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 non-linear 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.
Highlights
The study of point processes is one of the central topics in stochastic processes and has been widely used to model discrete events in continuous time
In this paper we propose a new framework for intensity estimation of a Poisson process with compositional noise
Twenty independent realizations of a non-homogeneous Poisson process were simulated with the intensity function λ(t) = 100(3 + 2 sin((8t − 1/2)π)) on [0, 1]
Summary
The study of point processes is one of the central topics in stochastic processes and has been widely used to model discrete events in continuous time. In order to use a Poisson process in applications, one key step is to estimate its intensity function from a given sequence of observed events. If the intensity can be assumed to have a known parametric form, likelihood-based methods can be used to estimate the model parameters [6]. In many cases, the shape of the intensity is unknown and estimation requires the implementation of non-parametric methods. Non-parametric estimations provide more flexibility than parametric methods and can better characterize the underlying intensity function. In the case where prior knowledge about the process or shape of the intensity is known, Bayesian methods can be adopted and they often lead to a more accurate estimation [11,12,13]
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