A novel GIS-based ensemble technique for flood susceptibility mapping using evidential belief function and support vector machine: Brisbane, Australia

Evidential belief function (EBF)

The Dempster–Shafer technique is a statistical procedure that is used to recognize spatial integration between dependent and independent factors (Smets, 1994). The Dempster-Shafer theory (DST) of evidence, developed by Dempster (2008), is a generalization of the Bayesian theory of subjective probability. Its major advantages are its relative flexibility in accepting uncertainty and the ability to combine beliefs from multiple sources of evidence (Tehrany et al., 2017).

Suppose that a set of flood conditioning factors C = (C_i, i = 1, 2, 3, …, n) that includes mutually exclusive and exhaustive factors of C_i is used in this research. C is known as the frame of discernment. A basic probability assignment is a function m: $P\left(C\right)\to \left[0,1\right].P\left(C\right)$ is the set of all subsets of C including the empty set and C itself. This function is also called a mass function and satisfies $m\left(\Phi \right)=0$ and ${\sum }_{AC}m\left(A\right)=1$, where Φ is an empty set and A is any subset of C. $m\left(A\right)$ measures the degree to which the evidence supports A, and it is denoted by Bel (A), a belief function. The degree of belief (Bel), degree of uncertainty (Unc), degree of disbelief (Dis), and degree of plausibility (Pls) are the main parameters of EBF (Althuwaynee, Pradhan & Lee, 2012). The dissimilarity among Bel and Pls is represented by Unc, which represents ignorance. Dis is the degree of belief of the hypothesis being incorrect for certain evidence. The relationships between these parameters have been previously described, and they include 1 − Unc − Bel, Dis = 1 − Pls and Bel + Unc + Dis = 1, where C_ij has no flood event, Bel will be zero, and Dis will be reset to zero (Awasthi & Chauhan, 2011).

Both BSA and MSA can be performed using EBF (Carranza, Woldai & Chikambwe, 2005). The number of pixels that represent flood or non-flood for each class of a flood conditioning factor are measured by overlapping the flood inventory layer of all flood conditioning factors. Assuming that N(L) and N(C) are the pixels that are inundated and that C_ij is the jth class of the flood contributing factor C_i(i = 1, 2, 3, …, n), N(C_ij) is the number of pixels in class C_ij, and N = (L∩C_ij) is the number of inundated pixels in C_ij. Hence, EBF can be calculated as follows (Carranza & Hale, 2003):

(3)${\displaystyle Bel\left(C_{ij}\right)=\frac{W_{C_{ij}\left(\text{Flood}\right)}}{\sum _{j=1}^n{W}_{C_{ij}\left(\text{Flood}\right)}}}$ (4)${\displaystyle W_{C_{ij}\left(\text{Flood}\right)}=\frac{\frac{N\left(L\cap C_{ij}\right)}{N\left(L\right)}}{\frac{\left[N\left(C_{{}_{ij}}\right)-N\left(L\cap C_{ij}\right)\right]}{\left[N\left(C\right)-N\left(L\right)\right]}}}$ (5)${\displaystyle Dis\left(C_{ij}\right)=\frac{W_{C_{ij}\left(\text{Non-flooded}\right)}}{\sum _{j=1}^n{W}_{C_{ij}\left(\text{Non-flooded}\right)}}}$ where

(6)${\displaystyle W_{C_{ij}\text{(Non-flooded)}}=\frac{\frac{\left[N\left(C_{ij}\right)-N\left(L\cap C_{ij}\right)\right]}{N\left(L\right)}}{\frac{\left[N\left(C\right)-N\left(L\right)-N\left(C_{ij}\right)+N\left(L\cap C_{ij}\right)\right]}{\left[N\left(C\right)-N\left(L\right)\right]}}.}$

The numerator in Eq. (4) is the proportion of flooded pixels in factor class C_ij; the numerator in Eq. (6) is the proportion of flooded pixels that do not occur in factor class C_ij; the denominator in Eq. (4) is the proportion of non-flooded pixels in factor class C_ij; the denominator in Eq. (6) is the proportion of non-flooded pixels in other attributes outside the factor class C_ij. Here, the weight of C_ij is represented by W_Cij(Flood), which supports the belief that floods are more likely to occur, and W_{Cij(Non-flood)} represents the weight of C_ij that supports the belief that floods are less likely to occur.

Excel and ArcGIS software were used to measure the EBF. Subsequently, Dempster’s rule of combination was applied using a raster calculator in ArcGIS to obtain the four integrated EBFs. The formulae for combining two flood conditioning factors C₁ and C₂ are as follows:

(7)${\displaystyle {Bel}_{C_1{C}_2}=\frac{{Bel}_{C_1}{Bel}_{C_2}+{Bel}_{C_1}{Unc}_{C_2}+{Bel}_{C_2}{Unc}_{C_1}}{1-{Bel}_{C_1}{Dis}_{C_2}-{Dis}_{C_1}{Bel}_{C_2}}}$ (8)${\displaystyle {Dis}_{C_1{C}_2}=\frac{{Dis}_{C_1}{Dis}_{C_2}+{Dis}_{C_1}{Unc}_{C_2}+{Dis}_{C_2}{Unc}_{C_1}}{1-{Bel}_{C_1}{Dis}_{C_2}-{Dis}_{C_1}{Bel}_{C_2}}}$ (9)${\displaystyle {Dis}_{C_1{C}_2}=\frac{{Dis}_{C_1}{Dis}_{C_2}+{Dis}_{C_1}{Unc}_{C_2}+{Dis}_{C_2}{Unc}_{C_1}}{1-{Bel}_{C_1}{Dis}_{C_2}-{Dis}_{C_1}{Bel}_{C_2}}}$

Integrated EBF of the flood conditioning factors are implemented sequentially using Eqs. (7)–(9).

Support vector machine (SVM)

Among the data-driven techniques, machine learning methods produce promising viewpoints in natural hazard mapping, and they are suitable for nonlinear multi-dimensional modeling problems (Yilmaz, 2010). SVM is based on the statistical learning concept. It contains a stage wherein the model is trained using a training dataset of related input and target output values. After the model is trained, it is used to assess the testing data. There are two main procedures underlying SVM for solving problems (Yao, Tham & Dai, 2008). First, a linear separating hyper-plane is created that splits the data based on their patterns. Second, mathematical functions (kernels) are used to transform the nonlinear data into a linearly distinguishable format (Micheletti et al., 2011).

Separating hyper-plane formations from a training dataset is the basis for this method. The separated hyper-plane is generated in the original space of n coordinates (x_i: parameters of vector x) between the points of two distinct classes (Shao & Deng, 2012). Values of +1 and −1 are assigned to the pixels that are above and below the hyper-plane, respectively. The training pixels that are closest to the hyper-plane are called support vectors. Modeling of the rest of the data can be undertaken after deriving the decision surface (Pradhan, 2013). The maximum margin of separation between the classes is discovered by SVM; therefore, it builds a classification hyper-plane in the center of the maximum margin.

Consider a training dataset of instance-label pairs (x_i, y_i) with x_i ∈ Rⁿ, y_i ∈ {1, −1} and i = 1, …, m. In this case study, x represents slope, aspect, elevation, curvature, TWI, geology, SPI, soil, LULC, rainfall, distance from roads, and distance from rivers. The classes of 1 and −1 show the flooded and non-flooded pixels, respectively. Finding the best hyper-plane is the goal of the SVM, which separates pixels into different classes, namely, flooded and non-flooded. A separating hyper-plane can be defined as:

(10)${\displaystyle y_i\left(w.x_i+b\right)\ge 1-{\xi }_i,}$

where the orientation of the hyper-plane in the feature space is shown by w, the offset of the hyper-plane from the origin is represented by b, and the positive slack variable is ξ_i (Cortes & Vapnik, 1995). The following optimization problem using Lagrangian multipliers was solved through the determination of an optimal hyper-plane (Samui, 2008).

(11)${\displaystyle \mathrm{Minimize}\sum _{i=1}^n{\alpha }_i-\frac{1}{2}\sum _{i=1}^n\sum _{j=1}^n{\alpha }_i{\alpha }_j{y}_i{y}_j\left(x_i{x}_j\right),}$ (12)${\displaystyle \mathrm{subject to}\sum _{i=1}^n{\alpha }_i{y}_j=0,0\le {\alpha }_i\le C,}$

where α_i are Lagrange multipliers, C is the penalty, and the slack variables ξ_i allow the penalized constraint violation. Then, the decision function that is used to classify the new data can be written as:

(13)${\displaystyle g\left(x\right)=\mathrm{sign}\left(\sum _{i=1}^n{y}_i{\alpha }_i{x}_i+b\right).}$

In the case where the hyper-plane cannot be separated by the linear kernel function, the original input data may be shifted into a high-dimension feature space through some nonlinear kernel functions. Then, the classification decision function is written as (Pradhan, 2013):

(14)${\displaystyle g\left(x\right)=\mathrm{sign}\left(\sum _{i=1}^n{y}_i{\alpha }_{j}K\left(x_i,x_j\right)+b\right)}$ where K(x_i, x_j) is the kernel function.

All of the conditioning factors were reclassified using the obtained EBF weights and entered into SPSS Modeler to implement the SVM modeling. Selection of the kernel function is very important in SVM modeling (Pradhan, 2013). SPSS Modeler offers four types of SVM kernels: LN, PL, RBF, and SIG. RBF is the most popular kernel because it works well in most cases (Yao, Tham & Dai, 2008). RBF has high interpolation capability and less extrapolation capability (Kavzoglu & Colkesen, 2009). PL has an inverse situation, which has better extrapolation capabilities compared to RBF. SIG and RBF perform in a similar manner for certain parameters. However, RBF offers more accuracy (Song et al., 2011). The LN kernel is less popular because it is based on a linear assumption. Using different kernels results in different outcomes. Therefore, in this study, all the kernels were used in ensemble modeling to find the optimal results and compare the outputs. The mathematical representation of each kernel is listed below (Pourghasemi et al., 2013a):

where γ (gamma) is a common parameter for all kernels except LN; d shows the polynomial degree term in the polynomial kernel function; r represents the bias term in the polynomial and sigmoid kernel functions. The parameters γ, d and r are defined by the user. The accuracy of these parameters directly influences the reliability and correctness of SVM outcomes (Ballabio & Sterlacchini, 2012). The two other SVM parameters of the cost of constraint violation (C) and epsilon (ε) were considered constant throughout the analysis.

Because of the importance of kernel parameter selection, listed in Table 2, a cross-validation method was used instead of the trial and error method (Zhuang & Dai, 2006). This process commenced with the division of the flood inventory into n folds: one fold was kept for accuracy assessment purposes, and the rest were saved to run the model (Yao, Tham & Dai, 2008). In this study, the dataset was split into five random folds for which every group had an equal number of flood points (Table 3). The average of the parameters was used for the final training.

Ensemble modeling

To perform the ensemble modeling, all the flood conditioning factors were reclassified based on the acquired EBF weight of C_ij. This stage represents the BSA. The next stage denotes the MSA by reclassifying the conditioning factors using the derived weights from EBF and using them in the SVM analysis. The ensemble method was applied using all four SVM kernels and the parameters obtained from the cross-validation. Consequently, four flood probability indices were derived. In addition, another flood probability map was generated using an individual SVM and an RBF kernel. A stand-alone SVM analysis was performed using the original flood conditioning factors that were not classified by the EBF results. The conditioning factors used in the individual SVM modeling were unclassified and were all in a continuous data format. The reason for this was to examine whether the data format or the use of classified factors can reduce the data variability (Sajedi Hosseini et al., 2018). In addition, individual EBF results were modeled, and a flood probability index was calculated.

Spatial sensitivity analysis

Uncertainty is an unavoidable factor in every analysis (Rahmati, Pourghasemi & Melesse, 2016). Considering these uncertainties helps in obtaining better interpretations of the model outcomes. Although it is not possible to achieve 100% accuracy, there are several approaches that can be implemented to reduce the uncertainty (Refsgaard et al., 2007). These include uncertainty engine (Brown & Heuvelink, 2007), inverse modeling (predictive uncertainty) (Friedel, 2005), and Monte Carlo analysis (Yang, 2011). Sensitivity analysis (SA) evaluates the impact of conditioning factor variations on model outputs, thereby allowing the quantitative assessment of the relative importance of uncertainty sources (Chen, Yu & Khan, 2010). In this study, the Jackknife test was used to assess the uncertainty among the conditioning factor datasets. This SA technique examines the impact of repeatedly removing every conditioning factor from the dataset on the final outcomes. This means that by using this process, the conditioning factor contribution in the analysis can be recognized. In addition, the percentage of relative decrease (PRD) of the AUC values was measured to investigate the dependency of the model output on the influence of conditioning factors using the following equation:

(15)${\displaystyle \mathrm{PRD}=\frac{{\mathrm{AUC}}_{\mathrm{all}}-{\mathrm{AUC}}_i}{{\mathrm{AUC}}_i}\times 100}$

where AUC_all indicates the AUC value derived using the full conditioning factor dataset, and AUC_i shows the prediction power of the method when the i th conditioning factor has been excluded from the dataset.

Validation

To evaluate the efficiency and reliability of the analytical outcomes, the popular AUC method was used. AUC is a popular assessment technique in natural hazard analysis because it provides an understandable and comprehensive way for performing validation (Beguería, 2006; Hand & Till, 2001). It commences with the arrangement of the probability index in descending order. Classification of the probability index into hundred categories on the y-axis with cumulative 1% breaks is the second step. Then, the flood occurrence in each class is examined, and prediction and success rates are derived. The prediction rate is the accuracy that is achieved using flood testing points. It shows how successful the applied technique was in predicting the flood-prone areas that were already inundated. Conversely, the success rate is produced using the flood training points, and this shows the model performance (Tehrany et al., 2015). The range of the AUC is between zero and one. The maximum accuracy is represented by the value 1, and 0 indicates the failure of the analysis. In this study, 106 flood locations were used for training and 53 locations were used for testing purposes.

Analyzing the weights derived from each method

EBF was applied, and the weight for each class of the flood conditioning factors was determined. The areas with high values of Bel and low values of Dis are the most susceptible to floods. Table 4 lists the EBF calculated for the twelve flood parameters. A range of 0.22 to 23.75 m in altitude received the highest Bel (77) and lowest Dis (2) values, thereby indicating the highest susceptibility to floods. All the altitude classes except the second one had considerably low Bel values, thereby indicating low susceptibility to floods. EBF results acquired for altitudes confirmed that most flooding occurred at low altitudes because the water flowed to and met in the lower areas, thereby indicating that flooding of areas at higher altitudes is almost impossible. The correlation between landslide occurrence and slope shows that steep slopes accelerated water flows. A range of 0–0.21° in the slope map attained the highest Bel value of 29 and a low Dis value of 8, followed by the slope range 0.62–1.25°. The aspect map received the highest Bel value of 52 and the lowest value of 8 for the class that was flat; this shows that floods occur in flat areas because water cannot infiltrate the saturated soil. Moisture preservation and vegetation density are affected by this aspect, which also influences flood occurrence. The morphology of the topography is indicated by curvature, which has three categories: concave, convex, and flat. A pixel with a negative curvature value denotes upward concave ground; a pixel with a positive curvature value denotes upward convex ground. A pixel with value zero represents flat ground. The Bel values for the convex and concave categories in the curvature map were low; this condition implies lower flood potential compared with the flat curvature class. The flat class received Bel and Dis values of 54 and 17, respectively.

The SPI range 157700.42–315400.84 had the highest flood susceptibility with a Bel value of 100. With regard to TWI, the highest flood potential was observed in the range 12.31–25.76 because this range showed the highest Bel (31) and lowest Dis (7) values. As the value of TWI increases, the water infiltration decreases, which can cause floods. Soil and geology also control water penetration and infiltration. The class of “Hard acidic yellow and red mottled soils” in soil and the class of “Andesitic to rhyolitic flows and volcaniclastic rocks” in geology received the highest Bel values of 40 and 71, respectively. The first three classes of distance from river, which were 0–489.65 m, 489.65–1305.74 m, and 1305.74–2285.05 m, received the highest Bel values. River proximity is one of the main factors in flood studies. The results showed that the areas closer to the river had higher chances of inundation. Heavy precipitation causes the ground to quickly become saturated and flood. This was confirmed by the acquired weight from EBF for the rainfall map. The highest rainfall class, 3.31–3.42, received the highest Bel value of 19 and a low Dis value of 9.

As described in ‘Ensemble Modeling’, every conditioning factor was reclassified based on the derived EBF weights and used in SVM analysis to implement ensemble modeling. The ensemble method was applied using all four SVM kernels. The kernel parameters were derived from the cross-validation (Table 3). The final step involved the derivation of four flood probability indices. In addition, another flood probability map was generated using an individual SVM and an RBF kernel. The stand-alone SVM was undertaken using the original flood conditioning factors, which were not classified by the EBF results. Figure 4 illustrates the six flood probability index maps.

Creations of flood susceptibility maps

To produce the flood susceptibility maps, the flood probability index has to be classified into different zones of susceptibility (Pradhan, 2013; Tehrany, Jones & Shabani, 2019). Natural break, equal interval, and quantile are some of the most commonly used methods in natural hazard probability index classification (Ayalew & Yamagishi, 2005). Two factors of data nature and data application influence the choice of classification method (Tehrany, Jones & Shabani, 2019). For instance, quantile is a technique that, without affecting the data, groups the pixels into same-size classes. This means that it groups equal numbers of pixels (area) into each susceptibility zone (Nampak, Pradhan & Manap, 2014). Therefore, it appears to be the most suitable method for classifying the flood probability index. To facilitate a reliable assessment of the impact of each class of a flood conditioning factor on flood occurrence, we attempted, where possible, to reduce the influence of the classification algorithm on the classes of the conditioning factor. However, natural break and equal interval might lead to a class with a large number of pixels and a class with few values (Chung & Fabbri, 2003). The flood susceptibility maps were produced by dividing each flood probability index into five susceptible classes of very low, low, moderate, high, and very high using a quantile method as seen in Fig. 5. The selected number of classes was based on the literature (Pham et al., 2017b; Termeh et al., 2018)

Accuracy assessment

To evaluate the reliability of the derived susceptibility maps, an accuracy assessment was performed using the AUC method (Fig. 6). The AUC results showed that the highest prediction (92.11%) and success (94.32%) rates were achieved by the ensemble EBF-SVM–RBF method. The individual methods produced lower accuracies (EBF: 82.60% success rate and 89.56% prediction rate; SVM: 86.91% success rate and 83.53% prediction rate) compared to all the ensemble methods except the ensemble EBF-SVM–LN method (81.21% success rate and 74.70% prediction rate). The reason is that the linear kernel is not appropriate for use in non-linear phenomena such as flooding. Based on the achieved accuracies, the ensemble EBF and SVM method can be used instead of the individual methods to improve the accuracy of the final maps. This can help planners to recognize the most susceptible areas with higher certainty. Using the ensemble technique improved the success rate by 12% and 7% and the prediction rate by 3% and 9% over the individual EBF and SVM methods, respectively.

According to the results obtained in this study, ensemble modeling provided considerable advantages compared to the traditional methods. For example, the processing time for SVM was significantly reduced due to the pre-analysis of the flood conditioning factors. Hence, the factors were assessed and reclassified based on the EBF analysis and then used as an input for SVM. This quickened the machine learning process. In terms of cost, there are no direct differences among the methods in terms of performance; however, reducing the processing time in a large-scale analysis may speed up the management process, thereby reducing the damage costs in hazardous areas.

Sensitivity analysis

As described earlier in the methodology section, every dataset includes an inevitable amount of uncertainty. The SA in this study was performed using the Jackknife test, and its outcomes are summarized in Table 5. The highest loss of performance or PRD ≈ 8.23 of the AUC method was achieved when slope was omitted from the conditioning factor dataset. This was followed by SPI (PRD ≈ 8.11) and geology (PRD ≈ 7.33). A higher PRD indicates that those conditioning factors provide specific information to the model that cannot be found in other factors. On the contrary, some of the conditioning factors did not represent strong contributions to the spatial prediction of flood occurrence such as distance from road (PRD ≈ 0.22), soil (PRD ≈ 0.65), and distance from river (PRD ≈ 0.71). These outcomes show that flood susceptibility mapping is highly sensitive to slope, SPI, geology, altitude, and LULC. Such an SA assists researchers in recognizing the most influential parameters in flood analysis. It is important to consider that these factors might be different in each study area.