Projection sparse principal component analysis: An efficient least squares method

Abstract

We propose a new sparse principal component analysis (SPCA) method in which the solutions are obtained by projecting the full cardinality principal components onto subsets of variables. The resulting components are guaranteed to explain a given proportion of variance. The computation of these solutions is very efficient. The proposed method compares well with the optimal least squares sparse components. We show that other SPCA methods fail to identify the best sparse approximations of the principal components and explain less variance than our solutions. We illustrate and compare our method with the analysis of a real dataset containing socioeconomic data and the computational results for nine datasets of increasing dimension with up to 16,000 variables.

SIMPCA: a framework for rotating and sparsifying principal components

“…(6) when some of the original variables are perfectly correlated, these methods fail to find the sparse representation of the standard pcs [27].…”

“…Merola [27] proposes to compute sparse pcs by simply projecting each pc, p j = Xv j , onto a block of variables, . X j .…”

Using sparse categorical principal components to estimate asset indices: new methods with an application to rural southeast asia

“…Merola () developed least squares SPCA, a method in which the sparse components explain the maximum variance of all the variables in the set. The optimization problem is solved by a backward elimination algorithm or by projection.…”

“…Since an optimal SPCA solution cannot be found in reasonable time, these methods differ in how suboptimal solutions to the problems are computed (see Trendafilov, 2014, for a review of these methods). Merola (2015Merola ( , 2018 developed least squares SPCA, a method in which the sparse components explain the maximum variance of all the variables in the set. The optimization problem is solved by a backward elimination algorithm or by projection.…”

“…In poorer Houaphanh province we find common assets, such as cellphones, satellite dishes, and fridges; in richer Thanh Hoa we find more sophisticated assets, such as stoves, dining tables, and other household equipment entering the index. As explained in Merola (2018), the SPCA approach we take is analogous to selecting a subset of the variables which explain, in a regression sense, the first PC with the required percentage of variability. This means that the variables excluded either have low correlation with the full-cardinality AI or they are highly (multiply) correlated with other variables in the model and, therefore, are redundant for explaining the variance of the AI.…”