## 3 Methods
### Inundation Mapping Based on Modest AdaBoost
#### 3.2.2 Inundation Mapping  
Boosting is a technique that combines several weak classifiers to generate a powerful one. The first proposed boosting algorithm, AdaBoost, was created by [PERSON] and [PERSON] in 1996 [34], which is regarded as the basis for all other kinds of boosting method. Due to its good generalisation ability, low computational complexity and high execution efficiency, boosting has become one of the most popular and effective classification tools in computer vision [35] and pattern recognition [36]. A number of algorithms are derived from the boosting method, such as the Discrete AdaBoost (DADB), Real AdaBoost (RADB) and Gentle AdaBoost (GADB). DADB is a boosting method that mainly employs binary weak classifiers, and RADB is a generalisation version of the basic AdaBoost algorithm [37]. On the basis of RADB, GADB is designed with better performance and higher resistance to outliers [38]. Here, we adopt a different boosting method called Modest AdaBoost, which proves to outperform GADB in generalisation error and overfitting. Its natural stopping criterion is also an advantage, which other boosting techniques lack [36]. The flowchart of mapping inundation using Modest AdaBoost is shown in Figure 4.  
Modest AdaBoost is a variant of boosting proposed by [PERSON] et al. [36]. The basic idea of this method is that in every iteration for computing the new distribution, more importance is given to the samples that are misclassified in the previous step (with low margins). In every step, the method is committed to improve the lowest margins of samples. While those training samples that already have high margins may be misclassified with the new distribution and the margins are decreased, this forces the weak classifier to work only in its domain and be'modest', which is the origin of the name MADB. Through this strategy, some regions of the input space have fewer chances to become overconfident,and the generalisation ability of the method benefits from this. The open source GML AdaBoost Matlab Toolbox [39] is used in the experiments to implement the Modest AdaBoost algorithm. It is a collection of classes and functions of several boosting algorithms. More details about the mapping procedure are presented below. First, each permanent pixel is set as a training sample point. Thus, the training dataset \((x_{1},y_{1}),(x_{N},y_{N})\) can be obtained. \(x_{i}\in X\) is the input vector, which consists of several feature values of the permanent pixel, and \(y_{i}\in\{-1,+1\}\), which is the corresponding class label of the permanent pixel. Here we define \(y=+1\) when the pixel is water, and \(y=-1\) when the pixel is non-water. \(N\) is the number of permanent pixels. At the beginning, we initialise the weight distribution on the input data as \(D_{0}(i)=1/N\), \(i=1,2,\ldots,N\). For each iteration \(t=1,\ldots,T\), with the weight distribution \(D_{t}(i)\), the weak classifier \(s_{t}(x)\in S\) can be trained by weighted least squares:  
\[s_{t}=\underset{s}{\text{argmin}}\left(\sum_{i=1}^{N}D_{t}(i)\cdot(y_{i}-s(x_ {i}))^{2}\right), \tag{12}\]  
In addition, the 'inverted' distribution of the data weights is calculated by  
\[D_{t}(i)=(1-D_{t}(i))\overline{a}_{t}, \tag{13}\]  
where \(\overline{a}_{t}\) is the normalisation coefficient. Then we compute the probabilities:  
\[P_{t}^{+1}(x)=P_{D_{t}}(y=+1\cap s_{t}(x)), \tag{14}\]  
\[\overline{P}_{t}^{+1}(x)=P_{\overline{D}_{t}}(y=+1\cap s_{t}(x)), \tag{15}\]  
\[P_{t}^{-1}(x)=P_{D_{t}}(y=-1\cap s_{t}(x)), \tag{16}\]  
\[\overline{P}_{t}^{-1}(x)=P_{\overline{D}_{t}}(y=-1\cap s_{t}(x)). \tag{17}\]  
Set  
\[f_{t}(x)=(P_{t}^{+1}(1-\overline{P}_{t}^{+1})-P_{t}^{-1}(1-\overline{P}_{t}^{ -1}))(x), \tag{18}\]  
and update the distribution by:  
\[D_{t+1}(i)=D_{t}(i)\exp(-y_{i}f_{t}(x_{i}))\alpha_{t}, \tag{19}\]  
Figure 4: Flowchart of inundation mapping procedure. where \(\alpha_{t}\) is the normalisation coefficient.