# Machine Learning-Based Tools for Wind Turbine Acoustic Monitoring
## 2 Materials and Methods
Therefore, assuming m > n, for the map we use the function expressed in Equation (1):  
\[f:\ R^{n}\ \rightarrow\ R^{m} \tag{1}\]  
In Equation (1):  
* x and y are input vectors
* \(f\) is a transformation function  
The technique of image spaces is particularly interesting for algorithms that use training data only through scalar products. In this case, in the space \(R^{m}\) we do not have to find f (x) and f (y) explicitly but it is enough to calculate their scalar product f (x) \(\cdot\) f (y). The function f is used to map the input from the n-dimensional space to the m-dimensional space, at a higher level than the n level. To make this last calculation simple--as it becomes very complicated in large spaces--a function called a kernel is used, which directly returns the scalar product of the images (Equation (2)). \[K\left(x,y\right)=f(x)\cdot f(y) \tag{2}\]  
In Equation (2), the \"\(\cdot\)\" denotes the dot product operation that takes the dot product of the transformed input vectors. The goal of the kernel function is to use the data as an input and transform it into the required form if it is not possible to determine a linearly separable hyperplane, as is the case in most cases [34]. The most used kernels are:  
* Linear: \[K(x,\ y)=\ \mathrm{x}\cdot\mathrm{y}\] (3) This is the simplest type of kernel. * Polynomial: \[K(x,\ y)=(\mathrm{x}\cdot\mathrm{y})^{d}\] (4)  
This kernel contains a parameter: a degree of freedom d. A d value of 1 represents a linear kernel. A value greater than d will make the decision limit more complex and could lead to data overfitting. * Radial Basis function (RBF):  
\[K(x,y)=\ \mathrm{exp}\left(-\frac{\left\|\mathrm{x}\cdot\mathrm{y}\right\|^{2}} {2\sigma^{2}}\right) \tag{5}\]  
The value of this type of kernel decreases with distance and varies between zero and one; for this reason, it can be considered a measure of similarity. * Sigmoid:  
\[K(x,\ y)=\ \mathrm{tan}\,\mathrm{h}(\alpha\cdot\mathrm{y}-\delta) \tag{6}\]  
There are two adjustable parameters in the sigmoid kernel, the alpha slope and the intercept constant \(\delta\). A common value for alpha is 1/N, where N is the size of the data. For SVMs, both training and error evaluation are fundamental. For this reason, the data are divided into training and testing sets. The first group is used for algorithm training and contains the correct inputs and outputs, as supervised training is required. It usually consists of about 70% of the total data. The second is used for testing, and consists of a set of data containing 30% of the unused data, that is, to evaluate the accuracy of the neural network by calculating the prediction error chosen. This phase also allows the testing the algorithm; thus, it consists in the real use of the designed model. One of the advantages of a support vector machine-based classifier is the ability to solve difficult and non-linear classification problems and ensure a high level of classification accuracy. As for solving simple problems, however, the accuracy is comparable to that of a classification technique based on a decision tree. Among the disadvantages are the high model creation time, although it is lower than that used by a neural network, and the non-interpretability of the model.