Modeling spatial compositional data: Dirichlet likelihoods and cross-entropy estimation

Compositional data find broad application across diverse fields, including ecology, geology, and economics, due to their efficacy in representing proportions of various components within a whole. Spatial dependencies often exist in compositional data, particularly when the data represent different land uses or ecological variables. Spatial autocorrelation can arise from shared environmental conditions or geographical proximity, and ignoring these autocorrelations in modelling of compositional data may lead to incorrect estimates of parameters, emphasizing the need for spatially aware statistical models. In this work, we investigate two complementary frameworks for modeling spatial compositional data. The first is a probabilistic approach based on a spatial autoregressive Dirichlet regression model, which extends classical Dirichlet regression by explicitly incorporating spatial dependence through a likelihood-based formulation. This model enables formal statistical inference, including uncertainty quantification and hypothesis testing. The second is a loss-based approach, referred to as the spatial cross-entropy compositional model (S-CECM), which directly estimates the conditional mean composition by minimizing cross-entropy and can incorporate spatial structure in similar manner than the Dirichlet framework. Through experiments on both synthetic and real datasets, we show that the two approaches achieve comparable predictive performance under standard evaluation metrics. However, they differ substantially in their interpretability and inferential capabilities. While the S-CECM is well suited for prediction-oriented tasks, the spatial Dirichlet model provides a principled probabilistic framework that facilitates uncertainty assessment and parameter interpretation. These results highlight a trade-off between predictive focus and inferential richness and offer practical guidance for selecting appropriate models for spatial compositional data analysis. • A spatial autoregressive Dirichlet model for compositional data is proposed. • A spatial cross-entropy compositional model is proposed for prediction-oriented tasks. • The models outperform their nonspatial alternative. • The approaches generate new insights for three environmental and social case studies. • The developed models are available on a public GitHub repository.