Measures of generalized magnitude-squared coherence: Differences and similarities

The magnitude-squared coherence (MSC) is a measure that estimates the extent to which one real- or complex-valued signal can be predicted from another real- or complex-valued signal using a linear model. It is also used as a measure of the similarities in the frequency content of two signals. The measure is widely used in signal analysis, especially in fields such as biomedical, where a large number of signals must be processed simultaneously. It is natural to wish to generalize this idea to compare two vector-valued signals, and a variety of approaches have been proposed. This paper reviews these generalizations, presents new relationships among the measures, and demonstrates a series of results that show the similarities and dissimilarities among these measures. Some of the measures have a clear link with total interdependence; some are related to the mutual information rate. Basic results such as the various Sandwich Theorems show how the measures relate, and understanding these properties is key to an informed use of any vector generalization of MSC.