Asymptotic properties of Principal Component Analysis and shrinkage-bias adjustment under the Generalized Spiked Population model
This work addresses the limitations of PCA for high-dimensional datasets with local correlations, which is incremental as it extends existing theoretical frameworks to more realistic scenarios.
The authors tackled the problem of principal component analysis (PCA) in high-dimensional data where traditional spiked models fail due to local correlations, by investigating asymptotic behaviors under a generalized spiked population model and proposing methods for consistent estimation and shrinkage bias adjustment. They demonstrated that these methods greatly reduce bias and improve prediction accuracy in numerical experiments and real genetic data.
With the development of high-throughput technologies, principal component analysis (PCA) in the high-dimensional regime is of great interest. Most of the existing theoretical and methodological results for high-dimensional PCA are based on the spiked population model in which all the population eigenvalues are equal except for a few large ones. Due to the presence of local correlation among features, however, this assumption may not be satisfied in many real-world datasets. To address this issue, we investigated the asymptotic behaviors of PCA under the generalized spiked population model. Based on the theoretical results, we proposed a series of methods for the consistent estimation of population eigenvalues, angles between the sample and population eigenvectors, correlation coefficients between the sample and population principal component (PC) scores, and the shrinkage bias adjustment for the predicted PC scores. Using numerical experiments and real data examples from the genetics literature, we showed that our methods can greatly reduce bias and improve prediction accuracy.