## 3 Results and Discussions  
Several tests were conducted to identify the optimal smoothing parameter \(\lambda\) on a total of 100 trials from 20 subjects (five trials per subject). The noise or measurement error might not be obviously visible in the raw data, but they will be amplified in the derivatives. For this reason, the first and second derivatives were used as indicators for determining the optimal smoothing parameter \(\lambda\). The right hip F/E angle and its respective first and second derivatives were plotted side by side, and only five trials of right hip F/E angle from a subject during AKS were displayed for a clear view of the effect of smoothing parameter \(\lambda\). Figure 2a shows the raw right hip F/E angle in a discrete form during AKS. Noise did exist, but it was not very noticeable (Figure 2a). The transformation of discrete data into a functional form and smoothing occurred simultaneously. Therefore, by default, a functional data object with class name _fl_ was created. The graph of the functional data object is plotted in Figure 2b. Since the second-derivative curves plotted in Figure 2b are unclear, several trials were performed to find the smallest value of \(\lambda\) that preserved the original pattern of the curves referred to in Figure 2b. As a result, the smallest value of \(\lambda\) that preserved the original pattern of the curves were found to be  
Figure 2: (**a**) Raw right hip flexion/extension (F/E) angle in a discrete form during American kettlebell swing (AKS), (**b**) functional data object by default, and (**c**) \(\lambda=1\times 10^{-21}\) with their first and second derivative curves. \(\lambda=1\times 10^{-21}\), as shown in Figure 2c. The first and second-derivative curves in Figure 2c were used as references in choosing the optimal smoothing parameter \(\lambda\). The automatic methods, CV and GCV, were applied to the raw right hip F/E angle data to obtain \(\lambda\). First, the data were tested with the CV method. The CV plot of mean squared error (MSE) corresponding to the smoothing parameter \(\lambda\) that was generated by default using the _glinnet_ package in R was plotted, as shown in Figure 3a. By default, the number of folds is 10 and the alpha is 1. The other CV plot of MSE of the range of \(\lambda\) sequences--seq(0, 1, 0.0001), seq(0.0001, 0.1, 0.0001), and seq(0.0001, 0.01, 0.0001)--were plotted, as shown in Figure 3b-d, respectively. The red dotted line is the CV curve and the error bars along the \(\lambda\) sequences are upper and lower standard deviation curves (95% confidence interval). The \(y\)-axis represents the MSE for respective values of \(\lambda\), while the upper \(x\)-axis represents the number of predictors. The two vertical dotted lines are two choices of smoothing parameter \(\lambda\), which are the value of \(\lambda\) that gives the minimum cross-validated error and the largest value of \(\lambda\) that gives the most regularized model where the error is within one standard error of the minimum. The first choice of \(\lambda\) would give the most accurate model, while the second choice would also give a good accurate model with the smallest number of predictors. Thus, both \(\lambda\) values were considered for optimal \(\lambda\). As the number of predictors decreased, the MSE  
Figure 3: Cross-validation (CV) plot of mean squared error (MSE) corresponding to smoothing parameter \(\lambda\) (**a**) by default using the _glinnet_ package in R, and with a range of \(\lambda\) sequences: (**b**) seq(0, 1, 0.0001), (**c**) seq(0.0001, 0.1, 0.0001), and (**d**) seq(0.0001, 0.01, 0.0001). increased gradually. The number of predictors stabilized after a certain number of predictors was added. However, the MSE of Figure 3a started to make the increasing pattern less steep at around \(\log_{10}(\lambda)=-3\), as there were fewer predictors. The MSE in Figure 3a started to stabilize upon reaching \(\log_{10}(\lambda)<-6\).