# The Value of Sentinel-2 Spectral Bands for the Assessment of Winter Wheat Growth and Development
## Materials and Methods
### Band Analysis and Model Evaluation Approaches  
The analysis of the eight MAIA/Sentinel-2 bands was carried out using the machine learning algorithm of Gaussian processes regression (GPR; [41]). Specifically, we applied GPR in order to determine the most informative MAIA/Sentinel-2 bands for the retrieval of LAI, LCC and CCC from the mean of the spectral data extracted at each trial plot. GPR provides a non-parametric and probabilistic modelling approach to establishing relationships between inputs (i.e. MAIA/Sentinel-2 bands) and outputs (vegetation variables), allowing for both the predictive mean and variance to be obtained [20; 41; 42]. Studies have demonstrated the calibration of GPR for the estimation of biophysical variables from satellite and airborne sensors [43; 44] and has been shown to perform favourably in comparison to alternative machine learning algorithms [45]. Using the ARTMO (automated radiative transfer models operator) machine learning regression algorithms (MLRA) toolbox [46], we applied GPR as a scaled Gaussian kernel function (Equation (2); [20]):  
\[k\Big{(}x_{i},x_{j}\Big{)}=v\,\exp(-\sum\limits_{b=1}^{B}\frac{\left(x_{i}^{b }-x_{j}^{b}\right)^{2}}{2\sigma_{b}^{2}})+\sigma_{n}^{2}\delta_{ij}, \tag{3}\]  
where for a given covariance function relating two observations, \(k\Big{(}x_{i},x_{j}\Big{)}\), the GPR model hyper parameters include the scaling factor, \(v\), a standard deviation describing the variance of the estimates, \(\sigma_{n}\), and the length-scale, \(\sigma_{b}\), for each of the MAIA/Sentinel-2 bands, \(b\). These hyper parameters, along with the model weight, are automatically optimised by maximising the marginal likelihood when training the GPR model using the MAIA/Sentinel-2 dataset and corresponding ground measurements for LAI, LCC and CCC. The inverse of \(\sigma_{b}\) represents the importance of each spectral band on \(k\), accordingly, a higher \(\sigma_{b}^{-1}\) value indicates a higher information content when developing a GPR model for estimating a variable of interest. To quantify the sensitivity of the MAIA/Sentinel-2 bands on the GPR model estimates, we trained the model using all 250 ground measurements recorded during the 2018 field campaign at the experimental trail plots and applied it within the ARTMO MLRA sequential backward band removal algorithm. This band removal algorithm entails an iterative procedure, whereby a GPR model is first developed using all eight MAIA/Sentinel-2 input bands. The least informative band (i.e., lowest \(\sigma_{b}^{-1}\)) is then removed and a new GPR model is developed with the remaining bands, with this process repeating until the single most sensitive band remains. At each iteration of the backward band removal, we used the same input data for training and validating the GPR model. However, in order to ensure a robust analysis of each band, we applied a three-fold cross-validation sampling scheme. The GPR band analysis results, therefore, included the mean and standard deviation of the cross-validation statistics, including the coefficient of determination (R\({}^{2}\)) and the NRMSE. From the statistical outputs of the sequential backward band removal procedure, we analysed the GPR models for each of the crop variables based on Akaike information criterion (AIC) [47]. AIC is an approach for model selection based on relative performance and works by balancing the trade-offsbetween model complexity (i.e., number of MAIA/Sentinel-2 bands) and goodness-of-fit against the validation data. We calculated the AIC based on the R\({}^{2}\) value for each GPR model [48]:  
\[AIC_{gpr\_n}=-2\log\bigl{[}L\bigl{(}\hat{\theta}\bigr{)}\bigr{]}+2K, \tag{4}\]  
where the AIC of a GPR model, \(AIC_{gpr\_n}\), is calculated based on the maximum likelihood of the parameter vector, \(L\bigl{(}\hat{\theta}\bigr{)}\), comprised of a number of bands, \(K\). For each variable, the GPR model with the lowest AIC value was selected for further analysis.