A parametric empirical Bayes (PEB) approach for estimating maize progress percentage at field scale

The Crop Progress Percentage (CPP) in a given phenology stage reflects growth status in the crop life cycle. Generally, routine field measurements of this variable are missing, hence various alternative approaches have been proposed for its estimation. Hidden Markov Models (HMMs) which follow the Bayesian structure are helpful tools for this aim. In the current study, an approach based on the parametric empirical Bayes (PEB) method is used for more accurate estimation of the maize CPP at field scale. The CPP information recorded in three experiment sites i.e. Karaj, Darab and Zarghan were used to validate the performance of the PEB method and to test the robustness. The PEB method includes a non-homogeneous HMM along with an empirical method based on fitting a gamma probability density function (PDF) on prior probabilities. Temporal sequence of phonological stages is regarded as the hidden layer and temporal sequence of NDVI and AGDD indices as the observable layer. The procedure of CPPs estimation is based on calculation of prior and posterior probabilities and an inverse normalization. The overall RMSE of the non-homogeneous HMM before applying the empirical method was 15.1, 11.5 and 7.8% for Karaj, Darab and Zarghan, respectively. However, it was found that the applied HMM fails to estimate the CPPs of final phenological stages accurately. The averaging process of the prior probabilities does not include the errors produced by factors such as climate variability or farming practices. To overcome these problems, we used an empirical Bayes method to estimate the hyperparameters of a gamma density function which was applied as a prior density. This simple approach to maintain the inter-annual variability due to non physiological factors, made the prior probabilities more flexible. The applied empirical Bayes approach (PEB) had significantly smaller RMSE (8, 7.5 and 6.4%, respectively); especially in final phenological stages, and led to more accurate prediction of the phenological dates. The findings derived by PEB are more consistent with those obtained by HMM when the inter-annual variability, mainly date of sowing, is minimum (Specifically, for Zarghan station). The proposed modified approach can be recommended for use at the field scale and serve as a promising tool especially in the regions which suffer the inter-annual variability.