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<p>Interval-valued data provides a&#xa0;natural representation for inherent uncertainty in domains ranging from environmental monitoring to predictive analytics. This paper introduces a&#xa0;novel attention-based framework for interval-valued regression that reformulates the task as an imprecise classification problem over “atomic” intervals derived from intersections of all observed training intervals. In contrast to many interval-valued models, predicted target outcome in the framework is represented as a&#xa0;discrete probability distribution over these “atomic” intervals, which are assumed to be constrained to truncated normal distributions with trainable standard deviations. The architecture employs a&#xa0;kernel-based aggregation layer, inspired by dot-product attention, where attention weights computed from input features aggregate probability distributions across instances to produce predictive distributions. We develop two parametric training models: IVT-MC (Interval-Valued Truncated normal-Monte Carlo), which approximates the expected log-likelihood loss via Monte Carlo sampling, and IVT-JL (Interval-Valued Truncated normal-Joint Learning), a&#xa0;more efficient approach that jointly optimizes model parameters and distributional standard deviations. Numerical experiments on synthetic and real datasets illustrate the models. Codes of the models are publicly available.</p>

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Interval-Valued attention-Based regression with truncated normal distributions over target intervals

  • 期刊论文

  • 发表日期：

    2026年5月8日

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  • Stanislav K. Kogan,
  • Lev V. Utkin,
  • Andrei V. Konstantinov

摘要

Interval-valued data provides a natural representation for inherent uncertainty in domains ranging from environmental monitoring to predictive analytics. This paper introduces a novel attention-based framework for interval-valued regression that reformulates the task as an imprecise classification problem over “atomic” intervals derived from intersections of all observed training intervals. In contrast to many interval-valued models, predicted target outcome in the framework is represented as a discrete probability distribution over these “atomic” intervals, which are assumed to be constrained to truncated normal distributions with trainable standard deviations. The architecture employs a kernel-based aggregation layer, inspired by dot-product attention, where attention weights computed from input features aggregate probability distributions across instances to produce predictive distributions. We develop two parametric training models: IVT-MC (Interval-Valued Truncated normal-Monte Carlo), which approximates the expected log-likelihood loss via Monte Carlo sampling, and IVT-JL (Interval-Valued Truncated normal-Joint Learning), a more efficient approach that jointly optimizes model parameters and distributional standard deviations. Numerical experiments on synthetic and real datasets illustrate the models. Codes of the models are publicly available.