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 transformation. The resulting approach is simple, fast and widely applicable. We study its robustness by deriving influnce functions and breakdown values, and computing the mean squared error on contaminated data. Using these results we select a method that performs well overall, which we call wrapping [1] and which is available in the R package cellWise [2]. Wrapping allows a very substantial speedup of the DetectDeviatingCells [3] technique for flagging cellwise outliers, which is applied to genomic data with 12,000 variables. Wrapping is able to deal with even higher dimensional data, which is illustrated on color video data with 920,000 dimensions.
Keywords: Anomaly detection, Covariance matrix, Data transformation.
References
[1] J. Raymaekers and P.J. Rousseeuw (2018). Fast robust correlation for high dimensional data, arXiv:1712.05151.
[2] J. Raymaekers, P.J. Rousseeuw and W. Van den Bossche (2018), Package cellWise version 2.0.8, CRAN.
[3] P.J. Rousseeuw and W. Van den Bossche (2018), Detecting deviating data cells, Technometics, 60:2, 135-145.