Author: Jakub Nowosad
Date: 2022-08-17

# sabre: Spatial Association Between REgionalizations - raster data

#### Jakub Nowosad

#### 2022-08-17

The sabre (Spatial Association Between REgionalizations) is an R package for calculating a degree of spatial association between regionalizations or categorical maps. This package offers support for `sf`, `RasterLayer`, `SpatRaster`, and `stars` spatial objects, and the following methods:

- the V-measure method (Nowosad and Stepinski, 2018)
- the MapCurve method (Hargrove et al., 2006)

# Basic example - SABRE

Two simple regionalizations would be used to show the basic concept of sabre.

    library(sabre)
    library(raster)
    data("partitions1")
    data("partitions2")

The first map, `partitions1`, has four regions (“r1”, “r2”, “r3”, “r4”) of the same size and shape. The second one, `partitions2`, contains three irregular regions where “z1” is the smallest and “z3” being the largest. Our goal is to compare these two regionalizations and calculate a degree of spatial association between them.

It can be done with `vmeasure_calc()`, which calculates “V-measure”, “Homogeneity”, and “Completeness” and returns two preprocessed input maps. For raster data, this function requires, at least, two arguments:

- `x` - an `RasterLayer`, `SpatRaster`, or `stars` object containing the first regionalization
- `y` - an `RasterLayer`, `SpatRaster`, or `stars` object containing the second regionalization

Importantly, both `x` and `y` must have the same coordinate reference system.

There are also one additional argument - `B`. If `B` > 1 then completeness is weighted more strongly than homogeneity, and if `B` < 1 then homogeneity is weighted more strongly than completeness. By default this value is 1.

    partitions_vm = vmeasure_calc(x = partitions1, y = partitions2)
    partitions_vm
    #> The SABRE results:
    #>
    #>  V-measure: 0.36
    #>  Homogeneity: 0.32
    #>  Completeness: 0.42
    #>
    #>  The spatial objects can be retrieved with:
    #>  $map1 - the first map
    #>  $map2 - the second map

The result is a list with three metrics of spatial association - `V-measure`, `Homogeneity`, `Completeness` - and two `RasterLayer` objects with preprocessed input maps - `$map1` and `$map2`. All of the above metrics are between 0 and 1, where larger values are desired. `V-measure` is a measure of an overall spatial correspondence between input maps. `Homogeneity` shows an average homogeneity of the regions in the second map with respect to the regions in the first map. `Completeness` is a function of homogeneity of the regions in the first map with respect to the regions in the second map. The spatial outputs, `$map1` and `$map2`, have two layers. The first one contains regions’ names and the second one (`rih`) describes regions’ inhomogeneities.

For example, “Map1” shows that three regions have the same inhomogeneity of 0.48. This is due a fact that all of these three have two regions from the second map. The upper left region has a larger inhomogeneity of 0.86 as its area “belongs” to three different regions in the second map. More information about this method and its applications can be found in Nowosad and Stepinski (2018).

# Basic example - MapCurves

The sabre also allows for calculating a degree of spatial association between regionalizations using the MapCurve method (Hargrove et al., 2006). The `mapcurves_calc()` function also requires two arguments, `x` and `y`. All of these arguments are explained in the previous section.

    partitions_mc = mapcurves_calc(x = partitions1, y = partitions2)
    partitions_mc
    #> The MapCurves results:
    #>
    #>  The goodness of fit: 0.61
    #>  Reference map: x
    #>
    #>  The spatial objects can be retrieved with:
    #>  $map1 - the first map
    #>  $map2 - the second map

The `mapcurves_calc()` returns a list with a value of the goodness of fit (GOF), the map used as a reference, and two `raster` objects with preprocessed input maps - `$map1` and `$map2`. Read Hargrove et al. (2006) to learn more about this method.

# References

- Nowosad, Jakub, and Tomasz F. Stepinski. “Spatial association between regionalizations using the information-theoretical V-measure.” International Journal of Geographical Information Science (2018). https://doi.org/10.1080/13658816.2018.1511794
- Rosenberg, Andrew, and Julia Hirschberg. “V-measure: A conditional entropy-based external cluster evaluation measure.” Proceedings of the 2007 joint conference on empirical methods in natural language processing and computational natural language learning (EMNLP-CoNLL). 2007.
- Hargrove, William W., Forrest M. Hoffman, and Paul F. Hessburg. “Mapcurves: a quantitative method for comparing categorical maps.” Journal of Geographical Systems 8.2 (2006): 187.