# Similarity Analysis for Time Series-Based 2D Temperature Measurement of Engine Exhaust Gas in TDLAT
## 4 Statistical-Based Mean Difference Test and Similarity Analysis
### [Step 2] Similarity Analysis
#### 4.2.4 Results of Similarity Analysis in [Step 2]  
Two-sample \(t\) test, RMSE, and TMD analysis were performed to select the cycles that indicate a \(T_{Rec}\) similar to that of \(T_{Thc}\). The results of a normality test conducted on \(T_{Thc}\) and \(T_{Rec}\) for 18 candidate cycles prior to the Two-sample \(t\) test showed that all were satisfactory for normality. However, the equivalence test between \(T_{Thc}\) and \(T_{Rec}\) showed that some results did not satisfy the equivalence. The results of the equivalence test are summarized in Appendix A. In general, [PERSON]'s \(t\)-test is used if the two groups do not meet equal variances. However, if the very small samples (\(n<10\)), the Two-sample \(t\) test performs slightly better than [PERSON]'s \(t\)-test [32]. The first Two-sample \(t\)-test results are shown in Table 4 and the null hypothesis is rejected when the \(p\)-value is less than 0.05 (see Section 4.2.1). The analysis results are explained based on the cycle, all \(p\)-values from 1 to 9 s for 31, 51, 91, 111, and 251 cycles were above 0.05. For the 71 cycles, the \(p\)-value was close to 1 at 2 s, 5 s, 6 s, and 8 s. In other words, the probability of selecting null hypothesis was high. For 151 and 201 cycles, the \(p\)-value was over 0.05 for all seconds except 5 s. On the other hand, the 451 cycles showed \(p\)-value less than 0.05 at 5 s and 9 s. The \(p\)-value of the 901 cycles was less than 0.05 at 8 s and 9 s. In the 501, 601, 701, and 801 cycles, the \(p\)-values were less than 0.05 for 5 s, 8 s, and 9 s. In the case of the 1001 cycles, the \(p\)-value of 3 s, 8 s, and 9 s was found to be below 0.05. The result of RMSE analysis is shown is Table 5. The smaller the RMSE values, the more the similarity between \(T_{Thc}\) and \(T_{Rec}\). First, the analysis results were based on 10 temperatures per group from 1 to 9 s (see Table 3). In terms of time, the RMSE value (1.951) of 401 cycles in 1 s was the smallest. In 2 s, the RMSE value of 1001 cycles was the smallest at 2.270. RMSE values for 601 cycles were the smallest in 3 s and 4 s. The RMSE values for 251, 201, and 701 cycles were the lowest in 5 s, 6 s, and 7 s. RMSE values for 1001 cycles were the smallest in 8 s and 9 s. The analysis using 90 temperatures from 1 to 9.9 s (see Table 3) showed that the RMSE value of 601 cycles was the smallest. Lastly, the analysis results of TMD are shown is Table 6. The smaller the TMD value, the more the similarity between \(T_{Thc}\) and \(T_{Rec}\). The analysis results were based on 10 temperatures per group from 1 to 9 s (see Table 3). In terms of time, the TMD value (1.626) of 351 cycles in 1 s was the smallest. In 2 s, the TMD value of 111 cycles was the smallest at 0.770. The TMD value for 151, 201, 251, and 801 cycles were the lowest in 3 s, 4 s, 5 s, and 6 s. TMD values for 201 cycles were the smallest in 7 s, 8 s, and 9 s. Next, the analysis using 90 temperature readings from 1 to 9.9 s showed that the TMD value (0.933) of 201 cycles was the smallest. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Cycles**} & \multicolumn{10}{c}{**Time (s)**} & \multicolumn{10}{c}{**Total**} \\ \cline{2-10}  & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & \\ \cline{2-10}  & & \multicolumn{10}{c}{**RMSE**} & & & & **RMSE** & **Rank** \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 4: Results of Two-sample \(t\)-test.