转换高斯相关性。任意分布的长程幂律相关时间序列的生成
Transforming Gaussian correlations. Applications to generating long-range power-law correlated time series with arbitrary distribution
摘要：
许多复杂动力系统的可观测输出包括在时间序列中表现出各种行为的自相关函数，包括长程幂律自相关函数，作为在许多时间或空间尺度上操作的相互作用的签名。通常，能够生成再现实时时间序列的属性的相关噪声的算法产生\textsl{Gaussian}输出，而真实的、实验观察到的时间序列通常是非高斯的，并且可能遵循具有关于支持、对称性或尾部属性的多种行为的分布。在这里，我们研究了两个高斯变量的相关性如何变化，当它们被转换为遵循不同的目标分布。具体来说，我们考虑有界和无界分布，对称和非对称分布，分布与不同的尾部属性，从衰减速度比指数重尾的情况下，包括幂律，我们发现这些属性如何影响最终变量的相关性。我们将这些结果扩展到高斯时间序列，将其转换为具有不同的边缘分布，并显示最终非高斯时间序列的自相关函数如何取决于高斯相关性和最终的边缘分布。作为我们的研究结果的应用，我们提出了如何推广标准算法产生高斯幂律相关的时间序列，以创建合成的时间序列与任意分布和控制幂律相关。最后，我们通过生成模仿实时时间序列的自相关函数的边际分布和幂律尾部的时间序列：股票价格的绝对收益率，展示了该算法的一个实际例子。
英文摘要：
The observable outputs of many complex dynamical systems consist in time series exhibiting autocorrelation functions of great diversity of behaviors, including long-range power-law autocorrelation functions, as a signature of interactions operating at many temporal or spatial scales. Often, algorithms able to generate correlated noises reproducing the properties of real time series produce \textsl{Gaussian} outputs, while real, experimentally observed time series are often non-Gaussian, and may follow distributions with a diversity of behaviors concerning the support, the symmetry or the tail properties. Here, we study how the correlation of two Gaussian variables changes when they are transformed to follow a different destination distribution. Specifically, we consider bounded and unbounded distributions, symmetric and non-symmetric distributions, and distributions with different tail properties, from decays faster than exponential to heavy tail cases including power-laws, and we find how these properties affect the correlation of the final variables. We extend these results to Gaussian time series which are transformed to have a different marginal distribution, and show how the autocorrelation function of the final non-Gaussian time series depends on the Gaussian correlations and on the final marginal distribution. As an application of our results, we propose how to generalize standard algorithms producing Gaussian power-law correlated time series in order to create synthetic time series with arbitrary distribution and controlled power-law correlations. Finally, we show a practical example of this algorithm by generating time series mimicking the marginal distribution and the power-law tail of the autocorrelation function of a real time series: the absolute returns of stock prices.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
