## 2 Materials and Methods
### Extreme Gradient Boosting (XGBoost)  
The XGBoost model is a machine-learning algorithm implemented in a gradient-boosting framework. The integration algorithm summarizes the modeling results of the sum of all the weak learners (classification and regression tree, CART). The XGBoost model adopts the training method of continuous accumulation of multiple weak learners to optimize the objective function; that is, the XGBoost model builds one CART at a time, and the newly established CART score will be accumulated with all previous CART scores. The \(t\)-th objective (\(O^{(t)}\)) function of the model can be expressed as follows:  
\[O^{(t)}=\sum_{i=1}^{n}I(y_{i},\,\hat{g}_{i}^{t-1}+f_{t}(x_{i}))+\Omega(f_{t})+C \tag{1}\] \begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline
**Factors/Sites** & **IT-Cas** & **JP-Mse** & **KR-CRK** & **PH-RiF** & **US-HRA** & **US-HRC** & **US-Twt** \\ \hline TA & 18.7 & 22.0 & 15.7 & 27.4 & 26.8 & 26.6 & 19.1 \\ SW\_IN & 314.9 & 380.8 & 313.0 & 273.1 & 373.1 & 375.5 & 356.1 \\ LW\_OUT & & 436.1 & 401.8 & 469.8 & 465.8 & 464.0 & \\ VPD & 15.7 & 8.9 & 7.4 & 11.1 & 8.1 & 10.0 & 12.7 \\ PA & 100.5 & 101.0 & 99.1 & 100.6 & 100.8 & 100.8 & 101.3 \\ WS & 1.2 & 2.3 & 2.4 & 2.1 & 2.5 & 2.5 & 4.3 \\ WD & 184.9 & 163.3 & 216.5 & 164.0 & 180.7 & 191.2 & 268.6 \\ SWC & 59.1 & 43.4 & 40.0 & 62.3 & 42.1 & 47.7 & \\ USTAR & 0.2 & 0.2 & 0.2 & 0.2 & 0.2 & 0.2 & 0.4 \\ NETRAD & 192.8 & & 195.4 & 195.1 & 246.6 & 253.6 & 191.4 \\ RECO & 4.3 & 3.1 & 3.7 & 4.0 & 5.4 & 5.2 & 4.4 \\ H & 25.8 & 20.2 & 23.6 & 23.0 & & & 9.2 \\ GPP & 9.1 & 7.9 & 6.6 & 7.5 & 14.5 & 13.3 & 7.1 \\ G & \(-\)21.7 & & 4.9 & 10.9 & 20.6 & 23.7 & 6.7 \\ TS & 17.7 & 20.3 & 14.8 & 28.6 & & & 18.7 \\ NDVI & 0.6 & 0.5 & 0.4 & 0.6 & 0.6 & 0.6 & 0.6 \\ LE & 101.5 & 163.4 & 101.7 & 120.8 & 154.9 & 148.8 & 148.4 \\ NEE & \(-\)4.5 & \(-\)4.7 & \(-\)2.6 & \(-\)3.4 & \(-\)9.0 & \(-\)7.8 & \(-\)2.5 \\ DeltaTA & \(-\)0.6 & \(-\)0.6 & \(-\)0.4 & \(-\)0.2 & \(-\)0.3 & \(-\)0.4 & \(-\)0.4 \\ Number & 8258 & 3594 & 18,366 & 9720 & 1326 & 1634 & 49,401 \\ \hline \hline \end{tabular} Note: the blank indicates missing data. Input factors are air temperature (TA), \({}^{\circ}\)C, incoming shortwave radiation (SW_IN), W m\({}^{-2}\), outgoing longwave radiation (LW_OUT), W m\({}^{-2}\), vapor pressure deficit (VPD), hPa, atmospheric pressure (PA), kPa, wind speed (WS), m s\({}^{-1}\), wind direction (WD), soil water content (SWC), %, friction velocity (USTAR), m s\({}^{-1}\), net radiation (NETRAD), W m\({}^{-2}\), ecosystem respiration (RECO), umolCO\({}_{2}\) m\({}^{-2}\) s\({}^{-1}\), soil heat turbulent flux (H), W m\({}^{-2}\), gross primary productivity (GPP), umolCO\({}_{2}\) m\({}^{-2}\) s\({}^{-1}\), soil heat flux (G), W m\({}^{-2}\), soil temperature (TS), \({}^{\circ}\)C, normalized difference vegetation index (NDVI), latent heat turbulent flux (LE), W m\({}^{-2}\), net ecosystem carbon exchange (NEE), umolCO\({}_{2}\) m\({}^{-2}\) s\({}^{-1}\), and the temperature difference between the previous day and the current day (DeltaTA), \({}^{\circ}\)C.  
\end{table} 
Table 4: The mean values and number of samples of the input factors for \(\mathrm{FCH_{4}}\) prediction at seven rice paddy sites.