*3.2. Appropriate Mother Wavelet Selection Based on Quantitative Criteria*

Additionally, the detection of gait events was also performed using two quantitative criteria, the cross-correlation coefficient and the energy-to-Shannon entropy ratio measures, to obtain an appropriate mother wavelet. The analysis results of the mother wavelets based on the cross-correlation coefficient criterion for gait event detection are presented in Figure 8. It can be seen from Figure 8a that amongst the Daubechies family, "db2" and "sym2" achieved the highest average correlation coefficient of 0.34 between the acceleration and the underlying mother wavelet followed by "gaus1" in the Gaussian family with an average value of 0.33. In other words, the "db2", "sym2", and "gaus1" mother wavelets have high similarity with the anterior-posterior acceleration signals from which the participants' gaits were detected. Meanwhile, we found that "db3", "sym3", and "gaus3" achieved the best correlation coefficients between the acceleration and the underlying mother wavelet function in their respective categories for the hemiplegic patients (Figure 8b).

The analysis results of mother wavelet characteristics based on the energy-to-Shannon entropy ratio criterion for gait event detection are presented in Figure 9. It can be observed that the average energy-to-Shannon entropy ratios for most of the investigated mother wavelets lies in the range of −0.15 to −0.13 with the "morl" mother wavelet achieving the highest ratio for healthy subjects (blue line). On the other hand, except for the "morl" mother wavelet that also provided the highest average energy-to-Shannon entropy ratio across the hemiplegic subjects (red line), the other mother wavelets had the average energy-to-Shannon entropy ratios that were in the range of −0.16 to −0.14. Although the "morl" mother wavelet function achieved a relatively higher mean energy-to-Shannon entropy ratio, however there was no significant difference between the "morl" and the other investigated mother wavelet functions based on ANOVA statistical analysis (*p* > 0.05).

**Figure 8.** Averaged cross-correlation coefficients between the anterior–posterior knee acceleration signal and specific mother wavelet function for healthy subjects (**a**) and hemiplegic subjects (**b**). Vertical dotted line indicates the standard deviation.

(**b**)

**Figure 9.** Averaged energy-to-Shannon entropy ratios of CWT coefficients for healthy subjects (blue line) and hemiplegic subjects (red line) across different mother wavelets (*p* > 0.05). Vertical dotted line indicates the standard deviation.
