Institute of Physics, University of Oldenburg, D-26111 Oldenburg, Germany.
Phys Rev E. 2018 Jan;97(1-1):012113. doi: 10.1103/PhysRevE.97.012113.
A scalar Langevin-type process X(t) that is driven by Ornstein-Uhlenbeck noise η(t) is non-Markovian. However, the joint dynamics of X and η is described by a Markov process in two dimensions. But even though there exists a variety of techniques for the analysis of Markov processes, it is still a challenge to estimate the process parameters solely based on a given time series of X. Such a partially observed 2D process could, e.g., be analyzed in a Bayesian framework using Markov chain Monte Carlo methods. Alternatively, an embedding strategy can be applied, where first the joint dynamics of X and its temporal derivative X[over ̇] is analyzed. Subsequently, the results can be used to determine the process parameters of X and η. In this paper, we propose a more direct approach that is purely based on the moments of the increments of X, which can be estimated for different time-increments τ from a given time series. From a stochastic Taylor expansion of X, analytic expressions for these moments can be derived, which can be used to estimate the process parameters by a regression strategy.
由 Ornstein-Uhlenbeck 噪声 η(t) 驱动的标量 Langevin 型过程 X(t)是非马尔可夫的。然而，X 和 η 的联合动力学由二维中的马尔可夫过程描述。尽管存在多种分析马尔可夫过程的技术，但仅根据给定的 X 时间序列来估计过程参数仍然具有挑战性。这种部分观测的 2D 过程可以例如在贝叶斯框架中使用马尔可夫链蒙特卡罗方法进行分析。或者，可以应用嵌入策略，首先分析 X 及其时间导数 X[over ̇]的联合动力学。随后，可以使用这些结果来确定 X 和 η 的过程参数。在本文中，我们提出了一种更直接的方法，该方法纯粹基于 X 的增量的矩，可以从给定的时间序列中为不同的时间增量 τ 对其进行估计。通过对 X 进行随机泰勒展开，可以推导出这些矩的解析表达式，然后可以通过回归策略使用这些表达式来估计过程参数。
