# Influence of Particulate Matter and Carbon Dioxide on Students' Emotions in a Smart Classroom
## 3 Results
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline
**Model** & **R** & **R\({}^{2}\)** & **Adjusted** & **RMSE** & **R\({}^{2}\)** & **df1** & **df2** & \(p\)** & **Autocorrelation** & **Statistic** & \(p\) \\ \hline Anger & 0.014 & 0.000 & \(-0.000\) & 2.373 \(\times\) 10\({}^{7}\) & 0.000 & 7 & 24,435 & 0.661 & \(-2.045\times 10^{-4}\) & 2.000 & 0.995 \\ \hline Disgust & 0.270 & 0.073 & 0.072 & 0.056 & 0.073 & 7 & 24,450 & \(<\)0.001 & 0.020 & 1.960 & 0.001 \\ \hline Fear & 0.742 & 0.551 & 0.551 & 0.118 & 0.551 & 7 & 24,449 & \(<\)0.001 & 0.075 & 1.849 & \(<\)0.001 \\ \hline Happiness & 0.603 & 0.364 & 0.364 & 0.159 & 0.364 & 7 & 24,450 & \(<\)0.001 & 0.042 & 1.917 & \(<\)0.001 \\ \hline Sadness & 0.734 & 0.538 & 0.538 & 0.151 & 0.538 & 7 & 24,450 & \(<\)0.001 & 0.116 & 1.769 & \(<\)0.001 \\ \hline Surprise & 0.701 & 0.491 & 0.491 & 0.204 & 0.491 & 7 & 24,450 & \(<\)0.001 & 0.072 & 1.857 & \(<\)0.001 \\ \hline Neutral & 0.789 & 0.622 & 0.622 & 0.198 & 0.622 & 7 & 24,450 & \(<\)0.001 & 0.068 & 1.865 & \(<\)0.001 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Summary of regression analyses for basic emotions. Figure 2: [PERSON]’s correlations among environmental factors (PM and CO\({}_{2}\)) and emotions with significance levels. * \(p\)\(<\) 0.05, ** \(p\)\(<\) 0.01, *** \(p\)\(<\) 0.001. To visually summarize these findings, Figure 3 presents the explained variance (R\({}^{2}\)) and predictive accuracy (RMSE) for each emotion. The neutral emotion model was the most robust in terms of explained variance, with an R\({}^{2}\) of 0.622, indicating that over 62% of the variance in this emotion was explained by the environmental factors analyzed. Additionally, the root mean square error (RMSE = 0.198) suggested notable accuracy in the predictions, further supported by a highly significant \(p\)-value (\(p\)! 0.001). The fear model showed strong results, with an R\({}^{2}\) of 0.551, explaining more than 55% of the variance in fear, with an RMSE of 0.118. The \(p\)-value (!0.001) indicated the high reliability of the model's predictions. Similarly, the sadness model exhibited an R\({}^{2}\) of 0.538, explaining over 53% of the variance in this emotion. The RMSE was 0.151, and the model was statistically significant (\(p\)! 0.001), making it a reliable predictor of sadness. The surprise model explained 49% of the variance (R\({}^{2}\) = 0.491), with an RMSE of 0.204. It was statistically significant (\(p\)! 0.001), though slightly less predictive compared to the sadness and fear models. For happiness, the model explained 36% of the variance (R\({}^{2}\) = 0.364) with an RMSE of 0.159. The model was also statistically significant (\(p\)! 0.001), though it explained a smaller proportion of variance relative to the other emotions. The disgust model had a lower R\({}^{2}\) of 0.073, indicating that only 7.3% of the variance in disgust could be attributed to the environmental factors. The RMSE was 0.056, and although the model was statistically significant (\(p\)! 0.001), it was less effective in predicting this emotion. Lastly, the anger model was ineffective, with an R\({}^{2}\) of 0.000 and an adjusted R\({}^{2}\) of \(-0.000\). The RMSE was extremely high, and the model did not reach statistical significance (\(p\) = 0.661), suggesting that the environmental predictors did not explain variance in anger. This section analyzes how environmental factors influence students' emotions. The standardized regression showed the impact of PM and CO\({}_{2}\), both directly and in quadratic effects, on students' emotions. Table 4 summarizes the regression coefficients and their effects. In this analysis, the largest effects of each variable are highlighted:  
PM\({}_{1}\) and its quadratic term (sq_PM\({}_{1}\)) have significant nonlinear effects on emotions like happiness and surprise. Negative coefficients in the linear term and positive coefficients in the quadratic term suggest a U-shaped relationship. Initially, these emotions decrease with increasing PM\({}_{1}\) levels, but then increase at higher levels.