When studying some process or development in different subjects or units--be it biological, chemical or physical--we usually see a typical pattern, common to all curves. Yet there is variation both in amplitude and dynamics between curves. Following some ideas of structural analysis introduced by Kneip and Gasser, we study a method--dynamic time warping with a proper cost function--for estimating the shift or warping function from one curve to another to align the two functions. For some models this method can identify the true shift functions if the data are noise free. Noisy data are smoothed by a nonparametric function estimate such as a kernel estimate. It is shown that the proposed estimator is asymptotically normal and converges to the true shift function as the sample size per subject goes to infinity. Some simulation results are presented to illustrate the performance of this method.
"Alignment of curves by dynamic time warping." Ann. Statist. 25 (3) 1251 - 1276, June 1997. https://doi.org/10.1214/aos/1069362747