# Development of a Predictive Tool for the Parametric Analysis of a Turbofan Engine
## 4 Discussion
### Machine Learning  
Four models were generated in MATLAB's regression learner app, with the basis of selection being the least root mean square error (RMSE). Mean squared error (MSE) is the average of the squared difference of the dataset's actual and forecasted values. It computes the residual variance. RMSE is the square root of MSE and evaluates the residuals' standard deviation. Other evaluation metrics are mean absolute error (MAE), which is the absolute value of the error, and R-squared, which is the proportion of variation covered in the regression model. MSE and RMSE are the most common and prominent indicators to assess the predictive ability of a model, with lower values signifying higher accuracy. They are easy to interpret and greatly penalize larger errors. Table 5 summarizes the model and RMSE for each input variable and their respective response variables. For every input variable, a total of 24 models were trained and the model with the least RMSE was selected, as shown in Figure 8. The actual vs. predicted plot of the linear regression model and quadratic SVM are shown in Figure 9 for comparison. \begin{table}
\begin{tabular}{c c c c} \hline
**Predictor** & **Response** & **Model** & **RMSE** \\ \hline Altitude & Specific thrust & Matern 5/2 Gaussian Process Regression & 0.0044 \\  & Specific fuel consumption & Matern 5/2 Gaussian Process Regression & 0.3514 \\  & Propulsive efficiency & Matern 5/2 Gaussian Process Regression & 0.0005 \\  & Total fuel flow & Wide Neural Network & 0.0027 \\ \hline Bypass ratio and fan pressure & Specific thrust & Matern 5/2 Gaussian Process Regression & 0.0887 \\  & Specific fuel consumption & Matern 5/2 Gaussian Process Regression & 0.0022 \\  & Propulsive efficiency & Matern 5/2 Gaussian Process Regression & 9.076 \(\times\) 10\({}^{-5}\) \\  & Total fuel flow & Matern 5/2 Gaussian Process Regression & 0.0005 \\ \hline Mach number and altitude & Specific thrust & Matern 5/2 Gaussian Process Regression & 0.4207 \\  & Specific fuel consumption & Matern 5/2 Gaussian Process Regression & 0.0131 \\  & Propulsive efficiency & Matern 5/2 Gaussian Process Regression & 0.0004 \\  & Total fuel flow & Wide Neural Network & 0.0016 \\ \hline Turbine inlet temperature & Specific thrust & Matern 5/2 Gaussian Process Regression & 0.2970 \\  & Specific fuel consumption & Squared Exponential Gaussian Process & 0.0133 \\  & Propulsive efficiency & Matern 5/2 Gaussian Process Regression & 0.0004 \\  & Total fuel flow & Matern 5/2 Gaussian Process Regression & 0.0005 \\ \hline \end{tabular}
\end{table} 
Table 5: Machine learning results. Figure 9 shows the data points being located far away from the unity line, as the RMSE for linear regression and quadratic SVM was greater than that for Gaussian process regression. Hence, these models were not selected. The predicted vs. actual response graphs for each model are shown in Figures 10-13. The data points were located close to the unity line, proving that the regression models have successfully predicted the response values. Gaussian process regression and wide neural network models performed better than the rest of the regression models as the RMSE was closer to zero and the predicted values were distributed on the unity line. Figure 10: Predicted response vs. actual response plots for (**a**) specific thrust vs. altitude; (**b**) specific fuel consumption vs. altitude; (**c**) propulsive efficiency vs. altitude; and (**d**) total fuel flow vs. altitude. Figure 9: Predicted response vs. actual response plots for the (**a**) linear regression model; and (**b**) quadratic SVM. Figure 11: Predicted response vs. actual response plots for (**a**) specific thrust vs. design bypass ratio and outer fan pressure ratio; (**b**) specific fuel consumption vs. design bypass ratio and outer fan pressure ratio; (**c**) propulsive efficiency vs. design bypass ratio and outer fan pressure ratio; and (**d**) total fuel flow vs. design bypass ratio and outer fan pressure ratio.