Fast robust correlation for high dimensional data
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The product moment covariance is a cornerstone of multivariate data analysis, from which one can derive correlations, principal components, linear regression, sparse modeling, variable screening, and many other methods. 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, and can be used in ultrahigh dimensional settings. It is illustrated on a genomic data set.
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