Symbolic Quantile Regression for the Interpretable Prediction of Conditional Quantiles

2025-08-11

AI Summary

In high-stakes applications requiring simultaneous modeling of multiple conditional quantiles while preserving interpretability, this paper proposes Symbolic Quantile Regression (SQR)—the first framework extending symbolic regression to conditional quantile modeling. SQR employs a tree-based evolutionary algorithm to directly synthesize closed-form quantile functions and optimizes them via quantile loss, enabling distinct characterization of covariate effects on both distributional centers and tails. Its core innovation lies in the principled integration of symbolic regression and quantile regression, jointly ensuring global interpretability, sensitivity to extreme values, and cross-quantile comparability of effect estimates. Experiments demonstrate that SQR outperforms conventional interpretable models across multiple benchmarks, matches the predictive accuracy of state-of-the-art black-box methods, and successfully identifies heterogeneous influential factors at different quantiles in an aviation fuel demand forecasting case study.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Cognitive Modeling & Cognitive Systems: Symbolic Representations

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Symbolic Regression (SR) is a well-established framework for generating interpretable or white-box predictive models. Although SR has been successfully applied to create interpretable estimates of the average of the outcome, it is currently not well understood how it can be used to estimate the relationship between variables at other points in the distribution of the target variable. Such estimates of e.g. the median or an extreme value provide a fuller picture of how predictive variables affect the outcome and are necessary in high-stakes, safety-critical application domains. This study introduces Symbolic Quantile Regression (SQR), an approach to predict conditional quantiles with SR. In an extensive evaluation, we find that SQR outperforms transparent models and performs comparably to a strong black-box baseline without compromising transparency. We also show how SQR can be used to explain differences in the target distribution by comparing models that predict extreme and central outcomes in an airline fuel usage case study. We conclude that SQR is suitable for predicting conditional quantiles and understanding interesting feature influences at varying quantiles.

Problem

• Predicting conditional quantiles with symbolic regression
• Enhancing interpretability in high-stakes quantile estimation
• Understanding feature influences across different distribution quantiles

Innovation

• Symbolic regression for conditional quantiles prediction
• Outperforms transparent models with comparable black-box performance
• Maintains model transparency while explaining distribution differences