Kriging prior Regression: A Case for Kriging-Based Spatial Features with TabPFN in Soil Mapping

Machine learning and geostatistics are two fundamentally different frameworks for predicting and spatially mapping soil properties. Geostatistics leverages the spatial structure of soil properties, while machine learning captures the relationship between available environmental features and soil properties. We propose a hybrid framework that enriches ML with spatial context through engineering of 'spatial lag' features from ordinary kriging. We call this approach 'kriging prior regression' (KpR), as it follows the inverse logic of regression kriging. To evaluate this approach, we assessed both the point and probabilistic prediction performance of KpR, using the TabPFN model across six fieldscale datasets from LimeSoDa. These datasets included soil organic carbon, clay content, and pH, along with features derived from remote sensing and in-situ proximal soil sensing. KpR with TabPFN demonstrated reliable uncertainty estimates and more accurate predictions in comparison to several other spatial techniques (e.g., regression/residual kriging with TabPFN), as well as to established non-spatial machine learning algorithms (e.g., random forest). Most notably, it significantly improved the average R2 by around 30% compared to machine learning algorithms without spatial context. This improvement was due to the strong prediction performance of the TabPFN algorithm itself and the complementary spatial information provided by KpR features. TabPFN is particularly effective for prediction tasks with small sample sizes, common in precision agriculture, whereas KpR can compensate for weak relationships between sensing features and soil properties when proximal soil sensing data are limited. Hence, we conclude that KpR with TabPFN is a very robust and versatile modelling framework for digital soil mapping in precision agriculture.

Modern Neural Networks for Small Tabular Datasets: The New Default for Field-Scale Digital Soil Mapping?

In the field of pedometrics, tabular machine learning is the predominant method for predicting soil properties from remote and proximal soil sensing data, forming a central component of digital soil mapping. At the field-scale, this predictive soil modeling (PSM) task is typically constrained by small training sample sizes and high feature-to-sample ratios in soil spectroscopy. Traditionally, these conditions have proven challenging for conventional deep learning methods. Classical machine learning algorithms, particularly tree-based models like Random Forest and linear models such as Partial Least Squares Regression, have long been the default choice for field-scale PSM. Recent advances in artificial neural networks (ANN) for tabular data challenge this view, yet their suitability for field-scale PSM has not been proven. We introduce a comprehensive benchmark that evaluates state-of-the-art ANN architectures, including the latest multilayer perceptron (MLP)-based models (TabM, RealMLP), attention-based transformer variants (FT-Transformer, ExcelFormer, T2G-Former, AMFormer), retrieval-augmented approaches (TabR, ModernNCA), and an in-context learning foundation model (TabPFN). Our evaluation encompasses 31 field- and farm-scale datasets containing 30 to 460 samples and three critical soil properties: soil organic matter or soil organic carbon, pH, and clay content. Our results reveal that modern ANNs consistently outperform classical methods on the majority of tasks, demonstrating that deep learning has matured sufficiently to overcome the long-standing dominance of classical machine learning for PSM. Notably, TabPFN delivers the strongest overall performance, showing robustness across varying conditions. We therefore recommend the adoption of modern ANNs for field-scale PSM and propose TabPFN as the new default choice in the toolkit of every pedometrician.

An Efficient Model-Agnostic Approach for Uncertainty Estimation in Data-Restricted Pedometric Applications

This paper introduces a model-agnostic approach designed to enhance uncertainty estimation in the predictive modeling of soil properties, a crucial factor for advancing pedometrics and the practice of digital soil mapping. For addressing the typical challenge of data scarcity in soil studies, we present an improved technique for uncertainty estimation. This method is based on the transformation of regression tasks into classification problems, which not only allows for the production of reliable uncertainty estimates but also enables the application of established machine learning algorithms with competitive performance that have not yet been utilized in pedometrics. Empirical results from datasets collected from two German agricultural fields showcase the practical application of the proposed methodology. Our results and findings suggest that the proposed approach has the potential to provide better uncertainty estimation than the models commonly used in pedometrics.