Deep Symbolic Learning for Histogram‐Valued Regression Data

ABSTRACT This paper proposes the Deep Symbolic Learning (DSL) model, a deep learning‐based framework for robust regression, specifically designed when both the response and predictors are histogram‐valued variables. DSL utilizes cumulative distribution functions (CDFs) of covariate histograms within a one‐dimensional convolutional neural network (1D‐CNN) to transform the conditional density estimation problem into a multi‐class classification task, optimized using the joint binary cross‐entropy (JBCE) loss function. Extensive simulations and real‐world applications, including air quality, traffic volume, and climate data, demonstrate that the DSL model outperforms existing methods across three key evaluation metrics: CDF distance, empirical coverage of the 90% prediction interval, and average quantile loss. This work contributes to the field of symbolic data analysis and conditional density estimation.