## 3 Method
### Shapley Additive Explanations (SHAP)  
In this study, SHapley Additive exPlanations (SHAP) were applied to elucidate the contribution of each input variable towards the prediction of vegetation indices. Originating from the Shapley value in game theory, SHAP effectively delineates the impact of individual features in a machine learning model on the predicted outcomes. The essence of the SHAP approach is the analogy of machine learning features to players in a game, with each feature's contribution analogous to its role across various combinations. Diverging from conventional feature importance assessments in machine learning, SHAP's distinctiveness resides in its model-agnostic characteristic and its consistency. This implies that SHAP furnishes consistent explanations irrespective of the machine learning model employed,assuring uniform interpretation of identical input features across varied instances. The mathematical underpinning of the SHAP algorithm can be summarized as follows:  
\[\text{SHAP}(X_{i})=\sum_{S\subseteq\{12,\ldots p\}\setminus\{i\}}\frac{|S|!(p-|S| -1)!}{p!}(f\Big{(}S\bigcup\{X_{i}\}\Big{)}-f(S)) \tag{4}\]  
In the SHAP algorithm, \(X_{i}\) represents the feature to be explained; \(S\) is a subset of other features; \(f(S\bigcup\{X_{i}\})\) is the model output prediction for the input set \(S\) that includes the feature \(X_{i}\); and \(p\) is the total number of features. The SHAP algorithm calculates the SHAP value for each feature by iterating over all possible subsets of features and then summing the weighted contributions of each subset. These SHAP values are used to explain the model predictions, providing a quantified impact assessment for each feature. Since global importance is needed in our study, the absolute values of each feature's SHAP values are averaged across the data to indicate their importance, which is expressed as:  
Within the SHAP algorithm framework, \(X_{i}\) denotes the feature under examination. The subset of other features is represented by S, where \(f(S\bigcup\{X_{i}\})\) signifies the predicted output of the model for the input set S inclusive of the feature \(X_{i}\). The algorithm operates on a premise where \(p\) is the total number of features. The SHAP value for each feature is computed by iterating over all potential feature subsets and aggregating the weighted contributions from each subset. These SHAP values play a pivotal role in interpreting the model's predictions, offering a quantifiable impact analysis for each feature. In our study, a global importance metric is essential. Therefore, we compute this by averaging the absolute SHAP values of each feature across all data points. This average signifies the overall importance of each feature and is mathematically articulated as:  
\[I_{i}=\frac{1}{n}\sum_{j=1}^{n}\left|\text{SHAP}(X_{i})^{(j)}\right| \tag{5}\]  
Furthermore, we also determine the direction of the effect of features on the prediction outcome by calculating the covariance of SHAP values. If the covariance is positive, it indicates a positive correlation between the feature and the SHAP value, implying a positive effect on the predicted value. Conversely, if the covariance is negative, it indicates a negative correlation between the feature and the SHAP value, leading to a negative impact on the predicted value. In this study, the SHAP algorithm is applied to explain the LightGBM regression model, thereby exploring the variations in the influence of climatic factors on vegetation indices. Additionally, this research incorporates an analysis of the directional impact of features on the prediction outcome by computing the covariance of SHAP values. A positive covariance suggests a positive correlation between the feature and its SHAP value, denoting a favorable influence on the prediction. In contrast, a negative covariance indicates an inverse relationship, signifying that the feature adversely affects the predicted outcome. In the context of this study, the SHAP algorithm is adeptly applied to decipher the LightGBM regression model.