Signal and noise in correlation matrix

Abstract

Using random matrix technique we determine an exact relation between the eigenvalue spectrum of the covariance matrix and of its estimator. This relation can be used in practice to compute eigenvalue invariants of the covariance (correlation) matrix. Results can be applied in various problems where one experimentally estimates correlations in a system with many degrees of freedom, like for instance those in statistical physics, lattice measurements of field theory, genetics, quantitative finance and other applications of multivariate statistics.

Economic sector identification in a set of stocks traded at the New York Stock Exchange: a comparative analysis

“…Multivariate time series are detected and recorded both in experiments and in the monitoring of a wide number of physical, biological and economic systems. The study of the properties of the correlation matrix has a direct relevance in the investigation of mesoscopic physical systems [1], high energy physics [2], information theory and communication [3,4,5], investigation of microarray data in biological systems [6,7,8] and econophysics [9,10,11,12,13,14,15].…”

A Random Matrix Approach to Dynamic Factors in Macroeconomic Data

“…The matrices are then diagonalized,provided T > N, M , and the empirical spectrum is compared to the theoretical Marčenko-Pastur spectrum [10,[26][27][28]in order to unravel statistically significant factors.The eigenvalues,which lie much below the lower edge of the Marčenko-Pastur spectrum represent the redundant factors, rejected by the system, so one can exclude them from further study and in this manner reduce somewhat the dimensionality of the problem, by removing possibly spurious correlations. Having found all eigenvectors and eigenvalues, one can then construct a set of uncorrelated unit variance input variablesX and output variablesŶ .…”

Multiscaled Cross-Correlation Dynamics in Financial Time-Series

“…It has been suggested recently by several authors, [14][15][16][17][18] that there may, in fact, be some real correlation information hidden in the RMT defined part of the eigenvalue spectrum. A technique, involving the use of power mapping to identify and estimate the noise in financial correlation matrices was described, [14], allowing the suppression of those noise eigenvalues, to reveal different correlation structures buried underneath.…”

“…A technique, involving the use of power mapping to identify and estimate the noise in financial correlation matrices was described, [14], allowing the suppression of those noise eigenvalues, to reveal different correlation structures buried underneath. The relationship, between the eigenvalue density c of the true correlation matrix, and that of an empirical correlation matrix C, was derived, [15,16], to show that correlations can be measured in the random part of the spectrum. The bulk of the spectrum was shown to deviate from the Wishart RMT class through the use of a Kolmogorov test, [17], where the existence of factors such as an firm size, industry type and the overall market effect was due to collective influence of the assets.…”