# Highway Travel-Time Forecasting with Greenshields Model-Based Cascaded Fuzzy Logic Systems
## 1 Introduction
The extended Kalman filter (EKF) has also been applied in second-order traffic flow models to enhance traffic status estimation using only fixed sensors. In [24; 25], three mixed estimation methods are compared, respectively, to highlight their respective strengths. A traffic optimization decision system in [26] demonstrated iterative updating of model parameters to improve predictions. However, many existing highway travel-time prediction methods still rely heavily on historical data and static models, which often struggle to maintain accuracy when traffic flow conditions change significantly. Non-parametric models for travel-time prediction typically involve various machine learning techniques and neural network models [10]. These models often require longer training times due to the large volumes of data they process. Many machine learning approaches for travel-time prediction are relatively straightforward--for example, using support vector regression (SVR) for travel-time estimation [27]. A method that enhances travel-time prediction through linear regression is demonstrated in [28]. A genetic algorithm (GA)-based scheme for predicting travel-time intervals is introduced in [29], while a hybrid model combining GA and SVM is proposed in [30] to improve bus arrival time accuracy. Based on large datasets, a dynamic K-nearest neighbors (K-NN) model is presented in [31] to enhance travel-time prediction accuracy. In [32], the K-NN method is integrated with clustering and principal component analysis to create a framework for bus arrival time prediction. Neural networks have also been applied in various contexts. For short-term travel-time prediction on interurban highways [33]. A graph neural network estimator using Google Maps data and meta-gradient-based training schedules is proposed for travel-time prediction in [34]. In addition, time-series linear regression has been combined with eXtreme Gradient Boosting (XGBoost), and Gated Recurrent Unit (GRU) neural networks have been used to develop several high-performance vehicle travel-time prediction methods [4; 35]. A neural network incorporating a fuzzy logic scheme for travel-speed prediction is presented in [36]. Fuzzy logic is a valuable approach for situations where information is imprecise and the relationships among various factors are complex. In fuzzy logic systems, rules based on expert knowledge are used to define connections between these factors. As a result, fuzzy logic is well-suited for systems where precise mathematical modeling is difficult or impractical. Its ability to manage uncertainty and vagueness makes it particularly effective for capturing the complexities of traffic flow in ITS [37]. In general, fuzzy logic systems offer advantages over traditional traffic modeling methods, often providing more accurate and robust predictions. Several fuzzy logic systems based on traffic models have been proposed for assessing traffic congestion levels [38; 39]. In [40], a fuzzy logic scheme combined with an SVM-based model is used for real-time traffic congestion prediction on road segments. For rating traffic congestion levels, a machine learning method integrated with a fuzzy comprehensive evaluation scheme (MF-TCPV) is presented in [5]. Additionally, a fuzzy inference system is developed in [41] to select appropriate modes for a dynamic and intelligent traffic light control system based on real-time traffic information. Traffic conditions are inherently unpredictable and influenced by many dynamic factors. Fuzzy logic provides an effective approach to incorporating expert knowledge and experience into traffic models while reducing modeling complexity. This leads to more realistic and effective predictions. Both fuzzy logic and neural networks are widely utilized in modern ITS for tasks such as pattern recognition, classification, and predictive modeling. The selection between these approaches depends on the complexity of the problem, the nature and availability of data, and the need for interpretability.