Variable selection and covariance structure identification using loadings

We provide sparse principal loading analysis which is a new concept that reduces dimensionality of cross sectional data and identifies the underlying covariance structure. Sparse principal loading analysis selects a subset of existing variables for dimensionality reduction while variables that have a small distorting effect on the covariance matrix are discarded. Therefore, we show how to detect these variables and provide methods to assess their magnitude of distortion. Sparse principal loading analysis is twofold and can also identify the underlying block diagonal covariance structure using sparse loadings. This is a new approach in this context and we provide a required criterion to evaluate if the found block-structure fits the sample. The method uses sparse loadings rather than eigenvectors to decompose the covariance matrix which can result in a large loss of information if the loadings of choice are too sparse. However, we show that this is no concern in our new concept because sparseness is controlled by the aforementioned evaluation criterion. Further, we show the advantages of sparse principal loading analysis both in the context of variable selection and covariance structure detection, and illustrate the performance of the method with simulations and on real datasets. Supplementary material for this article is available online.