Approximating the universal thermal climate index using sparse regression with orthogonal polynomials

This article explores novel data-driven modeling approaches for analyzing and approximating the Universal Thermal Climate Index (UTCI), a physiologically-based metric integrating multiple atmospheric variables to assess thermal comfort. Given the nonlinear, multivariate structure of UTCI, we investigate symbolic and sparse regression techniques as tools for interpretable and efficient function approximation. In particular, we highlight the benefits of using orthogonal polynomial bases-such as Legendre polynomials-in sparse regression frameworks, demonstrating their advantages in stability, convergence, and hierarchical interpretability compared to standard polynomial expansions. We demonstrate that our models achieve significantly lower root-mean squared losses than the widely used sixth-degree polynomial benchmark-while using the same or fewer parameters. By leveraging Legendre polynomial bases, we construct models that efficiently populate a Pareto front of accuracy versus complexity and exhibit stable, hierarchical coefficient structures across varying model capacities. Training on just 20% of the data, our models generalize robustly to the remaining 80%, with consistent performance under bootstrapping. The decomposition effectively approximates the UTCI as a Fourier-like expansion in an orthogonal basis, yielding results near the theoretical optimum in the L2 (least squares) sense. We also connect these findings to the broader context of equation discovery in environmental modeling, referencing probabilistic grammar-based methods that enforce domain consistency and compactness in symbolic expressions. Taken together, these results illustrate how combining sparsity, orthogonality, and symbolic structure enables robust, interpretable modeling of complex environmental indices like UTCI - and significantly outperforms the state-of-the-art approximation in both accuracy and efficiency.