### Modelling Light Use Efficiency
#### 2.5.2 Random Forest for Light Use Efficiency  
Random forests [66] is a machine learning algorithm which was used to develop a predictive model for LUE using the randomForest package in R [67]. The method uses an ensemble approach to build a large number of regression trees on bootstrap samples (typically 500) of the original dataset with varying inputs. Predictions from regression trees are aggregated to estimate \(L\tilde{U}E_{\text{RF}}\) and uncertainties can be quantified through the assessment of quantiles from the bootstrap distribution of outputs.  
Three predictive models for LUE were trained and evaluated: one trained on \(\text{LUE}_{\text{in situ}}\) with all 17 variables in Table 3 and one without crop type, as this may not always be known to users. A third model was also trained on \(\text{LUE}_{\text{max in situ}}\) using all 17 predictors to determine whether this parameter could be accurately modelled. Variable importance was extracted from the model and expressed as the percentage increase in mean squared error (MSE). The out-of-bag (OOB) predictions for \(L\tilde{U}E_{\text{RF}}\) were used to calculate GPP using  
\begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Group** & **Variable** & **Definition** & **Source** \\ \hline \multirow{8}{*}{Satellite bands} & \multirow{2}{*}{Red} & Landsat-7: 0.45–0.52 μm & [58] \\  & & Landsat-8 0.45–0.51 μm & & \\  & & Landsat-7: 0.52–0.60 μm & [58] \\  & & Landsat-8 0.53–0.59 μm & \\  & & Landsat-7: 0.63–0.69 μm & [58] \\  & & Landsat-8 0.64–0.67 μm & \\  & & Landsat-7: 0.77–0.90 μm & [58] \\  & & Landsat-8 0.85–0.88 μm & \\  & & Landsat-7: 1.55–1.75 μm & [58] \\  & & Landsat-8 1.57–1.65 μm & \\  & & Landsat-7: 2.09–2.35 μm & [58] \\  & & Landsat-8 2.11–2.29 μm & \\ \hline \multirow{8}{*}{Vegetation indices} & \multirow{2}{*}{Normalised difference} & \multirow{2}{*}{\(\frac{(NIR-Rad)}{(NIR+Rad)}\)} & \multirow{2}{*}{[59]} \\  & & & & \\  & & & \\  & & Green Chlorophyll Vegetation Index (GCVI) & & \\  & & Leaf Area Index (LAI) & From PySEBAL & [12; 23] \\  & & Soil-Adjusted Vegetation Index (SAVI) & & \\ \hline \multirow{8}{*}{Site and image information} & \multirow{2}{*}{Crop Type} & \multirow{2}{*}{Landsat-7 or Landsat-8} & \multirow{2}{*}{Table 1} \\  & & Sensor & \\  & Latitude & & \\ \cline{1-1}  & Day length & & Calculated on latitude & \\ \cline{1-1}  & & and DOY using geo-sphere R package & \\ \hline \multirow{8}{*}{Agroclimatic variables} & Daily Average Temperature & \multirow{8}{*}{Approximate sowing dates taken from regional crop calendars} & Section 2.3 \\  & & Daily Average Relative Humidity & Section 2.3 \\ \cline{1-1}  & & Days after Sowing (DAS) & \\ \cline{1-1}  & & & \\ \cline{1-1}  & & & \\ \cline{1-1}  & & & \\ \hline \hline \end{tabular}
\end{table}
Table 3: Definition of predictor variables used to train predictive models for light use efficiency.  
Equation (1). We refer to this as \(G\hat{\mathit{PP}}\). An OOB prediction for an observation, \(\mathrm{LUE_{i}}\), is based on trees which do not use the in situ value of \(\mathrm{LUE_{i}}\) in the bootstrapped training sample. Evaluations based on OOB predictions are therefore not prone to overfitting and are an alternative to traditional cross validation. The accuracy of \(G\hat{\mathit{PP}}\) against \(\mathrm{GPP_{in situ}}\) was evaluated using the root mean square error (RMSE) and mean error (ME). These formed the latter accuracy assessment shown in Figure 1, which was compared with the accuracy for \(\mathrm{GPP_{CASA}}\) (Equation (2)), the foremost accuracy assessment in Figure 1. Prediction uncertainty was calculated for each random forest model. This was presented as the standard error on the OOB prediction and calculated using the randomForestCI package [68].