Functional Sufficient Dimension Reduction for Functional Data Classification

We consider two novel functional classification methods for binary response and functional predictor. We extend the most popular functional sufficient dimension reduction methods such as functional sliced inverse regression (FSIR) and functional sliced average variance estimation (FSAVE) by introducing a regularized estimation procedure and incorporating the localized information of the functional predictor in the analysis. Compared to the existing FSIR and FSAVE, the proposed methods are appealing because they are capable of estimating more than one effective dimension reduction direction, whereas FSIR detects only one such direction and FSAVE produces inefficient estimation in the case of binary response. Moreover, our methods make use of the localized information of the functional predictor, thereby more efficiently capturing the nonlinear relation between the binary response and the functional predictor. Furthermore, the proposed methods can be extended to incorporate the ancillary unlabeled data in semi-supervised learning. The empirical performance and the applications of the proposed methods are demonstrated by simulation studies and real applications.