  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.
