# An Alternative Procedure for the Description of Seismic Intensity Parameter-Based Damage Potential
## 2 Methods
### Multiple Regression Analysis  
Properly selecting the most appropriate IMs for characterizing a seismic event is crucial in relating IMs with the sustained damage grade in a structure. A random selection of 30 records out of the 180 representing all degrees of estimated damage were selected as a test set and the remaining 150 were used as a training set. As there is no strict methodology for dividing the dataset into training and testing subsets and given the fact that all three damage grades (minor, moderate, or severe) should be equally represented in both the training and the testing sets, the data were split into 50-10 (training-test) instances for each damage grade subset. Considering that the usual testing set is around 20% of the initial data and the small number of available acceleration records, it was deemed appropriate that the training subset should contain enough cases to avoid jeopardizing the models' generalization capabilities. After some trial-and-error tests, it was found that the 50-10 cases split is the ideal ratio, as it provides enough instances at both training and testing procedures. Each set of 10 and 50 instances was randomly selected to ensure that the models were able to generalize and were not biased by any sequential cases. A correlation study [25] between IMs and the previously computed DIs was conducted on the training set to assess the selected Intensity Measures' predictive capacity and eliminate any irrelevant predictors. All statistical analyses were performed using the computer program STATGRAPHICS [52]. The remaining IMs, i.e., those with the highest correlation with structural damage, should all be able to differentiate between different intensity levels when different scaling factors are applied to the same input signal. Thus, when a scale factor is applied to the accelerogram, the IMs should change values accordingly. A threshold limit \(\rho_{\rm lim}\) of 90% rank correlation for the selected IMs was adopted, and only those that exhibited at least 90% rank correlation with all three DIs were used in the subsequent analysis steps. With the remaining 10 IMs and the studied 3 DIs, a series of multiple regression analyses [25] were performed. In a multiple regression analysis, the dependent variable (y) is expressed as a linear combination of a constant term (a), (n)-independent explanatory variables (x\({}_{i}\), where i = 1, \(\ldots\), n) with the corresponding coefficients (b\({}_{i}\)), and an error term (e), via the least squares method. The mathematical expression and its form in the current application are shown in Equation (5). \[\rm DI=b_{1}\cdot IM_{1}+b_{2}\cdot IM_{2}+\cdots+b_{n}\cdot IM_{n}+e \tag{5}\]  
Here, the Damage Index (DI) is the dependent variable, and the Intensity Measures (IMs) are the (n)-independent explanatory variables. Note that the constant term (a) was omitted (a = 0), as the models should not indicate any level of induced damage when a sufficiently weak (or no) excitation is provided to predict the corresponding Damage Index. Three multiple regression methods were tested to determine the best results: Ordinary Least Squares, Forward Stepwise Selection, and Backward Stepwise Selection [25]. The first is a direct method that incorporates all independent variables to generate the fitted model, thus resulting in models with many explanatory variables. Forward and Backward Stepwise Selection methods are also utilized to overcome this drawback. The former begins with an empty model and iteratively adds all relevant predictors to the model. This method usually produces models with the lowest possible number of predictors, generating simpler models. The latter achieves the same goal by starting with a model containing all the available independent variables and iteratively eliminating all irrelevant ones from the model.