Anticipated and Adaptive Prediction in Functional Discriminant Analysis

Abstract

Linear discriminant analysis with binary response is considered when the predictor is a functional random variable $X=\{X_{t},t\in [0,T]\}$, $T \in\mathbb{R}$. Motivated by a food industry problem, we develop a methodology to anticipate the prediction by determining the smallest $T^{*}$, $T^{*} \leq T$, such that $X^{*} = \{X_{t}, t\in [0,T^{*}]\}$ and X give similar predictions. The adaptive prediction concerns the observation of a new curve ω on $[0, T^{*}(\omega)]$ instead of [0, T] and answers to the question “How long should we observe ω ($T^{*}(\omega)=?$) for having the same prediction as on [0,T] ?”. We answer to this question by defining a conservation measure with respect to the class the new curve is predicted.