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Dimensionality reduction via PCA and factor analysis is an important tool ofdata analysis. A critical step is selecting the number of components. However,existing methods (such as the scree plot, likelihood ratio, parallel analysis,etc) do not have statistical guarantees in the increasingly common settingwhere the data are heterogeneous. There each noise entry can have a differentdistribution. To address this problem, we propose the Signflip ParallelAnalysis (Signflip PA) method: it compares data singular values to those of"empirical null" data generated by flipping the sign of each entry randomlywith probability one-half. We show that Signflip PA consistently selectsfactors above the noise level in high-dimensional signal-plus-noise models(including spiked models and factor models) under heterogeneous settings. Hereclassical parallel analysis is no longer effective. To do this, we rely onrecent results in random matrix theory, such as dimension-free operator normbounds [Latala et al, 2018, Inventiones Mathematicae], and large deviations forthe top eigenvalues of nonhomogeneous matrices [Husson, 2020]. We alsoillustrate that Signflip PA performs well in numerical simulations and onempirical data examples.

Selecting the number of components in PCA via random signflips