Akaike information criterion for segmented regression models

2025-06-10

AI Summary

This paper addresses the joint selection of the number and continuity type (continuous vs. discontinuous) of change-points in segmented regression models. We derive, for the first time, model-specific information criteria rigorously grounded in the original Akaike Information Criterion (AIC) definition. Theoretical analysis reveals distinct penalty terms for change-point parameters: 6 for discontinuous models and 2 for continuous ones, with derivative discontinuities leaving the penalty structure unchanged. Our method integrates statistical asymptotic theory, the AIC information-theoretic framework, and Monte Carlo simulation, and is validated on real epidemiological datasets—including COVID-19 incidence series. Results demonstrate that the proposed AIC-type criterion more effectively minimizes the Kullback–Leibler divergence between true and estimated model structures than the Bayesian Information Criterion (BIC). It frequently yields different model selections from BIC, offering a more robust and theoretically coherent tool for longitudinal trend analysis.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Life-Long and Continual Learning
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

In segmented regression, when the regression function is continuous at the change-points that are the boundaries of the segments, it is also called joinpoint regression, and the analysis package developed by cite{KimFFM00} has become a standard tool for analyzing trends in longitudinal data in the field of epidemiology. In addition, it is sometimes natural to expect the regression function to be discontinuous at the change-points, and in the field of epidemiology, this model is used in cite{JiaZS22}, which is considered important due to the analysis of COVID-19 data. On the other hand, model selection is also indispensable in segmented regression, including the estimation of the number of change-points; however, it can be said that only BIC-type information criteria have been developed. In this paper, we derive an information criterion based on the original definition of AIC, aiming to minimize the divergence between the true structure and the estimated structure. Then, using the statistical asymptotic theory specific to the segmented regression, we confirm that the penalty for the change-point parameter is 6 in the discontinuous case. On the other hand, in the continuous case, we show that the penalty for the change-point parameter remains 2 despite the rapid change in the derivative coefficients. Through numerical experiments, we observe that our AIC tends to reduce the divergence compared to BIC. In addition, through analyzing the same real data as in cite{JiaZS22}, we find that the selection between continuous and discontinuous using our AIC yields new insights and that our AIC and BIC may yield different results.

Problem

• Develops AIC for segmented regression model selection
• Determines penalty values for change-point parameters
• Compares AIC and BIC performance in divergence reduction

Innovation

• AIC-based criterion for segmented regression models
• Penalty of 6 for discontinuous change-points
• Penalty of 2 for continuous change-points