## 3 Results and Discussions
Since the MSE stopped at \(\log_{10}(\lambda)\) not more than approximately \(-6.7\) (Figure 3a), thus, a wider range of \(\lambda\) sequences was tested to determine the optimal \(\lambda\) with a wider \(\lambda\) interval. Several range of lambda with various sequences were tested in order to determine the optimal \(\lambda\) with a wider \(\lambda\) interval, as shown in Figure 3b-d. Both choices of \(\lambda\) (vertical dotted lines) were invisible, as both were 0 using seq(0, 1, 0.0001) (Figure 3b). Therefore, a smaller sequence, seq(0.001, 0.1, 0.0001) (Figure 3c) was tested and resulted in a wider range of \(\lambda\) with the two choices of \(\lambda\) as 0.0002 and 0.0003. The range of \(\lambda\) sequence in Figure 3d shows a well-adjusted \(\log_{10}(\lambda)\) range approximately less than \(-4\) and greater than \(-8.5\), although the MSE leveled off at around \(\log_{10}(\lambda)\) not more than \(-9\). The MSE increased steadily as the number of predictors decreased. For the two choices of \(\lambda\) from Figure 3d, 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 are 0.0002 and 0.0004, respectively. These values were tested, and their first and second-derivative curves were plotted side by side, as shown in Figure A1. By default, the order of a spline is four, which implies a cubic spline basis [16]. It clearly shows that the CV method could not be utilized to determine a suitable \(\lambda\), as both values of \(\lambda\) oversmoothed the first and second-derivative curves (Figure A1). Further testing with the quintic spline basis for \(\lambda\) driven from the CV method would be unnecessary, because testing with quintic, a higher polynomial than the cubic with the same \(\lambda\) driven from the CV method, would yield data oversmoothing. Subsequently, the other automatic method, GCV, was tested for its relevancy. Several values of \(\lambda\) were tested, and the graphs of degrees of freedom (_df_), root mean squared error (RMSE), and GCV were illustrated in Figure 4 and the data collected were tabulated in Table 1.  
Figure 4: Graphs of degrees of freedom (_df_), root mean squared error (RMSE), and generalized cross-validation (GCV) vary with various values of \(\lambda\). According to Figure 4 and Table 1, the value of \(\lambda=1\times 10^{-8}\) recorded the smallest GCV, 0.032, among the other \(\lambda\) values tested. At this smoothing level, the degrees of freedom measure has a value of 86, which is not far from the number of basis functions that were used, 100. The standard deviation of RMSE over the range of \(\lambda\) values studied was small. The value of \(\lambda=1\times 10^{-8}\) that gave the minimum GCV value was tested by plotting its derivative curves (Figure A2). Some spikes and noise in the first and second-derivative curves could be observed by smoothing with \(\lambda=1\times 10^{-8}\), which was provided by the GCV method using the cubic spline basis (Figure A2a). Additional testing with the quintic spline basis resulted in oversmoothed first and second-derivative curves (Figure A2b). Apparently, smoothing with \(\lambda\) driven by the automatic methods, (CV and GCV), failed to generate a suitable smoothing parameter \(\lambda\). These results were proven to comply with the recommendation from [PERSON] and [PERSON] (2016) that an automatic method such as CV should be used as a guide rather than a fixed rule prior to making a subjective assessment in choosing \(\lambda\). For these reasons, subjective assessment was applied to the raw data. The raw data were tested with smoothing parameter \(\lambda\) with the values \(1\times 10^{-6},1\times 10^{-5},1\times 10^{-4}\), and \(1\times 10^{-3}\) with order four (cubic spline basis) (Figure A3). These values were chosen because testing with \(\lambda=1\times 10^{-8}\) by GCV and the cubic spline basis produced undersmoothed first and second-derivative curves (Figure A2a).