## 2 Materials and Methods
### Datasets and Evaluation Metrics
The dataset consists of multiple txt files, with each file recording three-axis data (total of six axes) from the accelerometer and gyroscope of an intelligent wristband worn by different experimental subjects. The accelerometer and gyroscope data are sampled at a rate of 25 Hz. The accelerometer has a range of \(\pm\)8 G with a resolution of 4096/1 G (one gravity is represented by the value 4096). The gyroscope has a range of \(\pm\)2000 \({}^{\circ}\)/s with a resolution of 16.4/(1 \({}^{\circ}\)/s) (one degree per second is represented by the value 16.4). The dataset includes six types of motions: walking, jogging, sitting, waving, squats, and jumping jacks. For jumping jacks, due to variations in the physical abilities of the data collectors, some files contain rest phases between actions (performing action, then rest, then performing action), where the subject remains still during the rest phase. This experiment only uses data for jumping jacks and jogging, containing 74,314 samples, with a balanced class distribution. In this paper, the Area Under the ROC Curve (AUC) is used as the evaluation metric. The Receiver Operating Characteristic (ROC) curve is a graphical representation of the classification model's performance at different thresholds. It plots the relationship between False Positive Rate (FPR) and True Positive Rate (TPR). The confusion matrix for binary classification [25] is defined as follows: \begin{tabular}{c|c|c} Actual Predict & Positive (P) & Negative (N) \\ _ine_Positive (P) & TP (True Positive) & FN (False Negative) \\ Negative (N) & FP (False Positive) & TN (True Negative) \\ \end{tabular} \begin{table}
\begin{tabular}{c c} \hline \hline
**Feature Name** & **Meaning** \\ \hline x & x-coordinate of the sensor \\ \hline y & y-coordinate of the sensor \\ \hline z & z-coordinate of the sensor \\ \hline
010-000-024-033 & Sensor 1 data \\ \hline
010-000-030-096 & Sensor 2 data \\ \hline
020-000-032-221 & Sensor 3 data \\ \hline
020-000-033-111 & Sensor 4 data \\ anomaly & Anomaly label (whether a fall occurred) \\ \hline \hline \end{tabular} The data include multi-axis sensor measurements to detect falls.  
\end{table} 
Table 3: Fall Detection Data Features and Their Meanings. Then, the formulas for calculating \(FPR\) and \(TPR\) are  
\[TPR(\text{True Positive Rate}) =\frac{TP}{TP+FN} \tag{11}\] \[FPR(\text{False Positive Rate}) =\frac{FP}{FP+TN}\]  
AUC is the area under the ROC curve, typically ranging between 0 and 1. The closer the AUC is to 1, the better the classification performance of the model. If the AUC is close to 0.5, it indicates the model's performance is near random guessing, and if AUC is below 0.5, the model performs poorly. AUC is a robust performance metric for binary classification, especially in cases with imbalanced data, as it is unaffected by changes in the class distribution and provides a more stable performance evaluation standard. The AUC calculation method is as follows [26]:  
\[\text{AUC}=\frac{\sum{I(P_{pos},P_{neg})}}{M\times N} \tag{12}\] \[I(P_{pos},P_{neg})= \begin{cases}1,&P_{pos}>P_{neg}\\ 0.5,&P_{pos}=P_{neg}\\ 0,&P_{pos}<P_{neg}\end{cases} \]