Which statistical methods to compare the spatial distribution of two different variables in two raster maps?

Question

I am working with two global raster maps using the R Software (original file: netCDF for map 1 and GeoTIFF for map 2), with a resolution of 5 min of arc.

• Map 1 contains data on the share of the land in each grid-cell which is cultivated -i.e. cropland area in each grid-cell. Values range from 0 to 1.
• Map 2 contains data on the livestock density in each grid-cell -i.e. number of animals in each grid-cell. values range from 0 to thousands.

My objective is to calculate/obtain an indicator of segregation of livestock production in comparison of crop production -i.e. an indicator of the spatial co-presence of livestock and crop production. My final aim could be to get a numerical indicator of e.g. how much crop production and livestock are co-present within a certain territory, as instance in each grid-cell. In other words, I am interested in getting a sort of indicator estimating the spatial distance between livestock and crop at a certain spatial scale. Finally, I would be interested in get a sort of "average" value of such indicator at Country level. My idea is to use this indicator in estimating the amount of livestock excreta that could be available at a Country scale (i.e. if the there is a high co-presence of livestock and crop production within a certain Country, this is reflected in a high availability of excreta for such crop production).

Which kind of statistical method or set of tools could be suitable to reach such result?

Here some information on how the two dataset were generated, which I retrieved from the original sources:

• Map 1: The author used two different high-resolution (1-km) satellite based, global land cover classification data sets that are available for circa 2000, and applied a simple set of climatic parameters to mask obviously non-agriculture areas within the satellite datasets. Additional information can be found here http://www.earthstat.org/wp-content/uploads/METADATA_CroplandPastureArea2000.pdf
• Map 2: The datased is an improvement of a previous dataset (Gridded Livestock of the World - GLW). The estimated densities were based on statistical relationships between observed densities within administrative units derived from survey and census data, and several explanatory variables, including a time-series of remotely sensed satellite data relating to climate and the environment, and other spatial data relating to demography, land cover and terrain. Reported sub-national statistics were thus spatially disaggregated and gaps where none were available filled with predictions to provide a complete global distribution map for each species. More information are available here: http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0096084

Answer 1

tl;dr: pick a segregation measure from the seg package, probably the isp() function as an implementation of White (1983).

I'd recommend using an entropy segregation measure. Probably you need some sort of conditional entropy, telling you how much of the information in one spatial distribution is contained in the other.

Segregation is not easily expressed on a single dimension. E.g. Massey & Denton (1988) try to split it into evenness, exposure, concentration, centralization, and clustering. That's not a big problem, you just need to know what the measure you use does exactly and whether that's what matters in your case.

There's a lot of literature on the subject, here's a (far from comprehensive) overview. If you have a look at only the publications marked in italic you should get a rough idea of what's been going on in the field.

Mesuring segregation literature

• Jahn, J., Schmid, C. F. & Schrag, C. The mea- surement of ecological segregation. American Sociological Review 293–303 (1947).
• Williams, J. J. Another commentary on so-called segregation indices. American Sociological Re- view 298–303 (1948).
• Duncan, O. D. & Duncan, B. A methodological analysis of segregation indexes. American sociological review 210–217 (1955).
• Taeuber, K. E. & Taeuber, A. F. Negroes in cities: Residential segregation and neighborhood change (Atheneum, 1969).
• Henri Theil, A. J. F. A note on the measure- ment of racial integration of schools by means of informational concepts. The Journal of Mathematical Sociology 1:2, 187–193 (1971).
• White, M. J. The measurement of spatial segregation. American journal of sociology 1008–1018 (1983)
• Morgan, B. S. An alternate approach to the development of a distance-based measure of racial segregation. American Journal of Sociology 1237– 1249 (1983)
• Massey, D. S. & Denton, N. A. The dimensions of residential segregation. Social forces 67, 281–315 (1988).
• Wong, D. W. Spatial dependency of segregation indices. The Canadian Geographer/Le Géographe canadien 41, 128–136 (1997)
• Leibovici, D. G. Defining spatial entropy from multivariate distributions of co-occurrences. In International Conference on Spatial Information Theory, 392–404 (Springer, 2009).
• Leibovici, D. G., Claramunt, C., Le Guyader, D. & Brosset, D. Local and global spatio-temporal entropy indices based on distance-ratios and co- occurrences distributions. International Journal of Geographical Information Science 28, 1061– 1084 (2014)
• Vranken, I., Baudry, J., Aubinet, M., Visser, M. & Bogaert, J. A review on the use of entropy in landscape ecology: heterogeneity, unpredictability, scale dependence and their links with thermodynamics. Landscape Ecology 30, 51– 65 (2015). URL http://dx.doi.org/10.1007/ s10980-014-0105-0. DOI 10.1007/s10980-014- 0105-0.

White 1983 may be directly applicable in your case. Duncan & Duncan 1955 is a beautiful paper that sums up the literature to that point perfectly. Massey & Denton 1988 give you really good insight into the different dimensions of segregation. Vranken et al. 2015 is an insane overview over entropy measures in landscape ecology.

Mesuring segregation in R

• have a look at the seg package. It's an implementation of a number of indicies mentioned above and a few other important ones. You should check the publications mentioned in the documentation.
• Also, Leibovici et al. published some code, it is for point data though I think. Maybe it's overly complicated for what you are trying to achieve, but I included them anyway because they deal specifically with co-occurances, which closely matches your question.

Please be aware that a lot of these measures are highly sensitive to the resolution / zones that you use!