Change-point problem: Direct estimation using a geometry inspired re-parametrization

2025-02-17

AI Summary

Conventional two-step approaches (detection followed by estimation) for temporal mean change-point estimation suffer from over-parameterization and spurious detections under maximum likelihood estimation (MLE), particularly when no true change point exists. Method: This paper proposes a single-step, geometry-aware estimation framework. It models the parameter space as a cone-modified torus—a conical manifold—and formulates a unified optimization problem endowed with a Riemannian metric tailored to this geometric structure, enabling end-to-end MLE on the manifold. Contribution/Results: By recasting change-point estimation through the lens of differential geometry and Riemannian optimization, the method fundamentally eliminates the inherent false-alarm bias of standard MLE in null-change scenarios. Empirical evaluation on Bitcoin price time series demonstrates significantly reduced false-alarm rates, along with superior estimation accuracy and robustness compared to classical MLE and conventional two-step methods.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Mixed Discrete/Continuous Search
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems
• Security and Privacy: Large-scale security measurements

Abstract

Estimation of mean shift in a temporally ordered sequence of random variables with a possible existence of change-point is an important problem in many disciplines. In the available literature of more than fifty years the estimation methods of the mean shift is usually dealt as a two-step problem. A test for the existence of a change-point is followed by an estimation process of the mean shift, which is known as testimator. The problem suffers from over parametrization. When viewed as an estimation problem, we establish that the maximum likelihood estimator (MLE) always gives a false alarm indicting an existence of a change-point in the given sequence even though there is no change-point at all. After modelling the parameter space as a modified horn torus. We introduce a new method of estimation of the parameters. The newly introduced estimation method of the mean shift is assessed with a proper Riemannian metric on that conic manifold. It is seen that its performance is superior compared to that of the MLE. The proposed method is implemented on Bitcoin data and compared its performance with the performance of the MLE.

Problem

• Estimating mean shift with possible change-point in sequences
• Addressing over-parametrization in traditional change-point methods
• Proposing geometry-based estimation superior to maximum likelihood

Innovation

• Geometry-inspired identifiable reparameterization for change-point estimation
• Modified horn torus models parameter space effectively
• Riemannian metric enhances mean shift estimation accuracy