A Proposed Definition for Vector Correlation in Geophysics: Theory and Application

A universally accepted definition for vector correlation in oceanography and meteorology does not presently exist. To address this need, a generalized correlation coefficient, originally proposed by Hopper and later expanded upon by Jupp and Mardia, is explored. A short history of previous definitions is presented. Then the definition originally proposed by Hooper is presented together with supporting theory and associated properties. The most significant properties of this vector correlation coefficient are that it is a generalization of the square of the simple one-dimensional correlation coefficient, and when the vectors are independent, its asymptotic distribution is known; hence, it can be used for hypothesis testing. Because the asymptotic results hold only for large samples, and in practical situations only small samples are often available, modified sampling distributions are derived using simulation techniques for samples as small as eight. It is symmetric with respect to its arguments and has a simple interpretation in terms of canonical correlation. It is invariant under transformations of the coordinate axes, including rotations and changes of scale.