Multi-Stage Methods for Cost Controlled Data Compression Using Principal Component Analysis

Several online principal component analysis (PCA) methodologies exist for data arriving sequentially that focus only on compression risk minimization. Recent work in this realm revolves around minimizing the cost-compression risk, which takes into account compression loss and sampling costs using a two-stage PCA procedure. Even though the procedure enjoys first-order efficiency, the authors could not mathematically verify the existence of the second-order efficiency property. In this article, we minimize cost-compression risk using a modified two-stage PCA procedure, which takes into account the data compression loss as well as the sampling cost when the smallest eigenvalue of the population variance–covariance matrix or its positive lower bound is known when the data is assumed to follow a multivariate normal distribution. The modified two-stage PCA procedure is shown to possess the second-order efficiency property, among others, including the second-order risk efficiency property under some conditions. The proposed method is novel but also fast and efficient, as illustrated by extensive data analyses through simulations and real data analysis.