5.1. Experimental Results of The GWR-PSO-SVM Model
As mentioned in
Section 3.3, the classical PPMCC is used to weaken the correlations of the selected environmental factors and
T1 = 0.5. For simplicity, correlations of geomorphological and hydrological factors are listed in
Table 5 and
Table 6 and 10 factors are excluded for all the models used here. As a result, the remaining 19 environmental factors are relatively independent and can be further screened based on their importance values ranging from 0 to 0.205, as illustrated in
Figure 10, obtained using SPSS Clementine 12 software (IBM, Armonk, NY, USA). To this end, we set
T2 = 0.02 and exclude the environmental factor whose importance value is less than
T2. Finally, 12 environmental factors are selected for the construction of the coupling model,
i.e., catchment slope, distance from drainage, NDVI, bedding structure, slope angle, topographic wetness index, precipitation, lithology, NDWI, vertical distance to channel network, land-use and elevation.
According to the selection criterion mentioned in
Section 3.2, the most important environmental factors,
i.e., catchment slope, distance from drainage and NDVI, are selected as the regional division factors, whose GWR coefficients are obtained by exploiting an adaptive bi-square kernel and AIC in the GWR method. The GWR coefficient values of catchment slope are shown in
Figure 11. It can be easily observed from the figure that different clusters with respect to GWR are spatially developed. Based on the relationship between GWR and spatial autocorrelation mentioned in
Section 1, we can easily infer that the GWR coefficients in each cluster are very close. Consequently, spatial dependency are greatly reduced if each cluster is considered as a spatial variable. Therefore, it is possible that the study area can be partitioned into different prediction regions while spatial autocorrelations are very limited.
In this work, we set
N = 3,
i.e., these selected environmental factors are clustered into three classes by the natural breaks method and the corresponding classification maps are shown in
Figure 12a–c. For convenience, the slope-unit without landslide is named as the non-landslide slope-unit, while the slope-unit including landslide is named as the landslide slope-unit. The result of simple superposition is shown in
Figure 13a. According to the three rules for merging regions mentioned in
Section 3.2, the study area is finally divided into 34 prediction regions by superposing all classification maps. For simplicity, each prediction region is assigned to a unique label, as shown in
Figure 13b. It can be observed from this figure that 25 regions contain landslides in the study area. The numbers of the slope-units and the landslide slope-unit are listed in
Table 7.
For the GWR-PSO-SVM prediction model, all of prediction regions must be sampled as input variables. For each prediction region in
Figure 13b, the label of the landslide slope-unit is assigned as “1”, while the label of the non-landslide slope-unit is assigned as “0”. In our experiment, we use the same number of landslide slope-units and non-landslide slope-units in each prediction region to form training and verification samples. It can be observed from
Figure 13 that the total number of non-landslide slope-units in each prediction region is always more than that of the landslide slope-units. Therefore, all of the landslide slope-units and the same number of the randomly selected non-landslide slope-units form the required samples. Meanwhile, the proposed GWR-PSO-SVM model is a local model, which generates the optimal
C and
γ of the SVM model for each prediction region by using the PSO algorithm, as shown in
Table 8. It should be noted that the prediction regions without landslides are not included in this table. Meanwhile, we perform the SVM classifier to estimate the likelihood that each slope-unit contains the existing landslides and demonstrate the corresponding probability maps in
Figure 14. The probability value in the map ranging from 0% to 100% represents the different degrees of landslide susceptibility.
5.2. Methods to Assess Models Performance
To objectively evaluate the performance of the models considered, three methods are utilized. The first measure is overall prediction accuracy, which is used to evaluate prediction correctness and can be defined as:
where
a and
b are the numbers of correctly predicted landslide and non-landslide slope-units in the landslide susceptibility maps, respectively.
S is the total number of slope-units in the study area. According to (11), this measure can be appropriately applied to evaluate the global models, such as the SVM, PSO-SVM, RS-SVM models, by taking into account the entire study area. While it is used for the GWR-based models, the measure can be computed in each prediction region. In this work, the final measure of overall prediction accuracy is defined as follows:
where
i = 1,2,…,
npr (
npr is total number of prediction regions),
ai and
bi are the numbers of correctly predicted landslide and non-landslide slope-units in the
ith prediction region, respectively.
Si is the number of slope-units involved in the current prediction region.
The second measure is exploited to evaluate prediction accuracy of landslide areas in each class of landslide susceptibility maps obtained by the mentioned models according to the distribution of our study area. This measure is named as class-specific accuracy and is defined as follows:
where
j = 1,2,⋯,
M (
M is total number of landslide susceptibility zones),
Aj and
Bj are the numbers of landslide slope-units and total slope-units in the
jth landslide susceptibility zone, respectively. To perform this measure, our study area is classified into
M landslide susceptibility zones. In this work, the fixed interval method is used to achieve this aim and it is based on previous studies to segment study areas by the predefined thresholds, which is widely used for comparison of multiple models [
7,
46,
61].
The third measure is the classical receiver operation characteristic (ROC) curve and its area under curve (AUC). In a ROC curve the true positive rate (sensitivity) is plotted in function of the false positive rate (100-specificity) for different cut-off points. Each point on the ROC curve represents a sensitivity/specificity pair corresponding to a particular decision threshold. A test with perfect discrimination (no overlap in the two distributions) has a ROC curve that passes through the upper left corner (100% sensitivity, 100% specificity). Therefore, the closer a curve is to the upper left corner, the better are the prediction results [
62].
5.3. Comparison with Further Models
To better demonstrate the performance of our model, several models are compared to our method, including: (1) the SVM model, in which the study area are globally used for sampling and prediction; (2) the PSO-SVM model, in which the PSO algorithm is used to obtain the optimal
C and
γ to improve prediction accuracies; (3) the landslide susceptibility mapping method based on rough set (RS) and SVM proposed by Peng
et al. [
46]. RS theory is an effective tool introduced by Pawlak [
63] and discussed in many review papers [
64,
65,
66,
67,
68,
69,
70]. This technique can deal with vagueness and uncertainty information and identify cause-effect relationships in databases as a form of data mining and knowledge discovery [
46,
63,
71]. Meanwhile, it has been widely used in various disciplines of science [
72], including remote sensing [
73], geographic information science [
74], and landslide susceptibility mapping [
71],
etc. In the work of [
46], it was employed to select key environmental factors for landslide prediction; (4) the GWR-SVM model, which is a local model and similar to our coupling model, without the PSO step to obtain the optimal
C and
γ.
For a fair comparison, the same mapping unit and original environmental factors are used for all models used here. It should be noted that the RS-SVM model is different from the other models due to the fact that its input environmental factors are determined based on the RS theory after the PPMCC analysis. In our experiments, all of the remaining 12 factors are used for input variables for the SVM, PSO-SVM, GWR-SVM and our models, while 14 factors are selected based on the RS theory in the RS-SVM model, excluding land-use, mid-slope position, plane curvature, stream power index, terrain surface convexity from the remaining 19 factors.
It is well-known that the selection of samples for training and verification is a key step for the SVM prediction model. As mentioned above, the classical SVM, PSO-SVM and RS-SVM models can be considered as global ones due to the fact that the entire study area is taken into account for selecting samples,
i.e., all of the landslide slope-units in the study area and the same number of the randomly selected non-landslide slope-units are used for training their respective SVM models, while all of the slope-units in the study area are utilized for verification. Nevertheless, the selection scheme of the remaining GWR-based models is performed for each prediction region, instead of the entire study area, as mentioned in
Section 5.1. Therefore, the sample size of each model in this work is measured using the number of slope-units in the study area or each prediction region.
Table 9 depicts the training and verification sample sizes of all the models. In addition, the PSO algorithm is used for the PSO-SVM and GWR-PSO-SVM models to obtain the optimal
C and
γ to improve prediction performance of the SVM model.
To make probability maps more readable, we can divide probability values by using fixed interval method in ArcGIS software into five susceptibility categories,
i.e., very low, low, medium, high and very high, corresponding thresholds are fixed to 0.1, 0.35, 0.75 and 0.9, respectively, as shown in
Figure 15. It can be observed from
Figure 15 that all of the models can achieve the purpose of landslide prediction. Meanwhile, the very high-susceptibility zones are apparently mapped in the main urban area of Wanzhou district in all the susceptibility maps, which accords with the fact that the previously investigated landslides are mainly distributed in this area. The distribution of high and very high-susceptibility zones is greatly different for each model.
For instance, most of the previously investigated landslides are located in high or very high-susceptibility zones in the maps of the SVM, RS-SVM and GWR-SVM models. However, a large number of slope-units are unreliably classified by these models as high or very high-susceptibility zones as well. Landslides are typically a minority class in the study area, the PSO algorithm always results in local optima of the SVM model, when it is applied to the entire study area. As a consequence, the previously investigated landslides in the southwest of the study area cannot effectively be predicted by the PSO-SVM model. In contrast, the map by our model is consistent to the ground truth of landslide distribution. Although the PSO algorithm is used in our method to optimize the parameters in the SVM model, the division of our study area into prediction regions with appropriate sizes can greatly overcome trapping in local optimum. The high and very high-susceptibility zones mainly concentrate in the previously investigated landslide areas, while most of non-landslide areas are classified as low and very low-susceptibility zones, which guarantee the reliability of prediction results of landslide susceptibility. The overall accuracies of landslide susceptibility mapping by all the models used here are listed in
Table 10.
In this table, the item of “Correct” indicates the number of slope-units that are correctly predicted in prediction regions, while the item of “Total” means the number of slope-units in prediction regions. It should be noted that this “total” number in the GWR-SVM and GWR-PSO-SVM models are calculated using the prediction regions including landslides. It is obvious that the GWR-PSO-SVM model can achieve the best prediction accuracy of 91.10%, which is 7.8%–19.1% higher than the traditional SVM-based models. To further compare the performance of all the models, the class-specific accuracies are shown in
Figure 16. It can be clearly seen that the class-specific accuracy of the very high-susceptibility zone achieved by our model is highest (96.27%) when compared with the other models, which means that our model can detect the very high-susceptibility zones mainly including the previously investigated landslides.
The ROC curves of all the methods are plotted in
Figure 17. It is known that the closer the ROC curve is to the upper left corner, the higher the overall accuracy of the test is. As can be observed from
Figure 17, we can obtain similar conclusions as for the two previous evaluation measures,
i.e., the GWR-PSO-SVM model can achieve the best prediction result. Meanwhile, the ROC plots of the GWR-SVM and the RS-SVM models are pretty close to each other. Since the PSO algorithm is not very robust when it is applied to the whole study area, the ROC plot of the PSO-SVM model is not continuous and is close to the upper left corner when the value (of the 1-specificity) is 0.2, but worse than the RS-SVM model, GWR-SVM and our models when the value is larger than 0.2. In addition, the corresponding AUC is listed in
Table 11. The larger the value of AUC, the better the performance of the prediction model. As shown in this table, our model can produce the largest area of 0.971, when compared with the other models.
It should be noted that there are a few non-landslide regions in the prediction region map (
Figure 12b), since landslides are typically a minority class in the study area. To compare the performance of our model with the global models, we assume in this work that the overall prediction accuracies of these non-landslide regions are 100%, which may improve the overall accuracy of the entire study area. Meanwhile, our experiments not reported here confirm that the AUC value of our model can still reach 0.962 by removing these non-landslide regions from the study area. Furthermore, all the prediction models were applied to Zigui to Badong section in the Three Gorges Reservoir for landslide susceptibility mapping. The experimental results demonstrated that the GWR-PSO-SVM model can obtain the best prediction result as well and the AUC value of 0.965, which is highest among all the models. Therefore, the universality of our model can be validated. Finally, to objectively compare our model with the other models, we select the same number of landslide slope-units and non-landslide slope-units in each prediction region. Although the number of training samples is relatively small in certain prediction regions, the influence on the overall prediction accuracy is very limited.