Anomaly detection of time series correlations via a novel Lie group structure

Abstract

We investigate a natural Lie group structure that correlations possess, interpreted as a special case of the quotient manifold structure for correlation matrices. The one‐dimensional setting of the general theory gives rise to additional structure in the form of a natural associative multiplication of correlations making the space of correlations into a Lie group. We explicitly compute left‐invariant vector fields, exponential and logarithmic mappings, and ultimately, a closed‐form distance formula between correlations. The Lie group formalism is then applied to an application in anomalous correlation detection. The advantage of the Lie group method illustrated with time series data of stock prices.