Uncertainty-Aware Principal Component Analysis

• Published in: IEEE transactions on visualization and computer graphics Vol. 26; no. 1; pp. 822 - 831
• Main Authors: Gortler, Jochen, Spinner, Thilo, Streeb, Dirk, Weiskopf, Daniel, Deussen, Oliver
• Format: Journal Article
• Language: English
• Published: United States IEEE 01.01.2020; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Covariance matrices; Covariance matrix; Data visualization; Dimensionality reduction; linear projection; machine learning; Mathematical analysis; Matrix methods; Multivariate analysis; Principal component analysis; Principal components analysis; Probability distribution; Random variables; Reduction; Sensitivity analysis; Uncertainty; Uncertainty analysis
• ISSN: 1077-2626; 1941-0506; 1941-0506
• DOI: 10.1109/TVCG.2019.2934812

We present a technique to perform dimensionality reduction on data that is subject to uncertainty. Our method is a generalization of traditional principal component analysis (PCA) to multivariate probability distributions. In comparison to non-linear methods, linear dimensionality reduction techniques have the advantage that the characteristics of such probability distributions remain intact after projection. We derive a representation of the PCA sample covariance matrix that respects potential uncertainty in each of the inputs, building the mathematical foundation of our new method: uncertainty-aware PCA. In addition to the accuracy and performance gained by our approach over sampling-based strategies, our formulation allows us to perform sensitivity analysis with regard to the uncertainty in the data. For this, we propose factor traces as a novel visualization that enables to better understand the influence of uncertainty on the chosen principal components. We provide multiple examples of our technique using real-world datasets. As a special case, we show how to propagate multivariate normal distributions through PCA in closed form. Furthermore, we discuss extensions and limitations of our approach.