## 4 Method
### ML Methods Used for Prediction  
We use several ML methods in this study. Firstly, Linear Regression [68] is a model that uses a linear equation of input values (\(x\)) to predict output data (\(y\)) (see Equation (7)):  
\[y=b+wx \tag{7}\]  
where \(x\) is the predictor (feature), \(y\) is the output value (target), \(w\) is the slope of the regression line, and \(b\) is the value of \(y\) when \(x=0\). The goal of the Linear Regression model is to minimize the sum of square errors (Equation (8)). \[\sum\limits_{i=1}^{m}\ (y_{i}-wx_{i})^{2}\rightarrow\min \tag{8}\]  
where the meaning of \(y_{i}\), \(w_{i}\) and \(x_{i}\) are the same as in Equation (7). SVR [69] is a supervised ML model, during the training of which we aim to find the \"maximum-margin hyperplane\", i.e., to minimize the \(l2\)-norm of the coefficient vector (Equation (9)). SVR allows us to define the error acceptability of our model and find the right line to fit the data. The error term is constrained by the absolute error margins, called the maximum error (\(\varepsilon\)-epsilon) (Equation (10)). We can adjust the epsilon to obtain the performance we want from our model. Consequently, we should choose which is more appropriate: a more accurate prediction (but larger margin of error) or a smaller margin of error (but more prediction error) [70]. \[\text{MIN}\frac{1}{2}\|w\|^{2} \tag{9}\]  
\[|y_{i}-w_{j}x_{i}|\leq\varepsilon \tag{10}\] \begin{table}
\begin{tabular}{c c c c c} \hline  & **Surface** & **Ground** & **Oil Density** & **Spilled Oil** \\  & **Spreading** & **Coefficient** & **Thickness** & **Volume** \\ \hline Surface spreading & 1 & 0 & 0 & 0 \\ coefficient & 0 & 1 & 0 & 0 \\ Ground thickness & 0 & 0 & 1 & 0 \\ Oil density & 0 & 0 & 0 & 1 \\ Spilled oil volume & 0 & 0 & 0 & 1 \\ \hline \end{tabular}  
* Note that the Pearson correlation coefficient is calculated since the data is continuous. Moreover, we have zero values for the Pearson correlation coefficient, since we have only a small dataset that presents the prediction input. It presents clean data without noise and outliers.  
\end{table} 
Table 3: Correlations among variables (Pearson) *. This optimisation task can be formulated as follows: the minimum deviation \(\xi\), of the value outside of \(\varepsilon\) should be found as presented by Equations (11) and (12). \[\text{MIN}\frac{1}{2}\|w\|^{2}+C\sum_{i=1}^{n}|\xi_{i}| \tag{11}\]  
\[|y_{i}-w_{i}x_{i}|\leq\varepsilon+|\xi_{i}| \tag{12}\]  
where C is the hyper-parameter. The Decision Tree [71] is developed by recursive partitioning of the root node (i.e., the first parent) to the left and right child nodes. Note that in the literature, we can find binary and n-ary Decision Trees, whereas in this research we emphasize binary Decision Trees. Consequently, the data is divided into subsets, which results in information gain (IG) [72]. This partitioning process is repeated until we achieve pure leaves, i.e., all samples in the leaf have a certain feature. To divide the nodes according to the most informative features, we have to solve an optimization task by applying the tree-learning algorithm (Equations (13) and (14)). \[IG(D_{p}f)=I_{E}(D_{p})-(\frac{N_{left}}{N_{p}}I_{E}(D_{left})+\frac{N_{right }}{N_{p}}I_{E}(D_{right})) \tag{13}\]  
\[\text{where }I_{E}(z_{1},\,z_{2},\ldots,z_{j})=-\sum_{i=1}^{j}z_{i}\log_{2}z_{i} \tag{14}\]  
where \(f\) is the split feature; \(D_{p}\), \(D_{left}\), and \(D_{right}\) are the parent, left child, and right child nodes, respectively; \(Np\) is the total number of samples in the parent node; \(N_{left}\) and \(N_{right}\) are the numbers of samples in the left child and right child nodes, respectively; \(IE\) is the impurity measure; and \(z_{1}\),..., \(z_{j}\) represent the degree to which each class is present in the child node, resulting from a tree split. Consequently, the lower the IG of the child nodes, the larger the information gain. Depending on the recursive partitioning, in the literature we can find different Decision Tree algorithms, such as ID3, C4.5 [25], and C5.0 [26].