## 2 Materials and Methods
### Data Analysis  
As shown in Figure 2, during the collection process of the smartwatch in-air signatures, all of the data gathered from the accelerometer and gyroscope were time-series signals; the three dimensions were x, y, and z-axis data. The time stamps are shown as deep blue to red colors, with blue representing an earlier time stamp and red representing a later time stamp. Figure 2 depicts the 10 genuine signatures of the same participant. The dynamic time warping algorithm (DTW) [14] can be used to verify the validity of the dataset. The DTW algorithm can detect distances among various time-series signals [6; 31]. The distance measurement does not simply follow the Euclidean distance. The traditional Euclidean distance can only measure the static distance between two points while the DTW algorithm can be used to measure the distance of signals of different lengths. The DTW algorithm has many variations [32] and is widely applied to various time-series signal applications [33], including natural language processing [34]. \[c_{p}(X,Y):=\sum_{\ell=1}^{L}c(x_{n_{\ell}},y_{m_{\ell}}) \tag{1}\]  
As shown in Algorithm 1[35], the DTW [14] algorithm can create one-to-many alignments among two different signals as shown in Figure 3, in which the two signals are of different lengths. Those two accelerometer signals are samples taken from the same participant but of different categories, genuine signatures, and forgery signatures. The blue and the orange lines represent the fluctuations of two signals while the yellow line connecting the two signals is the alignment between the valley and peaks of two signatures. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & & & \multicolumn{6}{c}{**Smartwatch Data Format**} \\ \hline
**WatchGyroX** & **WatchGyroY** & **WatchGyroZ** & **WatchAccX** & **WatchAccY** & **WatchAccZ** & **WatchAtX** & **WatchAtY** & **WatchAtZ** \\ \hline
0.02570195 & \(-\)0.03224608 & \(-\)0.0120639 & \(-\)0.01228451 & \(-\)0.00830266 & 0.00186165 & 0.21137402 & 0.21104763 & \(-\)0.75536253 \\
0.02712557 & \(-\)0.03940424 & \(-\)0.0088675 & \(-\)0.00624303 & \(-\)0.00330171 & 0.00035724 & 0.2120575 & 0.21096044 & \(-\)0.75540958 \\
0.03478016 & \(-\)0.0445323 & \(-\)0.0054437 & \(-\)0.00450777 & 0.00784102 & \(-\)0.00614287 & 0.21283413 & 0.21098173 & \(-\)0.75545861 \\
0.05405451 & \(-\)0.04379743 & \(-\)0.01255126 & \(-\)0.00808688 & 0.01724837 & \(-\)0.0044408 & 0.21344206 & 0.21110055 & \(-\)0.7555888 \\ \hline... &... &... &... &... &... &... &... &... \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Smartwatch in-air signature data format. Figure 2: Smartwatch accelerometer data of 10 genuine signature signals from the same participant. The dynamic time warping cost matrix can be constructed as shown in Figure 4. The right and the top parts of the figure show the two in-air signatures; one of the signatures is from the genuine category of the participant while the other is from the forgery category of the participant. The dynamic time warping cost matrix shows the graduate changing from a dark blue color to a light yellow color; the dark blue color represents a lower alignment cost between the specific segments of the two matrices while the yellow color represents a high cost of alignment. Following the lower cost of the matrix, a warping path, shown as the red-colored line in the figure, goes from the origin to the last element of the matrix, and the total cost for the wrapping path can be calculated based on Formula (1) [36], in which \(c_{p}\) represents the total cost while \(c\) represents the local cost. The last element of the matrix presents the accumulative alignment cost of the smartwatch's in-air signatures, and the specific cost value is shown at the top-right corner of the figure. Figure 3: Dynamic Time Warping Alignment of two signature signals. The blue line and the orange line represent two signatures respectively while the yellow line shows alignment.