dorsal/arxiv
View SchemaRandom matrix ensembles of time-lagged correlation matrices: Derivation of eigenvalue spectra and analysis of financial time-series
| Authors | Christoly Biely, Stefan Thurner |
|---|---|
| Categories | |
| ArXiv ID | physics/0609053 |
| URL | https://arxiv.org/abs/physics/0609053 |
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.
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"abstract": "We derive the exact form of the eigenvalue spectra of correlation matrices\nderived from a set of time-shifted, finite Brownian random walks (time-series).\nThese matrices can be seen as random, real, asymmetric matrices with a special\nstructure superimposed due to the time-shift. We demonstrate that the\nassociated eigenvalue spectrum is circular symmetric in the complex plane for\nlarge matrices. This fact allows us to exactly compute the eigenvalue density\nvia an inverse Abel-transform of the density of the symmetrized problem. We\ndemonstrate the validity of this approach by numerically computing eigenvalue\nspectra of lagged correlation matrices based on uncorrelated, Gaussian\ndistributed time-series. We then compare our theoretical findings with\neigenvalue densities obtained from actual high frequency (5 min) data of the\nS\u0026P500 and discuss the observed deviations. We identify various non-trivial,\nnon-random patterns and find asymmetric dependencies associated with\neigenvalues departing strongly from the Gaussian prediction in the imaginary\npart. For the same time-series, with the market contribution removed, we\nobserve strong clustering of stocks, i.e. causal sectors. We finally comment on\nthe time-stability of the observed patterns.",
"arxiv_id": "physics/0609053",
"authors": [
"Christoly Biely",
"Stefan Thurner"
],
"categories": [
"physics.soc-ph",
"q-fin.ST"
],
"title": "Random matrix ensembles of time-lagged correlation matrices: Derivation of eigenvalue spectra and analysis of financial time-series",
"url": "https://arxiv.org/abs/physics/0609053"
},
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