## 3 Method
### Self-Training Method  
The self-training method falls under the pseudo-labeling approach of SSL [59]. This method iteratively trains its supervised classifier with both labeled data and data which have been pseudo-labeled in the previous iterations. Initially, the self-training method trains its supervised classifier only with labeled data. Then, it employs this trained classifier to predict labels for the unlabeled data points. The predictive labels, having the highest confidence, are added to the training dataset. This extended training dataset, consisting of both initial labeled data and new pseudo-labeled data, is utilized to retrain the supervised classifier. This iteration typically continues until all unlabeled data are pseudo-labeled [60].  
In line with this self-training method, we propose a semi-supervised explainable BRBES framework in this paper. This framework trains the BRBES with both labeled and pseudo-labeled data to provide more accurate prediction and explanation. As shown in Figure 1, this framework consists of the following three major components:  
* **Prediction Model:** For the prediction model, we use the BRBES, a symbolic AI model, to provide domain knowledge based prediction while handling associated uncertainties (Figure 1a). We then optimize the initial BRBES with supervised labeled training data using JOPS. Such optimization makes the initial BRBES supervised. We optimize the supervised BRBES unless its accuracy reaches the confidence threshold. If the accuracy reaches this threshold, the supervised BRBES will become a confident BRBES. This confident BRBES will be able to predict pseudo-labels for the unlabeled data points with due confidence. (Figure 1b).
* **Data Augmentation:** To overcome the shortage of labeled data, we synthetically generate unlabeled data. For this purpose, we leverage two kinds of augmentations: \"weak\" and \"strong\". Theses two types of augmentations are briefly mentioned below.  
**Weak Augmentation:** For weak augmentation, we generate unlabeled integer values for relevant antecedent attributes of BRBES. We predict labels for these weakly augmented unlabeled data points with confident BRBES. We then train the confident BRBES with both labeled and weakly augmented pseudo-labeled datasets. If training with this weakly augmented pseudo-labeled dataset improves the accuracy, the confident BRBES will transition to a weakly semi-supervised BRBES (Figure 1c-g).  
**Strong Augmentation:** A dataset has two types of noise: class noise and attribute noise [61]. Class noise is triggered against incorrectly assigned class label to a data instance. On the other hand, attribute noise reflects erroneous or missing values for one or more input features (independent variables) of the dataset [61]. To reduce both the class and attribute noise of the weakly augmented pseudo-labeled dataset, we apply strong augmentation. For strong augmentation, we synthetically generate unlabeled float values for the same antecedent attributes of BRBES as weak augmentation. We then predict labels for these strongly augmented unlabeled data points with a weakly semi-supervised BRBES. We optimize the weakly semi-supervised BRBES with labeled, as well as weakly and strongly augmented pseudo-labeled, datasets. If training with  
Figure 1: System Architecture of semi-supervised explainable BRBES framework to provide energy consumption prediction.  
the strongly augmented pseudo-labeled dataset improves the accuracy, weakly semi-supervised BRBES will transition to a semi-supervised BRBES (Figure 1h-l).
* **Prediction and Explanation:** This semi-supervised BRBES provides more accurate prediction, along with an explanation and counterfactual, through a user interface (Figure 1m,n).