Curve registration by nonparametric goodness-of-fit testing

TL;DR

This paper introduces a nonparametric likelihood ratio test for curve registration, demonstrating its asymptotic distribution, consistency, and practical effectiveness through simulations and an image analysis application.

Contribution

It develops a novel generalized likelihood ratio test for curve registration with proven asymptotic properties and practical validation.

Findings

Test statistic follows chi-squared distribution under null hypothesis

The test is consistent with power approaching 1 asymptotically

Finite sample performance confirmed by numerical simulations

Abstract

The problem of curve registration appears in many different areas of applications ranging from neuroscience to road traffic modeling. In the present work, we propose a nonparametric testing framework in which we develop a generalized likelihood ratio test to perform curve registration. We first prove that, under the null hypothesis, the resulting test statistic is asymptotically distributed as a chi-squared random variable. This result, often referred to as Wilks' phenomenon, provides a natural threshold for the test of a prescribed asymptotic significance level and a natural measure of lack-of-fit in terms of the $p$-value of the $\chi^2$-test. We also prove that the proposed test is consistent, \textit{i.e.}, its power is asymptotically equal to $1$. Finite sample properties of the proposed methodology are demonstrated by numerical simulations. As an application, a new local descriptor for digital images is introduced and an experimental evaluation of its discriminative power is conducted.