A robust algorithm for template curve estimation based on manifold embedding

The problem of finding a template function that represents the common pattern of a sample of curves is considered. To address this issue, a novel algorithm based on a robust version of the isometric featuring mapping (Isomap) algorithm is developed. When the functional data lie on an unknown intrinsically low-dimensional smooth manifold, the corresponding empirical Fréchet median function is chosen as an intrinsic estimator of the template function. However, since the geodesic distance is unknown, it has to be estimated. For this, a version of the Isomap procedure is proposed, which has the advantage of being parameter free and easy to use. The feature estimated with this method appears to be a good pattern for the data, capturing the inner geometry of the curves. Comparisons with other methods, with both simulated and real datasets, are provided.