# Moran's I
A measure of clustering for spatial data, defined as
$$
I = \frac{N}{W} \frac{\sum_{i=1}^N \sum_{j=1}^N w_{ij}(x_i-\overline{x})(x_j - \overline{x})}
{\sum_{i=1}{N}(x_i - x)^2} 
$$
where
- $N$ is the number of spacial units indexed by $i$ and $j$
- $x$ is the variable of interest
- $\overline{x}$ is the mean of $x$
$w_{ij} are the elements of a matrix of spatial weights that denote adjacency
- $W = \sum_{i=1}^N \sum_{j=1}^N w_{ij}$ is the sum of all $w_{ij}$

It may be considered time series stationarity-agnostic as the calculation does not make assumptions about temporal behavior of the underlying data.
The deviation from the global mean $\overline{x}$ is calculated at a snapshot in time and weighted by $w_{ij}$,
resulting in a value that ranges from $[-1;1]$.

## Sources
- [https://doi.org/10.2307/2332142](http://www.stat.ucla.edu/~nchristo/statistics_c173_c273/moran_paper.pdf)
- https://en.wikipedia.org/wiki/Moran%27s_I