Semiparametric Curve Alignment and Shift Density Estimation for Biological Data

We observe a large number of signals, all of them with identical, although unknown, shape, but with a different random shift. The objective is to estimate the individual time shifts and their distribution. Such an objective appears in several biological applications like neuroscience or ECG signal processing, in which the estimation of the distribution of the elapsed time between repetitive pulses with a possibly low signal-noise ratio, and without a knowledge of the pulse shape is of interest. We suggest an M-estimator leading to a three-stage algorithm: we first split our data set in blocks, then the shift estimation in each block is done by minimizing a cost function based on the periodogram; the estimated shifts are eventually plugged into a standard density estimator. We show that under mild regularity assumptions the density estimate converges weakly to the true shift distribution. The theory is applied both to simulations and to alignment of real ECG signals. The proposed approach outperforms the standard methods for curve alignment and shift density estimation, even in the case of low signal-to-noise ratio, and is robust to numerous perturbations common in ECG signals.