## 2 Materials and Methods
### Classification Methods
#### 2.6.2 Support Vector Machines (SVMs) Classification  
Support vector machines (SVMs) are a non-parametric supervised classification method that can develop multiband images of optimal resolution from information from remote-sensing tools. The essential basis of SVMs is the development of a hyperplane. By forming a hyperplane, the linked data are classified into different coverages [40]. An SVM produces a concentrated boundary that divides the different classes along the hyperplane (Figure 3). If the points representing the categories are located above the hyperplane, they will be categorized as plus one, and if they are located below the hyperplane they will be categorized as minus one. Control zones that are close to the optimal hyperplane are called support vectors. The best way to train the SVMs algorithm is by means of linearly separable classes, as detailed in Equations (4) and (5) [41]. Training points \((Y_{1},X_{1}),(Y_{2},X_{2})\)\(\ldots\)\((Y_{k},X_{k})\)\(\ldots\)\(Y_{k}\in[-1,+1]\) are recognized as linearly separable if there is a vector \(W_{k}\) and a scalar \(B_{k}\) for the inequalities [42], as described in Equations (4) and (5). \[W_{k}\cdot X_{k}+B_{k}\geq+1\text{ for all }Y_{k}=+1,\text{ i.e., a member of class }1 \tag{4}\]  
\[W_{k}\cdot X_{k}+B_{k}\leq-1\text{ for all }Y_{k}=-1,\text{ i.e., a member of class }2 \tag{5}\]  
These inequalities are valid for the whole training set \((Y_{k},X_{k})\); \(k\) is the proportion of samples denoted as \((Y_{k},X_{k})\); \(x\in Rn\), an n-dimensional space; \(y\in[-1,+1]\) represents the class label; \(W_{k}\) is the linear hyperplane. To avoid multiple evaluations of the objects to be analyzed in the support vector machines (SVMs) algorithm, the error in the performance of the method was minimized in relation to the evolutionary algorithms of the hyperparameters and the bounds of the validation error bounds were improved. The SVMs algorithm was processed by the following expressions [42]:  
\[f(x,\alpha)=w\cdot\Phi\cdot(x) \tag{6}\]  
\[w=\sum_{i=1}^{l}\alpha_{i}\cdot y_{i}\cdot\Phi\cdot(x_{i}) \tag{7}\]  
\[L(\alpha)=\sum_{i=1}^{l}\alpha_{i}-\frac{1}{2}\sum_{i,j=1}^{l}\alpha_{i}\cdot \alpha_{j}\cdot y_{i}\cdot\Phi\cdot(x_{i})\cdot\Phi\cdot(x_{j}) \tag{8}\]  
where \(x\in R^{d}=\ell\) is a vector composed of elements to be analyzed; \(w\) is the vector of adjustable parameters computed from a set of training points; \(\Phi\) is the mapping function over space; \(x_{i}\) represents the training data set. The signs of the vectors evaluated by means of Equation (6) were essential to guarantee a correct classification of the different classes. The settings of the parameter vector, \(w\), were performed by training the classifier. The training sets \((x_{i})\) and the associated classes were represented separately by \((y_{i})\in(-1,+1)\) for i = 1, \(\ldots\), l; the vector of weights, \(w\), was expressed by means of the linear combination of the different classes [43]. The term \(\alpha\) was found through the quadratic function, by means of Equation (8). Equation (8) is subject to the constraint \(C\geq\alpha_{i}\geq C\), for i = 1, \(\ldots\), l, where C is the value to be selected by the user, and was used to make the classifier tolerant to the lack of separability of the training data in space. The values \(\alpha_{i}\geq 0\) were defined as the support vectors and represented the closest points to the decision boundary hyperplane in the space H. The inner product \(\Phi\cdot(x_{i})\cdot\Phi\cdot(x_{j})\) in H was performed by using the function \(K(x_{i},x_{j})\), known as kernel. Through this function it was possible to choose the high-level parameters,  
Figure 3: Linear support vector machines (SVMs). usually called hyperparameters [44]. The selection of the hyperparameters was verified by means of the evolutionary algorithm of covariance matrix adaptation (CMA); each individual represented a dimensional vector with real values, which were updated by recombinations.