This paper introduces a robust approach to functional principal component analysis (FPCA) for relative data, particularly density functions. While recent papers have studied density data within the Bayes space framework, there has been limited focus on developing robust methods to effectively handle anomalous observations and large noise. To address this, we extend the Mahalanobis distance concept to Bayes spaces, proposing its regularized version that accounts for the constraints inherent in density data. Based on this extension, we introduce a new method, robust density principal component analysis (RDPCA), for more accurate estimation of functional principal components in the presence of outliers. The method's performance is validated through simulations and real-world applications, showing its ability to improve covariance estimation and principal component analysis compared to traditional methods.

翻译：本文针对相对数据（特别是密度函数）提出了一种稳健的函数主成分分析（FPCA）方法。尽管近期研究已在贝叶斯空间框架下探讨了密度数据，但开发能够有效处理异常观测和大噪声的稳健方法仍关注有限。为此，我们将马氏距离概念扩展至贝叶斯空间，提出了考虑密度数据固有约束的正则化版本。基于此扩展，我们引入了一种新方法——稳健密度主成分分析（RDPCA），以在存在异常值时更准确地估计函数主成分。通过仿真和实际应用验证了该方法的性能，结果表明与传统方法相比，其在协方差估计和主成分分析方面具有改进能力。