NeuroMemFPP: A recurrent neural approach for memory-aware parameter estimation in fractional Poisson process

TL;DR

This work tackles parameter estimation for the fractional Poisson process, a memory-aware generalization of the Poisson model. It proposes a direct LSTM-based framework that maps sequences of inter-arrival times to the FPP parameters $μ$ and $β$, leveraging long-range dependencies. On synthetic data, the approach achieves a substantial MSE reduction (~55.3%) over the method of moments and offers faster inference. The method is validated on real high-frequency datasets (emergency calls and finance), demonstrating its ability to track time-varying parameters and daily patterns in complex temporal data.

Abstract

In this paper, we propose a recurrent neural network (RNN)-based framework for estimating the parameters of the fractional Poisson process (FPP), which models event arrivals with memory and long-range dependence. The Long Short-Term Memory (LSTM) network estimates the key parameters $μ>0$ and $β\in(0,1)$ from sequences of inter-arrival times, effectively modeling their temporal dependencies. Our experiments on synthetic data show that the proposed approach reduces the mean squared error (MSE) by about 55.3\% compared to the traditional method of moments (MOM) and performs reliably across different training conditions. We tested the method on two real-world high-frequency datasets: emergency call records from Montgomery County, PA, and AAPL stock trading data. The results show that the LSTM can effectively track daily patterns and parameter changes, indicating its effectiveness on real-world data with complex time dependencies.

Figures (8)

• Figure 1: LSTM-based architectures for parameter estimation of the FPP: (top) main architecture for estimating $\mu > 0$ and $\beta \in (0, 1)$; (bottom left/right) two variants of the LSTM-based design.
• Figure 2: Comparison of parameter estimation accuracy for the FPP using RNN and MOM. The RNN model significantly performance well the method of moments, achieving a much lower Mean Squared Error
• Figure 3: Effect of training epochs (left) and sample sizes (right) on the RMSE, MAE, and $R^2$ of RNN (LSTM)-based FPP parameter estimation.
• Figure 4: Effect of sequence length (left) and learning rate (right) on the RMSE, MAE, and $R^2$ of RNN (LSTM)-based FPP parameter estimation.
• Figure 5: Effect of hidden size (left) and batch size (right) on the RMSE, MAE, and $R^2$ of RNN (LSTM)-based FPP parameter estimation.

Theorems & Definitions (2)

• Definition 1: Homogeneous Poisson process
• Definition 2: Fractional Poisson Process (FPP) lask