inequality: Inequality Measures

In genieclust: Genie: Fast and Robust Hierarchical Clustering

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 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).

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.

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

See Also

The official online manual of genieclust at https://genieclust.gagolewski.com/

Gagolewski, M., genieclust: Fast and robust hierarchical clustering, SoftwareX 15:100722, 2021, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/j.softx.2021.100722")}

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. the 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 Feb. 24, 2026, 1:06 a.m.