Optimal functional supervised classification with separation condition.(English)Zbl 1441.62930

The paper deals with binary supervised classification of trajectories of stochastic processes. Basing on the finite training set of trajectories one has to determine to which of two unknown processes the observed trajectory is related. In this paper the authors consider the trajectories of solutions of stochastic differential equations $dX_t = \mu(t) dt +dW_t, \qquad t \in [0,1],\quad \mu \in \{f,g\},$ where the drifts $$f$$ and $$g$$ belong to the Sobolev-Hilbert space $$H_s (0,1)$$, $$s\geq 0$$, and are significantly different $$\|f-g\| \geq \Delta >0$$.
The goal is to provide a natural classifier such that asymptotically no other classifier is significantly outperforming it.

MSC:

 62R10 Functional data analysis 62H30 Classification and discrimination; cluster analysis (statistical aspects) 68T05 Learning and adaptive systems in artificial intelligence 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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