Semiparametric curve alignment and shift density estimation for biological data

Assume that we observe a large number of curves, 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 split our data set in blocks, on which the estimation of the shifts is done by minimizing a cost criterion based on a functional of the periodogram; the estimated shifts are then 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 estimator of the shift distribution performs well, even in the case of low signal-to-noise ratio, and is shown to outperform the standard methods for curve alignment.Comment: 30 pages ; v5 : minor changes and correction in the proof of Proposition 3.

Semiparametric Curve Alignment and Shift Density Estimation: ECG Data Processing Revisited

We address in this contribution a problem stemming from functional data analysis. Assuming that we dispose of a large number of shifted recorded curves with identical shape, the objective is to estimate the time shifts as well as their distribution. Such an objective appears in several biological applications, for example in ECG signal processing. We are interested in the estimation of the distribution of elapsed durations between repetitive pulses, but wish to estimate it with a possibly low signal-to-noise ratio, or without any knowledge of the pulse shape. This problem is solved within a semiparametric framework, that is without any knowledge of the shape. We suggest an M-estimator leading to two different algorithms whose main steps are as follows: we split our dataset in blocks, on which the estimation of the shifts is done by minimizing a cost criterion, based on a functional of the periodogram. The estimated shifts are then plugged into a standard density estimator. Some theoretical insights are presented, which show that under mild assumptions the alignment can be done efficiently. Results are presented on simulations, as well as on real data for the alignment of ECG signals, and these algorithms are compared to the methods used by practitioners in this framework. It is shown in the results that the presented method outperforms the standard ones, thus leading to a more accurate estimation of the average heart pulse and of the distribution of elapsed times between heart pulses, even in the case of low Signal-to- Noise Ratio (SNR)