Anticipated and Adaptive Prediction in Functional Discriminant Analysis

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.