## 2 Materials and Methods
### System Modeling Process
#### 2.1.3 Classification Techniques
We obtained the hyper plane separation equation using the following formulae:  
\[\mathrm{w.x_{i}+b}=0. \tag{1}\]  
Once the linear separation and support vector were obtained, this equation changed to:  
\[\mathrm{w.x_{i}}\ +b=\pm 1. \tag{2}\]  
For nonlinear problems, finding the optimal hyper plane depends on solving this minimization problem by introducing penalty parameter and slack variables an introducing the scoring function formula:  
\[\mathrm{Min}\ \frac{1}{2}\parallel\mathrm{w}\parallel^{2}+\mathrm{C.}\sum_{i=1 }^{n}\varepsilon_{i}\ ;\ \sum_{i=1}^{m}(\alpha_{i}\mathrm{y_{i}}\.\mathrm{K(x_{i},x)+b}). \tag{3}\]  
The classification decision function is as follows:  
\[\mathrm{f(x)}=\mathrm{sign}\ (\sum_{i=1}^{N}(\alpha_{i}\mathrm{y_{i}}. \mathrm{K(x_{i},x)+b})). \tag{4}\]  
Finally, \(\mathrm{w}\) and \(\mathrm{b}\) can be obtained by mapping data points into the higher-dimensional feature space using a transformation \(\varphi\) where \(w\) and \(b\) are expressed as:  
\[\mathrm{w}=\sum_{i=1}^{L}(\alpha_{i}\mathrm{y_{i}}\.\varphi(x_{i}))\ ;\ \mathrm{b} =\frac{1}{N_{sv}}\sum_{i}^{N_{sv}}\ (y_{i}\ -\sum_{j}^{N_{sv}}\ (\alpha_{i} \mathrm{y_{j}}\.\mathrm{K(x_{i},x)})). \tag{5}\]  
Once this process is ready, it will therefore provide a learning model that can separate data into different classes. We simulated the classification process using different kernels, as shown in Table A2 of Appendix A. In general, a decision has to be made as to whether to use a linear, polynomial or radial basis function (RBF) (aka Gaussian) kernel. The RBF is by far the most popular choice of kernel type used in SVMs, mainly due to its localized and finite response over the entire feature-vector axis. RBF is typically used when the number of observations is larger than the number of features (which is our case), and the best predictive performance is seen for a nonlinear kernel (as demonstrated in this study). Another major reason for choosing this kernel is its method of scaling the data. During the training phase, the search for optimal parameters (c and gamma) requires us to first normalize the feature in order to select the support vectors. Rescaling all features to the range [0, 1] optimizes the maximum margin calculation between classes. This process is based on the principles of the second classification layer based on the probabilistic approach that was used to perform the broking process explained in the next section. This makes the transition between the two classification methods very encouraging given the homogeneous nature of data values. We therefore expected that this scaling data method would produce better results. (b) Probabilistic based approach:  
Many machine learning and data mining systems assume that data fed to these models follow a normal (Gaussian) distribution, allowing us to making inferences from a sample to a parent population. We defined some terminology below. Numerous parameters or numbers, such as the mean and variance, can be used to describe a population. Let \(x\) represent the blood pressure (BP) of a patient in a hospital; the population consists of the history of the patient-specific BP records. The mean \(\mu_{x}\) and the variance \(\sigma_{x}{}^{2}\) are two parameters associated with this population that are constant and do not fluctuate. We could find \(\mu_{x}\) and \(\sigma_{x}{}^{2}\) by summing the BPs of the patients and calculating the mean and variance. A sample \(n\) represents a random selection from the parent population. If we measured the values (x) of a variable such as an item of medical information (heart rate, oxygen saturation, etc.