Consistency of functional learning methods based on derivatives

In some real world applications, such as spectrometry, functional models achieve better predictive performances if they work on the derivatives of order m of their inputs rather than on the original functions. As a consequence, the use of derivatives is a common practice in functional data analysis, despite a lack of theoretical guarantees on the asymptotically achievable performances of a derivative based model. In this paper, we show that a smoothing spline approach can be used to preprocess multivariate observations obtained by sampling functions on a discrete and finite sampling grid in a way that leads to a consistent scheme on the original infinite dimensional functional problem. This work extends Mas and Pumo (2009) to nonparametric approaches and incomplete knowledge. To be more precise, the paper tackles two difficulties in a nonparametric framework: the information loss due to the use of the derivatives instead of the original functions and the information loss due to the ...

• Type: Journal
• Year: 2011
• Where: PRL
• Authors: Fabrice Rossi, Nathalie Villa-Vialaneix
• Keywords: Computer Networks, Original Functions, PRL 2011, Real World, Smoothing Spline