NeuroMem-FHP: A Likelihood-Free Deep Learning Framework for Parameter Estimation of Fractional Hawkes Process
Authors: Neha Gupta, Aditya Maheshwari
Organizations: Operations Management and Quantitative Techniques Area, Indian Institute of Management Indore, Indore 453556, India.
Abstract
In this paper, we propose deep learning based NeuroMem-FHP framework for estimating the parameters of the fractional Hawkes process (FHP), a self-exciting point process that captures long-range dependence through a fractional Mittag-Leffler excitation kernel. Two neural architectures, namely a Long Short-Term Memory (LSTM) network and a Transformer, are developed to estimate the model parameters (μ,γ,α,β) directly from sequences of inter-arrival times without requiring computationally intensive likelihood optimization. Experiments on synthetic data that both neural models significantly outperform the classical Maximum Likelihood Estimation (MLE) method, with the Transformer achieving the highest estimation accuracy (MSE = 0.1634), followed by the LSTM (MSE = 0.1752), compared to MLE (MSE = 2.8032). An ablation study further examines the effects of key hyperparameters on model performance. The proposed framework is also on two real-world high-frequency datasets, namely AAPL NBBO transaction data and Montgomery County 911 emergency call records. Using a predictive validation approach, event sequences simulated from the estimated parameters closely reproduce the empirical distribution, tail behavior, and temporal dependence structure of the observed data. These results demonstrate that Transformer-based parameter estimation provides an accurate and efficient alternative to conventional estimation techniques for FHP and offers a promising framework for modeling event-driven systems with long-memory dynamics.
We investigate a forecasting framework based on a simple discrete-time dynamic model with coefficients varying in time. The parameters of the model are recovered within a deep learning framework, which makes it possible to retain a transparent parametric structure while simultaneously accounting for complex and nonstationary patterns in the observed phenomenon. Our analysis covers two specifications of the noise process. Besides the standard Gaussian setting, we also consider Laplace-distributed noise, which can offer a more adequate description in the presence of heavier tails and sharper local fluctuations. For both cases, we formulate the predictive scheme of the model and analyze the associated uncertainty quantification, including the construction of prediction intervals. The results illustrate that a relatively simple model, when combined with time-dependent parameter estimation, can serve as a mathematically tractable and practically flexible tool for forecasting complex dynamics under different noise assumptions. The general model is stated for TVAR(p), while the prediction-interval formulas and the numerical experiments are developed for the TVAR(1) case.
Agnieszka Kopeć, Paweł Przybyłowicz, Martyna Wiącek
Heteroscedastic forecasting, where a network estimates a scale parameter alongside a location parameter, is normally motivated by uncertainty quantification. This paper shows it also improves the point estimate, in contrast to reported negative results for heteroscedastic estimation outside time series. Halo is a modification that reuses an existing deep forecaster's architecture, giving it a second output for the scale of its implied distribution and training it under the matching negative log likelihood. Adapting three state-of-the-art models --- a transformer, a graph network paired with a variational autoencoder, and a single-layer convolutional network --- under both Gaussian and Laplacian losses demonstrates the phenomenon. On the five electricity price markets of a standard forecasting benchmark, Halo improves MSE and MAE in 28 of 30 model-market-metric comparisons, cutting average MSE by 2.6% to 16.5% and average MAE by 1.7% to 11.0%. Two findings emerge: (1) whether the scale estimate comes from a second projection head or from a full parallel network matters far less than whether the network estimates scale, and (2) the improvement holds under the hyperparameters already tuned for the point-estimate baseline, so retuning is optional.
Long-term time-series forecasting needs models that are accurate yet efficient enough for commodity hardware. Lightweight linear forecasters are remarkably strong in this regime, yet they leave two openings: reversible instance normalization (RevIN) de-normalizes the entire horizon with a single lookback statistic, which is inaccurate under non-stationarity, and time-domain trend/seasonal decomposition relies on a fixed, non-adaptive filter. We present FreqLite, an ultra-lightweight, channel-independent frequency-decomposed linear forecaster: a learnable, lossless, partition-of-unity spectral filter splits the input into bands that are forecast by per-band linear heads and, unlike low-pass-truncation approaches, the high-frequency band is retained and modeled. FreqLite is the best lightweight model on the standard long-term forecasting benchmarks and, at long lookback (L=336), attains a lower average error than a PatchTST Transformer (0.3244 vs. 0.3587 MSE) while using 4x fewer parameters, 2.2x less memory, and 2.2x less time per epoch on a single 4 GB laptop GPU; although modest in magnitude, its improvements are statistically significant under paired Wilcoxon tests across all matched cells (p < 1e-5). We further introduce Adaptive Reversible Instance Normalization (A-RevIN), a regime-adaptive reversible normalization that strictly generalizes RevIN (recovered exactly when its gate is closed), engages under non-stationarity, and reduces to RevIN without harm on stationary data. We validate this on both a real strongly non-stationary dataset (ILI, up to ~5% MSE reduction) and a controlled synthetic drift sweep in which A-RevIN's benefit and its learned gate both rise monotonically with injected non-stationarity. Every component is independently ablatable (Linear and RLinear are special cases of FreqLite), and all results are reproducible on commodity hardware.