A Nonlinear Differential Equation for Generating Warping Function

TL;DR

This paper introduces a nonlinear differential equation approach to estimate warping functions for aligning data sequences, effectively reducing phase variability and improving data analysis.

Contribution

It presents a novel differential equation model for warping function estimation and demonstrates its effectiveness on synthetic data.

Findings

Effective alignment of curves with phase variability

Aligned curves show only amplitude variation

Method reduces phase differences efficiently

Abstract

Given set of functions $y_i(t)$ and $x(t)$ such that $y_i(t) = a_i x\left[h_i(t)\right]$ with $a_i$ being an unknown amplitude with low changes in time (or $\frac{\Delta a_i}{a^2_i} << 1$) and $h_i(t)$ an unknown warping function, the paper shows that $h_i(t)$ can be described using a non-linear differential equation. The differential equation then can be utilized to estimate the warping function $h_i(t)$ using a nonlinear least-squares optimization. This differential equation can also be useful for reducing and analyzing phase variability in data sequences. Results, obtained on synthetic curves, showed that the proposed method is effective in aligning the curves. The obtained aligned curves exhibit variation only in amplitude, and phase variation can be removed efficiently.