## 2 Methods
### Data Analyses  
The data were analyzed to assess the differences between the younger and older groups by using the tested features. In addition to the proposed features, the tested features included COP velocity features. Conventionally, the COP velocity is characterized by its mean velocity (\(MV\)), its mean velocity in the ML direction (\(MV_{ML}\)), and its mean velocity in the AP direction (\(MV_{AP}\)). This study chose \(MV_{AP}\) and \(MV_{ML}\) as the benchmark reference features because many studies have shown that they are reliable and informative COP measures for postural steadiness [37; 38; 39; 40; 41]. This is particularly true for \(MV_{AP}\), which has been considered the most sensitive measure for assessing postural balance performance [37; 42; 43]. To compare the efficacy of the proposed features, we first used independent two-sided \(t\)-tests to compare the means of the tested features of the younger and older age groups. A statistically significant difference was indicated if \(p<0.05\). The statistical results were obtained using the mean values of the tested features over twelve trials. The sample size for both age groups was 20. The results are summarized in Table 1.  
To further compare the efficacy of the tested features, we used each of the tested features to classify the two age groups. Because the generalization capability of a classifier depends strongly on the size of the training set, this study employed a simple three-trial averaging technique to increase the number of samples. Therefore, because we have 12 trial results (4 sessions \(\times\) 3 measurements/session) for each test subject and because the number of possible combinations for selecting three objects from 12 objects \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Features**} & \multicolumn{2}{c}{**Groups**} & \multirow{2}{*}{\(p\)**-Values**} \\ \cline{2-2} \cline{4-4}  & **Younger Group** & & **Older Group** \\ \hline \(\alpha_{ML}\) & 0.995 \(\pm\) 0.003 & 0.994 \(\pm\) 0.005 & 0.635 \\ \(\beta_{ML}\) & 0.387 \(\pm\) 0.101 & 0.427 \(\pm\) 0.118 & 0.256 \\ \(MV_{ML}\) & 4.466 \(\pm\) 1.055 & 4.233 \(\pm\) 0.825 & 0.443 \\ \(\alpha_{AP}\) & 0.982 \(\pm\) 0.009 & 0.995 \(\pm\) 0.003 & 6.62E–07 \\ \(\beta_{AP}\) & 0.523 \(\pm\) 0.056 & 0.677 \(\pm\) 0.068 & 1.88E–09 \\ \(MV_{AP}\) & 5.447 \(\pm\) 0.973 & 7.278 \(\pm\) 1.990 & 6.82E–04 \\ \hline \hline \end{tabular} Values of the second and third columns are mean \(\pm\) standard deviation. Feature units: dimensionless, \(\alpha_{x}\), \(\beta_{x}\), \(\alpha_{y}\), and \(\beta_{y}\); mm/s, \(MV_{AP}\) and \(MV_{ML}\).  
\end{table} 
Table 1: Statistical results for the tested features. is 220, by using the average of three trial results as the sample to be classified, this three-trial averaging technique generated 220 samples for each subject. Consequently, this simple three-trial averaging technique increased the size of the data set from 240 samples (12 measures/person \(\times\) 20 persons) to 4400 samples (220 measures/person \(\times\) 20 persons) for both age groups. Finally, in addition to accuracy, we also computed the area under the curve (AUC) of the receiver operating characteristic (ROC) curve. The AUC is one of the most effective performance measures for binary classification problems [44]. To obtain the ROC curve for each of the tested features, we assigned the older age group to the positive class when the sample mean of the tested feature was higher in the older group than in the younger group. Otherwise, the older group was assigned to the negative class. When classifying any given sample data point, if its feature value was higher than the decision threshold, then the binary classifier assigned this data point to the positive class. Otherwise, this data point was classified as a negative class sample. If a positive class sample was correctly classified as positive, it was considered a true positive. Conversely, if a negative class sample was incorrectly classified as positive, it was considered a false positive.