## 2 Materials and Methods
### Machine Learning Analyses
#### 2.3.2 Models  
To predict Fv\({}^{\prime}\)/Fm\({}^{\prime}\), seven machine learning algorithms were compared: Random Forest (RF), Support Vector Regression (SVR), Elastic net (EN), K-Nearest Neighbors (KNN), Gradient Boosting Machine (GBM), Decision Tree (DT), and Linear Model (LM). RF, developed by [PERSON] [57], is a powerful ensemble learning method widely applied for agricultural purposes [58]. It combines numerous decision trees, each trained on random samples of data (bagging). The final prediction is the average of the results of all trees, reducing overfitting; the use of multiple trees increases robustness and overall accuracy. Support vector machine (SVM) models were introduced by [PERSON] et al. [59], and it has been applied to solve many classification remote sensing problems [60]. Regularization methods as EN have emerged as useful regression methods in a wide range of fields, including environmental sciences [61; 62; 63]. It combines L1 (Lasso) and L2 (Ridge) penalties in a single framework. It uses a mixing parameter to balance the sparse nature of the coefficients with the stability of the model. KNN predicts new values by examining the k-closest data points in the training set and using their known values for the prediction [64]. The \"k\" parameter in KNN determines the number of nearest neighbors considered during the prediction process. While KNN is employed for its easy methodology, the performance and accuracy could decrease when it is applied to large datasets or with noisy data [65]. GBM combines the outputs of multiple weak learners, typically decision trees, to train a stronger predictive model [66]. Unlike bagging methods, such as RF, which build trees independently, gradient boosting builds trees sequentially to minimize residual errors. DTs divide the feature space into gradually smaller, homogeneous regions. At each node, the model identifies an optimal subdivision of one of the predictors that minimizes the variance of the target variable, based on a splitting criterion (e.g., Gini Index or Gain Ratio) [67]. Although decision trees are appreciated for their interpretability and ease of implementation, they often risk overfitting. The CART (Classification and Regression Tree) algorithm is a widely used decision tree method. With a loop, the algorithm splits the dataset based on specific features and thresholds; for each split, new binary branches are created following criteria for minimizing the sum of squared errors (SSE) [68].  
In order to identify the optimal hyperparameter configurations for each model, a systematic grid-based search was conducted using the expand.grid() function in R. Table 4 reports all the machine learning algorithms involved in this study, with the relative hyperparameters fine-tuned and the R packages used.  
To evaluate and compare the performance of the models, we used a 10-fold cross-validation approach. Specifically, the dataset was divided into 10 folds; thus, 9 folds were used for training, and the remaining 1-fold was used as a validation set (\(n\) = 144: 36 data points \(\times\) 4 sampling dates). In this procedure, all data points are rotated systematically, ensuring that each data point is used for validation. This approach is widely adopted in statistical modeling to provide a reliable estimate of the generalizability of the model [69]. Based on validation results we calculated the mean and standard deviation of the following performance parameters: coefficient of determination (R\({}^{2}\); Equation (1)), Root Mean Squared Error (RMSE; Equation (2)), Relative Root Mean Squared Error (RRMSE; Equation (3)) Mean Absolute Error (MAE; Equation (4)), and Mean Bias Error (MBE; Equation (5)). These parameters were determined as follows:  
\[R^{2}=1-\frac{SS_{res}}{SS_{tot}} \tag{1}\]