## 5 Results  
### Comparative Performance Analysis of the Examined MLAs  
The selection of the proper MLA in modelling the seismic demand prediction of the examined RC frame under single and multiple ground motion records is of outmost importance. This selection has to be made by taking into account the particularities of the current data, the EDA and restrictions of the examined algorithms. In order to obtain the most efficient algorithm, a thorough comparatively investigation among 10 different MLAs was performed. Their performance was assessed by conducting sensitivity and accuracy analysis regarding the estimated errors obtained by the provided data. The ten examined MLAs are: the Adaboost regressor (ABR) [65], the Bayesian ridge (BR) [66], the decision tree regressor (DTR) [67], the extra trees regressor (ETR) [68], the gradient boosting regressor (GBR) [69], the K-nearest neighbors (KNN) [70; 71], the light gradient boosting machine (LGBM) [72], the linear regressor (LR) [73], the multi-layer feed-forward neural network (MLNN) [74], and the random forest regressor (RFR) [75]. In this sense, an extensive and detailed comparison of 10 different MLAs on the two provided datasets was carried out. The MLAs are implemented using the Scikit-learn [76] and LightGBM [72] Python packages, while they are evaluated with Yellowbrick library [77; 78]. It should be said that the following metrics [73] for the comparison and cost analysis of the correct regression errors were taken into account and are listed below:  
Figure 7: Heatmap of Predictive Power Score (PPS) for every pair of the examined variables including input features and targets. Mean absolute error (MAE) is a measure of errors between the estimated and the observed values, and it is given by the following expression: \[\text{MAE}=\frac{1}{n}\sum_{i=1}^{n}\ |\ \hat{y}_{i}-y_{i}\ |\] (7) where \(\hat{y}_{i}\) is the predicted value, \(y_{i}\) the real value of the ith observation and n is the total number of observations; 2. Mean square error (MSE): \[\text{MSE}=\frac{1}{n}\sum_{i=1}^{n}(\hat{y}_{i}-y_{i})^{2}\] (8)
3. Root-mean-squared error (RMSE) calculates the average error between the estimated values and the observed values: \[\text{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^{n}(\hat{y}_{i}-y_{i})^{2}}\] (9)
4. The coefficient of determination, \(R^{2}\), expresses the variation in the dependent variable that is predictable from the independent variables: \[R^{2}=1-\frac{\sum_{i=1}^{n}(y_{i}-\hat{y}_{i})^{2}}{\sum_{i=1}^{n}(y_{i}- \overline{y}_{i})^{2}}\] (10) where \(\overline{y}\) is the average of the observed values; 5. Root-mean-squared-log-error (RMSLE) is an extension of MSE that is used mainly when the predicted values display high deviation: \[\text{RMSLE}=\sqrt{\frac{1}{n}\sum_{i=1}^{n}(\log(\hat{y}_{i}+1)-\log(y_{i}+1 ))^{2}}\] (11)
6. Mean absolute percentage error (MAPE) calculates the accuracy, as a ratio, and is defined by the following formulation: \[\text{MAPE}=\frac{1}{n}\sum_{i=1}^{n}\left|\frac{y_{i}-\hat{y}_{i}}{y_{i}}\right|\] (12)  
In order to thoroughly assess the ML methods, 30 percent of the total dataset was withheld and was employed as a test set for the final assessment. In the remaining part, we trained and evaluated all of the examined algorithms using the K-fold cross-validation strategy [76]. During this method, the above part is divided in K subsets. Each randomly defined subset consists of different observations. One subset is used as the cross-validation subset, while the K-1 others are merged and used as the training set. This process is performed K times using different sets as the validation set and the K-1 rest of them as the training set. The performance of the MLA is evaluated for each case and on average. By this, the performance of the method in relation to the prediction error is determined. Specifically, the statistical properties, the bias and the variance of the regression prediction error are recorded and analyzed using a 10-fold cross-validation procedure.