# Correlation Matrix

## Contents

## Definition

A **Correlation Matrix** denotes both a measure and a statistical of *dependency* (whether causal or not) between different stochastic processes

NB: This article focuses on aspects of correlation matrices relevant for risk management applications. More general statistical considerations are available in the wikipedia entry and links therein

### Alternative Correlation Measures

There are several alternative measures of correlation:

- The Pearson Correlation also referred to as Pearson's r, the Pearson product-moment correlation coefficient or the bivariate correlation, a measure of the
*linear*correlation between two variables X and Y - The Spearman's Rank Correlation a nonparametric measure of rank correlation (statistical dependence between the rankings of two variables)
- The Kendall Rank Correlation as Kendall's tau coefficient a statistic used to measure the ordinal association between two variables

## Properties

### Valid Range

All correlation matrix elements must satisfy

The diagonal elements must be identical to unity

### Symmetry

The matrix elements must satisfy the symmetry relation

### Positive Semi-Definiteness

A correlation matrix must be positive semi-definite, namely

When a matrix fails this requirement (usually due to Data Quality issues) one usually aims to find a *nearby* matrix that is valid. One approach to fix an invalid correlation matrix is provided here^{[1]}

An equivalent requirement is that all eigenvalues of the matrix are non-negative (See Definiteness of a matrix)

## Characterization

Correlation matrices are in general characterised according to the distribution of eigenvalues

## See Also

## References

- ↑ Higham, Nick (2002) Computing the nearest correlation matrix - a problem from finance; IMA Journal of Numerical Analysis 22