NeuroMem-FHP: A Likelihood-Free Deep Learning Framework for Parameter Estimation of Fractional Hawkes Process

Traditional Hawkes processes are limited by exponential decay kernels, which fail to capture long-range dependence in real-world event data. The Fractional Hawkes Process (FHP) addresses this by using a Mittag-Leffler excitation kernel. However, estimating the parameters of an FHP is computationally expensive using classical Maximum Likelihood Estimation (MLE). This paper investigates whether deep learning models can provide a faster, likelihood-free alternative for parameter estimation.

The authors propose NeuroMem-FHP, a framework that treats parameter estimation as a regression problem. They develop two neural architectures—a Long Short-Term Memory (LSTM) network and a Transformer—to map sequences of inter-arrival times directly to the four FHP parameters: baseline intensity (μ), decay (γ), branching ratio (α), and the fractional memory parameter (β). The models are trained on synthetic data generated via the thinning algorithm and validated against real-world datasets, including financial transaction data (AAPL) and emergency call records.

Both neural models significantly outperformed the classical MLE method in terms of estimation accuracy. On synthetic datasets, the Transformer achieved the lowest Mean Squared Error (MSE = 0.1634), followed by the LSTM (MSE = 0.1752), while the traditional MLE method performed substantially worse (MSE = 2.8032). The Transformer's self-attention mechanism proved particularly effective at capturing the long-range temporal dependencies inherent in the FHP. Furthermore, the framework successfully reproduced empirical distributions and tail behaviors when applied to real-world high-frequency data, demonstrating its practical utility for modeling complex, event-driven systems.

This research provides a scalable, efficient alternative to likelihood-based inference for non-Markovian point processes. By bypassing the need for computationally intensive numerical optimization, the NeuroMem-FHP framework enables real-time parameter estimation in fields where long-memory dynamics are critical, such as finance, seismology, and social network analysis.