shannonZ {SpatEntropy}

Shannon's entropy of the transformed variable Z.

Description

This function computes Shannon's entropy of variable Z, where Z identifies pairs of realizations of the variable of interest.

Usage

shannonZ(data)

Arguments

data

A data matrix or vector, can be numeric, factor, character, ... Alternatively, a marked ppp object.

Details

Many spatial entropy indices are based on the trasformation Z of the study variable, i.e. on pairs (unordered couples) of realizations of the variable of interest. 'Unordered couples' means that the relative spatial location is irrelevant, i.e. that a couple where category i occurs at the left of category j is identical to a couple where category j occurs at the left of category i. When all possible pairs occurring within the observation areas are considered, Shannon's entropy of the variable Z may be computed as

H(Z)=\sum p(z_r)\log(1/p(z_r))

where p(z_r) is the probability of the r-th pair of realizations, here estimated by its relative frequency. Shannon's entropy of Z varies between 0 and \log(R), R=binom(n+1,2) (where n is the number of observations) being the number of possible pairs of categories of the variable under study. The function is able to work with lattice data with missing data, as long as they are specified as NAs: missing data are ignored in the computations.

Value

a list of three elements:

• shannZ Shannon's entropy of Z

• range The theoretical range of Shannon's entropy of Z, from 0 to \log(R)

• rel.shannZ Shannon's relative entropy of Z

• probabilities a table with absolute frequencies and estimated probabilities (relative frequencies) for all Z categories (data pairs)

Examples

#NON SPATIAL DATA
shannonZ(sample(1:5, 50, replace=TRUE))

#POINT DATA
data.pp=runifpoint(100, win=square(10))
marks(data.pp)=sample(c("a","b","c"), 100, replace=TRUE)
shannonZ(marks(data.pp))

#LATTICE DATA
data.lat=matrix(sample(c("a","b","c"), 100, replace=TRUE), nrow=10)
shannonZ(data.lat)