Prediction of multivariate responses with a select number of principal   components
  This paper proposes a new method and algorithm for predicting multivariate responses in a regression setting. Research into classification of High Dimension Low Sample Size (HDLSS) data, in particular microarray data, has made considerable advances, but regression prediction for high-dimensional data with continuous responses has had less attention. Recently Bair et al (2006) proposed an efficient prediction method based on supervised principal component regression (PCR). Motivated by the fact that a larger number of principal components results in better regression performance, this paper extends the method of Bair et al in several ways: a comprehensive variable ranking is combined with a selection of the best number of components for PCR, and the new method further extends to regression with multivariate responses. The new method is particularly suited to HDLSS problems. Applications to simulated and real data demonstrate the performance of the new method. Comparisons with Bair et al (2006) show that for high-dimensional data in particular the new ranking results in a smaller number of predictors and smaller errors. 
Inge Koch, Kanta Naito