[Paper Review] Measures of Linear Correlation for Multiple Variables

Jianji Wang, Nanning Zheng|arXiv (Cornell University)|Jan 20, 2014

Random Matrices and Applications3 references3 citations

TL;DR

This paper introduces a novel pair of coefficients—multivariate linear correlation coefficient (LCC) and linear incorrelation coefficient (LIC)—to quantify multivariate linear correlation and irrelevance. The method generalizes Pearson’s correlation to multiple variables, enabling compact decomposition and offering a new geometric interpretation of the determinant in terms of multivariate linear dependence.

ABSTRACT

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. Pearson's correlation coefficient is a special case of the proposed multivariate LCC for two variables. Based on the proposed multivariate LIC, a compact formula of LIC for linear decomposition is also presented in this paper. The experiment results show that the proposed multivariate LCC is an effective measure for multivariate linear correlation, and a new explanation of determinant is also made from the view of multivariate linear correlation.

Motivation & Objective

• To address the lack of a compact, principled formulation for measuring multivariate linear correlation in statistical and data analytic applications.
• To define a symmetric, interpretable measure of linear dependence among multiple variables, extending bivariate correlation to higher dimensions.
• To develop a complementary measure of linear irrelevance (LIC) that enables decomposition of multivariate linear structure.
• To provide a new geometric interpretation of the determinant in terms of multivariate linear correlation.

Proposed method

• Proposes the multivariate linear correlation coefficient (LCC) as a generalization of Pearson’s correlation to multiple variables, based on covariance structure.
• Introduces the linear incorrelation coefficient (LIC) as a dual measure to quantify linear irrelevance, enabling decomposition of multivariate linear relationships.
• Derives a compact analytical formula for LIC that facilitates linear decomposition of multivariate data structures.
• Uses multivariate normal distribution assumptions to derive the theoretical properties of LCC and LIC.
• Applies the coefficients to real data to validate their effectiveness in measuring multivariate linear dependence.
• Reinterprets the determinant of a covariance matrix as a measure of multivariate linear correlation strength via the proposed framework.

Experimental results

Research questions

• RQ1 How can multivariate linear correlation be formally and compactly measured beyond pairwise correlations?
• RQ2 What is the mathematical relationship between the proposed LCC and existing correlation measures like Pearson’s r?
• RQ3 Can a dual measure of linear irrelevance (LIC) be defined to enable decomposition of multivariate linear structure?
• RQ4 How does the proposed framework reinterpret the determinant of a covariance matrix in terms of multivariate correlation?
• RQ5 What empirical evidence supports the effectiveness of the LCC in capturing multivariate linear dependence?

Key findings

• The proposed multivariate LCC generalizes Pearson’s correlation coefficient to multiple variables and provides a consistent, symmetric measure of linear dependence.
• The linear incorrelation coefficient (LIC) enables a compact decomposition of multivariate linear structure, offering a new analytical tool for data decomposition.
• The determinant of a covariance matrix is reinterpreted as a measure of multivariate linear correlation strength through the proposed framework.
• Experimental results confirm that the multivariate LCC effectively captures the strength of linear relationships in multivariate data.
• The theoretical formulation establishes a clear link between multivariate linear correlation and the geometric properties of covariance matrices.