Global Moran's I:
Moran index statistic is a kind of widely used spatial autocorrelation statistic, which is in the form as follows:

\( I=\frac{n}{S_0}.\frac{\sum_{i}^{n}\sum_{j=1}^{n}w_{ij}(x_i-x_{average})(x_j-x_{average})}{\sum_{i}^{n}(x_i-x_{average})^2} \)

\( x_i \) repesents the observed value at the ith spatial position ,\( x_p=\frac{1}{n}\sum_{i=1}^{n}x_i \) ,\( w_{ij} \) repesents whether i element and j element are adjacent. In general, adjacent elements are 1, and non-adjacent elements are 0. \( S_0 \) represents the sum of all the elements of the spatial weight matrix W.

It's easy to see by looking at this that \( (x_i-x_{average}) \) measure the difference between the i element and the mean of the whole region. and \( \sum_{j=1}^{n}w_{ij}(x_j-x_{average}) \) measure the difference between the elements which are near the i element and mean of the whole region. so if this region is spatial clustered, \( (x_i-x_{average}) \) and \( (x_j-x_{average}) \) will have the same sign, which will lead that Moran's I is greate than 0. and if this region is spatial alien, \( (x_i-x_{average}) \) and \( (x_j-x_{average}) \) will have the different sign, which will lead that Moran's I is less than 0.

Therefore, when Moran's I is significantly positive, there is a significant positive correlation, and similar observed values (high and low values) tend to cluster in space.

When Moran's I is a significant complex value.There was a significant negative correlation, and similar observed values tended to be dispersed.

When Moran's I approaches the expected value (\( \frac{-1}{n-1} \), which tends to 0 with the increase of the number of samples), it indicates that there is no spatial autocorrelation and the observed values are arranged randomly in space.

Local Moran's I:

In the global correlation analysis, if the global Moran index is significant, we can consider that there is spatial correlation in this region.However, we still don't know exactly where spatial aggregation exists.In this case, the local Moran exponent is needed, and the formula for the local Moran's I is as follows.

\( I_i=\frac{n}{S^2}.(y_i-y_{average})\sum_{j\neq{i}}^nw_{ij}(y_j-y_{average}) \)

where \( S^2=\sum_{i=1}^{n}(y_i-y)^2 \).Through observation, we can find that \( (y_i-y_{average}) \) represents the difference between the observed value of the ith region and the average observed value of the whole region, while \( \sum_{j\neq{i}}^{n}(y_j-y_{average}) \) represents the difference between the surrounding region and the average observed value of the whole region. The local Moran's I can be used to determine which position in the region has spatial autocorrelation