时滞相关矩阵的随机矩阵集成：特征值谱的推导和金融时间序列的分析
Random matrix ensembles of time-lagged correlation matrices: Derivation of eigenvalue spectra and analysis of financial time-series
摘要：
我们推导出一组时移有限布朗随机游动（时间序列）的相关矩阵的特征值谱的精确形式。这些矩阵可以被看作是随机的、实的、非对称的矩阵，由于时移而具有特殊的叠加结构。我们证明了相关的特征值谱是圆对称的复平面上的大型矩阵。这一事实使我们能够通过对称化问题的密度的Abel逆变换精确地计算特征值密度。我们证明了这种方法的有效性，通过数值计算的特征值谱的滞后相关矩阵的基础上不相关，高斯分布的时间序列。然后，我们将我们的理论研究结果与从实际的高频（5分钟）的S & P500数据获得的本征值密度进行比较，并讨论观察到的偏差。我们确定了各种非平凡的，非随机的模式，并发现与特征值的非对称依赖关系强烈偏离高斯预测的虚部。对于相同的时间序列，去除市场贡献后，我们观察到股票的强烈聚集，即因果部门。最后，我们评论的时间稳定性所观察到的模式。
英文摘要：
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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