C., Variabilita e Mutabilita, Tipografia di Paolo Cuppini, Bologna, 1912.

Examples

gini_index(c(2, 2, 2, 2, 2))   # no inequality
gini_index(c(0, 0, 10, 0, 0))  # one has it all
gini_index(c(7, 0, 3, 0, 0))   # give to the poor, take away from the rich
gini_index(c(6, 0, 3, 1, 0))   # (a.k.a. Pigou-Dalton principle)
bonferroni_index(c(2, 2, 2, 2, 2))
bonferroni_index(c(0, 0, 10, 0, 0))
bonferroni_index(c(7, 0, 3, 0, 0))
bonferroni_index(c(6, 0, 3, 1, 0))
devergottini_index(c(2, 2, 2, 2, 2))
devergottini_index(c(0, 0, 10, 0, 0))
devergottini_index(c(7, 0, 3, 0, 0))
devergottini_index(c(6, 0, 3, 1, 0))

genieclust documentation built on Oct. 18, 2023, 5:08 p.m.