Entropy {wINEQ} | R Documentation |
Generalized entropy index
Description
Computes generalized entropy index of a given variable taking into account weights.
Usage
Entropy(X, W = rep(1, length(X)), power = 0.5, zeroes = "include")
Arguments
X |
is a data vector |
W |
is a vector of weights |
power |
is a entropy parameter |
zeroes |
defines what to do with zeroes in the data vector. Possible options are "remove" and "include". See Details for more. |
Details
Entropy coefficient with respect to parameter \alpha
is equal to Theil_L(X,W) whenever \alpha=0
,
is equal to Theil_T(X,W) whenever \alpha=1
, and whenever \alpha \in (0,1)
we have
GE(\alpha) = \frac{1}{\alpha(\alpha-1)W}\sum_{i=1}^{n}w_{i}\left(\left(\frac{x_{i}}{\mu}\right)^\alpha-1\right)
where W
is a sum of weights and \mu
is the arithmetic mean of x_{1},...,x_{n}
.
Entropy coefficient is not well-defined for data vector with zero values whenever parameter is zero or one.
In such case, entropy index coincides with the definition of Theil L index and Theil T index, respectively, and entropy index is calculated with corresponding Theil function.
Theil L always removes zeroes. Theil T enables two ways to deal with zeroes by parameter zeroes.
Option "remove" discard these X's and corresponding weights. Works for power>0.
Option "include" puts 0\log{0=}0
due to limiting property of p\log{p}
in zero preserving zero value in dataset. It is valid only for Theil T index, that is power=0.
Value
The value of generalized entropy index
References
Shorrocks A. F.: (1980) The Class of Additively Decomposable Inequality Measures. Econometrica
Pielou E.C.: (1966) The measurement of diversity in different types of biological collections. Journal of Theoretical Biology
Examples
# Compare weighted and unweighted result
X=1:10
W=1:10
Entropy(X)
Entropy(X,W)
data(Tourism)
# Generalized entropy index for Total expenditure with sample weights
X=Tourism$Total_expenditure
W=Tourism$Sample_weight
Entropy(X,W)