Case Study: Moran's I

In the previous section the value $\rho$ was introduced and then largely ignored. $\rho$, or the spatial autoregressive term is a measure of the level of spatial autocorrelation that ranges from -1 to 1. Below is an example of lattice data (a regular grid of polygons) that would be perfectly negatively spatially autocorrelated (so $\rho = -1$).

And here is another example with varying degrees of spatial autocorrelation:

The Moran's I statistics is a method for estimating $\rho$, the level of spatial autocorrelation. Formally, Moran's I is formulated as:

$I = \dfrac{N}{\sum_{i}\sum_{j}w_{ij}} \dfrac{\sum_{i}\sum_{j}w_{ij}(X_{i}-\bar{X})(X_{j} - \bar{X})}{\sum_{i}(X_{i} - \bar{X})^{2}}$,

where $N$ is the total number of spatial units indexed by $i$ and $j$, $X$ is some variable of interest associated with each $N$ and $\bar{X}$ is the mean of the variable, and $w_{ij}$ is the now familiar spatial weights object.

One question in the assignment this week will ask you to compute a Moran's I score for a toy data set.