Optimization and testing in linear non‐Gaussian component analysis

Abstract

Independent component analysis (ICA) decomposes multivariate data into mutually independent components (ICs). The ICA model is subject to a constraint that at most one of these components is Gaussian, which is required for model identifiability. Linear non‐Gaussian component analysis (LNGCA) generalizes the ICA model to a linear latent factor model with any number of both non‐Gaussian components (signals) and Gaussian components (noise), where observations are linear combinations of independent components. Although the individual Gaussian components are not identifiable, the Gaussian subspace is identifiable. We introduce an estimator along with its optimization approach in which non‐Gaussian and Gaussian components are estimated simultaneously, maximizing the discrepancy of each non‐Gaussian component from Gaussianity while minimizing the discrepancy of each Gaussian component from Gaussianity. When the number of non‐Gaussian components is unknown, we develop a statistical test to determine it based on resampling and the discrepancy of estimated components. Through a variety of simulation studies, we demonstrate the improvements of our estimator over competing estimators, and we illustrate the effectiveness of our test to determine the number of non‐Gaussian components. Further, we apply our method to real data examples and show its practical value.