A neural network based model for multi-dimensional nonlinear Hawkes processes

Paper: arXiv 2303.03073

Authors: Sobin Joseph, Shashi Jain

Abstract

This paper introduces the Neural Network for Nonlinear Hawkes processes (NNNH), a non-parametric method based on neural networks to fit nonlinear Hawkes processes. Our method is suitable for analyzing large datasets in which events exhibit both mutually-exciting and inhibitive patterns. The NNNH approach models the individual kernels and the base intensity of the nonlinear Hawkes process using feed forward neural networks and jointly calibrates the parameters of the networks by maximizing the log-likelihood function. We utilize Stochastic Gradient Descent to search for the optimal parameters and propose an unbiased estimator for the gradient, as well as an efficient computation method. We demonstrate the flexibility and accuracy of our method through numerical experiments on both simulated and real-world data, and compare it with state-of-the-art methods. Our results highlight the effectiveness of the NNNH method in accurately capturing the complexities of nonlinear Hawkes processes.

Complexity vs Empirical Score

• Math Complexity: 7.0/10
• Empirical Rigor: 6.5/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents a solid mathematical formulation by parameterizing multi-dimensional nonlinear Hawkes processes with neural networks and deriving an unbiased gradient estimator. The empirical validation combines both synthetic and real-world high-frequency data, offering a novel non-parametric flexible approach to exciting and inhibitive event modeling.

Research Flowchart

  flowchart TD
    A[Goal: Model complex multi-dimensional<br>nonlinear Hawkes processes] --> B[Input: Event Sequences<br>Simulated & Real-World Data]
    B --> C[Model Architecture: NNNH<br>Neural Networks for Kernels & Base Intensity]
    C --> D[Optimization: Maximize Log-Likelihood<br>via SGD with Unbiased Gradient Estimator]
    D --> E[Evaluation: Compare NNNH against<br>State-of-the-Art Baseline Models]
    E --> F[Outcome: Accurately captures complex<br>excitation & inhibition patterns]