karlstrom {SpatEntropy}

Karlstrom and Ceccato's entropy.

Description

This function computes Karlstrom and Ceccato's spatial entropy for a chosen neighbourhood distance, following Karlstrom and Ceccato (2002), see also Altieri et al (2017) and following works (references are under the topic SpatEntropy).

Usage

karlstrom(
  data,
  category = 1,
  cell.size = 1,
  partition = 10,
  win = NULL,
  neigh = 4,
  method = "number",
  plotout = T
)

battyLISA(
  data,
  category = 1,
  cell.size = 1,
  partition = 10,
  win = NULL,
  neigh = 4,
  method = "number",
  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.

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 and the area is not 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.

neigh

A single number. It can be either the number of neighbours for each sub-area (including the area itself). or the Euclidean distance to define which sub-areas are neighbours, based on their centroids. Default to 4 neighbours.

method

Character, it guides the interpretation of neigh. Either "number" (the default) or "distance".

plotout

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

Details

Karlstrom and Ceccato's spatial entropy measures the heterogeneity in the spatial distribution of a phenomenon of interest, with regard to an area partition and accounting for the neighbourhood. It is similar to Batty's entropy (see batty) discarding the sub-area size, with the difference that the probability of occurrence of the phenomenon over area g is actually a weighted sum of the neighbouring probabilities.

H_{KC}=\sum p_g \log(1/ \tilde{p}_g)

where p_g is the probability of occurrence of the phenomenon over sub-area g, and \tilde{p}_g is the averaged probability over the neighbouring areas (including the g-th area itself). 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 Karlstrom and Ceccato's entropy is also returned, i.e. Karlstrom and Ceccato's entropy divided by its maximum log(number of sub-areas). 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:

• karlstrom Karlstrom and Ceccato's entropy

• range The theoretical range of Karlstrom and Ceccato's entropy

• rel.karl Karlstrom and Ceccato's entropy divided by \log(G) (number og sub-areas) for comparison across observation areas.

• areas a dataframe giving, for each sub-area, the absolute and relative frequency of the points/pixels of interest, the weighted probabilities of the neighbours and the sub-area size

• 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)
KC.entropy=karlstrom(data, category="a")
KC.entropy=karlstrom(data, category="a", neigh=3.5, method="distance")
##to plot
data.binary=matrix(as.numeric(data=="a"), nrow(data))
plot(as.im(data.binary, W=KC.entropy$area.tess$window), main="",
     col=grDevices::gray(seq(1,0,l=length(unique(c(data.binary))))), ribbon=FALSE)
plot(KC.entropy$area.tess, add=TRUE, border=2)

#POINT DATA

#unmarked pp
data=ppp(x=runif(100, 0, 10), y=runif(100, 0, 10), window=square(10))
KC.entropy=karlstrom(data)
##to plot
plot(data)
plot(KC.entropy$area.tess, add=TRUE, border=2)

#marked pp
data=ppp(x=runif(100, 0, 10), y=runif(100, 0, 10), window=square(10),
         marks=(sample(1:5, 100, replace=TRUE)))
#if you want to compute the entropy on all points
KC.entropy=karlstrom(unmark(data))
#if you want to compute the entropy on a category, say 3
KC.entropy=karlstrom(data, category=3)
##to plot using the selected category
ind=which(spatstat.geom::marks(data)==3)
data.binary=unmark(data[ind])
plot(data.binary)
plot(KC.entropy$area.tess, add=TRUE, border=2)