Simultaneous Multivariate Outlier and Trend Detection

We develop a methodology that combines stable principal component pursuit (SPCP) and elastic net regression to perform multivariate outlier and trend detection simultaneously. The SPCP framework detects both univariate and multivariate outliers by decomposing a data matrix into its sparse and low-rank components. Elastic net regression applied to the low-rank matrix identifies the response variables that are well-explained by a set of covariates without being affected by influential outliers. By combining these techniques into a single objective function, we simultaneously detect univariate and multivariate outliers and trend with an accompanying estimate of magnitude for each. Our methodology is applied to both real and synthetic data to show its value and accuracy.