In the light of regularized dynamic time warping kernels, this paper re-considers the concept of time elastic centroid for a set of time series. We derive a new algorithm based on a probabilistic interpretation of kernel alignment matrices. This algorithm expresses the averaging process in terms of a stochastic alignment automata. It uses an iterative agglomerative heuristic method for averaging the aligned samples, while also averaging the times of occurrence of the aligned samples. By comparing classification accuracies for 45 heterogeneous time series datasets obtained by first nearest centroid/medoid classifiers we show that: i) centroid-based approaches significantly outperform medoid-based approaches, ii) for the considered datasets, our algorithm that combines averaging in the sample space and along the time axes, emerges as the most significantly robust model for time-elastic averaging with a promising noise reduction capability. We also demonstrate its benefit in an isolated gesture recognition experiment and its ability to significantly reduce the size of training instance sets. Finally we highlight its denoising capability using demonstrative synthetic data: we show that it is possible to retrieve, from few noisy instances, a signal whose components are scattered in a wide spectral band.