Functional k-means inverse regression

A new dimension reduction method is proposed for functional multivariate regression with a multivariate response and a functional predictor by extending the functional sliced inverse regression model. Naive application of existing dimension reduction techniques for univariate response will create too many hyper-rectangular slices. To avoid this curse of dimensionality, a new slicing method is proposed by clustering over the space of the multivariate response, which generates a much smaller set of slices of flexible shapes. The proposed method can be applied to any number of response variables and can be particularly useful for exploratory analysis. In addition, a new eigenvalue-based method for determining the dimensionality of the reduced space is developed. Real and simulation data examples are then presented to demonstrate the effectiveness of the proposed method.