%0 Journal Article %1 biely_random_2006 %A Biely, Christoly %A Thurner, Stefan %D 2006 %J arXiv:physics/0609053 %K \_tablet\_modified, analysis covariance eigenvalue, embedding, finance, matrix, spectral, time-series %T Random matrix ensembles of time-lagged correlation matrices: Derivation of eigenvalue spectra and analysis of financial time-series %U http://arxiv.org/abs/physics/0609053 %X We derive the exact form of the eigenvalue spectra of correlation matrices derived from a set of time-shifted, finite Brownian random walks (time-series). These matrices can be seen as random, real, asymmetric matrices with a special structure superimposed due to the time-shift. We demonstrate that the associated eigenvalue spectrum is circular symmetric in the complex plane for large matrices. This fact allows us to exactly compute the eigenvalue density via an inverse Abel-transform of the density of the symmetrized problem. We demonstrate the validity of this approach by numerically computing eigenvalue spectra of lagged correlation matrices based on uncorrelated, Gaussian distributed time-series. We then compare our theoretical findings with eigenvalue densities obtained from actual high frequency (5 min) data of the S&P500 and discuss the observed deviations. We identify various non-trivial, non-random patterns and find asymmetric dependencies associated with eigenvalues departing strongly from the Gaussian prediction in the imaginary part. For the same time-series, with the market contribution removed, we observe strong clustering of stocks, i.e. causal sectors. We finally comment on the time-stability of the observed patterns.

@article{biely_random_2006, abstract = {We derive the exact form of the eigenvalue spectra of correlation matrices derived from a set of time-shifted, finite Brownian random walks (time-series). These matrices can be seen as random, real, asymmetric matrices with a special structure superimposed due to the time-shift. We demonstrate that the associated eigenvalue spectrum is circular symmetric in the complex plane for large matrices. This fact allows us to exactly compute the eigenvalue density via an inverse Abel-transform of the density of the symmetrized problem. We demonstrate the validity of this approach by numerically computing eigenvalue spectra of lagged correlation matrices based on uncorrelated, Gaussian distributed time-series. We then compare our theoretical findings with eigenvalue densities obtained from actual high frequency (5 min) data of the S\&P500 and discuss the observed deviations. We identify various non-trivial, non-random patterns and find asymmetric dependencies associated with eigenvalues departing strongly from the Gaussian prediction in the imaginary part. For the same time-series, with the market contribution removed, we observe strong clustering of stocks, i.e. causal sectors. We finally comment on the time-stability of the observed patterns.}, added-at = {2017-01-09T13:57:26.000+0100}, author = {Biely, Christoly and Thurner, Stefan}, biburl = {https://www.bibsonomy.org/bibtex/2c159120cb4cc6cd5c303ddd3b813bf43/yourwelcome}, interhash = {140f0be7e8809e9a648b4352d8ae5612}, intrahash = {c159120cb4cc6cd5c303ddd3b813bf43}, journal = {arXiv:physics/0609053}, keywords = {\_tablet\_modified, analysis covariance eigenvalue, embedding, finance, matrix, spectral, time-series}, month = sep, shorttitle = {Random matrix ensembles of time-lagged correlation matrices}, timestamp = {2017-01-09T14:01:11.000+0100}, title = {Random matrix ensembles of time-lagged correlation matrices: {Derivation} of eigenvalue spectra and analysis of financial time-series}, url = {http://arxiv.org/abs/physics/0609053}, urldate = {2014-03-01}, year = 2006 }