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Abstract:
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We introduce a general theory for linear sufficient dimension reduction for functional data. Important features are (1) response and predictor can be functions, (2) they can be vector-valued functions as well (3) previous methods are based on slice; we rely on linear operators (4) we also developed functional SAVE and functional DR, which are the first to our knowledge. In the meantime, we also developed the rigorous theoretical framework for SDR for functional data. This new theoretical framework is parallel to the classical sufficient dimension reduction, but have special features with new construction of response space. Our development of inverse regression techniques is such that it does not have to be inverse moment. In this sense, it is also more general than their multivariate counterparts even in the multivariate setting. We demonstrate our methods with simulation, real data analysis and comparison with other methods.
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