Functional data; Non parametric regression; Semimetric; Wavelet; Misalignment; Cross-Validation
Abstract :
[en] Our goal is to predict a scalar value or a group membership from the discretized observation of curves with sharp local features that might vary both vertically and horizontally. To this aim, we propose to combine the use of the nonparametric functional regression estimator developed by Ferraty and Vieu (2006) [18] with the Bagidis semimetric developed by Timmermans and von Sachs (submitted for publication) [36] with a view of efficiently measuring dissimilarities between curves with sharp patterns. This association is revealed as powerful. Under quite general conditions, we first obtain an asymptotic expansion for the small ball probability indicating that Bagidis induces a fractal topology on the functional space. We then provide the rate of convergence of the nonparametric regression estimator in this case, as a function of the parameters of the Bagidis semimetric. We propose to optimize those parameters using a cross-validation procedure, and show the optimality of the selected vector. This last result has a larger scope and concerns the optimization of any vector parameter characterizing a semimetric used in this context. The performances of our methodology are assessed on simulated and real data examples. Results are shown to be superior to those obtained using competing semimetrics as soon as the variations of the significant sharp patterns in the curves have a horizontal component.
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