Out-of-distribution robustness for multivariate analysis via causal regularisation

We propose a regularisation strategy of classical machine learning algorithms rooted in causality that ensures robustness against distribution shifts. Building upon the anchor regression framework, we demonstrate how incorporating a straightforward regularisation term into the loss function of classical multivariate analysis algorithms, such as (orthonormalized) partial least squares, reduced-rank regression, and multiple linear regression, enables out-of-distribution generalisation. Our framework allows users to efficiently verify the compatibility of a loss function with the regularisation strategy. Estimators for selected algorithms are provided, showcasing consistency and efficacy in synthetic and real-world climate science problems. The empirical validation highlights the versatility of anchor regularisation, emphasizing its compatibility with multivariate analysis approaches and its role in enhancing replicability while guarding against distribution shifts. The extended anchor framework advances causal inference methodologies, addressing the need for reliable out-of-distribution generalisation.