Correlation Matrices in Cosine Space

Abstract

The correlation coefficient be can interpreted as the cosine of the angle between centered or standardized variable vectors in subject space. Using this interpretation of the correlation, the space occupied by 3x3 correlation matrices first demonstrated by Rousseeuw and Molenberghs (1994) can be re-portrayed. Once the cosine transformation is imposed on the space, the space occupied by 3x3 correlation matrices becomes a regular tetrahedron. Extensions of this space in higher dimensions are discussed and explored. Uniform sampling from the regular tetrahedron produces non-uniform sampling from the 3x3 correlation space, such that correlation matrices with more extreme elements are sampled more frequently. Simulations demonstrating this phenomenon are presented and compared to established generation methods.