## 4 Emotional Quantification Analysis
### Emotion Mapping  
In this section, the 21 emotions are quantified and mapped into valence-arousal domains based on the similarity matrixes. The parameters that involved in the quantification process are listed in Table 2.  
Firstly, the matrixes of questionnaire and semantic similarity as \(QS\) and \(SS\) are multiplicated to form the final similarity matrix \(FS\) according to Equation (1):  
\[FS_{(i,j)}=QS_{(i,j)}\times SS_{(i,j)}, \tag{1}\] \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Emotion**} & \multicolumn{8}{c}{**Category**} \\ \cline{2-9}  & **Pleasure** & **Excitement** & **Arousal** & **Distress** & **Misery** & **Depression** & **Sleepiness** & **Contentment** \\ \hline Joy & 87 & 40 & 7 & / & / & / & / & 17 \\ Intimacy & 47 & 63 & 8 & / & / & 1 & 2 & 30 \\ Trust & 52 & 6 & 1 & 1 & 2 & / & 6 & 83 \\ Confidence & 33 & 8 & 22 & 2 & 2 & 1 & / & 83 \\ Concentration & 11 & 9 & 20 & 2 & 2 & 3 & 65 & 39 \\ Anxiety & 3 & 3 & 44 & 47 & 14 & 40 & / & / \\ Insecurity & 2 & 6 & 56 & 40 & 12 & 33 & 1 & 1 \\ Fear & 2 & 6 & 73 & 34 & 24 & 11 & 1 & / \\ Surprise & 42 & 72 & 32 & 1 & 1 & 2 & / & 1 \\ Sadness & 2 & 1 & 1 & 12 & 55 & 78 & 2 & / \\ Pain & 1 & 1 & 3 & 69 & 42 & 32 & 3 & / \\ Despair & 1 & / & 1 & 33 & 57 & 57 & 2 & / \\ Tired & 2 & 1 & 1 & 9 & 2 & 36 & 100 & / \\ Shame & 3 & 3 & 5 & 52 & 58 & 27 & 3 & / \\ Disgust & 2 & 2 & 3 & 18 & 22 & 61 & 42 & 1 \\ Anger & 2 & 22 & 4 & 49 & 52 & 21 & / & 1 \\ Manic & 7 & 64 & 53 & 9 & 7 & 10 & / & 1 \\ Passion & 7 & 64 & 68 & 3 & 3 & 1 & / & 5 \\ Gratitude & 37 & 5 & 2 & / & 1 & / & 3 & 103 \\ Hope & 26 & 7 & 17 & 2 & / & / & 31 & 68 \\ Relaxation & 33 & 6 & 1 & / & / & 1 & 50 & 60 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Questionnaire Result of Emotional Similarity. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Definition** \\ \hline \(FS\) & The final similarity matrix. \\ \(QS\) & The questionnaire similarity matrix. \\ \(SS\) & The semantic similarity matrix. \\ \(MP_{j}\) & The top \(j\)th similarity value. \\ \(Ang\) & The angle of emotion in valence-arousal domains \\ \(OA\) & The angle of basic emotion category in valence-arousal domains. \\ \(RQ_{(x,y)}\) & Reference coordinate based on \(QS\). \\ \(RS_{(x,y)}\) & Reference coordinate based on \(SS\). \\ \(AQ_{(x,y)}\) & The actual coordinate of emotion in valence-arousal domains. \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters in Emotion Mapping Process. where \(FS_{(i,j)}\) represents the similarity value of \(i\)th emotion in \(j\)th categories from the final similarity matrix, \(QS_{(i,j)}\) represents the similarity value of \(i\)th emotion in \(j\)th categories from the questionnaire similarity matrix, and \(SS_{(i,j)}\) represents the similarity value of \(i\)th emotion in \(j\)th categories from the semantic similarity matrix. Secondly, since the valence-arousal domains is divided into 8 octants based on the basic categories, each emotion is distributed into one of octant area according to the top two maximum values in \(FS\), and the angle of each emotion as \(Ang\) in valence-arousal domains is calculated based on the octant angle of most similar category as \(OA\) and the similarity ratio of adjacent categories according to Equation (2):  
\[Ang=OA+\frac{45}{\left(\frac{MP_{1}}{MP_{2}}+1\right)}, \tag{2}\]  
where \(MP_{1}\) and \(MP_{2}\) are referred to the top two maximum similarity values of emotion of adjacent categories in \(FS\) clockwise.