## 2 Methods
### Use of Machine Learning Methods  
Machine learning is applied to establish prediction models, which are used in landslide potential and displacement prediction, by inputting the spatial factors and displacement observations. Widely used machine learning algorithms for classification prediction include naive Bayes, DT, random forest, adaptive boosting (AdaBoost), and extreme gradient boosting (XGBoost). * Naive Bayes  
As the probability model of naive Bayes assumes that the factors are independent of each other and conform to a Gaussian distribution, naive Bayes classification helps clarify a large number of complex classification problems. The early-stage spatial factors correspond to the landslide and nonlandslide slope units, and they are regarded as training samples to establish a prediction model. The later-observed spatial factors are inputted into the model to determine the landslide probability of each slope unit. The naive Bayes prediction model is based on the probability density function of the Bayesian classification method [33], as shown in Equation (16). \[\mathrm{P}(\mathrm{w}_{\mathrm{i}}\,|\,\mathrm{x})=\frac{\mathrm{P}(\mathrm{ x}|\mathrm{w}_{\mathrm{i}})\mathrm{P}(\mathrm{w}_{\mathrm{i}})}{\mathrm{P}(\mathrm{x})}, \left\{ \begin{array}{l}\mathrm{j}\
eq\mathrm{i}\\ \mathrm{j}=1,2\end{array} \right. \tag{16}\]  
where \(\mathrm{P}(\mathrm{w}_{\mathrm{i}}\,|\,\mathrm{x})\) is the probability of the classifying \(\mathrm{w}_{\mathrm{i}}\) occurring in the slope unit \(\mathrm{x}\), \(\mathrm{P}(\mathrm{x}|\mathrm{w}_{\mathrm{i}})\) is the probability of the slope unit \(\mathrm{x}\) occurring in the classifying \(\mathrm{w}_{\mathrm{i}}\), \(\mathrm{P}(\mathrm{w}_{\mathrm{i}})\) is the probability of classifying \(\mathrm{w}_{\mathrm{i}}\), and \(\mathrm{P}(\mathrm{x})\) is the probability of the slope unit \(\mathrm{x}\). * DT  
A DT assumes that the factors are independent of each other, and the category probability of the DT path is defined by the factor characteristics [34]. This algorithm adopts a dichotomy method, which is similar to a double-forked tree branch, to calculate the Gini coefficient value at the node. Finally, the gain value in each path is summed, and the largest accumulator will be predicted to belong to a category, as shown in Equation (17). \[\mathrm{gain}=\sum\mathrm{P}_{\mathrm{i}}\bigg{(}1-\sum\mathrm{P}_{\mathrm{i}} ^{2}\bigg{)} \tag{17}\]  
where \(\mathrm{P}_{\mathrm{i}}\) is the probability. If the node has only one category, \(\mathrm{P}_{\mathrm{i}}\) will be 0. If the numbers of two categories are the same, \(\mathrm{P}_{\mathrm{i}}\) is 0.5. * Random forest  
Random forest is a collection of multiple DTs and adds the use of bagging. The observation data are taken out of the number of samples and trained as n types of classifiers. According to the sample difference in each DT, the random uncertainty of the data is considered. Under the same weight, the classifier uses the summed majority as the best classification tree to predict the classification [35]. Equation (18) represents the probability of the c-th factor in the t-th DT, and the average probability value gc of the category is obtained according to the sum of multiple DTs. Finally, the category of the slope unit \(\mathrm{x}\) is determined according to the maximum \(\mathrm{g}_{\mathrm{c}}\) value (Equation (19)). \[\mathrm{P}(\mathrm{c}|\mathrm{v}_{\mathrm{i}}(\mathrm{x}))=\frac{\mathrm{P}( \mathrm{c}|\mathrm{v}_{\mathrm{i}}(\mathrm{x}))}{\sum_{\mathrm{l}=1}^{n-1} \mathrm{P}(\mathrm{c}_{\mathrm{l}}|\mathrm{v}_{\mathrm{i}}(\mathrm{x}))} \tag{18}\]  
\[\mathrm{g}_{\mathrm{c}}(\mathrm{x})=\frac{1}{\mathrm{t}}\sum_{\mathrm{i}=1}^{ \mathrm{t}}\hat{\mathrm{P}}(\mathrm{c}|\mathrm{v}_{\mathrm{i}}(\mathrm{x})) \tag{19}\]  
where \(\mathrm{P}\) is the probability, c-th is the category, \(\mathrm{v}\) is the node, \(\mathrm{l}\) is the number of categories, \(\mathrm{t}\) is the number of DTs, and \(\mathrm{g}_{\mathrm{c}}\) is the average probability of the c-th category.