fHill {OnomasticDiversity}

Calculate the Hill's diversity numbers

Description

This function obtains the Hill's diversity numbers introduced by M. O. Hill. It is a method for quantifying species biodiversity that can be adapted to the context of onomastic.

Usage

fHill(x, k, n, location, lambda)

Arguments

x

dataframe of the data values for each species.

k

name of a variable which represents absolute frequency for each species.

n

name of a variable which represents total number of individuals.

location

represents the grouping element.

lambda

free parameter.

Details

For a community i, the Hill's diversity numbers are defined by the expression J(\lambda) = \left(\sum \limits_{k\in S_i} p_{ki}^\lambda\right)^{\frac{1}{1-\lambda}} with the restriction \lambda \geq 0 where p_{ki} represents the relative frequency of species k and S_i are all species at the community, species richness, and \lambda is a free parameter. (This is equivalent to the exponential of Renyi's generalised entropy). The Renyi entropy of order \lambda, where \lambda \geq 0 and \lambda \neq 1, is defined as \mathrm{H}_{\lambda}(X)=\frac{1}{1-\lambda} \log \left(\sum \limits_{i=1}^{n} p_{i}^{\lambda}\right) Here, X is a discrete random variable with possible outcomes in the set \mathcal{A}=\left\{x_{1}, x_{2}, \ldots, x_{n}\right\} and corresponding probabilities p_{i} \doteq \operatorname{Pr}\left(X=x_{i}\right) for i=1, \ldots, n. The logarithm is conventionally taken to be base 2, especially in the context of information theory where bits are used. If the probabilities are p_{i}=1 / n for all i=1, \ldots, n, then all the Renyi entropies of the distribution are equal: \mathrm{H}_{\lambda}(X)=\log n. In general, for all discrete random variables X, \mathrm{H}_{\lambda}(X) is a non-increasing function in \lambda..

Particular cases of \lambda values: \lambda = 0, J(0)=S_i, it corresponds species richness; \lambda = 1, J(1)=e^{H_{t}}, it corresponds the exponential of Shannon's entropy; and \lambda = 2, J(2)= D_{S_i}, it corresponds the 'inverse' Simpson index.

In onomastic context, p_{ki} denotes the relative frequency of surname k in region (\approx community diversity context) i and S_i are all surnames in region i.

Value

A dataframe containing the following components:

location

represents the grouping element, for example the communities / regions.

hill

the value of the Hill's diversity index.

Author(s)

Maria Jose Ginzo Villamayor

References

Hill, M. O. (1973). Diversity and Evenness: a unifying notation and its consequences. Ecology, 54, 427–32.

See Also

fCressieRead.

Examples

data(surnamesgal14)
result = fHill (x= surnamesgal14, k="number", n="population",
location  = "muni", lambda= 0)
result

data(namesmengal16)
result = fHill (x= namesmengal16, k="number", n="population",
location  = "muni", lambda= 0)
result

data(nameswomengal16)
result = fHill (x= nameswomengal16, k="number", n="population",
location  = "muni", lambda= 0)
result