## 4 Experimental Design
### Mathematical Model
#### 4.4.1 Non-Parametric Regression Model  
According to [69], a non-parametric regression model is a form of regression analysis in which the predictor does not have a predetermined way, but it is constructed according to information derived from the data. Specifically, the goal of regression analysis is to produce a reasonable explanation of the relationship between a dependent variable (response variable \(y\)) and a set of independent variables (explanatory variables) \(x1,\ldots,Xn\). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Time** & **CPU\%** & **Disk-Read** & **Disk-Write** & **Memory-MB** & **Network-Rx** & **Network-Tx** \\ \hline
1 & 39.593 & 0 & 0 & 7621.167 & 0 & 0 \\
2 & 37.810 & 31.50 & 0 & 7621.167 & 0 & 0 \\
3 & 31.578 & 35.29 & 50600 & 7813.601 & 68100 & 68100 \\
4 & 32.338 & 37.56 & 10900 & 7894.582 & 72793 & 73509 \\
5 & 40.722 & 52.87 & 1179648 & 8006.667 & 64500 & 64500 \\
6 & 63.805 & 47.74 & 2994176 & 8648.839 & 423000 & 224000 \\
7 & 74.193 & 45.17 & 2781184 & 8997.964 & 360000 & 187000 \\
8 & 73.880 & 42.61 & 3366912 & 9471.820 & 498000 & 247000 \\
9 & 61.369 & 37.56 & 4349952 & 9304.417 & 430000 & 222000 \\
10 & 67.910 & 47.20 & 2727936 & 9316.832 & 444000 & 229000 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Example of data recorded in the performance measurement of microservices architecture [8]. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Time** & **CPU\%** & **Disk-Read** & **Disk-Write** & **Memory-MB** & **Network-Rx** & **Network-Tx** \\ \hline
1 & 31.313 & 0 & 0 & 867.207 & 0 & 0 \\
2 & 35.820 & 0 & 0 & 867.207 & 0 & 0 \\
3 & 36.616 & 1084 & 22.500 & 1025.601 & 84600 & 84600 \\
4 & 38.000 & 444 & 13.500 & 1047.164 & 6591820 & 659  
We assume that \(n\) pairs of data are observed \(\left(x_{i},y_{i}\right)\) then it is able to establish the non-parametric regression model (1):  
\[y_{i}=m(x_{i})+\varepsilon_{i} \tag{1}\]  
where \(\varepsilon_{i}\ldots\varepsilon_{n}\) is the random variable distributed identically with \(E(\varepsilon_{i})=0\) and \(E(\varepsilon_{i})=\sigma^{2}\), while the values of the variable \(x_{i}\ldots x_{n}\) are known, leading to a design that is considered fixed. Additionally, the variance has constant errors. Therefore, the model presents homoscedasticity (i.e., when the variance of the conditional error to the explanatory variables is constant throughout the observations). Given \(\left(X,Y\right)\) are bivariate random variables (i.e., a bivariate distribution represents the joint probability distribution of a pair of random variable) with joint density \(f\left(x,y\right)\), the regression function is defined as well:  
\(m\left(x\right)=E\left(Y/X=x\right)\), which is the expected value of Y when X takes the known value x. Thus \(E\left(Y/X\right)=m\left(X\right)\), and defining \(\varepsilon=Y-m\left(X\right)\) it has to, \(Y=m\left(X\right)+\varepsilon,E\left(\varepsilon/X\right)=0,V\left( \varepsilon/X\right)=\sigma^{2}\), Therefore, the non-parametric regression model is (2):  
\[Y_{i}=m(X_{i})+\varepsilon_{i},i=1,2,\ldots,n \tag{2}\]  
As we already established the model, it is now necessary to determine the estimate or to adjust it based on the n observations available in the exploration. Therefore, an estimator m \(\hat{m}\left(x\right)\) of the regression function and an estimator \(\sigma^{2}\) of the error variance need to be generated. These estimation procedures of \(m\left(x\right)\) are known as smoothed methods. Among the various non-parametric smoothing techniques, local adjustment is considered, which only considers data close to the point where it needs to estimate the function [70]. Therefore, we focus on research with the core function, which is reasonable to assume that the density function of the variable being estimated is continuous. These core type estimators (or Kernel) were designed to solve models with such characteristics within the defined variables.