## 3 Methods
### Development of Multivariate Control Charts  
To exemplify the construction of the multivariate control charts MCUSUM and MEWMA, 30 observations are shown for each of the variables from data taken from production, which will allow estimating the vector of means and covariance matrix. See Table 3.  
The vector of means and covariance matrix are defined in Equations (11) and (12), respectively. \[\overline{x}= \begin{bmatrix}0.07856&0.61838&0.12829\end{bmatrix}  \tag{11}\]  
\[S= \begin{bmatrix}0.00021095&0.00082909&0.00016761\\ 0.00082909&0.00336724&0.00064279\\ 0.00016761&0.00064279&0.00013834\end{bmatrix}  \tag{12}\]  
To verify the multivariate normality assumption, the Mardia test is applied. According to the results, the sample follows a multivariate normal distribution with a significance level of 0.05, skewness of \(p(0.0774)>\alpha\) (0.05), and kurtosis of \(p(0.8341)>\alpha\) (0.05). See Table 4.  
To continue with the development of the multivariate control chart, a new sample of measurements is extracted for each of the quality characteristics monitored in the process, which will allow verifying the statistical stability of the manufacturing system, in this case, the supervision in the application of solder in the assembly process of the electronic cards. Table 5 shows the sample that will be used to exemplify the development of the MCUSUM and MEWMA control chart. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Test** & **Statistic** & \(\mathbf{p}\)**Value** \\ \hline Skewness & 16.8636 & 0.0774 \\ Kurtosis & 0.2093 & 0.8341 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Multivariate normal distribution test. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multicolumn{3}{c}{**Volume**} & \multicolumn{5}{c}{**Area**} & \multicolumn{5}{c}{**Height**} \\ \hline
**1** & 0.06754 & **16** & 0.09632 & **1** & 0.55495 & **16** & 0.68841 & **1** & 0.12235 & **16** & 0.14114 \\
**2** & 0.06913 & **17** & 0.08188 & **2** & 0.58755 & **17** & 0.63986 & **2** & 0.12012 & **17** & 0.13004 \\
**3** & 0.05268 & **18** & 0.06934 & **3** & 0.52240 & **18** & 0.58669 & **3** & 0.10666 & **18** & 0.12059 \\
**4** & 0.08399 & **19** & 0.07786 & **4** & 0.61406 & **19** & 0.61000 & **4** & 0.13763 & **19** & 0.12829 \\
**5** & 0.07520 & **20** & 0.08447 & **5** & 0.58841 & **20** & 0.64836 & **5** & 0.13045 & **20** & 0.13539 \\
**6** & 0.10338 & **21** & 0.10945 & **6** & 0.71045 & **21** & 0.73764 & **6** & 0.14744 & **21** & 0.14902 \\
**7** & 0.06432 & **22** & 0.07958 & **7** & 0.55300 & **22** & 0.62316 & **7** & 0.11867 & **22** & 0.12830 \\
**8** & 0.08364 & **23** & 0.05461 & **8** & 0.64876 & **23** & 0.52422 & **8** & 0.13170 & **23** & 0.10522 \\
**9** & 0.08312 & **24** & 0.09452 & **9** & 0.64687 & **24** & 0.67790 & **9** & 0.12992 & **24** & 0.14276 \\
**10** & 0.07903 & **25** & 0.05670 & **10** & 0.62674 & **25** & 0.51833 & **10** & 0.12770 & **25** & 0.11294 \\
**11** & 0.08597 & **26** & 0.08965 & **11** & 0.65148 & **26** & 0.65259 & **11** & 0.13422 & **26** & 0.14157 \\
**12** & 0.06230 & **27** & 0.08022 & **12** & 0.55148 & **27** & 0.61960 & **12** & 0.11343 & **27** & 0.13046 \\
**13** & 0.05567 & **28** & 0.08602 & **13** & 0.53859 & **28** & 0.65743 & **13** & 0.10876 & **28** & 0.13223 \\
**14** & 0.08410 & **29** & 0.08708 & **14** & 0.65377 & **29** & 0.66809 & **14** & 0.13085 & **29** & 0.13435 \\
**15** & 0.06387 & **30** & 0.09528 & **15** & 0.57263 & **30** & 0.67785 & **15** & 0.11566 & **30** & 0.14074 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Quality characteristic measurements. The detailed procedure for calculating the MCUSUM control chart statistics is shown. We take, as an example observation number, one of the samples and start with the difference between the sample and the vector of means; see Equation (13). \[A= \begin{bmatrix}0.08568&0.64758&0.13428793\end{bmatrix} - \begin{bmatrix}0.07856&0. 61838&0.12829\end{bmatrix}