## 3 Results and Discussions
Therefore, sets of \(\lambda\) with higher values than \(1\times 10^{-8}\) were applied to the raw data (Figure A3). Smoothing using \(\lambda=1\times 10^{-6}\), resulted in undersmoothed first and second-derivative curves (Figure A3a). Noise and fluctuations in the derivative curves remained when smoothing using \(\lambda=1\times 10^{-5}\), although some noise was reduced tremendously as the value of \(\lambda\) increased (Figure A3b). The right hip F/E angle and its first and second-derivative curves were indeed smoother as higher \(\lambda\) values were used. However, as \(\lambda\) reached \(1\times 10^{-4}\), they started to lose the pattern and original traits of the raw data, which might have had a significant effect on interpretation of the data (Figure A3c). Using a higher \(\lambda\), \(1\times 10^{-3}\), resulted in oversmoothed first and second-derivative curves (Figure A3d). It appeared that for all \(\lambda\) tested with the cubic spline basis, the corresponding second-derivative curves, specifically angular acceleration curves, were all zero at endpoints. This occurrence is predictable, as cubic spline tends to have zero acceleration at endpoints ([PERSON] et al., 2017). Hence, none of the smoothing parameters demonstrated satisfactory results, forcing the use of a higher-order spline basis. Accordingly, the quintic spline basis (order six) was used for a better result, and several \(\lambda\) values were tested: \(1\times 10^{-13},1\times 10^{-12},1\times 10^{-11}\), and \(1\times 10^{-10}\) (Figure A4). These values were chosen because a higher polynomial (quintic spline basis) requires a smaller \(\lambda\) value than the cubic spline basis. Thus, sets of smaller values of \(\lambda\) than \(1\times 10^{-6}\) were applied to the raw data (Figure A4). The second-derivative curves still had a few fluctuations when the quintic spline basis and \(\lambda=1\times 10^{-13}\) were applied (Figure A4a), while the second-derivative curves showed well-fitted and smoothed curves and maintained the original traits of the curves when the quintic spline basis and \(\lambda=1\times 10^{-12}\) were applied (Figure A4b). As \(\lambda\) increased, the second-derivative curves became smoother but lost a bit of their pattern when \(\lambda=1\times 10^{-11}\) was used (Figure A4c). Testing with higher \(\lambda\), \(1\times 10^{-10}\) resulted in oversmoothed first and second-derivative curves (Figure A4d). Compared  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \(\mathbf{log}_{10}(\lambda)\) & **Degrees of Freedom (_df_)** & **RMSE** & **GCV** \\ \hline \(-12\) & 100 & 35.859 & 0.038 \\ \(-11\) & 100 & 35.859 & 0.038 \\ \(-10\) & 100 & 35.859 & 0.037 \\ \(-9\) & 98 & 35.859 & 0.036 \\ \(-8\) & 86 & 35.858 & 0.032 \\ \(-7\) & 60 & 35.858 & 0.046 \\ \(-6\) & 36 & 3 with the cubic spline basis in Figure 10, smoothing using the quintic spline basis for all \(\lambda\) tested in Figure 11 resulted in non-zero angular accelerations at endpoints, which was different from the cubic spline basis. In order to avoid endpoint error, quintic spline was chosen rather than cubic spline. Although the test showed that the quintic spline basis and \(\lambda\) value of \(1\times 10^{-12}\) were the optimal smoothing parameters by far, the effect of penalizing the curve using the roughness penalty was examined. In order to penalize the curvature of the second derivative, the curves should be penalized with the fourth-order derivative [16]. Using the roughness penalty approach, the curves were smoothed using the quintic spline basis, \(\lambda\) value of \(1\times 10^{-12}\) and penalized with second and fourth-order derivatives. Penalizing the curves with second-order derivative failed to yield smoothed first and second-derivative curves (Figure 11), whereas penalizing the curves with fourth-order derivative with the same parameters yielded smoothed first and second-derivative curves (Figure 5).