# An Examination of Thermal Features' Relevance in the Task of Battery-Fault Detection
## 3 Adjustments, Testing and Results
### Statistical Testing  
The correlation coefficient among multivariate variables or towards a categorical variable is not directly obtainable. What we can examine is the correlation between the obtained temperature and voltage data. The temperature of the battery is represented by six different time series, where each represents one measured surface spot. In case of fault, only some of them will increase their values, which will signify the battery overheat. The maximum values of these time series were therefore taken for the correlation estimation towards the voltage vector, because it is represented only by one time set and the battery malfunction is immediately readable from these data. In case of a high level of correlation, the implication about thermal features relevancy could arise. On the other hand, in case of uncorrelated behavior, we can only confirm no presence of redundancy among these variables, while relevance of the thermal data remains unconfirmed. In Figure 5, we can see that the maximum temperature values and the voltage vectors do not possess similar behavior, which is also confirmed by their correlation coefficients. They are listed in Table 2 and their values were calculated on each battery of all battery sets and both (dis)charging processes. For the following test, the raw data were transformed into time series of ranged values (the difference between the maximum and minimum value of the examined variables of the entire battery set during each measured time) in order to test the statistical difference of their values (ANOVA \begin{table}
\begin{tabular}{c c c c c c c} \hline  & & **Charging** & **Discharging** & & **Charging** & **Discharging** \\ \hline  & Batt1 & \(-\)0.429 & \(-\)0.691 & & Batt1 & 0.403 & \(-\)0.213 \\  & Batt2 & \(-\)0.278 & \(-\)0.749 & & Batt2 & 0.498 & \(-\)0.548 \\  & Batt3 & \(-\)0.094 & \(-\)0.787 & BS4 & Batt3 & \(-\)0.014 & \(-\)0.171 \\  & Batt4 & \(0.322\) & \(-\)0.634 & & & & \\  & Batt5 & \(-\)0.074 & \(-\)0.615 & & & & \\ \hline  & Batt1 & 0.665 & \(-\)0.482 & & Batt1 & 0.577 & \(-\)0.809 \\  & Batt2 & 0.255 & \(-\)0.639 & & Batt2 & 0.65 & \(-\)0.887 \\ BS2 & Batt3 & 0.262 & \(-\)0.905 & BS5 & Batt3 & \(-\)0.228 & \(-\)0.091 \\  & Batt4 & 0.387 & \(-\)0.654 & & & & \\  & Batt5 & 0.672 & \(-\)0.835 & & & & \\ \hline  & Batt1 & 0.297 & \(-\)0.73 & & Batt1 & \(-\)0.609 & \(-\)0.58 \\  & Batt2 & 0.408 & \(-\)0.653 & & Batt2 & \(-\)0.631 & \(-\)0.76 \\ BS3 & Batt3 & 0.144 & \(-\)0.897 & BS6 & Batt3 & 0.856 & \(-\)0.337 \\  & Batt4 & \(-\)0.11 & \(-\)0.915 & & & & \\  & Batt5 & 0.097 & \(-\)0.821 & & & & \\ \hline \end{tabular}
\end{table} 
Table 2: Correlation coefficients between pairs of variables (thermal range and voltage range) of all battery sets during both processes (charging and discharging). Figure 5: The maximum measured values on the given battery set at the given time. The temperature data are visualized on the left side and the voltage data on the right side. test). As it was mentioned previously, the result of ANOVA test is the rejection of null hypothesis, which states the statistical equality of means of the observed variables. In case the hypothesis is not rejected, it would be difficult to use variables without distinguishing ability. In all cases, the resulting p-values dropped under the level of significance (\(volt_{discharge}\): \(2.71\times 10^{-226}\), \(temp_{discharge}\): \(1.13\times 10^{-171}\), \(volt_{discharge}\): \(1.1\times 10^{-239}\), \(temp_{discharge}\): \(6.11\times 10^{-47}\)), which implies the rejection of the null hypothesis in all cases. The graphical representation of ANOVA means comparison is depicted in Figures 6 and 7. Because the statistical difference among the groups of data was observed, we are allowed to perform an additional post-hoc analysis that will uncover which group significantly differs from others. The graphical representations of the post-hoc analysis results are depicted in Figure 8 and 9.