Detection and characterization of changes of the correlation structure in multivariate time series

Abstract

We propose a method based on the equal-time correlation matrix as a sensitive detector for phase-shape correlations in multivariate data sets. The key point of the method is that changes of the degree of synchronization between time series provoke level repulsions between eigenstates at both edges of the spectrum of the correlation matrix. Consequently, detailed information about the correlation structure of the multivariate data set is imprinted into the dynamics of the eigenvalues and into the structure of the corresponding eigenvectors. The performance of the technique is demonstrated by application to N(f)-tori, autoregressive models, and coupled chaotic systems. The high sensitivity, the comparatively small computational effort, and the excellent time resolution of the method recommend it for application to the analysis of complex, spatially extended, nonstationary systems.

Cross-Correlation Dynamics in Financial Time Series

“…Additional analysis of high frequency data may also be useful in the characterisation of correlation dynamics, especially prior to market crashes. It would also be worthwhile to study the possible relationship between the dynamics of the small eigenvalues and additional correlation information which, according to some authors [10][11][12][13][14]17,20], may be hidden in the part of the eigenvalue spectrum normally classifed as noise. Similar to [17,20], this could be acheived through analysis of the relative dynamics of the small and large eigenvalues at times of extreme volatility (such as during market crashes).…”

“…There are two limiting cases for the distribution of the eigenvalues [17,18]. When all of the time series are perfectly correlated, C i ≈ 1, the largest eigenvalue is maximised with a value equal to N , while for time series consisting of random numbers with average correlation C i ≈ 0, the corresponding eigenvalues are distributed around 1, (where any deviation is due to spurious random correlations).…”

“…Equal-time cross-correlation matrices have been used, [17], to characterise dynamical changes in nonstationary multivariate time-series. It was shown that, as the synchronisation of k time series within an M −dimensional multivariate time series increases, this causes a repulsion between eigenstates of the correlation matrix, in which k levels participate.…”

“…The technique in [17] was applied to the dynamic analysis of the eigenvalue spectrum of the equal time cross-correlation matrix of multivariate Epileptic Seizure time series, using sliding windows. The authors demonstrated that information about the correlation dynamics is visible in both the lower and upper eigenstates.…”

Emerging spectra of singular correlation matrices under small power-map deformations

“…It is essential to suppress the corresponding noise in correlation matrices to reveal the actual correlations. Various techniques are available [14,[33][34][35][36]. Among these we shall use a recent and efficient one, namely the power map [35][36][37] both for noise reduction and for a purpose, quite different from the one for which it was designed and more * Electronic address: vinayaksps2003@gmail.com † Electronic address: rudi.schaefer@uni-due.de ‡ Electronic address: seligman@ce.fis.unam.mx central to our paper.…”