Intensity Estimation for Poisson Process with Compositional Noise

TL;DR

This paper introduces an alignment-based framework for estimating Poisson process intensities in the presence of compositional noise, leveraging area-preservation and phase difference metrics to improve accuracy.

Contribution

It proposes a novel method that accounts for nonlinear time shifts in observations, enabling consistent intensity estimation under compositional noise.

Findings

The method provides consistent normalized intensity estimates.

It improves classification accuracy in neural spike train data.

The approach is validated through simulations and real data applications.

Abstract

Intensity estimation for Poisson processes is a classical problem and has been extensively studied over the past few decades. Practical observations, however, often contain compositional noise, i.e. a nonlinear shift along the time axis, which makes standard methods not directly applicable. The key challenge is that these observations are not "aligned", and registration procedures are required for successful estimation. In this paper, we propose an alignment-based framework for positive intensity estimation. We first show that the intensity function is area-preserved with respect to compositional noise. Such a property implies that the time warping is only encoded in the normalized intensity, or density, function. Then, we decompose the estimation of the intensity by the product of the estimated total intensity and estimated density. The estimation of the density relies on a metric which measures the phase difference between two density functions. An asymptotic study shows that the proposed estimation algorithm provides a consistent estimator for the normalized intensity. We then extend the framework to estimating non-negative intensity functions. The success of the proposed estimation algorithms is illustrated using two simulations. Finally, we apply the new framework in a real data set of neural spike trains, and find that the newly estimated intensities provide better classification accuracy than previous methods.