LSTM Estimation of Hawkes Process Parameters

Hawkes Processes

Model Specification

  • Hawkes processes are a type of point process where the intensity is dependent on the history of events, and each event is assumed to have a triggering effect.
  • A hawkes process can be defined by its conditional intensity, which is the rate the events occur at time \(t\), given the history of events.
  • A temporal Hawkes process has the conditional intensity \[\lambda(t|\mathcal{H}_t) = \mu(t) + \sum\limits_{i:t_i<t} g(t-t_i).\]
  • Consider a Hawkes process of the form \[\lambda(t|\mathcal{H}_t) = \mu + \alpha \sum\limits_{i:t_i<t} \beta \, e^{-\beta t}.\]
  • \(\mu\) is the rate of background events.
  • \(\alpha\) is the expected number of offspring per event.
  • \(\beta\) is the expected time between parent and offspring events.

Example

Consider a simulated Hawkes process with parameter values:

  • \(\mu = 1\)
  • \(\alpha = 0.5\)
  • \(\beta = 2\)

Figure 1: The unobserved branching structure of a simulated Hawkes process.

Example

Figure 2: The conditional intensity of a simulated Hawkes process.

Maximum Likelihood Estimation

  • Maximum likelihood estimation is difficult, but possible to implement numerically.
  • The MLEs are known to be slow to compute and have poor asymptotic performance.
Parameter True T = 100 T = 250 T = 500 T = 1000 T = 2500 T = 5000
mu 1.000 1.045 1.198 1.082 1.003 0.952 0.981
alpha 0.500 0.463 0.353 0.473 0.485 0.496 0.506
beta 2.000 1.417 3.493 2.558 1.688 1.988 1.996
  • Note that the estimates require a large time window to converge (over 8000 events for \(T=5000\)).
  • For more complicated models, maximum likelihood estimation becomes much more difficult.

Neural Network Parameter Estimation

LSTM Hawkes Parameter Estimation

  • We consider an alternative to maximum likelihood estimation by training a neural network to estimate parameters.
  • Lee (2023) implemented a neural network to construct point estimates of Hawkes processes for high-frequency financial data.
  • The neural network had similar MSE and significantly improved computation time.
Fig.3 - Simulated MSE and average computation time for neural network and MLE estimators.
Fig.4 - Simulated estimates by paramter for the neural network and MLE estimators.

Project Plan

Simple Project:

  • Train a LSTM neural network on simulated Hawkes processes.
  • Evaluate the point estimates compared to the MLEs for various parameter values.

Advanced Project:

  • Extend the neural network to estimate parameters for a spatio-temporal Hawkes process.

Future Work

  • Confidence interval equivalent for neural network based estimation.
  • More complicated models, including multivariate processes and marked processes.
  • Compare point estimation from a transformer neural network to the LSTM network.

References

  • Cavaliere, Giuseppe, Ye Lu, Anders Rahbek, and Jacob Stærk-Østergaard. “Bootstrap Inference for Hawkes and General Point Processes.” Journal of Econometrics, March 30, 2022. https://doi.org/10.1016/j.jeconom.2022.02.006.
  • Hawkes, Alan G. “Spectra of Some Self-Exciting and Mutually Exciting Point Processes.” Biometrika 58, no. 1 (1971): 83–90. https://doi.org/10.2307/2334319.
  • Lee, Kyungsub. “Recurrent Neural Network Based Parameter Estimation of Hawkes Model on High-Frequency Financial Data.” Finance Research Letters 55 (July 2023): 103922. https://doi.org/10.1016/j.frl.2023.103922.
  • Reinhart, Alex. “Point Process Modeling with Spatiotemporal Covariates for Predicting Crime.” Dissertation, Carnegie Mellon University, 2018.

Caleb Fox