## 3 Methods
### Regression Analysis  
The relative importance of CO\({}_{2}\) fertilisation, temperature effect and water stress was determined through different combinations of these stress factors per site and crop type. Previous studies showed that NDVI and fAPAR were used to define site specific relations with crop yield [4; 5; 6; 13; 95; 96]. Therefore NDVI and fAPAR were compared with both CGLS derived and modified DMP.  
The regression analysis was done to relate remote sensing Biomass Proxies (BPs) and crop yields of silage maize and common wheat for the period 1999-2012 in Belgium, France and Morocco. A linear regression was calculated between official yield statistics and a regional BP value cumulated over  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **CGLS-DMP** & **Modified DMP** \\ \hline \hline
**R** & \multicolumn{2}{c}{Obtained from JRC-MARSOP on a daily basis at a 0.25\({}^{\circ}\) grid} \\ \hline \(\epsilon_{p}\) & \multicolumn{2}{c}{\(\epsilon_{p}=0.48\)} \\ \hline
**fAPAR** & \multicolumn{2}{c}{Derived from 10-daily SPOT VGT imagery at 1 km\({}^{2}\) resolution} \\ \hline \(\epsilon_{H_{2}O}\) & No water stress factor & Water stress factor based on AET \\ \hline \(\epsilon_{\text{RUE}_{\text{MAX}}}\) & 2.54 kgDM/GJ for all C3 plants & \
\begin{tabular}{c} C\({}_{3}\) plants (wheat) \\ 2.75 kgDM/GJ for wheat \\ \end{tabular} &
\begin{tabular}{c} C\({}_{4}\) plants (maize) \\ 3.5 kgDM/GJ for maize \\ \end{tabular} \\ \(\epsilon_{\text{T}}\) & Blue curve in Figure 3 & Blue curve in Figure 3 & Red curve in Figure 3 \\ \(\epsilon_{\text{CO2}}\) & Blue to red dots in Figure 4(a) & Blue to red lines in Figure 4(b) & Dashed green line in Figure 4(c) \\ \(\epsilon_{\text{AR}}\) & Figure 4(a), range: 0.65–0.85 for & Figure 4(b), range: 0.5–0.7 for study regions \\ \(\epsilon_{\text{AR}}\) & \multicolumn{2}{c}{study regions} \\ \hline \hline \end{tabular}
\end{table}
Table 3: Summary of unchanged and changed parameters in the DMP equation from CGLS as compared to the modified version.  
the different periods during the growing season [6] for each crop type and region. Different dekadal combinations of BPs were explored, e.g., the maximum BPs, maximum BPs plus number of dekads, end of season BPs and sum of BPs between flowering and ripening. Although the optimal temporal window varies with crop type and region, the cumulated period of the BPs in the majority of cases includes the reproductive stage. The model performance was assessed using the coefficient of determination (R\({}^{2}\)). The _p-value_, calculated by Pearson correlation, was also given to assess whether the relations between the crop yields and calibrated BPs were statistically significant.  
A leave-one-year-out cross validation enabled testing the model robustness and goodness of fit using the best combination of variables in the regression analysis. The root mean square error (RMSE), the relative RMSE (RRMSE), mean bias error (MBE) and the index of model performance (\(E_{1}\)) were presented to evaluate the model performance. \(E_{1}\) is a dimensionless index of model-observation agreement [97]. \(E_{1}\) takes the value of 1 for perfect agreement. Although not negatively bounded, the value of 0 indicates that such a model has no more ability to predict the observed values than the observed mean (_i.e._, a null model). All values below 0 reflect a model that performs worse than the null model.