The Moran coefficient (Moran's I)

The Moran coefficient, a measure of spatial autocorrelation (also known as Global Moran's I)

mc(x, w, digits = 3, warn = TRUE, na.rm = FALSE)

Source

Chun, Yongwan, and Daniel A. Griffith. Spatial Statistics and Geostatistics: Theory and Applications for Geographic Information Science and Technology. Sage, 2013.

Cliff, Andrew David, and J. Keith Ord. Spatial processes: models & applications. Taylor & Francis, 1981.

Arguments

x: Numeric vector of input values, length n.
w: An n x n spatial connectivity matrix. See shape2mat.
digits: Number of digits to round results to.
warn: If FALSE, no warning will be printed to inform you when observations with zero neighbors or NA values have been dropped.
na.rm: If na.rm = TRUE, observations with NA values will be dropped from both x and w.

Value

The Moran coefficient, a numeric value.

Details

The formula for the Moran coefficient (MC) is $$MC = \frac{n}{K}\frac{\sum_i \sum_j w_{ij} (y_i - \overline{y})(y_j - \overline{y})}{\sum_i (y_i - \overline{y})^2}$$ where $n$ is the number of observations and $K$ is the sum of all values in the spatial connectivity matrix $W$, i.e., the sum of all row-sums: $K = \sum_i \sum_j w_{ij}$.

If any observations with no neighbors are found (i.e. any(Matrix::rowSums(w) == 0)) they will be dropped automatically and a message will print stating how many were dropped. (The alternative would be for those observations to have a spatial lage of zero, but zero is not a neutral value.)

Examples

library(sf)
data(georgia)
w <- shape2mat(georgia, style = "W")
x <- georgia$ICE
mc(x, w)