Regression Analysis Under Interval-Valued Targets as an Imprecise Classification Problem

Regression analysis with interval-valued outcomes presents a fundamental challenge in modeling data where uncertainty is inherent rather than incidental. Such data, arising naturally in fields ranging from meteorology to finance, require methods that preserve information about both central tendency and dispersion. We introduce a novel class of attention-based regression models that reformulates interval-valued regression as a multiclass classification task. The key idea behind the model is in partitioning the outcome domain into basic intervals derived from training data intersections and representing each interval-valued observation as a set of feasible discrete probability distributions over these intervals. This imprecise probabilistic representation allows us to train a classification-style model by minimizing the expected log-likelihood over all consistent distributions. We propose two training algorithms: a Monte Carlo sampling approach and a more efficient joint optimization method that simultaneously updates both the constrained probability distributions and model parameters. The model incorporates a kernel-based aggregation mechanism using trainable dot-product attention, where attention weights are computed from input features but applied to the probability distributions over basic intervals. Numerical experiments with real datasets illustrate the approach. By introducing the class of attention-based models for interval-valued regression, this work offers a novel perspective on applying machine learning to uncertain data. Codes implementing the proposed models are publicly available.