## 2 Materials and Methods
### System Modeling Process
#### 2.1.3 Classification Techniques
), then the distribution of this variable (BP in our case) was based on the following equation:  
\[\mathrm{f(x)}=\frac{1}{\sigma\sqrt{2\pi}}\mathrm{e}^{\frac{-(\mathrm{x}-\mu)^{2 }}{2\sigma^{2}}}, \tag{6}\]  
where \(\sigma\) and \(\mu\) are expressed as:  
\[\mu=\frac{\sum_{i=1}^{n}x_{i}}{n}\ ;\ \sigma=\ \sqrt{\frac{1}{n}\sum_{i=1}^{n}(x_{i }-\mu)^{2}}. \tag{7}\]  
Table 11 of Appendix A groups and defines the variables used in the probabilistic approach. The value of the mean \(\mu\) is established as the point at which the curve is centered, with a symmetrical shape. The standard deviation cannot be negative, as its value determines the spread of the graph; the more spread out the data distribution, the greater its standard deviation \(\sigma\). At the same time, the area under the density curve is equal to 100% of all probabilities. A random variable X is distributed with mean \(\mu\) and variance \(\sigma^{2}\) is noted:  
\[\mathrm{X}\sim\mathrm{N}(\mu,\sigma^{2}). \tag{8}\]  
The central limit theorem defines both the parameters of the sampling distribution and the distribution of the sample means, based on the parameters of the initial population and the sample size. According to this, the means of independent samples from the same population \((\mu,\sigma^{2})\) follow a normal distribution of parameters \((\mu,\frac{\sigma^{2}}{n})\). A random variable Z follows the standard normal distribution (SND), denoted as \(\mathrm{X}\sim\mathrm{N}(\mu,\sigma^{2})\) such that \(\mu=0\) and \(\sigma=1\). Then \(Z=\frac{\mathrm{X}-\mu}{\sigma}\sim\mathrm{N}(0,1)\). These formulas allow us to propose an alert classification system using an SND-based approach. Usually, medical samples from different people (patients) are used to create general medical rules (e.g., if x \(>\) y and t \(<\) mean \(=>\) z is high). This probabilistic technique relies on the use of a sample that makes it possible to generate rules and therefore quantify common knowledge about all the patients. Our aim here was to use only patient-specific data and the behavior of vital variables to draw conclusions. We assumed that there were three classes of anomalies according to the severity of the observation. Therefore, the first class named Class 0 did not represent any health concern since it was squarely in a very close range to the normal and even ideal state of the mean values collected. The other two classes of generated alerts were Class 1 and Class 2 as:  
* Class 0: with probability density \(\mathrm{P0}\in[\mu\ -\ \sigma,\ \mu]\cup[\mu\,\ \mu\ +\ \sigma]\);
* Class 1: with probability density \(\mathrm{P1}\in[\mu\ -2\sigma,\ \mu\ -\ \sigma]\cup[\mu\ +\ \sigma,\ \mu\ +2\sigma]\);
* Class 2: with probability density \(\mathrm{P2}\in[\mu\ -4\sigma,\ \mu\ -2\sigma]\cup[\mu\ +2\sigma,\ \mu\ +4\sigma]\). These intervals were chosen based on the intrinsic nature of the normal distribution, namely the \"68-95-99.7\" rule. If the observation belongs to Class 1, an alert of Type 1 (Risky) is generated. Only the patient is therefore warned that these vital data are abnormal. Generally, observations of Class 1 were not critical, since the normal values were not far distant from the mean \(\mu\). If the observation belongs to Class 2 (critical), the system generates a Type 2 alert. Both the patient and the medical staff are thus warned that the patient is in an immediate danger, and requires an emergency intervention before its condition worsens, which can be fatal or have irreversible consequences for the patient's health condition. Since this approach classifies the anomalies, it is essential to reiterate that observations that range in \([\mu-\ \sigma,\ \mu+\ \sigma]\) do not represent a potential risk and are therefore not subject to this classification layer nor the previous classification layer using SVM. Figure 9 illustrates the generated classes (Class 1 and Class 2) of the standard normal distribution graph of a given data set.