Selecting the number of components in PCA via random signflips.
Existing principal component analysis (PCA) model selection methods lack statistical guarantees for determining the number of leading components under heteroscedastic noise—where observation-wise noise variances differ—in high-dimensional settings. Method: We propose Signflip Parallel Analysis (Signflip PA), a novel parallel analysis method that generates an empirical null distribution via random sign flips and adaptively calibrates singular value thresholds. Contribution/Results: Signflip PA is the first to integrate dimension-free operator norm bounds and large-deviation theory for eigenvalues of non-homogeneous matrices into PCA model selection, ensuring consistent factor recovery. We establish its theoretical consistency under a signal-plus-heteroscedastic-noise model. Empirical studies—including simulations and real-data analyses—demonstrate that Signflip PA significantly outperforms classical approaches such as scree plots and conventional parallel analysis, overcoming the fundamental limitation wherein heteroscedasticity causes traditional methods to fail.