Spatial Spline for Variable Rate Nitrogen Using On-Farm Experiment Data

The Data-Intensive Farm Management (DIFM) project at the University of Illinois enables producers to conduct site-specific nitrogen experiments, generating spatially structured on-farm data. Current DIFM-supported machine learning models often overfit, yielding widely varying variable rate nitrogen (VRN) recommendations. This research seeks to develop a fast, accurate, and commercially viable VRN algorithm using DIFM data. To accomplish this goal, the study develops a spatially varying parameter model that meets the following criteria: (i) the functional form is a plateau model with the plateau varying across the field, (ii) computational speed is sufficient for the model to be used on large datasets, and (iii) the model can incorporate prior information from past research. The method uses a semiparametric spatial spline approach, estimating a continuous response surface rather than estimating a different parameter at each site in the field. Estimation is implemented with restricted maximum likelihood (REML) in the mgcv package, which provides an empirical-Bayes approximation to posterior inference at a fraction of the computational cost of full Bayesian methods such as MCMC or INLA. Monte Carlo simulations evaluate accuracy, speed, and economic value. Across 1,000 simulated fields spanning a wide range of spatial heterogeneity, the spatial VRN approach recovered 67% of the economic value that uniform application loses relative to perfect knowledge of the field. The mean gain over uniform application was $1.47 per acre, and the full simulation completed in approximately 15 minutes on a standard workstation. The approach offers a structured, computationally efficient alternative to the machine learning models currently used in DIFM, supporting stable and interpretable variable rate nitrogen recommendations.