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Detecting Structural Shifts in Multivariate Hawkes Processes with Fréchet Statistics

Rui Luo, Vikram Krishnamurthy

TL;DR

The paper addresses change-point detection in multivariate Hawkes processes by reframing the problem as network hypothesis testing on causal kernels. It introduces kernel-integral networks estimated via the NPHC method and leverages a Fréchet-mean/variance framework on the space of signed Laplacians to detect structural shifts with a test statistic based on Fréchet variances. The approach demonstrates accurate change-point localization in synthetic data and reveals meaningful regime shifts in cryptocurrency markets, underscoring its applicability to complex, high-dimensional event data. The work contributes a novel, scalable methodology for detecting changes in the endogenous network dynamics driving multivariate point processes, with potential impact in finance and neuroscience.

Abstract

This paper proposes a new approach for change point detection in multivariate Hawkes processes using Fréchet statistic of a network. The method splits the point process into overlapping windows, estimates kernel matrices in each window, and reconstructs the signed Laplacians by treating the kernel matrices as the adjacency matrices of the causal network. We demonstrate the effectiveness of our method through experiments on both simulated and cryptocurrency datasets. Our results show that our method is capable of accurately detecting and characterizing changes in the causal structure of multivariate Hawkes processes, and may have potential applications in fields such as finance and neuroscience. The proposed method is an extension of previous work on Fréchet statistics in point process settings and represents an important contribution to the field of change point detection in multivariate point processes.

Detecting Structural Shifts in Multivariate Hawkes Processes with Fréchet Statistics

TL;DR

The paper addresses change-point detection in multivariate Hawkes processes by reframing the problem as network hypothesis testing on causal kernels. It introduces kernel-integral networks estimated via the NPHC method and leverages a Fréchet-mean/variance framework on the space of signed Laplacians to detect structural shifts with a test statistic based on Fréchet variances. The approach demonstrates accurate change-point localization in synthetic data and reveals meaningful regime shifts in cryptocurrency markets, underscoring its applicability to complex, high-dimensional event data. The work contributes a novel, scalable methodology for detecting changes in the endogenous network dynamics driving multivariate point processes, with potential impact in finance and neuroscience.

Abstract

This paper proposes a new approach for change point detection in multivariate Hawkes processes using Fréchet statistic of a network. The method splits the point process into overlapping windows, estimates kernel matrices in each window, and reconstructs the signed Laplacians by treating the kernel matrices as the adjacency matrices of the causal network. We demonstrate the effectiveness of our method through experiments on both simulated and cryptocurrency datasets. Our results show that our method is capable of accurately detecting and characterizing changes in the causal structure of multivariate Hawkes processes, and may have potential applications in fields such as finance and neuroscience. The proposed method is an extension of previous work on Fréchet statistics in point process settings and represents an important contribution to the field of change point detection in multivariate point processes.
Paper Structure (12 sections, 2 theorems, 20 equations, 3 figures, 1 table)

This paper contains 12 sections, 2 theorems, 20 equations, 3 figures, 1 table.

Table of Contents

  1. INTRODUCTION
  2. Multivariate Hawkes Process
  3. Kernel Matrix Estimation based on Integrated Cumulants
  4. Fréchet Statistics Based Change Point Detection
  5. Dynamic Causal Network
  6. Fréchet Mean of Signed Laplacian Matrices
  7. Estimating the Location of a Change Point
  8. Test Statistics Construction
  9. Empirical Analysis on Simulated and Practical Networks
  10. Simulation Study
  11. Cryptocurrency Price Data
  12. Conclusions

Key Result

Proposition III.1

Under the following assumptions (Assumptions 1-3 in dubey2019frechet): The following result is obtained: where $\sigma^2_F = \textrm{Var}\{{\textrm{d}}^2(\mu_F, {\tilde{L}}) \}$.

Figures (3)

  • Figure 1: Our algorithm effectively detected all change points in a synthetic 10-dimensional point process of length 600000 using Fréchet statistics. The data was split into 119 overlapping windows of length 10000, with the number of events in each window shown in green. The Fréchet test statistic ${nT_n(u): u\in \mathcal{I}_c}$ was also plotted for each analyzed segment due to the use of binary segmentation. The accuracy of our algorithm was validated by the ground truth, as the peak of the test statistics occurs around the significant changes in the number of events.
  • Figure 2: The sequence of symmetrized kernel matrices $A^{(t)}=(H^{(t)} + {H^{(t)}}^T)/2$ estimated by the NPHC algorithm (see Section \ref{['subsec: kernel matrix estimation']}) is displayed before and after the detected change point at time 33. The change in the underlying causal network dynamics is reflected in the shift of the kernel matrix patterns from the earlier (left) to the later (right) time periods.
  • Figure 3: Multiple change point detection results of the cryptocurrency price dataset. The number of events in each window is displayed in green.

Theorems & Definitions (2)

  • Proposition III.1: Fréchet Variance CLT
  • Proposition III.2: Weak Convergence of $nT_n(u)$