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Online Change Point Detection for Multivariate Inhomogeneous Poisson Processes Time Series

Xiaokai Luo, Haotian Xu, Carlos Misael Madrid Padilla, Oscar Hernan Madrid Padilla

TL;DR

This paper addresses online change point detection in multivariate inhomogeneous Poisson point process time series with temporal dependence. It develops a scalable detector that maps each PPP observation to a finite-dimensional intensity matrix via RKHS bases and exploits a low-rank structure through a restricted SVD in a CUSUM framework, achieving single-pass operation with per-observation cost independent of the long-time horizon. The authors establish nonasymptotic, finite-sample guarantees for false-alarm control and detection delay by introducing a Matrix Bernstein inequality for geometrically $\\beta$-mixing PPPs, and they validate the approach through synthetic 3D/4D simulations and a real Oklahoma earthquake dataset, showing superior detection speed at comparable false-alarm rates. The work provides a practical, theoretically sound tool for online nonparametric change detection in complex point-process time series with broad applications in seismology, climate monitoring, and epidemiology.

Abstract

We study online change point detection for multivariate inhomogeneous Poisson point process time series. This setting arises commonly in applications such as earthquake seismology, climate monitoring, and epidemic surveillance, yet remains underexplored in the machine learning and statistics literature. We propose a method that uses low-rank matrices to represent the multivariate Poisson intensity functions, resulting in an adaptive nonparametric detection procedure. Our algorithm is single-pass and requires only constant computational cost per new observation, independent of the elapsed length of the time series. We provide theoretical guarantees to control the overall false alarm probability and characterize the detection delay under temporal dependence. We also develop a new Matrix Bernstein inequality for temporally dependent Poisson point process time series, which may be of independent interest. Numerical experiments demonstrate that our method is both statistically robust and computationally efficient.

Online Change Point Detection for Multivariate Inhomogeneous Poisson Processes Time Series

TL;DR

This paper addresses online change point detection in multivariate inhomogeneous Poisson point process time series with temporal dependence. It develops a scalable detector that maps each PPP observation to a finite-dimensional intensity matrix via RKHS bases and exploits a low-rank structure through a restricted SVD in a CUSUM framework, achieving single-pass operation with per-observation cost independent of the long-time horizon. The authors establish nonasymptotic, finite-sample guarantees for false-alarm control and detection delay by introducing a Matrix Bernstein inequality for geometrically -mixing PPPs, and they validate the approach through synthetic 3D/4D simulations and a real Oklahoma earthquake dataset, showing superior detection speed at comparable false-alarm rates. The work provides a practical, theoretically sound tool for online nonparametric change detection in complex point-process time series with broad applications in seismology, climate monitoring, and epidemiology.

Abstract

We study online change point detection for multivariate inhomogeneous Poisson point process time series. This setting arises commonly in applications such as earthquake seismology, climate monitoring, and epidemic surveillance, yet remains underexplored in the machine learning and statistics literature. We propose a method that uses low-rank matrices to represent the multivariate Poisson intensity functions, resulting in an adaptive nonparametric detection procedure. Our algorithm is single-pass and requires only constant computational cost per new observation, independent of the elapsed length of the time series. We provide theoretical guarantees to control the overall false alarm probability and characterize the detection delay under temporal dependence. We also develop a new Matrix Bernstein inequality for temporally dependent Poisson point process time series, which may be of independent interest. Numerical experiments demonstrate that our method is both statistically robust and computationally efficient.
Paper Structure (17 sections, 17 theorems, 161 equations, 3 figures, 2 tables, 3 algorithms)

This paper contains 17 sections, 17 theorems, 161 equations, 3 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related Works
  3. Summary of Results
  4. Notations
  5. Online Change Point Detection for Poisson Point Processes
  6. Online change point detection
  7. Numerical Studies
  8. Selection of Tuning Parameters
  9. 3D Intensity with Temporally Dependent Latent Variables
  10. 4D Intensity with Autoregressive Scale
  11. A Real Data Example
  12. Conclusion
  13. Proof of \ref{['theorem:main']}
  14. Deviation Bounds
  15. Poisson Matrix Properties
  16. ...and 2 more sections

Key Result

Theorem 1

Let the univariate basis functions $\{\phi_k\}_{k =1}^\infty$ in eq:basis-blocks be the Legendre polynomials. Assume the PPP time series $\{X^{(i)}\}_{i = 1}^\infty$ satisfies Definition def:model_ppp_beta_mixing on a compact domain $\mathbb X\subset\mathbb R^d$, with $d=p+q\ge2$. Suppose the traini Let $\kappa=\|\lambda^*-\lambda_a^*\|_{{\mathbf L_2}}$ and define where $C_{\mathop{\mathrm{lag}}\

Figures (3)

  • Figure 1: FAP vs. ADD comparison among three detectors under the 3D simulation setting.
  • Figure 2: FAP vs. ADD comparison among three detectors under the 3D simulation setting.
  • Figure 3: Average earthquake intensity before and after the detected change point.

Theorems & Definitions (35)

  • Definition 1: PPP time series with temporal dependence
  • Remark 1: Coordinate split
  • Remark 2: Computational cost
  • Remark 3: PPP change point detection in 1D
  • Theorem 1: False-alarm control and detection delay
  • proof : Proof of \ref{['theorem:main']}
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 25 more