## 3 Method
### Proposed Semi-Supervised Explainable BRBES Framework
For the contrastive learning loss function, we use InfoNCE loss [82], as shown in Equation (3):  
\[InfoNCE_{loss}=-\frac{1}{N}\sum_{i=1}^{N}log\frac{exp(sim(z_{i},z_{i}^{+}))}{ \sum_{j=1}^{N}exp(sim(z_{i},z_{j}))} \tag{3}\]  
where \(z_{i}\) is an anchor, which represents a labeled data point with actual output in the test dataset; \(z_{i}^{+}\) is the predictive output that is similar (positive pair) to the anchor; \(z_{j}\) is the predictive output that is dissimilar (negative pair) to the anchor; N is the total number of anchors in the test dataset; and the negative sign is intended to minimize the loss. \(sim()\) is the similarity function which measures similarity and dissimilarity of the anchor with positive and negative predictive output, respectively, as shown below. \[sim(z_{i},z_{i}^{+})= \begin{cases}1&\text{if }|\text{(actual output) - (predictive output)}|\leq\text{(actual output) * 10\%}\\ 0&\text{otherwise}\end{cases} \]  
\[sim(z_{i},z_{j})= \begin{cases}-1&\text{if }|\text{(actual output) - (predictive output)}|>\text{(actual output) * 10\%}\\ 0&\text{otherwise}\end{cases} \]  
Here, we consider 10% as the similarity threshold of the \(sim()\) function, which returns 1 for a positive pair (similarity with anchor) and \(-1\) for a negative pair (dissimilarity with anchor). The exponential in Equation (3) keeps the similarity positive and amplifies the differences in similarity scores. The log scales the loss to ensure that the small values do not dominate the optimization process. Thus, it maximizes the probability of correctly identifying the positive pair. In Table 7, we show different values of the InfoNCE loss function against various levels of similarity. According to this table, higher similarity results in higher value of this loss function. In our proposed framework, we take minimum 80% similarity (InfoNCE loss 0.60 or above) as the accuracy threshold, which represents the confidence level of the predictive output of the supervised BRBES. If the accuracy reaches this confidence threshold, the supervised BRBES will transition to a confident BRBES. Otherwise, we review the rule base (Table 5) of initial BRBES and optimize the initial BRBES again using Equation (2). Transition from the supervised BRBES to the confident BRBES is shown below. \[BRBES_{supervised}= \begin{cases}BRBES_{confident}&\text{if (InfoNCE}_{loss}\geq 0.60)\\ \text{Table 5 and Equation (2)}&\text{otherwise}\end{cases} \]After transitioning to the confident BRBES, our next step is to generate unlabeled data through weak augmentation and assign pseudo-labels to the unlabeled data. **Weak Augmentation. ** In this step, we apply weak augmentation to generate unlabeled data for the antecedent attribute \"interior space\" of BRBES (Figure 1c). For interior space, we generate integer values from 10 (minimum interior space) to 200 (maximum interior space), with an interval of 1. With these generated values, we cover every possible whole number between the minimum and maximum interior space. For solar illumination and interior inhabitance, we do not generate any unlabeled data through augmentation because all possible time instances and seasonal variations are covered in Table 2 (solar illumination) and Tables 3 and 4 (interior inhabitance). Hence, for these two antecedent attributes, we generate float values from 0 to 1, as specified in Table 2 (solar illumination) and Tables 3 and 4 (interior inhabitance). With these generated values, we formulate a weakly augmented unlabeled dataset. This dataset combines every single possible integer value of interior space with specific values of solar illumination and interior inhabitance. **Pseudo-Label for Weakly Augmented Unlabeled Data. ** We then employ confident BRBES to predict the value of consequent attribute 'energy consumption' against each set of unlabeled antecedent attributes (Figure 1d,e). Thus, with these predictive values, we assign pseudo-labels, \(pl_{w}\), to the weakly augmented unlabeled dataset (Figure 1f).