batty {SpatEntropy}

Batty's entropy.

Description

This function computes Batty's spatial entropy, following Batty (1976), see also Altieri et al (2017 and following) (references are under the topic SpatEntropy).

Usage

batty(
  data,
  category = 1,
  cell.size = 1,
  partition = 10,
  win = NULL,
  rescale = T,
  plotout = T
)

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.

category

A single value matching the data category of interest for computing Batty's entropy. Default to 1. If the dataset is an unmarked point pattern, this argument must not be changed from the default. In the plot, only data belonging to the selected category are displayed.

cell.size

A single number or a vector of length two, only needed if data are lattice. It gives the length of the side of each pixel; if the pixel is rectangular, the first number gives the horizontal side and the second number gives the vertical side. Default to 1. Ignored if data are points.

partition

Input defining the partition into subareas. If an integer, it defines the number of sub-areas that are randomly generated by areapart; if a two column matrix with coordinates, they are the centroids of the subareas built by areapart. Alternatively, it can be the output of areapart, a tess object built by the user, a list object with arguments tiles, i.e. a list of owin objects defining the partition, and n, the number of subareas. Lastly, it can be an im object, i.e. a factor- or character-valued pixel image on the same observation window as the data, so that the partition is defined according to the values of the image. The default option is partition=areapart(data, G=10), which generates 10 random sub-areas.

win

Optional, the observation area given as a owin object. If data are a point pattern ppp object, this argument is ignored and the observation area is extracted from the object. If data are given as a matrix, the area should be specified; the default is a rectangle with x range from 0 to the number of columns of the data, and y range from 0 to the number of rows of the data.

rescale

Logical. Default to TRUE, checks whether the size of the observation area or of any of the sub-areas is smaller than 1. If so, computational issues may arise due to negative logarithms, therefore the function automatically performs a rescaling of all sub-areas, computes the entropy over the rescaled area and then transforms it back to the entropy of the original dataset. In such case, a warning is produced to make the user aware of such operation.

plotout

Logical. Default to TRUE, produces an informative plot as part of the function output.

Details

Batty's spatial entropy measures the heterogeneity in the spatial distribution of a phenomenon of interest, with regard to an area partition. It is high when the phenomenon is equally intense over the sub-areas, and low when it concentrates in one or few sub-areas. This function allows to compute Batty's entropy as

H_B=\sum p_g \log(T_g/p_g)

where p_g is the probability of occurrence of the phenomenon over sub-area g, and T_g is the sub-area size. When data are categorical, the phenomenon of interest corresponds to one category, which must be specified. If data are an unmarked point pattern, a fake mark vector is be created with the same category for all points. For comparison purposes, the relative version of Batty's entropy is also returned, i.e. Batty's entropy divided by its maximum \log(\sum T_g). Note that when the total observation area is 1, then \log(\sum T_g)=0, therefore in that case during the computation all T_gs are multiplied by 100 and a warning is produced. The function is able to work with grids containing missing data, specified as NA values. All NAs are ignored in the computation.

Value

A list of five elements:

• batty Batty's entropy

• range The theoretical range of Batty's entropy

• rel.batty Batty's entropy divided by \log(\sum Tg) for comparison across observation areas.

• areas a dataframe giving, for each sub-area of the partition, the absolute and relative frequency of the points/pixels of interest, the sub-area size and the intensity defined as pg/Tg

• area.tess a tess object with the area partition

Moreover, a plot is produced showing the data and the area partition.

Examples

#LATTICE DATA

data=matrix((sample(c("a","b","c"), 100, replace=TRUE)), nrow=10)
batty.entropy=batty(data, category="a")

#POINT DATA

#unmarked pp
data=ppp(x=runif(100, 0, 10), y=runif(100, 0, 10), window=square(10))
batty.entropy=batty(data)

#smaller window so that some areas' size are smaller than 1
data=ppp(x=runif(100, 0, 3), y=runif(100, 0, 3), window=square(3))
batty.entropy=batty(data)

#marked pp
data=ppp(x=runif(100, 0, 10), y=runif(100, 0, 10), window=square(10),
         marks=(sample(1:5, 100, replace=TRUE)))
plot(data) #see ?plot.ppp for options
#if you want to compute the entropy on all points
batty.entropy=batty(unmark(data))
#if you want to compute the entropy on a category, say 3
batty.entropy=batty(data, category=3)