| inequality {genieclust} | 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. doi:10.1016/j.ins.2016.05.003
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, 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. 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))