*3.4. Quantitative Comparison*

The Receiver Operating Characteristic (ROC) curve approach is used for the quantitative comparison of different models. The curve is a plot between the False Positive Rate (FPR) on the x axis and the True Positive Rate (TPR) on the y axis. These parameters are calculated using a conventional confusion matrix where true positives are correctly predicted landslide points, true negatives are correctly predicted, no landslide points, false positives are incorrectly predicted, no landslide points and false negatives are landslide points missed by the model. From these four values, TPR and FPR are calculated as follows:

$$TPR = \frac{True\ Positives}{True\ Positives + False\ Negatives} \tag{5}$$

$$FPR = \frac{False\ Positives}{False\ Positives + True\ Negatives} \tag{6}$$

The plot with maximum Area Under the Curve (AUC) had the best performance. The landslide susceptibility maps were then prepared using the probabilities predicted by the derived ML models. The model predicts the probability of the occurrence of landslides in each cell, varying from 0 to 1. Based on the probability, the district is categorized into five [45–47] (0.0 to 0.2, 0.2 to 0.4, 0.4 to 0.6, 0.6 to 0.8, and 0.8 to 1.0) and the corresponding susceptibility classes are defined as very low, low, medium, high, and very high. The classification based on equal interval was chosen over the other approaches such as natural break and quantiles, as this study focuses on the comparison of probabilities predicted by different approaches. By using equal interval, the susceptibility classes predicted by each approach can be compared directly to evaluate the agreemen<sup>t</sup> or disagreement between the predicted probability values. In other approaches, relative values predicted by each model are used separately for defining the classes and hence the comparison of predicted probabilities is difficult. The statistical attributes such as accuracy and AUC do not provide insights into the agreemen<sup>t</sup> and disagreement between the different landslide susceptibility maps prepared. Hence, another parameter, called the Empirical Information Entropy (EIE), or H index, is used to evaluate the agreemen<sup>t</sup> between different maps. H index can be calculated as:

$$H = -\sum\_{l=1}^{n} P(i) \log(P(i))\tag{7}$$

where, *P*(*i*) is the likelihood of the susceptibility class (very low, low, etc.) *i*, which is numbered from 1 to 5 in this study (1 is very low and 5 is very high), and *n* is the number of classes (5 in this case). When all the maps agree with each other, the value of *H* is zero and as the value increases; the disagreement also increases.

The value of the H-index can be used as an indication to quantify the mutual agreemen<sup>t</sup> between the landslide susceptibility maps considered [4]. When two landslide susceptibility maps are compared, there are two outcomes. When both the outcomes are same, the probability of occurrence of one susceptibility class becomes 1 and that of all the other classes are zero. Hence, the H-index becomes zero. In cases where both the outcomes are different, the probability of occurrence of two susceptibility classes is 0.5 and that of remaining classes are zero. The H-index value is the absolute value of twice the product of 0.5 and log(0.5); i.e., 0.30. When five landslide susceptibility maps are compared, the possible combinations of outcomes and H index values are given in Table 1 below. The number of landslide susceptibility maps predicting each class is interchangeable along the row, and all combinations result in the same value of H index.

From Table 1, it is clear that, as the value of H-index increases, the entropy increases [48], i.e., the disagreement between landslide susceptibility maps increases [4]. Hence, the value can be used to quantify the agreemen<sup>t</sup> amongs<sup>t</sup> the results. If more landslide susceptibility maps predict the same class for a cell, the predicted results can be considered to be highly reliable.


**Table 1.** Possible H index values while comparing the landslide susceptibility maps produced using five algorithms.

The numbers in rows three to nine can be interchanged among the first five columns. The resulting H-index will remain the same.
