## 4 Weighting Factor Estimation Method from Scheduling Results
### Feature Selection  
In this section, we describe how to combine the features extracted in Section 4.2. There are three main goals of feature selection. The three main goals are to improve prediction performance, to improve computational efficiency, and to understand the relationship between the results and the features. There are three main feature selection methods commonly used: filter methods, wrapper methods, and embedded methods [34]. The filter method statistically ranks each feature and decides whether to use it for prediction. It is computationally efficient, but because it confirms at features one by one, the combined effect of features cannot be checked. The wrapper method determines which features to select by adding (or deleting) features that will increase (or decrease) the accuracy the most from the already selected features. Compared to the filter method, the wrapper method improves accuracy, but the computational cost increases as the number of features increases. The embedded method determines the features to be selected based on the importance of each feature during model prediction. It is a well-balanced method between the filter and wrapper methods. In this study, the wrapper method is used. There are two types of wrapper methods: sequential forward selection (SFS), which adds features, and sequential backward selection (SBS), which removes features. In this study, SFS is used. Outline of wrapper feature selection method is shown in Figure 4. The procedure is as follows.  
Start with an empty set and add the feature \(z\) that improves accuracy the most to the subset \(S\).  
Add the feature \(z^{+}\) that improves accuracy the most when adding features to subset \(S\). The MSE is computed when the feature is added.  
If the state of Equation (11) is maintained, go to 2. Otherwise, exit.  
\(S\) is a subset of the selected features and \(S^{+}\) is a subset of the features when feature \(z^{+}\) is added. \(\varepsilon\) is constant, \(n_{s}\) is the number of selected features.  
\[\frac{\varepsilon}{n_{s}^{2}}>\text{MSE}\big{(}S^{+}\big{)}-\text{MSE}(S) \tag{11}\]  
Many sets of features are extracted as described in Section 4.2 and the weighting factors corresponding to each problem are prepared. Features and the corresponding weighting factors are divided into training data for training in a random forest, and test data for examining the accuracy of the machine learning model. When dividing, stratified sampling is performed so that the weighting factors are not biased. When dividing a node in random forest, the data is divided by the feature that minimizes the impure of the node. The MSE is used as an impure index when dividing a node. The division is performed so that this impure index is the smallest. After constructing a learning model from the training data, new test data that does not include the weighting factor is inputted into the model, and the weighting factor is estimated. In the regression type random forest method, the weighting factor is outputted as a continuous value. The model is evaluated by the MSE between the estimated weighting factor and the true weighting factor.