VIPurPCA: Visualizing and Propagating Uncertainty in Principal Component Analysis

Variables obtained by experimental measurements or statistical inference typically carry uncertainties. When an algorithm uses such quantities as input variables, this uncertainty should propagate to the algorithm's output. Concretely, we consider the classic notion of principal component analysis (PCA): If it is applied to a finite data matrix containing imperfect (i.e., uncertain) multidimensional measurements, its output—a lower-dimensional representation—is itself subject to uncertainty. We demonstrate that this uncertainty can be approximated by appropriate linearization of the algorithm's nonlinear functionality, using automatic differentiation. By itself, however, this structured, uncertain output is difficult to interpret for users. We provide an animation method that effectively visualizes the uncertainty of the lower dimensional map. Implemented as an open-source software package, it allows researchers to assess the reliability of PCA embeddings.