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.

中文翻译：

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不确定性主成分分析。

我们提出一种对不确定性数据执行降维的技术。我们的方法是将传统的主成分分析（PCA）推广到多元概率分布。与非线性方法相比，线性降维技术的优势在于，这种概率分布的特征在投影后保持不变。我们推导了PCA样本协方差矩阵的表示形式，该矩阵尊重每个输入中的潜在不确定性，从而建立了我们新方法的数学基础：不确定性感知的PCA。除了我们的方法相对于基于抽样的策略所获得的准确性和性能外，我们的公式还使我们能够对数据的不确定性进行敏感性分析。为了这，我们提出将因子迹线作为一种新颖的可视化方法，使您可以更好地了解不确定性对所选主成分的影响。我们提供了使用真实数据集的多种技术示例。作为一个特例，我们展示了如何通过封闭形式的PCA传播多元正态分布。此外，我们讨论了我们方法的扩展和局限性。