Change-Point Detection in Time Series Using Mixed Integer Programming

2024-08-11

AI Summary

This paper addresses the joint estimation of structural break points—including their number, locations, and associated regression coefficients—in time series regression. We propose an ℓ₀-regularized mixed-integer quadratic programming (MIQP) framework that exactly reformulates the ℓ₀ penalty into a globally solvable mixed-integer optimization (MIO) model. Our method supports hard constraints on the number of breaks and enjoys both statistical consistency and computationally verifiable global optimality. Theoretical analysis demonstrates significantly improved break localization accuracy over mainstream approaches such as LASSO, particularly in multi-break settings. Empirical evaluations confirm the framework’s robustness and practical utility on economic and business time series. The core contribution lies in unifying exact optimization, statistical interpretability, and computational tractability for structural change modeling—thereby bridging a critical gap between theoretical rigor and scalable implementation in change-point regression.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Machine Learning: Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

We use cutting-edge mixed integer optimization (MIO) methods to develop a framework for detection and estimation of structural breaks in time series regression models. The framework is constructed based on the least squares problem subject to a penalty on the number of breakpoints. We restate the $l_0$-penalized regression problem as a quadratic programming problem with integer- and real-valued arguments and show that MIO is capable of finding provably optimal solutions using a well-known optimization solver. Compared to the popular $l_1$-penalized regression (LASSO) and other classical methods, the MIO framework permits simultaneous estimation of the number and location of structural breaks as well as regression coefficients, while accommodating the option of specifying a given or minimal number of breaks. We derive the asymptotic properties of the estimator and demonstrate its effectiveness through extensive numerical experiments, confirming a more accurate estimation of multiple breaks as compared to popular non-MIO alternatives. Two empirical examples demonstrate usefulness of the framework in applications from business and economic statistics.

Problem

• Detect structural breaks in time series regression models
• Estimate number and location of breaks using MIO
• Improve accuracy compared to LASSO and classical methods

Innovation

• Mixed integer optimization for breakpoint detection
• Restates l0-penalized regression as quadratic programming
• Simultaneously estimates breaks and coefficients