Soybean and maize yield residuals in the Cerrado Biome

Based on the available information from the Municipal Agricultural Production Survey provided by the Brazilian Institute of Geography and Statistics [45], we obtained the average yield of soy-maize double cropping system (ton/ha) at the municipal level between 2006 and 2019. The tabular data were also converted into time-series maps with a gridcell resolution of 28×28 km using the Inverse Distance Weighting interpolation.The<b> </b>calculation of the Soybean and maize yield residuals is needed to eliminate the influence of other factors, such as technological advancements, which could bias our analyses. To achieve this, we calculated soybean and second-crop maize yield residuals using a three-step procedure (Section S1.8). Firstly, we applied generalized additive models (GAM) [29] to fit mathematical equations to the historical series of each grid cell as a function of time. Soybean and maize yields (Ys_t and Ym_t, in kg/ha) were the dependent variables, and time (t, in years) was the independent variable. Here, we assume that the yields follow a Poisson distribution. This is based on the consideration that the probability of a series of events occurring over a specific period is calculated assuming that these events are independent of the time of the last event. In the second step, the angular coefficients of the fitted equations were used to estimate the yield values (̂Ys_t 𝑒 ̂Ym_t), representing the trend for Ys_t and Ym_t. In the third step, we subtracted the fitted yield values from the observed values (Equations 2, 3).<br>Y′s_t = Ys_t− ̂Ys_t_𝑡 (Eq. 2)Y′m_t = Ym_t− ̂Ym_t (Eq. 3)<br>The resulting values are considered the yield residual, isolating the influence of other climate factors.

本研究依托巴西地理与统计研究所（Brazilian Institute of Geography and Statistics）提供的市级农业生产调查公开数据[45]，获取了2006年至2019年间市级行政单元尺度下的大豆-玉米复种系统平均单产（单位：吨/公顷）。同时，通过反距离权重（Inverse Distance Weighting）插值法，将表格型数据转换为格网分辨率为28×28 km的时序地图。为消除可能干扰分析结果的其他因素（如技术进步）的影响，需计算大豆与玉米的单产残差。为此，我们采用三步流程（附录S1.8）计算大豆及第二季玉米的单产残差：第一步，针对每个格网的历史时序数据，以时间为自变量，拟合广义可加模型（generalized additive models, GAM）[29]构建数学方程。本研究以大豆单产（$Y_{s,t}$，单位：kg/ha）与玉米单产（$Y_{m,t}$，单位：kg/ha）作为因变量，时间（$t$，单位：年）作为自变量。此处假设单产服从泊松分布，该假设基于"特定时段内一系列事件发生的概率，以各事件与上一事件的发生时间相互独立为前提进行计算"的考量。第二步，利用拟合方程的斜率系数，估算单产预测值（$hat{Y}_{s,t}$与$hat{Y}_{m,t}$），该值代表$Y_{s,t}$与$Y_{m,t}$的时序趋势。第三步，将实测单产减去拟合得到的预测单产，得到残差（式2、式3）：
$Y'_{s,t} = Y_{s,t} - hat{Y}_{s,t} ag{2}$
$Y'_{m,t} = Y_{m,t} - hat{Y}_{m,t} ag{3}$
最终得到的残差值可剥离其他非气候因素的影响，即为单产残差。