This vignette explains what a weighted co-occurrence matrix
(wecoma) representation is and how to calculate it using the
comat package. If you do not know what a co-occurrence
matrix is, it could be worth to read the first
package vignette first. The examples below assume the
comat package is attached, and the
raster_x and raster_w datasets are loaded:
The raster_x object is a matrix with three rows and
columns with values of 1, 2, and 3.
We can imagine that the value of 1 (blueish color) represents population A, the value of 2 (dark green) is population B, and the value of 3 (light green) represents population C.
The raster_w object is also a matrix of the same
dimensions. It has values between 2 and 9.
This object is different from the first one, as it does not represent categories. Its role is to provide some weights to the previous raster. We can think of it as a number of occurrences in each population.
The weighted co-occurrence matrix (wecoma) representation is a modification of the co-occurrence matrix (coma). In the co-occurrence matrix, each adjacency contributes to the output with the constant value 1. In the weighted co-occurrence matrix, on the other hand, each adjacency contributes to the output based on the values from the weight matrix. The contributed value is calculated as the average of the weights in the two adjacent cells.
We can use the get_wecoma() function to calculate this
weighted co-occurrence matrix (wecoma) representation:
In this representation, we do not count the neighbors but sum the
contributed values from the weight matrix. The smallest value (5)
represents the relation between the adjacent cells between the first and
the second category. This is due to the relatively small values of the
neighborhooding cells of these classes, but also because there is only
one case of adjacent cells of these classes. Central left cell (blueish,
category 1) has a value of 6, and the bottom left cell (dark green,
category 2) has a value of 4. The output value, 5, is an average of the
two adjacent weights. On the other hand, a light green region has the
largest values in the weight matrix. Therefore, the output of the
get_wecoma() function has the largest value (49) for the
relation between the adjacent cells of the third category.
This function allows for some parametrization using additional
arguments - fun and na_action. The
fun argument selects the function to calculate values from
adjacent cells to contribute to the output matrix. It has three possible
options: "mean" - calculate average values from adjacent
cells of the weight matrix, "geometric_mean" - calculate
geometric mean values from adjacent cells of the weight matrix, or
"focal" assign a value from the focal cell. The
na_action argument decides on how to behave in the presence
of missing values in the weight matrix. The default,
"replace", replaces missing values with 0,
"omit" does not use cells with missing values, and
"keep" keeps missing values.
get_wecoma(raster_x, raster_w, fun = "focal", na_action = "omit")
#> 1 2 3
#> 1 12 6 10
#> 2 4 12 16
#> 3 17 13 49Similarly to the co-occurrence matrix (coma), it is possible
to convert wecoma to its 1D representation. This new form is
called a weighted co-occurrence vector (wecove), and can be
created using the get_wecove() function, which accepts an
output of get_wecoma():
my_wecoma = get_wecoma(raster_x, raster_w)
get_wecove(my_wecoma, normalization = "pdf")
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7]
#> [1,] 0.08633094 0.03597122 0.0971223 0.03597122 0.08633094 0.1043165 0.0971223
#> [,8] [,9]
#> [1,] 0.1043165 0.352518You can see the weighted co-occurrence matrix (wecoma) concept, there described as an exposure matrix, in action in the vignettes of the raceland package (Nowosad, Dmowska, and Stepinski, 2020):