## 2 Materials and Methods
### Methodology  
The goal is to establish a series of methods that can be used to evaluate dam stability based on displacements or serve as one of the components in such an evaluation. Three aspects were selected for development: spatial correlation of surface displacements, prediction of shear hazards using surface displacements, and prediction of displacement trends over time. For each of these issues, a solution was devised and tested on actual data. The hot spots method is a type of local indicator of spatial association (LISA), which was originally proposed by [PERSON] in 1995 [13]. The aim of LISA is to decompose global spatial statistics into individual observations and identify the locations of correlation clusters. Global spatial statistics, such as Moran's I, test the null hypothesis of spatial randomness in favor of an alternative hypothesis of clustering across the entire area. In contrast, LISA provides a value for each point in the analyzed space. LISA is defined by two conditions:  
* For each point, return information about the spatial clustering of similar values around that point. * The sum is proportional to a statistic of global spatial association [13]. LISA can be written in a general sense as \[L_{i}=f\big{(}y_{i},y_{i}\big{)}\] (9) where \(f\) is some function, \(y_{i}\) is the value of some variable in location \(i\), and \(y_{i}\) are the values in a neighborhood of location \(i\)[13]. Therefore, it is necessary to define the neighborhood. There are many methods for this, but the focus is on one distance-based weights matrix. In the analyzed case, this method is appropriate because it is assumed that the closer the measurement is, the greater the impact on the result. In addition, this type of spatial weighting is suitable due to the uneven distribution of measurement points (located at different distances from each other). For example, using the alternative method based on nearest neighbors may result in combining measurement points that are far apart and are not correlated with each other. One of the methods that can be used for LISA is local Moran statistic (a generalization of Moran's I statistic) which can be defined as  
\[I_{i}=\frac{y_{i}-\overline{y}}{\frac{1}{n}\sum_{i}(y_{i}-\overline{y})^{2}} \sum_{i}w_{ij}\big{(}y_{j}-\overline{y}\big{)}, \tag{10}\]  
where \(y_{i}\) is observation in location \(i\), \(\overline{y}\) is the mean of \(y\), \(n\) is the number of observations, \(w_{ij}\) is spatial weights, and \(j\) are locations defined as neighboring for \(i\), i.e., \(j\in J_{i}\)[13]. Like [PERSON]'s I statistic, the hot spots method produces values ranging from \(-1\) to \(1\), where higher values indicate positive spatial autocorrelation, lower values indicate negative autocorrelation, and values near zero indicate no correlation. However, not all results are statistically significant, so a conditional randomization test can be used to obtain levels of pseudo-significance. This involves randomly permuting the locations of adjacent \(y_{j}\) values while keeping the location of \(y_{i}\) fixed, and determining the \(I_{i}\) statistic for each new dataset to obtain its empirical distribution and statistical significance. Hot spots, or local spatial clusters, are identified as those groups of locations where LISA is significant. To differentiate between different types of autocorrelation, a Moran scatterplot is used, which plots the values of a variable against the values of its neighbors (Figure 2). On the graph, the solid line is a linear regression, while the dashed lines are mean values. The scatterplot is divided into four rectangles that classify the values according to their spatial associations:  
* Upper right: spatial clustering of similar high values (hot spot). * Lower left: spatial clustering of similar low values (cold spot). * Upper left: spatial association of dissimilar values, where low values are surrounded by high values (doughnuts). * Lower right: spatial association of dissimilar values, where high values are surrounded by low values (diamonds).