## 2 Materials and Methods
### Construction and Evaluation of Decomposition Degree Predictive Models  
In this study, random forest (RF) [21] and extreme gradient boosting (XGBoost) [22] were used as machine learning algorithms to construct the predictive model for degradation. Mw and the weight-loss ratio were used as indicators of the degree of degradation. To construct a predictive model to estimate the degree of degradation by machine learning, Mw obtained by GPC was used as the objective variable. We constructed a decomposition degree predictive model using the weight-loss ratio as the objective variable and compared the predictive accuracy of the predictive models constructed by different machine learning algorithms. The NMR spectra were dimensionally compressed by principal component analysis (PCA) without centering and scaling the data. Explanatory variables in the datasetfor machine learning were collected from features of the NMR spectra and initial crystallinity as analytical data, and the degradation period and moisture content in the compost environment were obtained as the experimental conditions. Dimensional compression was performed to overcome the cost limitation of Bayesian optimization. If there are too many variables, the number of search items increases, and the calculation cost becomes expensive. To factor the moisture content in the compost environment, the moisture weight was estimated from the total weight of the sample, and the values at each measurement were obtained and averaged. Accuracy of the decomposition degree predictive model was evaluated by the coefficient of determination (\(R^{2}\)) and the root mean square error (RMSE) of experimental data and validation data with five-fold cross validation. The best hyperparameters in machine learning were explored using a three-grid search. In addition, the NMR spectra and experimental conditions at the optimal decomposition level were obtained by applying the proposed conditions to the predictive model and performing iterative calculations using Bayesian optimization. In addition, we compared and discussed the magnitude of the influence of the moisture content in the compost environment and other factors considered as experimental conditions based on the factor importance in the predictive model. The importance of each explanatory variable for prediction was determined using the filter-based method. The predicted \({}^{13}\)C-CP/MAS spectrum was constructed from the dot product calculation of the PCA loadings and the predicted PCA scores obtained from the predictive model and Bayesian optimization. The concept of these experiments in this study is summarized in Figure 1.  
PCA, machine learning, and Bayesian optimization were performed using the stats library, caret library [23], and rBayesianOptimization library [24] in R software. The method name \"rf\" in the caret library was chosen to use the RF algorithm. The hyperparameter \"mtry\" as randomly selected predictors in the RF algorithm was tuned by grid searching, and the method name \"xgbLinear\" in the caret library was chosen to use the XGBoost algorithm. The hyperparameters \"nrounds\" as boosting iterations, \"lambda\" as L2 regularization, \"alpha\" as L1 regularization, and \"eta\" as learning rate in the XGBoost algorithm were tuned by grid searching. In Bayesian optimization, the expected improvement was used as an acquisition function type. The Gaussian process updated the acquisition function. The search range was from \(-2\) times the minimum value to \(2\) times the maximum value of each explanatory variable. Ten points as initial variables were chosen randomly to sample the target function before Bayesian optimization of fitting the Gaussian process. The iterative calculation of Bayesian optimization was based on 20 i  
Figure 1: Concept of real and virtual degradable experiments in this study. Actual decomposition experiments were performed to obtain experimental conditions and various analytical data values (**left**).