Symbolic regression for discovering vegetation indices: a case study using alfalfa biomass estimation

Predicting crop yield accurately is crucial for agricultural management. Traditional methods often rely on predefined vegetation indices derived from remote sensing data, which may not fully capture the complex relationships present in high dimensional hyperspectral datasets. We explored the potential of symbolic regression (SR), a machine learning technique that discovers best fit mathematical expressions from data, to develop a novel and interpretable model for predicting alfalfa biomass. Using hyperspectral reflectance data with 274 bands in the visible to near infrared, and corresponding alfalfa biomass measurements from 87 distinct sampling points, the PySR library was employed to search for optimal equations. Analyses revealed a trade-off between model complexity and accuracy. A moderately complex equation (complexity 22) involving key wavelengths in the red-edge (~779 nm) and near-infrared (~932 nm, ~952 nm, ~968 nm) regions, associated with chlorophyll content, canopy structure, and water content, achieved a coefficient of determination (r²) of approximately 0.71 on the training data. While demonstrating the capability of SR to uncover potentially meaningful relationships, the results also highlighted limitations such as systematic under-prediction at lower yields and potential feature saturation at higher yields. Selected models tended to include only three or four spectral bands, and a search for simpler expressions that follow normalized difference formulations revealed multiple options that further improved fit. SR may be useful for discovery in hyperspectral remote sensing but must be balanced against the simplicity and explainability of existing formulae.