# Uncertainty-Aware Principal Component Analysis

## TL;DR

This paper introduces an uncertainty-aware PCA method that extends traditional PCA to handle data with uncertainty, enabling more accurate dimensionality reduction and sensitivity analysis on probabilistic data.

## Contribution

It generalizes PCA to multivariate probability distributions, providing a mathematically grounded approach that accounts for data uncertainty and offers new visualization tools.

## Key findings

- Improved accuracy over sampling-based methods
- Allows sensitivity analysis of data uncertainty
- Provides closed-form propagation of normal distributions

## Abstract

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.