## 2 Materials and Methods
### PLS and RF Regression  
In this study, two regression models, including PLS and RF regression, were used to establish the model for estimating plant density. These two methods are based on different analyses, where the PLS is a representative of the statistical approach, while RF is a representative of the machine learning technique. The main reason for using PLS and RF regression is because these approaches can deal with highly intercorrelated data such as the spectral reflectance data of satellite imagery. Conventional regression analysis, such as multiple linear regression, cannot handle multi-temporal satellite imagery data due to the fact of multi-collinearity problems [46]. Partial least squares regression is a generalized multiple linear regression that provides a multivariate approach to predicting one response variable using strongly collinear (correlated) predictor variables [47; 48]. Details of PLS regression can be found in [PERSON] et al. [49]. A partial least squares regression model was established using the 'pls' package implemented in R [50]. In the analysis of PLS regression, the number of components was determined using the'selectNcomp' function attached to the 'pls' package, according to the one-sigma method. The one-sigma heuristic chooses the fewest components for the model that has an accuracy of less than one standard error from the best model. Random forest regression is one of the techniques in machine learning that can be used to estimate the response variable, based on classification and regression tree (CART) [51]. This analysis is efficient in processing non-linear data, which can also be found in the data set [32]. Details about RF can be found in [PERSON] [51]. A random forest regression model was established using the 'randomForest' package [52] implemented in R. Two parameters were adjusted to optimize the RF regression model: _ntree_, the number of trees grown in the regression forest, was set at 100; _mtry_, the number of different predictors sampled at each node (default = the number of predictors divided by 3). The selection of predictors is an important process for the model's establishment, as it could greatly affect the prediction accuracy and facilitate a better understanding of the relationships between predictors and response variables [12]. Thus, the PLS and RF regression models were trained to estimate plant density using several possible predictor variable combinations, as follows: (1) using spectral reflectance values only; (2) using NDVI values only; (3) using a combination of spectral reflectance and NDVI; (4) using a combination of PlanetScope imagery and climate data. The variable combinations were used to test the impact of different predictors in estimating plant density. In addition, we also examined the variable importance measurement from RF regression as a method to determine the suitable variable predictors for predicting soybean plant density. This method has been widely used in many scientific fields to select and identify relevant predictors for predicting response variables [53]. Variable importance can be extracted from the regression trees by calculating _IncNodePurity_, which is the total decrease in node impurities from the splitting on the predictors, using the importance function in the 'randomForest' package. An increase in the value of _IncNodePurity_ implies a decrease in the mean square error, which means that the largest values represent the most important variables to the response [54]. Variable importance was evaluated from the RF model with all available predictor variables. We verified whether the variable predictors with high values of _IncNodePurity_ could improve the prediction capability of the developed regression model, since the predictors with low _IncNodePurity_ are less critical to the response variable. The performance of all developed regression models in this section was compared based