## 2 Methods and Materials
### Detail of Dataset and Analysis Method
#### 2.1.3 Machine Learning-Based Analysis Scheme  
In previous studies on the L-RFA evaluation of orthopedic implants, a correlation evaluation between implant stability and the strongest vibration frequency obtained by L-RFA was performed. From this correlation evaluation, it was found that the vibration frequency obtained via L-RFA increased as implant stability increased [9; 10; 11]. However, using only this correlation evaluation has been shown to reduce the estimation accuracy of cadaveric bones [10]. Multivariate analysis is effective because there are many modes of natural vibration, from basic to a higher order, each of which contains information on implant stability. In general, deep learning requires a large amount of high-quality training data. However, the amount of data managed in this study was limited because of the data requirements for each implant type and the repeatability limitations associated with clinical practices. Therefore, it is necessary to consider a machine learning scheme that can achieve high accuracy even with a small amount of training data. To date, methods were proposed using machine learning, the least absolute shrinkage and selection operator (LASSO), and support vector regression (SVR) for multivariate analyses [20]. Our proposed machine learning-based analysis process consists of two main schemes, the first of which introduces LASSO [25; 26] as an explanatory variable selection process. In the conventional method, the only explanatory variable evaluated for correlation with the embedded torque is the peak vibration frequency. However, many explanatory variables, such as the intensity and dispersion, are stored in the vibration frequency spectrum and spread over the entire measurement frequency range. Therefore, it was possible to divide the frequency range in which the analysis was performed and extract each explanatory variable for each range. Eight variables (peak frequency, peak intensity, frequency center of gravity, intensity center of gravity, average intensity, variance, kurtosis, and skewness) were extracted for eight frequency ranges (30-150 Hz, 150-500 Hz, 500-1000 Hz, 1000-5000 Hz, 5000-10,000 Hz, 10,000-15,000 Hz, 15,000-20,000 Hz, and 30-20,000 Hz). In this study, explanatory variables of total 64 explanatory variables were prepared from the L-RFA dataset reported in a previous study [10]. As a specific procedure, regularization was performed to smooth the orders of the explanatory and objective variables. The lasso was performed 2000 times with multiple regularization strengths, and 40 data points were extracted to determine the regularization strength at the best decision coefficient. The important explanatory variables were ranked by the regularization strength determined by sorting the data in the order of the number of times they were selected. The second scheme uses SVR to perform non-linear regression, which is an adaptation of the support vector machine (SVM) [27; 28], a pattern recognition method for regression that can be adapted to non-linear problems by incorporating a kernel function. In this study,a radial basis function (RBF) kernel, which is computationally fast and accurate, was used. Here, a promising regression technique in addition to SVR is multiple regression analysis. Multiple regression analysis provides an intuitive interpretation of the relationship between multiple explanatory and objective variables, indicating the magnitude and direction of the effect of the coefficient of each explanatory variable on the objective variable. However, multiple regression analysis assumes that a linear relationship exists between explanatory and objective variables. The predictive performance of the model may be affected for data with non-linear relationships. In addition, in multiple regression analysis, multicollinearity problems can occur when high correlations exist among the explanatory variables. This can make it difficult to accurately assess the impact of explanatory variables.