The Moran coefficient (Moran's I)

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.)

See also

Examples

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