diff --git a/ex23-24-25.md b/ex23-24-25.md new file mode 100644 index 0000000..b64ced0 --- /dev/null +++ b/ex23-24-25.md @@ -0,0 +1,20 @@ +# 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 \ No newline at end of file