# Wheat Yield Estimation Study Using Hyperspectral Vegetation Indices
## 3 Results  
### Modeling Analysis and Model Evaluation  
The IBM SPSSStatistics 23 software was used to construct the wheat yield estimation models based on 2/3 of the experimental data collected during the critical growth stage of wheat. This subset consisted of 20 randomly selected plots out of the 30 experimental plots. Five different methods, including Multiple Linear Regression (MLR), Stepwise Multiple Regression (SMLR), Principal Component Analysis (PCA), Artificial Neural Networks (ANN) and Random Forest (RF), were employed to select the appropriate independent variables (vegetation indices) for modeling the estimation of the dependent variable (yield). The remaining 1/3 of the experimental data (the remaining 10 plots out of the total 30 experimental plots) were used to evaluate the accuracy of the wheat yield estimation models. This evaluation aimed to assess the stability and precision of the model in estimating wheat yields.  
In the assessment of yield estimation model accuracy, three evaluation metrics are introduced: the coefficient of determination (R\({}^{2}\)), root mean square error (RMSE) and mean square error (MSE). These metrics are used to evaluate the precision of yield estimation models based on real yield measurements and predicted values. R\({}^{2}\) reflects the goodness of fit of the yield estimation model, with a higher value indicating a better fit. RMSE measures the dispersion between observed and predicted values and is particularly sensitive to outliers, making it valuable for assessing the accuracy of predicting both relatively low and high yields. Generally, higher R-squared values and lower values for RMSE and MSE indicate a higher model accuracy. The formulas for calculating these three evaluation metrics are as follows:  
\[R^{2}=1-\sum_{i=1}^{n}{(x_{i}-y_{i})^{2}}/\sum_{i=1}^{n}{(x_{i}-\overline{x})^{2}} \tag{2}\]  
\[RMSE=\sqrt{\sum_{i=1}^{n}{(y_{i}-x_{i})^{2}}/n} \tag{3}\]  
\[MSE=\sum_{i=1}^{n}{(y_{i}-x_{i})^{2}}/n \tag{4}\]  
where \(x_{i}\) is the yield of wheat, \(\overline{x}\) is the average yield of wheat, \(y_{i}\) is the wheat yield predicted by the model, and \(n\) is the number of data points.  
The model constructed using the Random Forest regression method can be evaluated using three key metrics. Firstly, the coefficient of determination (R\({}^{2}\)), which, when its value exceeds 0.6 but remains less than 1.0, signifies a higher level of model accuracy and yields a relatively ideal estimation performance. Secondly, the root mean square error (RMSE) can be used. A smaller RMSE value is indicative of heightened model accuracy. This metric gauges the degree of dispersion between observed and predicted values, revealing how closely the predictions align with the actual values. Lastly, the mean square error (MSE) is frequently selected as an accuracy evaluation metric. A smaller MSE indicates superior model precision. It quantifies the dispersion between observed and predicted values, serving as a measure of accuracy. In unison, these three metrics provide a comprehensive assessment of the model's performance, aiding in the determination of its suitability for the specific task.