# Energy Performance Forecasting of Residential Buildings Using Fuzzy Approaches
## 4 Results and Discussion
### Feature Selection Results and Discussion  
As described in the previous section, FIR finds that two of the eight input variables, RC and GA, have the highest causal relation with the outputs and are enough to predict the heating and cooling loads. This is an interesting result. On the one hand, it is consistent with [PERSON] and [PERSON]'s outcomes that claim the GA is the most important predictor for both HL and CL. On the other hand, it allows the conclusion that the remaining six variables, SA, WA, RA, OH, O and GAD, are redundant or irrelevant. Again, this is consistent with the previous work that concludes variables RC, SA, WA, RA, and OH appear reasonably strongly correlated to the output variables, and also finds that some input variables are highly correlated between them. Based on the FIR feature selection process, it is reasonable to think that the relative compactness variable, RC, includes the information of other relevant variables involved in the study, such as SA or RA. In fact, this makes sense because there is an analytic formula linking RC, SA and the volume [17]. It is clear that the WA variable is directly related to the GA, making it redundant. Therefore, the five variables that appear reasonably strongly correlated to the output variables contain redundant information if RC and GA are already selected. Based on these results, and with the goal to validate the FIR feature selection process, it has been decided to use the selected features, RC and GA, as the only input variables to find a new ANFIS model. If the rest of the variables are redundant, as the feature selection process of FIR states, ANFIS should be able to model the heating and cooling loads using only RC and GA. Note that the ANFIS models presented in Section 3 used the whole set of input variables. To perform this study, different parameters of the ANFIS methodology are evaluated. Now the number of input variables is reduced from 8 to 2, allowing experimentation with different parameter sets involved in ANFIS modelling: number of epochs (i.e., 50, 100, 500, 1000, 2000) and number of classes of the input variables or partition (i.e., 3-2 and 5-2). All of the experiments use Gaussian membership functions, linear output function, and the hybrid optimization method. Notice that in this experiment it is possible to work with a linear  
Figure 10: Real versus ANFIS and FIR prediction results for heating load (upper plot) and cooling load (lower plot) models. Each point in the plot represents a building. The results of the Cooling Load correspond to fold #1 in one of the 10 iterations. MAE and RMSE obtained by the ANFIS model in this fold are 1.08 and 1.76, respectively. MAE and RMSE obtained by the FIR model in this fold are 1.16 and 1.74, respectively. The results of the Heating Load correspond to fold #7 in one of the 10 iterations. MAE and RMSE obtained by the ANFIS model in this fold are 0.54 and 0.68, respectively. MAE and RMSE obtained by the FIR model in this fold are 0.42 and 0.61, respectively. output function since the number of rules for the rule set is 6 in the case that the chosen partition is 3-2, and 10 if the chosen partition is 5-2. The results obtained for the heating and cooling loads using the different set of parameters and performing an average of the errors obtained when executing the 10-fold CV 100 times are summarized in Tables 3 and 4, respectively. Comparing Table 2 vs. Tables 3 and 4 it is clear that there are ANFIS models that reach the same level of accuracy, or even a little higher, than the ANFIS models that use the whole set of 8 features. The best results (shaded cells) are obtained when RC and GA variables are discretized into 5 and 2 classes (5-2), respectively. Fifty epochs is enough to derive a good CL model; increasing the number of optimization steps do not improve the model. The HL models need at least 100 epochs to be optimized properly. Note that in these experiments a linear output function is always selected.