The flood prediction in this study is considered as a pattern classification problem within which “flood” and “nonflood” are the two class labels of interest. As a result, the probability (posterior probability) of pixels belonging to the flood class, which are derived from the model, will be used as susceptibility indices. These susceptibility indices of the pixels are then used to generate the flood susceptibility map. To cope with the complexity as well as the uncertainty of the problem of interest, a Bayesian framework is employed in this study to evaluate the flood susceptibility of each data sample. Figure 3 demonstrates the general concept of the Bayesian framework used for classification.

The Bayesian framework provides a flexible way for probabilistic modeling. This method features a strong ability for dealing with uncertainty and noisy data (Theodoridis, 2015; Cheng and Hoang, 2016). Nevertheless, previous studies have rarely examined the capability of this approach for inferring flood susceptibility. Basically, pattern classification aims at assigning a pattern to one of M=2 distinctive class labels Ck, in which k is either 1 or 2. C1=1 and C2=0 denote the flood class and the nonflood class, respectively. To recognize an input pattern based on the information supplied by its feature vector X, we need to attain the posterior probability P(Ck|X), which indicates the likelihood that the feature vector X falls into a certain group Ck. Based on such information, the pattern will be categorized to the group with the highest posterior probability. The posterior probability P(Ck|X) is calculated as follows (Webb and Copsey, 2011): P(Ck|X)=p(X|Ck)×P(Ck)p(X), where P(Ck|X) denotes the posterior probability. The term p(X|Ck) represents the likelihood, which is also called the class-conditional probability density function (PDF). P(Ck) denotes the prior probability, which implies the probability of the class before any feature is measured. The denominator p(X) is the evidence factor; this quantity is merely a scale factor for guaranteeing that the posterior probabilities are valid; it can be calculated as follows: P(X)=∑k=1Mp(X|Ck)×P(Ck).

Generally, the prior probabilities P(Ck) can be calculated by computing the ratio of training instances in each class. Thus, the bulk of establishing a Bayesian classification model is the calculation of the likelihood p(X|Ck). This likelihood expresses the density of input patterns in the learning space within a certain group of data. In most of situations, p(X|Ck) is unknown and must be estimated from the available data. In this research, the Gaussian mixture model is utilized for computing the class-conditional probability density function p(X|Ck).

In machine learning, the performance of a model may be enhanced if latent variables are used (Yu, 2011). Therefore, latent variable approach is employed in this research. Accordingly, radial-basis-function Fisher discriminant analysis (RBFDA) proposed Mika et al. (1999), an extension of the Fisher Discriminant Analysis for dealing with data nonlinearity, is used to generate a latent factor for flood analysis. Thus, RBFDA is utilized to project the feature from the original learning space to a projected space that expresses a high degree of class reparability (Theodoridis and Koutroumbas, 2009). Using this kernel technique, the data from an input space I is first mapped into a high-dimensional feature space F. Hence, discriminant analysis tasks can be performed nonlinearly in I.

Herein, φ(.) is defined as a transformation from an input space I to a high-dimensional feature space F; to compute w (the projecting vector), it is necessary to maximize the Fisher discriminant ratio as follows: J(w)=wTSBφwwTSWφw,whereSBφ=m1φ-m2φm1φ-m2φT,SWφ=∑k=1C∑i=1Nkφ(xi)-mkφφ(xi)-mkφT,mkφ=1Nk∑i=1Nkφxik.

To obtain w, the kernel trick is applied. Thus, one only needs to establish a formulation of the algorithm which only requires dot-product φ(x)⋅φ(y) of the training data and employ kernel functions which calculate φ(x)⋅φ(y). The widely employed radial-basis kernel function (RBKF) is expressed in the following formula (with σ denoting the kernel function bandwidth): K(x,y)=exp⁡-x-y22σ2.

Since a solution of the vector w lies in the span of all data samples in the projected space, the transformation vector w is shown in the following formula: w=∑i=1Nαiφ(xi).

From Eqs. (17) and (19), we have the following: wTmkφ=1Nk∑j=1N∑i=1Nkαjkxj,xik=αTMk,Mk=1Nk∑i=1Nkkxj,xik.

Taking into account the formulas of J(w), SBφ, as well as Eq. (20), we can restate the numerator of Eq. (14) in the following manner: wTSBφw=αTMα,whereM=(M1-M2)(M1-M2)T.

Based on the Eq. (17) that defines mkφ, the denominator of Eq. (14) can be demonstrated in the following way: wTSWφw=αTNα, where N=∑k=12Kk(I-1lk)KkT, Kk denotes a N×Nk kernel matrix with a typical element is kxn,xmk, and I represents the identity matrix and 1lk is a matrix within which all positions are 1/lk.

Considering all Eqs. (14), (21), and (22), the solution of RBFDA can be found by maximizing the following: J(α)=αTMααTNα.

The optimization problem with the objective function expressed in Eq. (23) is found by identifying the primal eigenvector of N-1M. Based on the optimization results, an input patter in I is projected on to a line defined by the vector w in the following manner: w⋅φ(x)=∑i=1Nαik(xi,x).

To formulate a flood assessment model, the first stage is to construct a GIS database (see Fig. 5) within which locations of past flood events, maps of topographic feature, Landsat-8 imagery, maps of geological features, and precipitation statistical records are acquired and integrated. In this study, the data acquisition, processing, and integration were performed with ArcGIS (version 10.2) and IDRISI Selva (version 17.01) software packages.

Furthermore, a C++ application has been developed by the authors to transform the flood susceptibility indices into a GIS format for ArcGIS implementation. Accordingly, the compiled outcomes are employed to form a database that includes the aforementioned flood-influencing features with two class outputs: flood and nonflood. As mentioned earlier, a total of 76 flood locations have been recorded. To balance the dataset and reliably construct the flood prediction model, 76 locations of nonflood areas are randomly sampled and included for analysis. Hence, the total database consists of 152 data samples.

The proposed model for flood susceptibility assessment that incorporates RBFDA, the Bayesian classification framework, and GMM is presented in this section of the study. The overall flowchart of the proposed Bayesian framework based on GMM and RBFDA for flood susceptibility prediction, named as BayGmmKda, is demonstrated in Fig. 6.

Firstly, the whole dataset, including 152 data samples, was separated into two sets: a training set (90 % or 137 samples), employed for model establishing, and a testing set (10 % or 15 samples), used for model testing. It is noted that the input variables of the dataset have been normalized using the minimum–maximum normalization; the purpose of data normalization was to hedge against the situation of unbalanced variable magnitudes.

Secondly, a latent input factor was generated using the RBFDA (explained in Sect. 3.4) and added to the training dataset, with the aim of enhancing the classification performance. Subsequently, the feature evaluation was performed to quantify the degree of relevance of each input factors with the flood inventories in the training set. Any nonrelevant factor should be eliminated from the modeling process to reduce noise and enhance the model performance (Tien Bui et al., 2016a, 2017). For this purpose, in this research, the Mutual Information Criterion (Kwak and Choi, 2002; Hoang et al., 2016), a widely employed techniques for feature selection in machine learning, was selected to express the pertinence of each influencing factors to the flood. It is noticed that the larger the mutual information, the stronger the relevancy between the influencing factor and flood.

In the next step, the BayGmmKda model was trained and established using the training set. The purpose of the training process was to find the best parameters for the mixture component (k) used in GMM and the kernel function bandwidth (σ) used in RBFDA of the BayGmmKda model. To determine the best k, the EM algorithm that employs Akaike information criterion (AIC; Akaike, 1974) was used. Thus, the value of k was varied from 1 to 20, and then AIC was estimated and used to select the model that exhibits the best fit to the data at hand. It is noted that a model with a number of mixture components (k) indicates a lesser degree of complexity (Olivier et al., 1999). In addition, the unsupervised GMM learning (Figueiredo and Jain, 2002) is also used for autonomously determining the best k. Accordingly, the model starts with a maximum component number (k) of 20; the algorithm carries out the model selection process by removing irrelevant mixture components if applicable. To determine the best σ, the grid search procedure is performed and the parameter σ corresponding to the highest classification accuracy rate was selected.

Using the best k and σ in the previous step, the final BayGmmKda model was finally constructed and the Bayesian classification framework was derived. The Bayesian framework was then used to estimate the posterior probability (flood susceptibility index) for all the pixels in the study areas. The flood susceptibility index was then transferred to a raster format to open in ArcGIS.

It is noted that the coupling of the GMM with the EM training algorithm is implemented with the MATLAB statistical toolbox (MathWorks, 2012a); meanwhile, the BayGmmKda performs the unsupervised algorithm with the program code provided by Mário A. T. Figueiredo (http://www.lx.it.pt/~mtf/, last access: 1 April 2016). The RBFDA algorithm and the unified BayGmmKda model have been coded in MATLAB by the authors. In addition, a software program with a graphical user interface (GUI; see Fig. 7) for the implementation of the BayGmmKda model has been coded in a MATLAB environment by the authors. The GUI development aims at providing a user-friendly system for performing flood susceptibility predictions.

As shown in Fig. 7, the program consists of three modules: data process and visualization, model training, and model prediction. The first module provides basic functions for data inspection and visualization, including data normalization, data viewing, and preliminary feature selection with mutual information. In the second module, the users simply provide model parameters, including the kernel function parameter and the GMM training method. The trained model is employed to carry out prediction tasks in the third module, within which the model prediction performance is reported.

The outcome of the preliminary examination on the pertinence of flood-influencing factors is reported in Fig. 8a. As mentioned earlier, the relevancies of influencing factors are exhibited by the mutual information criterion. Based on the outcome, IF5 (SPI) features the highest mutual dependence, followed by IF7 (stream density) and IF8 (NVDI). Influencing factors of IF4 (TWI) and IF10 (rainfall) exhibit comparatively low values of mutual information. Because all the mutual information values are not null, all influencing factors are deemed to be relevant and should be retained for the subsequent processes of model training and prediction.

It is worth keeping in mind that the BayGmmKda's training phase is executed in two consecutive steps, training RBFDA and training GMM. RBFDA analyzes the data in the training set to establish a latent factor which is a one-dimensional representation of the original input pattern. Figure 8b shows the resulted latent factor constructed by RBFDA. In the next step of the training phase, GMM is constructed by the original input patterns with their corresponding labels which consist of 10 input factors and with the RBFDA-based latent factor.

The classification accuracy rate (CAR) is employed to exhibit the rate of correctly classified instances. In addition, a more detailed analysis on the model capability can be presented by calculating true positive rate (TPR), false positive rate (FPR), false negative rate (FNR), and true negative rate (TNR). These four rates are also widely utilized to exhibit the predictive capability of a prediction model (Hoang and Tien-Bui, 2016). TPR=TPTP+FN,FPR=FPFP+TN,FNR=FNTP+FN,TNR=TNTN+FP, where TP, TN, FP, and FN represent the values of true positive, true negative, false positive, and false negative, respectively.

In addition to the four rates, the receiver operating characteristic (ROC) curve (van Erkel and Pattynama, 1998) is used to summarize the global performance of the model. The ROC curve basically demonstrates the trade-off between the two aforementioned TPR and FPR, when the threshold for accepting the positive class of flood varies. In addition, the area under the ROC curve (AUC) is employed to quantify the global performance. In generally, a better model is characterized by a larger value of the AUC.

As aforementioned, the dataset is randomly separated into the training set and the testing set which occupy 90 and 10 % of the data samples, respectively. The training set is employed to train the mode; meanwhile, the testing set is used for validating the model capability after being trained. Since one selection of data for the training set and the testing set may not truly demonstrate the model's predictive capability, this study carries out a repetitive subsampling procedure within which 30 experimental runs are carried out. In each experimental run, 10 % of the dataset is retrieved in a random manner from the database to constitute the testing set; the rest of the database is included in the training set.

The testing performance of the proposed Bayesian framework for flood susceptibility is reported in Table 2 and Fig. 9, which provides the average ROC curves of the proposed model framework, obtained from the random subsampling process, with two methods of GMM training. Herein, the two Bayesian models that employ the EM algorithm and the unsupervised learning (UL) algorithm for training GMM are denoted as BayGmmKda-EM and BayGmmKda-UL, respectively. It can be seen that the BayGmmKda-UL model demonstrates clearly better predictive performance (CAR = 89.58 %, AUC = 0.94, TPR =  0.96, TNR = 0.91) than that of the BayGmmKda-EM model (CAR = 86.67 %, AUC = 0.93, TPR = 0.95, TNR = 0.85). Although the performances of the BayGmmKda-EM model and the BayGmmKda-UL model are comparable in TPR, however, the BayGmmKda-UL model is deemed more accurate than the BayGmmKda-EM model when the two models predict samples with the nonflood class.

Because this is the first time the BayGmmKda model has been proposed for the measurement flood susceptibility, the validity of the proposed model should be assessed. Hence, the benchmarks were used for the comparison, including the support vector machine, adaptive neuro-fuzzy inference system, and the GMM-based Bayesian classifier. The above machine learning techniques were selected because SVM and ANFIS have been recently verified to be effective tools for predicting flood susceptibility (Tien Bui et al., 2016c; Tehrany et al., 2015b). It is noted that the GMM-based Bayesian classifier (BayGmm) is the Bayesian framework for classification which employs GMM for density estimation; however, BayGmm is not integrated with the RBFDA algorithm. BayGmm is used in the performance comparison section to confirm the advantage of the newly constructed BayGmmKda and to verify the usefulness of RBFDA in enhancing the discriminative capability of the hybrid framework.

To construct the SVM model, the model's hyperparameters of the regularization constant (C) and the parameter of the radial-basis kernel function (σ) need to be specified. Herein, a grid search process, which is identical to the one used to identify the kernel function bandwidth used in RBFDA, is employed to fine-tune such hyperparameters of the SVM model. It is noted that the SVM method is implemented in a MATLAB package (MathWorks, 2012b). Meanwhile, the ANFIS model is trained with the metaheuristic approach described in the previous work of Tien Bui et al. (2016c).

It is noted that a random subsampling with 30 runs is employed for all models in this experiment. The result comparison between the proposed BayGmmKda model and three benchmark models is shown in Table 3. The result shows that the proposed model yields the best results (CAR = 89.58 % and AUC = 0.94). It is followed by the ANFIS model (CAR = 85.63 %, AUC = 0.83); the BayGmm model (85.02 %, AUC = 0.92), and the SVM model (83.75 %, AUC = 0.82).

To confirm the performance of the proposed BayGmmKda model is significantly higher than that of the three benchmark model, the Wilcoxon signed-rank test is employed. The Wilcoxon signed-rank test is widely used to evaluate whether classification outcomes of prediction models are significantly dissimilar (Tien Bui et al., 2016e). Using this test, the p values that were obtained from experimental results of the four models can be computed using a threshold value of 0.05. The result of the Wilcoxon signed-rank test is shown in Table 4. It is noted that the signs “++”, “+”, “- -”, and “-” represent a significant win, a win, a significant loss, and a loss, respectively. The result confirms that the proposed BayGmmKda model achieves significant wins over the other models.

Experimental outcomes have indicated that the BayGmmKda model is the best for this study area, and therefore the model was used to compute the posterior probability for all the pixels of the study area. The posterior probability values that were used as flood susceptibility indices were further transformed to a raster format and open in ArcGIS 10.4 software package. Using these indices, the flood susceptibility map (see Fig. 10) was derived and visualized by the mean of five classes: very high (10 %), high (10 %), moderate (10 %), low (20 %), and very low (50 %). The threshold values for separating these classes were determined by overlaying the historical flood locations and the flood susceptibility indices map (Tien Bui et al., 2016c), and then a graphical curve (see Fig. 10) was constructed and the threshold values were derived.

Interpretation of the map shows that 10 % of the Tuong Duong district was classified into the very high class and this class covers 73.68 % of the total historical flood locations. Meanwhile, both the high class and the moderate classes cover 10 % of the region but account for only 15.79 and 7.9 % of the total historical flood locations, respectively, whereas the low class covers 20 % of the district but it contains only 2.63 % of the total historical flood locations. In particular, 50 % of the district, which is categorized to the very low class, contains no flood location. These results indicate that the proposed BayGmmKda model has successfully delineated susceptible flood-prone areas. In other words, the interpretation results confirm the reliability of the proposed Bayesian framework in this work.