## 3 Human Emotion Label Prediction Based on Fuzzy Learning (HELPFuL)
### Algorithm  
In this subsection, we propose two FLDL algorithms based on fuzzy rough sets. The first one is the un-weighted algorithm belonging to a lazy learning approach, and the other is the weighted algorithm; the weights are learned from the training data. Figure 4: Example for SLL, MLL, and FLDL. Figure 3: Label distribution task. Given a fuzzy label distribution system \(FMLS=<U,C,D>\), the fuzzy upper and lower approximations of a test instance \(x^{\prime}\) to the fuzzy label \(d_{j}\in D\) are defined as follows:  
\[ \begin{array}{l}\overline{R}d_{j}(x^{\prime})=\sup_{y\in U,x^{\prime}\
otin U} min(R(x^{\prime},y),d_{j}(y)),\\ \underline{R}d_{j}(x^{\prime})=\inf_{y\in U,x^{\prime}\
otin U}max(1-R(x^{ \prime},y),d_{j}(y)).\end{array}  \tag{5}\]  
The fuzzy upper approximation of the fuzzy label represents the possibility of the instance belonging to this label. The fuzzy lower approximation of the fuzzy label represents the necessity of the instance belonging to this label. Fuzzy rough sets are regarded as the measures of possibility and necessity. In modeling the uncertainty of a judgment, it is natural to expect that the boundary between an event and its contrary is not too rigid. In order to characterize the uncertainty, both possibility and necessity are required. Therefore, we use fuzzy upper and lower approximations to confirm the label of an instance. Given a fuzzy label distribution system \(FMLS=<U,C,D>\), for the test instance \(x^{\prime}\), the fuzzy membership degree of \(x^{\prime}\) to label \(d_{j}\in D\) is defined as follows:  
\[d_{j}(x^{\prime})=u_{j}*\overline{R}d_{j}(x^{\prime})+l_{j}*\underline{R}d_{j} (x^{\prime}), \tag{6}\]  
where \(u_{j}\) and \(l_{j}\) are the weights of the fuzzy upper and lower approximations of \(d_{j}\), and these weights satisfy \(u_{j}\geq 0\), \(l_{j}\geq 0\), \(u_{j}+l_{j}=1\). There are three special methods in assigning a label to an instance: pessimism assignment method, optimistic assignment method, and mean assignment method. 1. The optimistic assignment method \(D_{upper}(x^{\prime})=\{d_{j}(x^{\prime})\}_{j=1}^{L}\): If \(u_{j}=1\) and \(l_{j}=0\), then we have \(d_{j}(x^{\prime})=\overline{R}d_{j}(x^{\prime})\). The fuzzy upper approximation is regarded as the fuzzy membership degree of \(x^{\prime}\) to \(d_{j}\). 2. The pessimism assignment method \(D_{lower}(x^{\prime})=\{d_{j}(x^{\prime})\}_{j=1}^{L}\): If \(u_{j}=0\) and \(l_{j}=1\), then we have \(d_{j}(x^{\prime})=\underline{R}d_{j}(x^{\prime})\). The fuzzy lower approximation is regarded as the fuzzy membership degree of \(x^{\prime}\) to \(d_{j}\). 3. The mean assignment method \(D_{mean}(x^{\prime})=\{d_{j}(x^{\prime})\}_{j=1}^{L}\): If \(u_{j}=\frac{1}{2}\) and \(l_{j}=\frac{1}{2}\), then we have \(d_{j}(x^{\prime})=\frac{1}{2}(\overline{R}d_{j}(x^{\prime})+\underline{R}d_{j }(x^{\prime}))\). The mean of the fuzzy upper and lower approximation is regarded as the fuzzy membership degree of \(x^{\prime}\) to \(d_{j}\). The optimistic assignment method may lead to overestimation of the fuzzy membership degree. The pessimism assignment method easily results in the underestimation of the fuzzy membership degree. The mean assignment method can avoid the estimation problems mentioned above. However, a simple average method can easily ignore the important information included in the training data. Therefore, we propose the weighted average assignment method by learning the weights of the fuzzy upper and lower approximations of each label. The label set of \(x^{\prime}\), assigned by the weighted average method, is denoted as \(D_{weight}(x^{\prime})=\{d_{j}(x^{\prime})\}_{j=1}^{L}\), where \(d_{j}(x^{\prime})=u_{j}*\overline{R}d_{j}(x^{\prime})+l_{j}*\underline{R}d_{j }(x^{\prime})\). The fuzzy upper and lower approximation of each training instance to each label are computed. The label of each training instance is known. The weights \(u_{j}\) and \(l_{j}\) are learned by the least squares method. The optimization objective is defined as follows:  
\[