# Highway Travel-Time Forecasting with Greenshields Model-Based Cascaded Fuzzy Logic Systems
## 4 Simulation Results and Discussions
### Greenshields Models Based on Regression Analysis  
Many regression analysis methods can be applied to establish Greenshields models. In this study, polynomial regression is used to construct models based on traffic data collected from the segment between Dingjin and Rende on the Sun Yat-Sen Highway in Taiwan [42], for validation and comparison with the proposed fuzzy logic system. The regression process extracts key information from the collected data and smooths out noise present in the measurements. The resulting models in the non-congested mode include the relationship curve between vehicle speed and traffic density shown in Figure 21, the graph of traffic flow versus traffic density illustrated in Figure 22, and the diagram of vehicle speed versus traffic flow presented in Figure 23.  
Figure 21: Relation graphs between vehicle speed and traffic density in non-congested mode: collected data and regression curve. Figure 22: Relation graphs between traffic flow and traffic density in non-congested mode: collected data and regression curve. To evaluate the model fit and predictive performance of each regression model, statistical analyses were conducted for the regression plots shown in Figures 21-23. The corresponding results are summarized in Tables 9-11. While regression slope estimates can be significantly affected by high-leverage points, the collected data show no such influence (leverage point value: 0), indicating the absence of significant outliers. The quality of the slope estimator, as reflected by the standard deviation (SE Coef) of \(\beta_{1}\), is influenced by two key factors: the error variance of the mean square error (MS value) and the degree of variability in the independent variable. A higher error variance for a given level of the independent variable makes it more challenging to obtain an accurate estimate of the regression line. Notably, the standard deviation values (SE Coef) of \(\beta_{1}\) are small in each model. When compared with their corresponding slope parameter values (Coef), this confirms that the slope estimates are reliable and accurately reflect the underlying process. Similarly, the intercept estimates \(\beta_{0}\) also exhibit low standard errors (SE Coef) relative to their parameter values (Coef), indicating minimal extrapolation and enhancing the credibility of the models. \(T\)-tests were performed on each \(\beta_{1}\) parameter, and the resulting \(p\)-values for the two-tailed tests are effectively zero. This confirms that all slope coefficients are statistically significant. The 95% confidence intervals--specified by the lower (Lwr 95%) and upper (Upr 95%) bounds--for both the intercept and slope values, as reported in Tables 9-11, further demonstrate the precision of the estimates and suggest minimal uncertainty. All models also report high \(R\)-squared (\(R^{2}\)) values, approaching 1, indicating a strong fit between the regression line and the observed data. The standard error of the regression (SER) quantifies the average deviation between observed values and the predicted values on the regression line. Each model demonstrates a low SER, indicating a high prediction accuracy. Specifically, in Table 9, the SER for the estimated vehicle speed is 3.83 km/h, confirming a high degree of precision in the speed prediction. Although the regression analysis can offer a high predictive accuracy with statistically significant coefficients and low standard errors, it is generally a slow and time-consuming process and less adaptable to dynamic traffic conditions. When conditions or the environment change, new parameters or parameter updates require a substantial amount of additional data for retraining. Despite these limitations, regression models are valuable for examining and verifying the relationships among variables. According to [PERSON] theory, each curve provides useful insights into the relationships between vehicle speed and traffic density, traffic flow and density, and vehicle speed and traffic flow.