Multivariate Linear Correlation Analysis

Multivariate linear correlation analysis plays an important role in various fields such as statistics, economics, and big data analytics. However, there was no compact formulation to define and measure multivariate linear correlation. In this paper, we propose a pair of coupling coefficients, the multivariate linear correlation coefficient (LCC) and linear incorrelation coefficient (LIC), to measure the strength of multivariate linear correlation and linear irrelevance. We show that Pearson's correlation coefficient is a special case of the proposed multivariate LCC for two variables. Based on a set-partition-style inner product-determinant equation, the proposed multivariate LCC and LIC can be efficiently computed by the inner product or determinant group. Finally, we give a new explanation of determinant from the view of multivariate linear correlation.