1. Introduction

2. Materials and Methods

2.1 Study Area

The study area covers seven districts of the NLC: Los Olivos, San Martin de Porras, Independencia, Comas, Carabayllo, Ancon, and Puente Piedra covers an area of 793 km², ranging from -11.572° to -12.039° S latitude and -77.199° to -76.810° W longitude. The elevation of the study area ranges from 26.5 to 2748.0 m above sea level (m asl), with a maximum elevation difference of 2721.5 m asl. The NLC districts are located on the western-central boundary of the South American continent, in the subduction zone of the Nazca plate in relation to the South American. On the other hand, historically, during El Niño phenomenon, anomalies in precipitation triggering MM phenomena have been demonstrated on the coast of Lima [19,20].

Figure 1. Ubication map. High vulnerability of housing on the hillsides of Metropolitan Lima.

Figure 1. Ubication map. High vulnerability of housing on the hillsides of Metropolitan Lima.

2.3 Data

The following table shows the vector and raster data used in the research.

Table 1. Research data.

Table 1. Research data.

Type	Description	Type	Scale or spatial resolution	Year	Source	Link (last Access 02/02/2024)
Vectorial	Lithology	Discrete	1/100,000	-	INGEMMET	https://geocatmin.ingemmet.gob.pe/geocatmin/
	Geomorphology	Discrete	1/100,000	-	INGEMMET	https://geocatmin.ingemmet.gob.pe/geocatmin/
	Hydrogeology	Discrete	1/100,000		INGEMMET	https://geocatmin.ingemmet.gob.pe/geocatmin/
	Mass Movements Inventory	Discrete	1/50,000	2021	INGEMMET	https://geocatmin.ingemmet.gob.pe/geocatmin/
Raster	Vegetation Cover	Discrete	1/100,000		INGEMMET	https://www.datosabiertos.gob.pe/dataset/cobertura-vegetal-ministerio-del-ambiente
	Digital Elevation Model (DEM)	Continuous	12.5m	2010	USGS	https://earthexplorer.usgs.gov/
	Seismic Microzonation	Continuous	-	-	IGP/CISMID	https://www.igp.gob.pe/servicios/infraestructura-de-datos-espaciales/componentes/webservice
	Precipitation Anomalies in El Niño phenomenon	Discrete	100m	2021	SENAMHI	Information provided by the institution.

In addition, Table 2 presents the DEM-derived topographic products, geological and environmental variables of study for use in the susceptibility and hazard models for the NLC, their representation is presented in Figure A1 and A2.

Table 2. Research variables names.

Table 2. Research variables names.

	Class		Name	Variable	PCA	Type of variable
Conditioning factor
	Geological and environmental		Lithology	X1	-	Categorical
			Geomorphology	X2	-	Categorical
			Hydrology	X3	-	Categorical
			Vegetation cover	X4	-	Categorical
	Topographical		Slope	T1	PCA1 PCA2 PCA3	Continuous
			Aspect	T2		Continuous
			Topographic wetness index (TWI)	T3
			Terrain roughness index (TRI)	T4		Continuous
			Flow direction	T5		Continuous
			Profile curvature	T6		Continuous
			General curvature	T7		Continuous
Triggering factors
Seismic 8.8Mw (seismic microzonation) Precipitation anomalies in El Niño phenomenon				D1	-	Continuous
				D2	-	Continuous

2.4 Methods

Figure 2 shows the flowchart of this study, which is divided into five steps: the first step is data downloading and preparation, followed by variable exploration, geospatial modeling of the MMS, evaluation of MMS models using the area under the curve (AUC), and estimation of MM hazard under two scenarios.

Figure 2. Flowchart of the study.

Figure 2. Flowchart of the study.

2.4.1. Exploratory Variable Methods

Before applying the MMS model, it is important to ensure that there is no dependence between variables (variable correlation) and that the variables are not influenced by multicollinearity.

2.4.1.1. Pearson Correlation

To discard the linear correlation of the variables, the Pearson correlation was applied, which is considered an effective method for this purpose [11]. The following formula was used.

$$r={\sigma }_{xy}/{\sigma }_x{\sigma }_y$$

(1)

Where ${\sigma }_{xy}$ is the covariance of variables "x" and "y", ${\sigma }_x$ and ${\sigma }_y$ are the standard deviations of variables "x" and "y" respectively. The value of the Pearson coefficient can vary from -1 to 1, where if r<0, the correlation is negative and stronger as it approaches r=-1; if r>0, the correlation is positive and stronger as it approaches r=1; finally, if r=0, then there is no relationship between the variables. Additionally, as indicated by [22], a value greater than ±0.8 could lead to multicollinearity issues.

2.4.1.2. Multicollinearity

Multicollinearity indicates the mutual independence of each influential factor. To verify multicollinearity, the variance inflation factor (VIF) was used, where, in practical terms, a VIF value greater than 5 or 10 [23] indicates that the association of regression coefficients is poorly estimated due to multicollinearity issues [24].

$$VIF=1/(1-R_j^2)$$

(2)

Where $R_j^2$ is the coefficient of determination for the regression of $x_j$ on other explanatory variables.

2.4.1. Principal Component Analysis (PCA)

PCA is a multivariate mathematical procedure that performs the orthogonal transformation of a set of correlated variables into a smaller number of uncorrelated variables or Principal Components (PCs) [25], which are mutually independent [26]. Once multicollinearity among variables has been assessed and confirmed, it is common practice to exclude highly correlated variables that influence each other. However, PCA allows for addressing the multicollinearity issue among influential factors without the need to eliminate variables [8]. This is considered a good option because natural processes are integral, and excluding a variable could lead to loss of information in the analysis of the studied phenomenon. Additionally, PCA allows for evaluating the impact of different influencing factors on MMS. The majority of information from input variables is found in the first PC, which can be expressed as shown in the following equation [27].

$$PCm=a_{m1}{y}_1+a_{m2}{y}_2+...+a_{mn}{y}_n$$

(3)

Where PC is the principal component in the “m-th” place, and $a_{mn}$is the weight for the m-th PC for the n-th variable.

The number of PC was determined based on the level of variance explanation, which ranges from 0 to 1. In this study, a minimum threshold of 0.6 for variance explanation was used, although in some similar studies, values as low as 0.40 [26] or as high as 0.68 [28] for variance explanation based on the number of PC have been found. The choice of the variance explanation value is based on the objective of reducing the correlation and multicollinearity of the variables.

2.4.2. Weights of Evidence (WoE)

Quantitative techniques based on statistics establish functional relationships between instability factors and the past and present distribution of mass movements [26]. The WoE method proposed by [29] is a bivariate statistical technique based on Bayesian probability theory. The stability or instability of certain regions can be estimated through a set of conditioning factors, which are measured by the relationship and spatial distribution of areas known and affected by MM [28]. In this study, WoE was used to determine the relationship between factors influencing the occurrence of MM, to map the MMS, and as input variables for the LR, MLP, SVM, RF, and NB models.

The first step in determining the WoE for each class involves obtaining the prior probability of finding MM, which is estimated as the area affected by MM (L) in the past over the total study area (A).

$$P\left\{L\right\}=N\left\{L\right\}/N\left\{A\right\}$$

(4)

[29,30], present a mathematical development of the methodology's fundamentals. The method assigns positive (W+) or negative (W-) weights to each class of conditioning variables based on the degree of association between the variable class and the spatial distribution and density of evidence of MM. $B$y $\overline{B}$ represent the presence and absence of the conditioning factor in potential mass movements, respectively, and $\overline{L}$ indicates the absence of mass movements.

$${Wi}^{+}=\ln (P\left\{B|L\right\}/P\left\{B|\overline{L}\right\})$$

(5)

$${Wi}^{-}=\ln (P\left\{\overline{B}|L\right\}/P\left\{\overline{B}|\overline{L}\right\})$$

(6)

Where Wi+ with a positive value (>0) indicates that the variable is present where there is the presence of mass movements; additionally, its magnitude represents the positive correlation between the presence of the factor and MM. On the other hand, Wi+ with a negative value (<0) indicates that the absence contributes to the generation of MM. Wi- is used to evaluate the importance of the absence of the factor in the occurrence of MM, when Wi- is positive (>0) the absence of the variable is favorable in the generation of MM, the opposite (<0) is not [30]. Both Wi+ and Wi- are estimated for each class of variables.

The contrast value or final weight $W_f$ (Equation 7) indicates the measure of correlation between the conditioning factor and the MM. If $W_f=0$, then the spatial distribution of MM is independent of the considered variable. If $W_f>0$, then there is a positive association between the variable and the generation of MM. Lastly, if $W_f<0$, there is a negative association, meaning that the absence of the factor contributes to the generation of MM [30, 31].

$$W_{f=}{W^{+}-W}^{-}$$

(7)

Finally, to obtain the MMS map of the conditioning factors by WoE, the algebraic sum of all the contrast values of each variable is calculated. This means that the weights of each class of the variables are summed pixel by pixel, and the sum is the susceptibility map.

$${SMM}_{WoE}=W_{f,Variable1}+W_{f,Variable2}+\dots W_{f,Variable"n"}$$

(8)

2.4.3. Logistic Regression (LR)

LR is considered one of the most popular statistical methods for multivariate regression analysis used to investigate binary response from a set of measurements [32] in the earth sciences. In other words, it estimates the relationship between a dependent variable and multiple independent variables [33]. The variables can be continuous, or discrete, or a combination of both; they can have a normal or non-normal distribution and the dependent variable is dichotomous [34].

In the analysis of MMS, the dependent variable is the absence (probability of occurrence, 0) or presence (probability of occurrence, 1) of MM, which for the purposes of this study were taken from the MMI of INGEMMET. LR transforms the dichotomous dependent variable into a logit variable, which can be used to form a multivariate regression relationship between the dependent variable and the independent variables, which in this case are the multiple conditioning factors [35]. The results of the LR estimate the probability of the presence or absence of MM based on the predictor variables according to the following equation.

$$p=1/(1+e^{-z})$$

(9)

Where p is the probability that the dependent variable will be 1 (maximum probability of MM presence) or 0 (minimum probability of MM presence), forming an “S-curve”; “e” is the neperian number, and “z” is the linear combination of the independent variables. The linear combination “z” can be expressed by the following formula [32].

$$z=B_0+B_1{X}_1+B_2{X}_2+B_3{X}_3+\dots B_n{X}_n$$

(10)

Where X is the independent variable (conditioning factors), which can be represented as $X(x_1,x_2,\dots x_n)$, and B is the estimated coefficient for each independent variable, $B(b_1,b_2,\dots ,b_n)$. B₀ is the intercept of the model, and “n” is the number of independent variables [36]. Based on Equations 9 and 10, the LR model can be expressed in its extended form as Equation 11. It is worth mentioning that the statsmodels module of Python 3.6 was used.

$$Logit(P)=(\frac{1}{1+e^{-(B_0+B_1{X}_1+B_2{X}_2+B_3{X}_3+\dots B_n{X}_n)}})$$

(11)

2.4.4. Multilayer Perceptron (MLP)

The MLP is an artificial neural network composed of multiple layers of interconnected neurons, commonly used in supervised learning. It is used to solve complex classification and regression problems, thanks to its ability to learn nonlinear representations of data based on mathematical algorithms to mimic the learning process of the human brain [37]. Structurally, the network consists of input layers, hidden layers with different numbers of neurons, and an output layer, the latter being the MMS model. The neurons in the layers are connected through weight values, which are trained and tested to form a stable network structure with decision-making capabilities [6]. The MLP model was implemented using the scikit-learn module of Python 3.6.

2.4.5. Support Vector Machine (SVM)

Proposed by [38], it is a supervised ML method based on the concept of an optimal separating hyperplane in the sample space, such that the distance to the classification hyperplane of the two class groups is a maximum function between the margins of the class boundaries [39, 40, 41]. This classification capability makes SVM used for solving non-linear classification and regression problems; thus, it is one of the most used ML techniques in assessing MMS. The SVM model was implemented using the scikit-learn module of Python 3.6.

2.4.6. Random Forest (RF)

Proposed by [42], it is a supervised classification algorithm that is based on classification and regression trees used to create each decision tree. It utilizes a random subset of variables at each node based on a Bootstrap sample, and RF generates thousands of random binary trees to form a forest [17]. In cases with large amounts of sample data, the more feature elements present, the fewer errors and overfitting are generated [40, 43]. RF is frequently used to determine the MMS [17, 41, 43, 44], in this study the Python Scikit-learn module was applied to implement the RF model.

2.4.7. Naive Bayes (NB)

NB is an important algorithm in the field of ML and data mining [40], applied to various fields, and is based on the Bayes probability theorem, Bayes rule, or Bayes formula [45], which is suitable when the data has a high dimension, is not affected by the distribution of the data, and all variables are considered independent of each other [46]. The main objective of NB is to determine the a priori probability of an event based on the proportion of observed cases relative to a specific output class [47]. In this study, the Python Scikit-learn module was applied to implement the NB model.

2.4.8. Validation and Testing of the Models

Curva ROC y AUC

The Receiver Operating Characteristic (ROC) curve is a graphical representation of sensitivity (true positive rate) on the y-axis and 1-specificity (false positive rate) on the x-axis [8]. The area under curve (AUC) of the ROC allows assessment of model fit and prediction [26] and is commonly applied as a criterion for selecting the most appropriate model to determine MMS [8]. The AUC value indicates the percentage of observed positive pixels that are correctly predicted, quantifying the probability that susceptibility models correctly classify the presence or absence of mass movement phenomena based on a set of independent variables or conditioning factors (Figure A1).

The AUC ranges from 0 to 1. AUC values between 0.5 and 0.6 are considered poor models. AUC values between 0.6 and 0.7 are considered average models. AUC values between 0.7 and 0.8 are considered good models. AUC values between 0.8 and 0.9 can be considered very good models, and an AUC value greater than 0.9 is considered an excellent model [9].

F-1 Score

F-1 score shows the performance of the model, high values are optimal [40]. This indicator unifies accuracy and sensitivity [15], accuracy is determined by dividing the true positives by the total number of pixels classified as MM. Sensitivity is the proportion of true positives predicted in the real positive class sample. Therefore, the F-1 score is the harmonic mean of precision and sensitivity [48].

2.4.9. Machine Learning Hyperparameters

In this research, there is no intention to evaluate the influence of hyperparameter optimization on the results of the MMS mapping. The choice of hyperparameters has been made based on the literature, specifically on the values and configurations most commonly used in susceptibility mapping studies. As mentioned by [39], most ML models can achieve excellent accuracy with their default set of hyperparameters, as it is nearly impossible to manually search through combinations due to the countless possibilities of trial and error.

Table 3. Hyperparameters used in this research.

Table 3. Hyperparameters used in this research.

Model	Hyperparameters
LR	method='bfgs',
MLP	lr=0.1, arquitectura [4,4,4,1], epochs=1000, activation 'relu'
SVM	Kernel='linear'
RF	n_estimators=360, max_depth=11, criterion='gini', min_samples_split=5, min_samples_leaf=1
NB	priors=None, var_smoothing=1e-9

3. Results

3.1. Exploration of Variables

To determine the MMS using the WoE, LR, MLP, SVM, RF and NB methods, the correlation of the topographic variables was analyzed and then the multicollinearity of the variables was analyzed using VIF. Figure 3 shows the Pearson correlation of topographic variables derived from a DEM.

Figure 3. Pearson correlation of topographic variables.

Figure 3. Pearson correlation of topographic variables.

From the graph, it is observed that there are variables that are correlated, as there are variables with a correlation coefficient ≥ 0.8. The correlation coefficient between the TRI (T4) and the slope (T1) was 0.99, and the correlation coefficient between the general curvature (T7) and the profile curvature (T6) was 0.8. This indicates a high correlation between the influencing factors. Therefore, we proceed to conduct multicollinearity analysis using the VIF statistic, where a value greater than 5 indicates multicollinearity issues in the fitting of susceptibility models, which are sensitive to the linear correlation of influencing factors.

Table 4. Multicollinearity analysis of topographic variables.

Table 4. Multicollinearity analysis of topographic variables.

Name	Variable	VIF
Intercept	-	10.1
Slope	T1	67.3
Aspect	T2	1.5
Topographic Wetness Index (TWI)	T3	-
ÍTerrain Roughness Index (TRI)	T4	67.3
Flow direction	T5	1.4
Profile curvature	T6	3
General curvature	T7	3

From the multicollinearity result, it is observed that the slope and TRI are affected by multicollinearity. Therefore, dimensionality reduction of the variables was performed using PCA to exclude variable correlation and multicollinearity issues. In Figure 4, the contribution of the variance of the seven PC is observed. Assuming a variance explanation level of ≥0.65, 3 PC are required, resulting in 0.85 variance explanation, with each component being independent of the others.

Table 5 shows the importance and contribution of each influential variable in the three PC. The higher the value (in absolute terms), the greater the contribution to the PC. For PC-1, the most relevant variable was slope. For PC-2, the general curvature variable showed the highest relevance. Lastly, orientation was the most relevant factor for PC-3. Figure 5 displays the selected PC as input variables for the MMS.

Finally, the PCs were reclassified into five quantiles, and contrast values were determined by applying WoE to both PCs and topographic factors to determine the relationship between the study variables and MM phenomena; these variables were used in the WoE, LR, MLP, SVM, RF and NB models.

3.2 Mass Movement Susceptibility (MMS)

The results of the MMS models were reclassified into five quintiles, very low, low, medium, high, and very high in QGIS. The spatial distribution expressed in area is shown in Table 6.

The highest MMS values (very high and high) were generated by the RF models, followed by the NB, SVM, and LR models, all with 20% of the surface of the study area in the very high and high susceptibility levels. The MMS levels by the heuristic method were generated by INGEMMET based on lithology, hydrogeology, slope, land use, and vegetation cover.

Figure 6. MMS models by the method WoE, LR, MLP, SVM, RF y NB.

Figure 6. MMS models by the method WoE, LR, MLP, SVM, RF y NB.

4. Discussion

Figure 8. ROC curve and AUC value for training (a) and test (b) data.

Figure 8. ROC curve and AUC value for training (a) and test (b) data.

Figure 9. Predictive capability of methods, heuristics, WoE, LR, MLP, SVM, RF and NB.

Figure 9. Predictive capability of methods, heuristics, WoE, LR, MLP, SVM, RF and NB.

Table 8. Hazard levels for MM under an El Niño phenomenon and earthquake greater than 8.8Mw.

5. Conclusions

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