## 3 Method
### Proposed Semi-Supervised Explainable BRBES Framework
BRBES, we selected the optimum number of referential values of its antecedent attributes through Structure Optimization based on Heuristic Strategy (SOHS) algorithm [49]. The new structure's learning parameters are then used to optimize the parameters of BRBES. Until the structure does not change further, these iterations continue. Thus, we jointly optimize both the parameters and structure of the BRBES with the JOPS algorithm [49]. The flowchart of this joint optimization is presented in Figure 3. To determine the optimal parameters and structure of the BRBES, JOPS uses the generalization error instead of the training error [49]. A BRBES with a small training error on its training dataset may not be able to generate ideal inference output on its testing dataset due to overfitting [49]. On the other hand, the generalization capability, quantified by generalization error, is an important criterion to evaluate a prediction model's effectiveness in delivering stable results on data not encountered during training [77; 78; 79]. Using the Hoeffding inequality theorem of probability theory [80], it has been demonstrated that the generalization error is a more useful criterion than the training error for identifying the ideal parameters and structure of the BRBES [49]. An upper bound on the likelihood that the sum of random variables will differ from its expected value is provided by this Hoeffding inequality theorem. Hence, the JOPS selects the BRBES with the minimum generalization error as the most optimal BRBES. Thee generalization error, ge, is shown in Equation (1):  
\[ge=\frac{\sum_{i=1}^{T}\left|f(x_{i})-y_{i}\right|}{T}+\sqrt{\frac{(d\ln 2- \ln r)(u(D_{N})-u(D_{1}))^{2}}{2T}} \tag{1}\]  
Here, the predictive output of the BRBES for the i\({}^{th}\) input data is represented by f(x\({}_{i}\)), i\({}^{th}\) input data's actual output is y\({}_{i}\), the total number of independent input-output data pairs (x\({}_{i}\), y\({}_{i}\)) is quantified by T, the total number of referential values of all antecedent attributes is symbolized by T, r is the probability value within the interval (0, 1], and the interval of the predictive output f(x) is within [u(D\({}_{1}\)), u(D\({}_{N}\))]. We utilize labeled training dataset to optimize BRBES with JOPS. With this optimization, the initial BRBES transitions to a supervised BRBES, as shown in Equation (2):  
\[BRBES_{supervisd}=JOPS(BRBES_{initial}(il)) \tag{2}\]  
Here, _il_ defines the initial labeled training data. In the next step, we ensure that the accuracy of this supervised BRBES reaches a certain confidence threshold. Figure 3: Flowchart of joint optimization of BRBES. **Confident BRBES. ** As part of self-training-based semi-supervised regression, we generate unlabeled data through weak and strong augmentation. Unlabeled data refer to a set of antecedent attributes, where the actual value of the corresponding consequent attribute is unknown. In this research, we generate data through augmentation for the antecedent attribute \"interior space\" of the BRBES. These augmented data are unlabeled because the actual values of their corresponding consequent attributes (crisp value of energy consumption in kWh) are unknown. We then predict the value of energy consumption against unlabeled data of the antecedent attributes using a supervised BRBES. This predictive value of energy consumption is the pseudo-label, which we assign to the unlabeled data. However, before predicting this pseudo-label, the accuracy of the supervised BRBES has to reach the confidence threshold. Otherwise, this BRBES will not be confident enough to predict the pseudo-label with reasonable accuracy. To evaluate this confidence level of the accuracy of supervised BRBES, we use contrastive loss [81]. This contrastive loss measures how similar a predictive output is to the actual output.