Chen Lijian, Bassler Kevin E, McCauley Joseph L, Gunaratne Gemunu H
Department of Physics, University of Houston, Houston, Texas 77204, USA.
Department of Mathematics, University of Houston, Houston, Texas 77204, USA.
Phys Rev E. 2017 Apr;95(4-1):042141. doi: 10.1103/PhysRevE.95.042141. Epub 2017 Apr 28.
The state of a stochastic process evolving over a time t is typically assumed to lie on a normal distribution whose width scales like t^{1/2}. However, processes in which the probability distribution is not normal and the scaling exponent differs from 1/2 are known. The search for possible origins of such "anomalous" scaling and approaches to quantify them are the motivations for the work reported here. In processes with stationary increments, where the stochastic process is time-independent, autocorrelations between increments and infinite variance of increments can cause anomalous scaling. These sources have been referred to as the Joseph effect and the Noah effect, respectively. If the increments are nonstationary, then scaling of increments with t can also lead to anomalous scaling, a mechanism we refer to as the Moses effect. Scaling exponents quantifying the three effects are defined and related to the Hurst exponent that characterizes the overall scaling of the stochastic process. Methods of time series analysis that enable accurate independent measurement of each exponent are presented. Simple stochastic processes are used to illustrate each effect. Intraday financial time series data are analyzed, revealing that their anomalous scaling is due only to the Moses effect. In the context of financial market data, we reiterate that the Joseph exponent, not the Hurst exponent, is the appropriate measure to test the efficient market hypothesis.
通常假定，在时间t上演变的随机过程的状态服从正态分布，其宽度按t的1/2次方缩放。然而，已知存在概率分布非正态且缩放指数不同于1/2的过程。寻找此类“反常”缩放的可能起源以及量化它们的方法，是本文所报道工作的动机。在具有平稳增量的过程中，即随机过程与时间无关时，增量之间的自相关以及增量的无穷方差会导致反常缩放。这些来源分别被称为约瑟夫效应和诺亚效应。如果增量是非平稳的，那么增量随t的缩放也会导致反常缩放，我们将这种机制称为摩西效应。定义了量化这三种效应的缩放指数，并将其与表征随机过程整体缩放的赫斯特指数相关联。提出了能够对每个指数进行准确独立测量的时间序列分析方法。使用简单的随机过程来说明每种效应。对日内金融时间序列数据进行了分析，结果表明其反常缩放仅归因于摩西效应。在金融市场数据的背景下，我们重申，检验有效市场假说的合适度量是约瑟夫指数，而非赫斯特指数。
