Deep Symbolic Learning for Histogram-Valued Regression Data

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.