## 3 Results
### Yield Estimation Using a Multiple Linear Regression Model  
A multiple linear regression model was established for county-level crop yield estimation using EVI2 and year as two independent variables. Results from previous sections show that the strongest correlation between crop yields and EVI2 were observed at the peak growth stages of the crops, with inter-annual variations of the best time within a time window. To establish a simple model applicable across different years, average EVI2 was derived within a narrow time window correspondent to the peak growth stages of the crops. The use of the average EVI2 for crop yield estimation will induce a loss of accuracy in some years, but can increase model robustness through a reduced variability associated with crop growth calendars. The time window for averaging was from DOY 201 to 217 using the general cropland mask, from DOY 121 to 145 using crop-specific mask for winter wheat, and from DOY 209 to 241 for corn and soybean using the two masks. In considering that cropping practices (cultivars and growth calendars) might be different among the three agricultural regions (Southern, Western, and Central Ontario), the regression model was established individually for each region. Performance was assessed on the results obtained using the three models independently, and on the results combined. Regression results and statistical performance indicators are reported in Table 1. Figure 7 shows the comparison between all the reported and estimated yields using general cropland mask and crop-specific masks. The following observations can be made:  
(1) Crop yields can be estimated reliably in all cases using the multiple linear regression model (\(p<0.001\)), except for winter wheat in Central Ontario, which had a low \(R^{2}\) (-0.10) and smaller \(F\)-value (\(p<0.025\)). The inferior performance for winter wheat in Central Ontario was also demonstrated by a smaller correlation coefficient for EVI2 (r_EVI2; 0.03 for SM and -0.06 for GM), and a larger uncertainty of the regression coefficients (\(\sigma(a_{1})\)) respective to the coefficient itself (\(a_{1}\)). The regression coefficient for EVI2 was \(0.909\pm 1.745\) using crop-specific mask and -1.611 \(\pm\) 1.539 using the general cropland mask. Trend of yields was the main factor of yield variability in this case (r_Year \(\sim 0.30\)). (2) Performance of yield estimation was always poorer in Central Ontario than in the other two regions, as shown by a relatively larger MRAE and a lower \(R^{2}\). (3) Models based on crop-specific masks achieved slightly better results than those based on a general cropland mask, according to MRAE and lower \(R^{2}\). (4) Yields of the three crops had a clear increasing trend over the 14 year period, as shown by the positive correlation coefficients between crop yields and the year (r_Year in Table 1). Partial \(t\)-test showed that regression coefficient \(a_{2}\) was at the significance level of \(p<0.001\) (\(t>3.0\)). According to the derived coefficient \(a_{2}\), annual increase of crop yields was between 0.125 to 0.166 t/ha for corn, 0.037 to 0.058 t/ha for soybean, and 0.054 to 0.094 t/ha for winter wheat over the 14 year period, dependent on regions. (5) Although EVI2 at peak growth stage accounted for yield variability effectively (Section 3.3), considerable temporal variability over the long period was due to the long term yield trends. Over the  
Figure 6: Example annual relationships between crop yields and EVI2 derived using crop specific masks for the three annual crops; only data from 3 years (2006, 2011, and 2016) are shown; DOY refers to MODIS nominal composite day-of-year with the strongest correlation between EVI2 and crop yields. study period, yield variability due to the long term trend and due to spatial heterogeneity were comparable, as demonstrated by the comparable correlation coefficients r_EVI2 and r_Year. (6) For all cases in Figure 7, there was an underestimation when yields were higher than average and an overestimation when yields were lower than average.