AUTHOR=Schluck Glenna , Wu Wei , Srivastava Anuj TITLE=Intensity Estimation for Poisson Process With Compositional Noise JOURNAL=Frontiers in Applied Mathematics and Statistics VOLUME=7 YEAR=2021 URL=https://www.frontiersin.org/journals/applied-mathematics-and-statistics/articles/10.3389/fams.2021.648984 DOI=10.3389/fams.2021.648984 ISSN=2297-4687 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.