Discretization of continuous stochastic processes is needed to numerically simulate them or to infer models from experimental time series. However, depending on the nature of the process, the same discretization scheme, if not accurate enough, may perform very differently for the two tasks. Exact discretizations, which work equally well at any scale, are characterized by the property of invariance under coarse-graining. Motivated by this observation, we build an explicit Renormalization Group approach for Gaussian time series generated by auto-regressive models. We show that the RG fixed points correspond to discretizations of linear SDEs, and only come in the form of first order Markov processes or non-Markovian ones. This fact provides an alternative explanation of why standard delay-vector embedding procedures fail in reconstructing partially observed noise-driven systems. We also suggest a possible effective Markovian discretization for the inference of partially observed underdamped equilibrium processes based on the exploitation of the Einstein relation.
翻译:连续随机过程的分解需要从数字上模拟这些过程或从实验时间序列中推断模型。 但是,根据过程的性质,同一离散办法,即使不够准确,也可能对这两项任务产生非常不同的效果。 精确的分解办法在任何规模上都同样运作良好,其特点是粗糙的偏差特性。 受这一观察的驱动,我们为自动递减模型产生的高斯时间序列建立了明确的重新定性小组办法。 我们表明,RG固定点与线性SDE的分解相对应,而只是以第一级Markov程序或非Markovian程序的形式出现。 这一事实提供了另一种解释,说明标准延迟定位程序为何在重建部分观测到的噪音驱动系统方面失败。 我们还建议,根据利用爱因斯坦关系的结果,对部分观测到的未充分平衡过程的推断,可采用有效的Markovian分解法。