Properties of a vector correlation coefficient with application to geophysical data

In many practical problems, one common question is to find a measure of vector associations. An example is marine surface wind data. Wind data are collected from buoys located in a certain areas on the sea surface. A typical observation of wind data is composed of two components, wind speed and wind direction. For two buoys or two groups of buoys, two sets of observations can be obtained. A measure of association of the two sets of observations will provide an additional tool for understanding the sea surface wind. Theoretical problems are raised in defining such a measure of association and in investigating its statistical properties. While the definition of a vector correlation coefficient is not unique, one definition has been proposed by Jupp & Mardia (1980). The sample distributional properties of such a vector correlation coefficient $r\sp2$ have been studied by many authors. This is because the statistic is a function of the canonical correlation coefficients. The exact distribution of the sample statistic $r\sp2$ under the alternative hypothesis of independence has not been well established. However, detailed asymptotic distributional properties are derived for both normal and nonnormal marginal distributions. The limiting distribution of ${nr}\sp2$ is shown to be a linear combination of noncentral chi-square variate with one degree of freedom and the noncentrality parameters determined by a function of the Pearson correlation coefficients of each pair of components of vectors, with weights being related to the fourth order moments of the marginal distributions. By appropriately selecting the weights in such a linear combination, the limiting distribution of ${nr}\sp2$ is expressed as a sum of a series of noncentral chi-square distributions. By utilizing additional independent observations from the marginal distributions, the power of the test of the independence of two vectors is higher. Simulation studies are applied to verify the theoretical results. The simulation results showed that the limiting distribution provides good fit for large sample sizes under normal marginal distributions. For dependent vectors with nonnormal marginal distributions, the results did not show uniformly a satisfactory fit. Possible reasons are truncating errors in computing the infinite series, and estimating procedures of the fourth order moments. Further studies are proposed to carry out. An application to the marine surface wind data is carried out using bootstrap techniques.