Scalable robust covariance and correlation estimates for data mining

Covariance and correlation estimates have important applications in data mining. In the presence of outliers, classical estimates of covariance and correlation matrices are not reliable. A small fraction of outliers, in some cases even a single outlier, can distort the classical covariance and correlation estimates making them virtually useless. That is, correlations for the vast majority of the data can be very erroneously reported; principal components transformations can be misleading; and multidimensional outlier detection via Mahalanobis distances can fail to detect outliers. There is plenty of statistical literature on robust covariance and correlation matrix estimates with an emphasis on affine-equivariant estimators that possess high breakdown points and small worst case biases. All such estimators have unacceptable exponential complexity in the number of variables and quadratic complexity in the number of observations. In this paper we focus on several variants of robust covariance and correlation matrix estimates with quadratic complexity in the number of variables and linear complexity in the number of observations. These estimators are based on several forms of pairwise robust covariance and correlation estimates. The estimators studied include two fast estimators based on coordinate-wise robust transformations embedded in an overall procedure recently proposed by [14]. We show that the estimators have attractive robustness properties, and give an example that uses one of the estimators in the new Insightful Miner data mining product.