Effects of likelihood function form on variety parameters during rice phenological model correction

Biased calibration can often be found in the model parameter calibration using the traditional Gaussian likelihood function (GLF). Particularly, there are measurement errors in the observation data and complex algorithmic structures in the model. The GLF is also limited to capturing the heteroscedastic characteristics of model residuals, leading to the inaccurate calibration in Markov chain Monte Carlo (MCMC). In this study, two modified likelihood functions were introduced: the Gaussian likelihood function with coefficient of variation transformation (GLF-CV), and the Gaussian likelihood function with Box-Cox transformation (GLF-BC). These modified likelihood functions were then incorporated the heteroscedasticity induced by both the observation data and the model structure. A more accurate representation was obtained for the complex characteristics of the model residuals. The performance of the modified likelihood functions was also evaluated to conduct the parameter calibration and uncertainty analysis using the uncertainty ratio (UR). The field experimental data was collected from three ecological sites: Gaoyao Xuehuanian (early-maturing) from 2004 to 2009, Xinghua Wuyujing3 (mid-maturing) from 2001 to 2004, and Liu'an Shanyou63 (late-maturing) from 1991 to 2004. The RiceGrow and Oryza2000 phenology models were selected to compare the effects of GLF-CV, GLF-BC, and GLF on the calibration. Model parameter calibration was also conducted using the ensemble sampling for affine-invariant MCMC (EMCEE). The research findings were summarized as follows: 1) The UR ranges were 2.66-4.54, and 2.30-4.41 days, respectively, for the RiceGrow and Oryza2000 phenology models under all three likelihood functions. There were some effects of all three likelihood functions on the parameter calibration, leading to reasonable UR values within the specified ranges. 2) The GLF-BC of the RiceGrow model was achieved in the smallest UR values for the Gaoyao Xuehuanian, Xinghua Wuyujing3, and Liu'an Shanyou63 varieties. Specifically, the predicted UR values with the GLF-BC were 0.09, 0.07, and 0.80 days smaller than those with the GLF, while 1.21, 0.20, and 0.07 days smaller than those with the GLF-CV. The superior adaptability of GLF-BC was then achieved in the RiceGrow phenology model, indicating the better-improved calibration. 3) The likelihood function with the smallest UR varied greatly among the different rice varieties in the Oryza2000 phenology model. The GLF obtained the smallest UR for the Gaoyao Xuehuanian variety, while the GLF-BC and GLF-CV were the smallest UR for the Xinghua Wuyujing3 and Liu'an Shanyou 63 rice varieties, with values of 2.30, 4.17, and 3.50 days, respectively. Consequently, the likelihood function depended mainly on the primary source of heteroscedasticity in the model residuals. The optimal model was achieved in the GLF-CV, when the main source was the observation data, whereas, the GLF-BC was preferred, when the main source was the model structure. The GLF was selected as the suitable likelihood function with the minimal heteroscedasticity of model residuals. In conclusion, the heteroscedastic characteristics of model residuals were captured from the measurement errors and complex algorithmic structures. Two modified likelihood functions (GLF-CV and GLF-BC) can also provide more accurate descriptions of heteroscedasticity induced by both observation data and model structure. The comparative analysis of parameter calibration and model uncertainty with the UR demonstrated that the modified likelihood functions can be expected to effectively improve the calibration. The likelihood function also depended on the main source of heteroscedasticity, with the preferable GLF-CV under the primary source of observation data, and the preferred GLF-BC under the primary source of the model structure. The GLF was suitable for the minimal heteroscedasticity of model residuals.