Cleaning large correlation matrices: Tools from Random Matrix Theory

This review covers recent results concerning the estimation of large covariance matrices using tools from Random Matrix Theory (RMT). We introduce several RMT methods and analytical techniques, such as the Replica formalism and Free Probability, with an emphasis on the Marčenko–Pastur equation that...

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Bibliographic Details
Published in:Physics reports Vol. 666; pp. 1 - 109
Main Authors: Bun, Joël, Bouchaud, Jean-Philippe, Potters, Marc
Format: Journal Article
Language:English
Published: Elsevier B.V 13.01.2017
Elsevier
Subjects:
Condensed Matter
Correlation matrix
High dimensional statistics
Physics
Quantitative Finance
Random Matrix Theory
Rotational invariant estimator
Spectral decomposition
Statistical Finance
Statistical Mechanics
Statistics
Statistics Theory
High dimensional statistics
Rotational invariant estimator
Spectral decomposition
Correlation matrix
Random Matrix Theory
ISSN:0370-1573, 1873-6270
Online Access:Get full text
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Summary:This review covers recent results concerning the estimation of large covariance matrices using tools from Random Matrix Theory (RMT). We introduce several RMT methods and analytical techniques, such as the Replica formalism and Free Probability, with an emphasis on the Marčenko–Pastur equation that provides information on the resolvent of multiplicatively corrupted noisy matrices. Special care is devoted to the statistics of the eigenvectors of the empirical correlation matrix, which turn out to be crucial for many applications. We show in particular how these results can be used to build consistent “Rotationally Invariant” estimators (RIE) for large correlation matrices when there is no prior on the structure of the underlying process. The last part of this review is dedicated to some real-world applications within financial markets as a case in point. We establish empirically the efficacy of the RIE framework, which is found to be superior in this case to all previously proposed methods. The case of additively (rather than multiplicatively) corrupted noisy matrices is also dealt with in a special Appendix. Several open problems and interesting technical developments are discussed throughout the paper.
ISSN:0370-1573
1873-6270
DOI:10.1016/j.physrep.2016.10.005