## 2 Materials and Methods
### Support Vector Machines (SVM) Algorithm
#### 2.6.1 Radial Basis Function (RBF)  
The RBF kernel of the SVM learning algorithm was used for the categorisation of Test Data within the scope of forest cover detection, which is presented in Equation (24) [71]:  
\[exp(-\gamma\|x-x^{\prime}\|^{2}) \tag{24}\]  
where \(\gamma\) is specified by parameter gamma and must be a positive value (greater than 0), and \(\|x-x^{\prime}\|^{2}\) is the square of the Euclidean distance between the points \(x\) and \(x^{\prime}\). The performance of SVMs, particularly those employing the RBF kernel, hinged significantly on the choice of parameters C and gamma (\(\gamma\)), the fundamental elements defining the SVM algorithm. This essential selection process, known as hyperparameter tuning, optimised the model's performance. Parameter C, the cost or regularisation parameter, mediated the delicate balance between reducing training data error and mitigating model complexity to evade overfitting. A smaller C fostered a broader margin, tolerating more misclassifications yet potentially yielding a more straightforward, less overfitted decision function. On the contrary, a larger C pursued a tighter margin and fewer misclassifications, which might engender a more complex model susceptible to overfitting [72]. The gamma parameter, integral to the RBF kernel, designated the reach of a single training example's influence. An example's influence was widespread if low, resulting in a broad, smooth decision boundary. Conversely, a high gamma suggested a limited influence, creating an irregular decision boundary that may capture finer detail and potential noise in the dataset [70]. This parameter could also be seen as the inverse of the standard deviation of the Gaussian function used in the RBF kernel, emphasising the closeness of data points as a similarity measure. The quest for optimal C and gamma values typically necessitates iterative testing of various parameter combinations, employing grid and randomised search techniques. The preferred model produced the highest mean test accuracy across all iterations. In a k-fold dataset division, each unique pair of C and gamma parameters trains the model k times, utilising different folds as the test set in each instance [71]. [PERSON]'s influential text _Pattern Recognition and Machine Learning_ underscored this procedure's import: \"The RBF kernel has two parameters: C and gamma. The optimum settings for these parameters are data  
Figure 7: Non-linear SVM kernel trick. dependent and must be determined through experimentation. Typically, a range of values are evaluated on a validation set, and the best performance parameters are chosen\" [73]. While grid and randomised search methods are standard and straightforward, it is crucial to correctly carry out the validation process to prevent model overfitting to training data, potentially compromising the model's generalisation ability. More sophisticated methods, such as Bayesian optimisation, might offer improved results in some instances [74; 75]. The optimisation of the SVM parameters process involves tuning the parameters to find the optimal values that yield the best performance. The grid search method used in this study is a commonly used technique for parameter optimisation. A grid search algorithm is a method for optimising the parameters of a machine learning model by exhaustively searching through a manually specified subset of the hyperparameter space. The algorithm evaluates the model's performance on a validation set and adjusts the hyperparameters to maximise the performance metric. The algorithm then repeats this process until it finds the optimal hyperparameter combination [76]. For the SVM, the grid search method involves selecting a range of values for the parameters C and gamma. The range of these parameters is usually chosen based on the problem at hand and the nature of the data. The grid search method then trains an SVM for each pair of (C, gamma) values in the Cartesian product of the two ranges. The optimal choice is the pair with the best cross-validation accuracy [77; 78].