Fast robust correlation for high-dimensional data

Abstract

The product moment covariance is a cornerstone of multivariate data analysis, from which one can derive correlations, principal components, Mahalanobis distances and many other results. Unfortunately the product moment covariance and the corresponding Pearson correlation are very susceptible to outliers (anomalies) in the data. Several robust measures of covariance have been developed, but few are suitable for the ultrahigh dimensional data that are becoming more prevalent nowadays. For that one needs methods whose computation scales well with the dimension, are guaranteed to yield a positive semidefinite covariance matrix, and are sufficiently robust to outliers as well as sufficiently accurate in the statistical sense of low variability. We construct such methods using data transformations. The resulting approach is simple, fast and widely applicable. We study its robustness by deriving influence functions and breakdown values, and computing the mean squared error on contaminated data. Using these results we select a method that performs well overall. This also allows us to construct a faster version of the DetectDeviatingCells method (Rousseeuw and Van den Bossche, 2018) to detect cellwise outliers, that can deal with much higher dimensions. The approach is illustrated on genomic data with 12,000 variables and color video data with 920,000 dimensions.

References in corpus (3)

• Measuring and testing dependence by correlation of distances
• Propagation of outliers in multivariate data
• MacroPCA: An all-in-one PCA method allowing for missing values as well as cellwise and rowwise outliers

Cited by in corpus (14)

• The Cellwise Minimum Covariance Determinant Estimator
• Challenges of cellwise outliers
• Handling cellwise outliers by sparse regression and robust covariance
• Real-time outlier detection for large datasets by RT-DetMCD
• Multivariate Singular Spectrum Analysis by Robust Diagonalwise Low-Rank Approximation
• The FEDHC Bayesian network learning algorithm
• MacroPARAFAC for handling rowwise and cellwise outliers in incomplete multi-way data
• Robust Distance Covariance
• Robust Variable Selection under Cellwise Contamination
• Comparison of correlation-based measures of concordance in terms of asymptotic variance
• Compatibility and attainability of matrices of correlation-based measures of concordance
• Casewise and Cellwise Robust Multilinear Principal Component Analysis
• Robust Principal Components by Casewise and Cellwise Weighting
• On a Generalization of the Average Distance Classifier