ex23-24-25

ex23-24-25.md

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+# 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