## 4 Discussion
In particular, linear regression and ElasticNet showed lower values as they are models that assume there is a linear relationship between variables x and y, as expressed by the formula y = a + bx, allowing the dependent variable y to be estimated based on the independent variable x [27; 45]. These findings suggest that decision-tree-based models, such as random forest and LightGBM, are particularly advantageous for capturing the complex, non-linear relationships inherent in pronation angle estimation. In contrast, linear models like linear regression and ElasticNet may be insufficient for such tasks due to their assumption of linearity between variables. This observation is crucial for researchers developing machine learning models for biomechanical data, as it highlights the importance of selecting models capable of handling non-linear dependencies to achieve higher accuracy. Consequently, future studies on motion analysis should consider leveraging decision-tree-based algorithms for their superior performance, especially when dealing with rotational or complex joint movements. Normally, the greater the abduction angle, the greater the pronation angle [28; 29]. Therefore, the characteristic value for pronation motion was expected to be high for norm_tip_size, as observed for radial abduction and palmar abduction. However, in this study, the characteristic value for pronation motion was highest for palm_angle, which is typically considered constant. This result may reflect the limitations of MediaPipe, which models the skeleton as a series of straight lines, making it less effective for evaluating pronation angles involving axial rotation. The estimation of pronation angles using MediaPipe presents unique challenges due to its limitations in capturing axial rotation. Specifically, MediaPipe models the bone structure of the hand as a series of straight lines, making it unsuitable for accurately evaluating rotational movements around the thumb's longitudinal axis. These findings highlight the need to introduce new parameters or adopt alternative modeling approaches to better capture axial rotations and improve estimation accuracy. Future research will likely require increasing the number of parameters that correlate with pronation, such as detecting the size of fingernails, rather than relying solely on parameters derived from skeletal coordinates provided by MediaPipe. Although the LightGBM model achieved a high correlation coefficient, namely 0.94, the strong performance of LightGBM can be attributed to the large amount of variation in the dataset. This suggests that LightGBM, a non-linear model, outperformed linear regression, ElasticNet, and SVM, which are linear models. However, its performance is expected to improve the level of palmar and radial abduction through the introduction of new parameters and an increase in the number of samples. The estimation model developed in this study can estimate thumb abduction, palmar adduction, and pronation angles in real time without the use of complex equipment, a feat that has not been documented in previous studies. Notably, the ability to easily quantify the pronation angle in real time, which is clinically significant but has traditionally been time-consuming and costly to measure, is an innovative feature. Furthermore, the use of an angle sensor mitigated the issues of examiner error and reproducibility identified in previous studies. Except for our previous research, no markerless range-of-motion study has achieved high accuracy in measuring thumb abduction, and no study has investigated pronation. Therefore, this is the first study to measure and estimate the range of motion of thumb abduction and adduction using markerless motion capture. The findings of this study have the potential to enable real-time remote diagnosis and measurement using a tablet as well as the analysis of thumb and finger movements from different positions using a smartphone.