## 4 Results  
### Data Description and Metrics  
In this section, extensive experiments were performed to verify the effectiveness of the proposed method. To compare with prior work, we evaluated our method in the United States and chose soybean as the target crop since it has been widely investigated in prior work [6; 14]. The input data are remote sensing data with nine channels. The first seven channels contain the remote sensing data on the surface reflectance from seven bands, which are shown in Figure 6. The remaining two-channel data contain the remote sensing data on the surface temperature from two bands, which are shown in Figure 7[20]. These data are all derived from the MODIS satellite, which are L3-level products of MODIS products. The maturity of these data is usually relatively high. They are generated based on calibrated L1B- or L2-level data. In addition, MODIS L3-level data is usually composed of multiple satellite observation data combinations, and the differences caused by sensors have been corrected accordingly. There is no need to further calibrate the data source to eliminate the differences caused by sensors. In addition, we also use cropland data derived from the USDA website, which is used as mask data to process MODIS remote sensing data to highlight the focus area. The above three types of data are resampled and cropped in counties within the United States administrative region to form the final dataset.  
We use multispectral images collected 32 times a year, from the 49 th day to the 305 th day at 8-day intervals. We discretize all the images using 32 bins to compute the pixel histogram. The resulting input histogram is \(\Psi=\Big{(}H^{(1)},\cdots,H^{(T)}\Big{)}\), \(H^{(t)}\in R^{b\times d}\) with b = 32, d = 9, and T = 32. The ground truth output data are the yearly average soybean yield at the county level measured in bushels per acre, publicly available on the USDA website [21]. We selected 11 states in the U.S. that account for over 75% of the national soybean production and used data from 2010 to 2021. All sources of remote sensing data were cropped according to county borders, while non-soybean pixels were removed with the help of a cropland data layer. To evaluate the generalization ability of the proposed method for unseen data in future years, the temporal nested validation approach was used. We conducted prediction experiments for 6 years, from 2016 to 2021, independently. When a year \(y\) was selected to collect the test data, data from year \(y-1\) were used for validation. And data collected from year \(y-y_{0}-1\) to \(y-2\) were used for training. \(y_{0}\) was used to control the size of the training data and the performance of a model. In the experiments, the hyperparameters of the proposed method were optimized in cross-validation. Accordingly, the grid search from reasonable hyperparameter combinations was adopted, and 300 epochs were run, with the best model saved based on validation performance. For the triplet margin \(\alpha\) in (7), a value too large would lead to overfitting, so we properly set this parameter to 0.2. And \(y_{0}\) was set to 6.  
In addition, the ground truth crop yield data were downloaded from the Quick Stats database. The Quick Stats database is the most comprehensive tool for accessing agricultural data published by NASS. We used the Root Mean Square Error (RMSE) as the evaluation metric for the experiment, which has been widely used to evaluate the performance of crop yield prediction in previous studies [22]. The smaller the RMSE, the better the performance of the network. When estimating crop yields for a certain year, \(\widetilde{y}_{i}\) is the prediction yield learned from the network for state \(i\). \(y_{i}\) is the corresponding true crop yield. For N states, the RMSE formula is presented as follows:  
\[RMSE=\sqrt{\frac{1}{n}\sum_{i=1}^{n}\left(y_{i}-\widetilde{y}_{i}\right)^{2}} \tag{9}\]  
Figure 6: The remote sensing data on the surface reflectance.  
Figure 7: The remote sensing data on the land surface temperature.