2.4.1. Hotspot Analysis

Moran's I is a popular index for globally assessing spatial autocorrelation; however, it does not efficiently recognize the grouping of spatial patterns [43]. Hotspot analysis was used to assess whether experimental plots with either high or low values clustered spatially. Hotspot analysis uses the Getis–Ord local statistic, given as:

$$\mathbf{G}\_{i}^{\*} = \frac{\sum\_{j=1}^{n} w\_{i,j} \mathbf{x}\_{j} - \overline{\mathbf{X}} \sum\_{j=1}^{n} w\_{i,j}}{\mathbf{S} \sqrt{\frac{\left[n \sum\_{j=1}^{n} w\_{i,j}^{2} - \left(\sum\_{j=1}^{n} w\_{i,j}\right)^{2}\right]}{n-1}}} \tag{3}$$

where *xj* is the disease severity value for an experimental plot *j*, *wi,j* is the spatial weight between the experimental plot *i* and *j*, *n* is the total number of experimental plots and

$$\overline{X} = \frac{\sum\_{j=1}^{n} x\_j}{n} \tag{4}$$

$$S = \sqrt{\frac{\sum\_{j=1}^{n} x\_j^2}{n}} - \left(\overline{X}\right)^2\tag{5}$$

The Getis–Ord Gi statistic assesses whether the neighborhood of each experimental plot is significantly different from the study area and can distinguish high-value clusters (hotspots) and low-value clusters (cold spots).

The Gi\* statistic returns a z-score, which is a standard deviation. For statistically significantly positive z-scores, higher values of the z-score indicate the clustering of high values (hotspot). For statistically significantly negative z-scores, lower values indicate the clustering of low values (cold spot).

#### 2.4.2. Cluster and Outlier Analysis

Anselin Local Moran's I was used to identify clusters and spatial outliers. The index identifies statistically significant (95%, *p* < 0.05) clusters of high or low disease severity and outliers. A high positive local Moran's I value implies that the experimental plot under study has values similarly high or low to its neighbors'; thus, the locations are spatial clusters. The spatial clusters include high–high clusters (high values in a high-value neighborhood) and low–low clusters (low values in a low-value neighborhood). A high negative local Moran's I value means that the experimental plot under study is a spatial outlier [44]. Spatial outliers are those values that are obviously different from the values of their surrounding locations [45]. Anselin Local Moran's I enables us to distinguish outliers within hotspots, because it excludes the value of the experimental plot under study, contrary to the hotspot analysis, which takes it into account.

The local Moran's I is given as:

$$I\_i = \frac{\mathbf{x}\_i - \overline{X}}{S\_i^2} \sum\_{\substack{j=1,\ j \neq i}}^n w\_{i,j} \left(\mathbf{x}\_i - \overline{X}\right) \tag{6}$$

where *xi* is an attribute for feature *I*, *X* is the mean of the corresponding attribute, *wi,j* is the spatial weight between feature *I* and *j,* and:

$$S\_i^2 = \frac{\sum\_{j=1, j \neq i}^n w\_{ij}}{n-1} - \overline{X}^2 \tag{7}$$
