# leibovici: Leibovici's entropy. In SpatEntropy: Spatial Entropy Measures

## Description

This function computes Leibovici's entropy according to a chosen distance d (with O'Neill's entropy as a special case) following Leibovici (2009), see also Altieri et al (2017). References can be found at `SpatEntropy`.

## Usage

 `1` ```leibovici(data, cell.size = 1, ccdist = cell.size, verbose = F) ```

## Arguments

 `data` If data are lattice, a data matrix, which can be numeric, factor, character, ... If the dataset is a point pattern, `data` is a `ppp` object. `cell.size` A single number. If data are lattice, the length of the side of each pixel. Default to 1. Ignored if data are points. `ccdist` A single number. The chosen distance for selecting couples of pixels/points within the observation area. Default to `cell.size`. `verbose` Logical. If `TRUE` an output is printed in order to follow the progress of the work (recommended for large dataset). Default set to `FALSE`.

## Details

This index is based on the transformed variable Z identifying couples of realizations of the variable of interest. A distance of interest is fixed, which in the case of O'Neill's entropy is the contiguity, i.e. sharing a border for lattice data. Then, all couples of realizations of the variable of interest lying at a distance smaller or equal to the distance of interest are counted, and their relative frequencies are used to compute the index with the traditional Shannon's formula. #'

H_L(Z)=∑ p(z_r|d) \log(1/p(z_r|d))

where z_r=(x_i, x_{i'}) is a generic couple of realizations of the study variable X. The conditioning on d means that only couples within a predefined distance are considered. The maximum value for Leibovici's entropy is \log(I^2) where I is the number of categories of the study variable X. The relative version of Leibovici's entropy is obtained by dividing the entropy value by its maximum, and is useful for comparison across datasets with a different number of categories. The function is able to work with grids containing missing data, specified as NA values. All NAs are ignored in the computation and only couples of non-NA observations are considered.

## Value

a list of three elements:

• `probabilities` - a table with the estimated probabilities (relative frequencies) for all couple categories;

• `leib` - Leibovici's entropy;

• `rel.leib` - Leibovici's relative entropy.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24``` ```#random grid data - high entropy data=matrix(sample(c("a","b","c"), 400, replace=TRUE), nrow=20) leibovici(data, cell.size=1, ccdist=2) #plot data plot(as.im(data, W=square(nrow(data))), col=gray(seq(1,0,l=length(unique(c(data))))), main="", ribbon=TRUE) #compact grid data - low entropy data=matrix(sort(sample(c("a","b","c"), 400, replace=TRUE)), nrow=20) leibovici(data, cell.size=1, ccdist=1.5) #plot data plot(as.im(data, W=square(nrow(data))), col=heat.colors(length(unique(c(data)))), main="", ribbon=TRUE) #point data data=ppp(x=runif(400), y=runif(400), window=square(1), marks=sample(1:4, 400, replace=TRUE)) leibovici(data, ccdist=0.1) #plot data plot(data) #nicer plot data.cat=data; marks(data.cat)=as.character(marks(data)) plot(data.cat, cols=1:length(unique(marks(data.cat))), main="", pch=16) ```

SpatEntropy documentation built on April 7, 2021, 5:09 p.m.