## 2 Algorithm and Methodology
### Sliding Window Model
The membership functions for each class are defined as follows:

Awake (left-shoulder trapezoidal function):

\[\mu_{\text{Awake}}(F_{\text{BYN}})=\left\{ \begin{array}{ll}1,&F\leq 0.15\\ \frac{0.30-F}{0.15},&0.15<F\leq 0.30\\ 0,&F>0.30\end{array} \right. \tag{10}\]

Mild Fatigue (triangular function):

\[\mu_{\text{Mild}}(F_{\text{BYN}})=\left\{ \begin{array}{ll}0,&F\leq 0.20 \;\;\text{or}\;F\geq 0.45\\ \frac{F-0.20}{0.10},&0.20<F\leq 0.30\\ \frac{0.45-F}{0.15},&0.30<F<0.45\end{array} \right. \tag{11}\]Moderate Fatigue (triangular function):

\[\mu_{\text{Moderate}}(F_{\text{BYN}})= \begin{cases}0,&F\leq 0.35\;or\;F\geq 0.65\\ \frac{F-0.35}{0.15},&0.35<F\leq 0.50\\ \frac{0.65-F}{0.15},&0.50<F<0.65\end{cases}  \tag{12}\]

Severe Fatigue (right-shoulder trapezoidal function):

\[\mu_{\text{severe}}(F_{\text{BYN}})= \begin{cases}1,&F\leq 0.55\\ \frac{F-0.55}{0.15},&0.55<F\leq 0.65\\ 0,&F>0.65\end{cases}  \tag{13}\]

The final fatigue level is determined using the maximum membership principle:

\[Y=\text{arg max }\mu_{l}(F) \tag{14}\]

\[l\in\{\text{Awake, Mild, Moderate, Severe}\}\]

This fuzzy classification mechanism significantly enhances the continuity and interpretability of fatigue detection. It reduces instability caused by hard thresholds and provides a more flexible and physiologically consistent approach to identifying driver fatigue states. The corresponding membership curves are illustrated in Figure 7.

## 3 Experimental Results and Analysis

This article evaluates the performance of our proposed facial multi-feature fusion for fatigue detection. Initially, the algorithm's feasibility and robustness were verified through online experiments with 12 participants, divided into three groups (A, B, and C), each consisting of 4 subjects. All participants completed a 30 min simulated driving task in a laboratory designed to replicate real-world conditions, including ambient noise and driving visuals. To simulate different fatigue levels, experiments were conducted during both afternoon and late-night hours. Within each group, two participants performed the task in the afternoon, and two others during late-night sessions. This schedule ensured variations in mental alertness and induced mild to moderate fatigue. Participants followed a controlled daily schedule with limited rest, and caffeine intake was prohibited 3 h before the test. The experimental protocol was reviewed and approved by the institutional ethics committee. Figure 7: Fuzzy membership functions used for classifying fatigue levels based on the normalized fatigue score \(F_{\text{BYN}}\). Additionally, 45 randomly selected video segments from the YawDD dataset were used to further assess the algorithm's accuracy in an offline setting, confirming the effectiveness of the personalized thresholds and the algorithm's overall performance. The data extraction and analysis platform used the programming language python 3.9 operated in a PyCharm environment based on the Windows 11 system. In addition, the Dlib toolkit was used to detect the face profile information and estimate the head posture for the 3D coordinate establishment. The camera provided an image resolution of 640 \(\times\) 480 pixels, meeting the image acquisition requirement. ### Changes in \(\eta_{NAE}\), \(\eta_{NAM}\), and \(HE\alpha\) Values for Fatigue Detection

Figure 8 presents the blinking, yawning, and head movement behavior recorded from a single subject (participant \"e\") among the 12 participants. This sample is used to illustrate the dynamic changes in \(\eta_{NAE}\), \(\eta_{NAM}\), and \(\text{HEA}\alpha\) values across different fatigue states. Figure 8a,d,h record the changes in \(\eta_{NAE}\), \(\eta_{NAM}\), and \(\text{HEA}\alpha\) values of an alert individual over a sliding window of 1800 test frames (approximately one min). Specifically, the \(\eta_{NAE}\) values remain mostly on a stable baseline, as shown in Figure 8a. When normal blinking occurs, a significant drop in the \(\eta_{NAE}\) values is observed.