Boltzmann entropy (also called configurational entropy) has been
recently adopted to analyze entropy of landscape gradients (Gao et
al. (2017), Gao et al. (2018)). The goal of belg is to
provide an efficient C++ implementation of this method in R. It also
extend the original idea by allowing calculations on data with missing
values.
Basic example
library(raster)
library(belg)
complex_land = raster(system.file("raster/complex_land.tif", package = "belg"))
simple_land = raster(system.file("raster/simple_land.tif", package = "belg"))
Let’s take two small rasters - complex_land representing
a complex landscape and simple_land representing a simple
landscape.
#> Assigning a new crs. Use 'project' to transform a SpatRaster to a new crs
#> Assigning a new crs. Use 'project' to transform a SpatRaster to a new crs
#> Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
#> ℹ Please use `linewidth` instead.
#> Assigning a new crs. Use 'project' to transform a SpatRaster to a new crs
#> Assigning a new crs. Use 'project' to transform a SpatRaster to a new crs
The main function in this package, get_boltzmann(),
calculates the Boltzmann entropy of a landscape gradient:
get_boltzmann(complex_land, method = "hierarchy")
#> Assigning a new crs. Use 'project' to transform a SpatRaster to a new crs
#> Assigning a new crs. Use 'project' to transform a SpatRaster to a new crs
#> [1] 191.1567
get_boltzmann(simple_land, method = "hierarchy")
#> Assigning a new crs. Use 'project' to transform a SpatRaster to a new crs
#> Assigning a new crs. Use 'project' to transform a SpatRaster to a new crs
#> [1] 104.8581
The results, unsurprisingly, showed that the complex landscape has a
larger value of the Boltzmann entropy than the simple one.
The get_boltzmann() function accepts a
RasterLayer, RasterStack,
RasterBrick, matrix, or array
object as an input. As a default, it uses a logarithm of base 10
(log10), however log and log2 are
also available options for the base argument.
get_boltzmann(complex_land, method = "hierarchy") # log10
#> Assigning a new crs. Use 'project' to transform a SpatRaster to a new crs
#> Assigning a new crs. Use 'project' to transform a SpatRaster to a new crs
#> [1] 191.1567
get_boltzmann(complex_land, method = "hierarchy", base = "log")
#> Assigning a new crs. Use 'project' to transform a SpatRaster to a new crs
#> Assigning a new crs. Use 'project' to transform a SpatRaster to a new crs
#> [1] 440.1546
get_boltzmann(complex_land, method = "hierarchy", base = "log2")
#> Assigning a new crs. Use 'project' to transform a SpatRaster to a new crs
#> Assigning a new crs. Use 'project' to transform a SpatRaster to a new crs
#> [1] 635.0089
It also allows for calculation of the relative (the
relative argument equal to TRUE) and absolute
Boltzmann entropy of a landscape gradient.
Relative Boltzmann entropy of a landscape gradient
The main idea behind the Boltzmann entropy of a landscape gradient is
to calculate an entropy in a sliding window of 2 x 2 pixels. The
relative configurational entropy is a sum of entropies for all windows
of the original data.
get_boltzmann(complex_land, method = "hierarchy", relative = TRUE)
#> Assigning a new crs. Use 'project' to transform a SpatRaster to a new crs
#> Assigning a new crs. Use 'project' to transform a SpatRaster to a new crs
#> [1] 88.55451
Absolute Boltzmann entropy of a landscape gradient
It is possible to calculate an average value for each sliding window
of 2 x 2 pixels and therefore create a resampled version of the original
dataset:
#> Assigning a new crs. Use 'project' to transform a SpatRaster to a new crs
#> Assigning a new crs. Use 'project' to transform a SpatRaster to a new crs
The absolute configurational entropy is a sum of relative
configurational entropies for all levels, starting from the original
data to the resampled dataset with at least two rows or columns.
get_boltzmann(complex_land, method = "hierarchy", relative = FALSE)
#> Assigning a new crs. Use 'project' to transform a SpatRaster to a new crs
#> Assigning a new crs. Use 'project' to transform a SpatRaster to a new crs
#> [1] 191.1567
Calculation of the configurational entropy in a sliding window
Determining the number of microstates belonging to a defined
macrostate in a crucial concept for calculation of the configurational
entropy. We explore this topic using five different cases of 2 x 2
windows:
win_1 = raster(matrix(c(1, 3, 3, 4), ncol = 2))
win_2 = raster(matrix(c(1, 3, 3, NA), ncol = 2))
win_3 = raster(matrix(c(1, 3, NA, NA), ncol = 2))
win_4 = raster(matrix(c(1, NA, NA, NA), ncol = 2))
win_5 = raster(matrix(c(NA, NA, NA, NA), ncol = 2))
Data without missing values
The configurational entropy for data without missing values is
calculated using the analytical method by Gao et al. (2018).
Twenty-four different microstate are possible in the above case. The
common (base 10) logarithm of 24 is equal to 1.380211. We can compare
this result to the get_boltzmann() output:
get_boltzmann(win_1, method = "hierarchy")
#> [1] 1.380211
The generalized (resampled) version of this window has one value,
3, which is a rounded average of the four original
values.
Data with missing values
The papers of Gao et al. (2017, 2018) only considered data without
missing values. However, the belg package provides a
modification allowing for calculation also for data with missing values.
Cells with NA are not considered when calculating
microstates.
For example, three microstates are possible for the above case:
The common (base 10) logarithm of 3 is equal to 0.477121.
get_boltzmann(win_2, method = "hierarchy")
#> [1] 0.6361617
The generalized (resampled) version of this window is 2.
The third window has two combinations. The common logarithm of 2 is
equal to 0.30103.
get_boltzmann(win_3, method = "hierarchy")
#> [1] 0.60206
The generalized (resampled) version of this window is also 2.
The fourth window has only one microstate, therefore its common
logarithm equals to 0.
get_boltzmann(win_4, method = "hierarchy")
#> [1] 0
The generalized (resampled) version of this window is the same as
only existing value - 1.
Finally, the last window consists of four missing values. In these
cases, the configurational entropy is zero.
get_boltzmann(win_5, method = "hierarchy")
#> [1] NaN
Importantly, the generalized version of this window is represented by
NA.
References
- Gao, Peichao, Hong Zhang, and Zhilin Li. “An efficient analytical
method for computing the Boltzmann entropy of a landscape gradient.”
Transactions in GIS (2018).
- Gao, Peichao, Hong Zhang, and Zhilin Li. “A hierarchy-based solution
to calculate the configurational entropy of landscape gradients.”
Landscape Ecology 32(6) (2017): 1133-1146.
These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.