## 3 Human Emotion Label Prediction Based on Fuzzy Learning (HELPFuL)
### Example  
Example 1. We apply an FLDL task in which we encounter four instances. The detailed information is shown in Table 1, where \(C\) is the condition attribute and \(D=d_{1},d_{2},d_{3}\) is a complete label set.  
We consider the Gaussian kernel function to compute the fuzzy equivalence relation between instances. The kernel parameter is set to 0.2. Given the Fuzzy label distribution training dataset and a test instance \(x^{\prime}\), the value of condition attribute \(C(x^{\prime})=0.65\), and the label set \(D(x^{\prime})\) of \(x^{\prime}\) is computed by FLDL-FRS.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Input:** Fuzzy label distribution system \(U=\{x_{1},x_{2},\cdots,x_{N}\}\), label set \(D\) \\
**for\(\forall d_{j}\in D\)do** & \(\forall x\in U\)do & \(\overline{R}d_{j}(x^{\prime})=\inf_{x\in U}max(1-R(x^{\prime},x),d_{j}(x))\) \\
**end** \\
**Algorithm 3**Fuzzy-rough-based Fuzzy label distribution classification  
\end{table}
Table 1: Fuzzy label distribution dataset.  
First, the fuzzy equivalence relation between the training and test data is computed by the Gaussian kernel function.  
\[R(x^{\prime},x_{1}) =0.7132,1-R(x^{\prime},x_{1})=0.2868;\] \[R(x^{\prime},x_{2}) =0.3314,1-R(x^{\prime},x_{2})=0.6686;\] \[R(x^{\prime},x_{3}) =0.6376,1-R(x^{\prime},x_{3})=0.3624;\] \[R(x^{\prime},x_{4}) =0.2085,1-R(x^{\prime},x_{4})=0.7915.\]  
Then, the fuzzy upper and lower approximations are computed by the Formula (5).  
\[\overline{R}d_{1}(x^{\prime}) =\sup_{x_{i}\in U}min(R(x^{\prime},x_{i}),d_{1}(x_{i}))=\{0.64,0. 3314,0.6376,0.2085\}=0.64;\] \[\overline{R}d_{2}(x^{\prime}) =\sup_{x_{i}\in U}min(R(x^{\prime},x_{i}),d_{2}(x_{i}))=\{0.31,0. 3314,0.42,0.2085\}=0.42;\] \[\overline{R}d_{3}(x^{\prime}) =\sup_{x_{i}\in U}min(R(x^{\prime},x_{i}),d_{3}(x_{i}))=\{0.7132,0.23,0.55,0.19\}=0.7132;\] \[\underline{R}d_{1}(x^{\prime}) =\inf_{x_{i}\in U}max(1-R(x^{\prime},x_{i}),d_{1}(x_{i}))=\{0.64, 0.6686,0.76,0.7915\}=0.64;\] \[\underline{R}d_{2}(x^{\prime}) =\inf_{x_{i}\in U}max(1-R(x^{\prime},x_{i}),d_{2}(x_{i}))=\{0.31,0.74,0.42,0.7915\}=0.31;\] \[\underline{R}d_{3}(x^{\prime}) =\inf_{x_{i}\in U}max(1-R(x^{\prime},x_{i}),d_{3}(x_{i}))=\{0.72,0.6686,0.55,0.7915\}=0.55.\]  
Finally, the label set of \(x^{\prime}\) can be presented by three methods.  
\[\begin{array}{l}D_{upper}(x^{\prime})=\{0.64,0.42,0.7132\};\\ D_{lower}(x^{\prime})=\{0.64,0.31,0.55\};\\ D_{mean}(x^{\prime})=\{0.64,0.365,0.6316\}.\end{array}\]  
Given the Gaussian kernel as follows:  
\[\left[\begin{array}{cccc}1&0.8021&0.9992&0.6376\\ 0.8021&1&0.8655&0.9603\\ 0.9992&0.8655&1&0.7132\\ 0.6376&0.9603&0.7132&1\end{array}\right]\]  
The fuzzy upper and lower approximations of the test instance \(x^{\prime}\) are computed by Formula (5). We obtain the label set of \(x^{\prime}\) by the weighted average assignment method.  
\[D_{weight}(x^{\prime})=\{d_{j}(x^{\prime})\}_{j=1}^{L}=\{0.64,0.3710,0.6301\};\]  
The real label is denoted as Original, and the four methods are denoted as FFRS-u, FFRS-l, FFRS-m, and FFRS-w. We can see that the top and bottom lines denote the upper and lower approximations, respectively. FFRS-m and FFRS-w, which are two lines in the middle, are close to the real label.