## 4 Experimental Design
### Mathematical Model
#### 4.4.2 Calculus and Bandwidth Estimation in a Non-Parametric Regression Model  
As noted in previous sections, the current study is innovative, being an extended development of our previous published work [8]. With this, we described the application design, the runtime environment, and conducted stress tests. Additionally, we described the tools such as JMeter, NewRelic, and ServerAgent, which allowed the collection of log files that were subsequently processed and analyzed for their proper interpretation. This finalized with the assessment and the performed comparison. These obtained results will be validated in this section by applying the Non-Parametric Regression Method.  
From data sets of the variables Microservices performance Table 2 and Monolithic performance Table 3, through applying the kernSmooth library (i.e., Smooth Core Functions), the bandwidth \(h\) is calculated, while the \(h\) is the so-called smoothing parameter, also known as bandwidth or window [70]. This parameter is of significant importance for the performance of the regression core estimator. Depending on the weight function's choice, there will be different subtypes of the general class of core estimators [69]. Then, in order to determine the appropriate value of the \(h\) window needs to follow the subsequent model (see Equation (6))  
\[h=\delta_{k}\left(\frac{3}{8}\right)\pi^{\frac{1}{10}}S_{n}n^{-\frac{1}{5}} \tag{6}\]  
where:  
* \(n\): is the width of the sample
* \(S_{n}=\left(\frac{1}{n}\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}\right)^{ \frac{1}{5}}\)standard deviation
* \(\delta_{k}\) depends on the coreK,and it is calculated as:  
\[\delta_{k}=\left(\frac{\int K^{2}(t)dt}{\left(\int u^{2}K(t)dt\right)^{2}} \right)^{\frac{1}{5}}\]  
So that:  
If K is the Gaussian core, then  
\[\delta_{k}=\left(\frac{1}{4\pi}\right)^{\frac{1}{10}} \tag{7}\]  
If K is the Epanechnikov core, then  
\[\delta_{k}=\left(15\right)^{\frac{1}{5}} \tag{8}\]  
For exploration, it is considered a sample of \(X_{1},\ldots,X_{n}\), i.e., for the CPU variable (%), with some 88 observations. Therefore, the standard deviation is: \(S_{n}=8.089328\). Subsequently, using the core function of Epanechnikov, the performance for \(h\) will be (9):  
\[h=\delta_{k}\left(\frac{3}{8}\right)\pi^{\frac{1}{10}}S_{n}n^{-\frac{1}{5}} \tag{9}\]  
\[h=\left(15\right)^{\frac{1}{5}}\left(\frac{3}{8}\right)\pi^{\frac{1}{10}}\left( 8.089328\right)\left(88\right)^{-\frac{1}{5}}=2.3877\]  
And for the Gaussian model, it has  
\[h=\left(\frac{1}{4\pi}\right)^{\frac{1}{10}}\left(\frac{3}{8}\right)\pi^{ \frac{1}{10}}\left(8.089328\right)\left(88\right)^{-\frac{1}{5}}=1.0786\]Hereby, a grid is generated for \(t\), between the interpolating values, with semi-spaced points. Thus, for each \(t_{j}\), \(K\left(\frac{t_{j}-X_{i}}{h}\right)\), K is calculated. Afterwards, the estimate of \(f\) is obtained in order to generate the graphic representation.  
\[\hat{f}(t)=\frac{1}{nh}\sum_{i=1}^{n}K\left(\frac{t-X_{i}}{h}\right) \tag{10}\]  
The resources may be considered to be a minimum for a test with these characteristics. The used values to generate the workload are moderate, and in a certain way, they limit the level of work of the resources.