Transformed partial least squares for multivariate data

The research described herein is motivated by a study of the relationship between agricultural meteorology and three major yields of crops in a province of China. To build a regression model for this data set with multivariate response and high-dimensional covariates, three issues are of particular interest: reducing the dimension of the covariates, avoiding the collinearity between the components of the covariates, and capturing the nonlinearity structure. To deal with these problems, we propose a method of nonparametric response transformation to build a single-index type model, and use partial least squares to reduce the dimension of covariates and to overcome the problem of collinearity. Our method is an alternative approach to sliced inverse regression when the underlying model is single-index type. To select the transformations, a new criterion based on maximizing the covariance matrix is recommended. The selected transformations are estimated by splines; here B-splines are used for general cases and I-splines with a penalty function are suggested when the transformations are monotonic. A modified BIC selection principle is proposed to determine the dimensionality of the space of spline transformations. The consistency of the estimators is proved and easily implemented algorithms are provided. Application to the agricultural data set is carried out.