Principal component analysis (PCA) is widely used in various fields to reduce high dimensional data sets to lower dimensions. Traditionally, the first a few principal components that capture most of the variance in the data are thought to be important. Tipping and Bishop (1999) introduced probabilistic principal component analysis (PPCA) in which they assumed an isotropic error in a latent variable model. Motivated by a general error structure and incorporating the novel idea of ‘‘envelope” proposed by Cook et al. (2010), we construct principal envelope models (PEM) which demonstrate the possibility that any subset of the principal components could retain most of the sample’s information. The useful principal components can be found through maximum likelihood approaches. We also embed the PEM to a factor model setting to illustrate its reasonableness and validity. Numerical results indicate the potentials of the proposed method.
