## 3 Surrogate Modeling
### Model Validation  
Before the predictor is used for inter-ply shear stress analysis, its fidelity should be verified. Model accuracy is often validated by comparing output data obtained through computer simulation and the data obtained from the developed model. To validate the developed model, computer-based simulations were conducted for the 25 sample points, and their corresponding maximum shear stress (output) was obtained. The simulation results and the model outputs, using Equation (6), were compared to verify the fidelity of the model. The following commonly known statistical methods were used for the validation of the model. * Residual (RSD): the measure of the difference between the computer based simulation results and the model outputs. The mathematical form of RSD is, \[\text{RSD}(x_{j})=f(x_{j})-\hat{f}(x_{j})\] (7) If the plot of the residual versus surrogate model shows a random distribution of points about the axis represented by the surrogate model, then this model is statistically considered as an appropriate model to be used for unknown function (inter-ply shear stress) approximation. The random distribution of points in Figure 2b shows that the model is valid for implementation. Furthermore, the linearity between the simulation results and model outputs, as shown in Figure 2a, tells us that the developed model is adequate to govern the evolution of the function to be approximated. * R-squared: the measure of how close the model versus simulation data are to the linear fit. The closeness of the data to the linear fit is shown in Figure 2a. The more the data are closer to the linear fit, the higher the value of R-squared is. The mathematical expression of R-squared is given as, \[R^{2}=1-\frac{\sum_{j=1}^{M}\left\{f(x_{j})-\hat{f}(x_{j})\right\}^{2}}{\sum_{ j=1}^{M}\left\{f(x_{j})-\hat{f}(x_{j})\right\}^{2}}\] (8) If the data lie on the linear fit, then the developed model is said to be the exact functional form of the unknown function. The R-squared value of the developed model, as shown in Table 4, indicates that the data are close enough to the linear fit. However, the value of R-squared depends on the number of independent variables. R-squared increases with the increase in the number of independent variables. Therefore, the R-squared value is not always reliable. To resolve this reliability issue, often, the number of independent variables is incorporated into the R-squared, and the modified expression is termed the \"adjusted R-squared\". This is mathematically given as: \[R_{adj}^{2}=1-\frac{(1-R^{2})(M-1)}{M-K-1}\] (9) where \(K\) represents the number of independent variables. * Root mean squared error (RMSE): measures the average model performance. The mathematical form of this measure is given as, \[\text{RMSE}=\sqrt{\frac{1}{M}\sum_{j=1}^{M}\left\{f(x_{j})-\hat{f}(x_{j}) \right\}^{2}}\] (10) The upper and lower limit of RMSE is zero and one, respectively. The value zero indicates the absence of residual variance. This means that the developed model is exactly equal to the unknown function. On the other hand, the value of one indicates that the developed model does not approximate the unknown function at all. RMSE is affected by the number of sample data points. To eliminate this effect, RMSE is normalized and hence named normalized root mean squared error (NRMSE), which is expressed as \(\frac{RMSE}{f_{max}-f_{min}}\) where \(f_{max}\) and \(f_{min}\) are the maximum and minimum values, respectively, of inter-ply shear stress (simulation results). The percentage of NRMSE is shown in Table 4. This error of 0.6 validates the adequacy of the model. * Standard deviation: measures the average amount of deviation of the predicted values from their mean.