Learning the continuous equations of motion from discrete observations is a common task in all areas of physics. However, not any discretization of a Gaussian continuous-time stochastic process can be adopted in parametric inference. We show that discretizations yielding consistent estimators have the property of `invariance under coarse-graining', and correspond to fixed points of a renormalization group map on the space of autoregressive moving average (ARMA) models (for linear processes). This result explains why combining differencing schemes for derivatives reconstruction and local-in-time inference approaches does not work for time series analysis of second or higher order stochastic differential equations, even if the corresponding integration schemes may be acceptably good for numerical simulations.
翻译:从离散观测中学习运动的连续方程式是所有物理领域的一项共同任务。然而,在参数推论中,不能采用高斯连续时间随机学过程的任何分离。我们表明,产生一致估计的离散性具有“粗重重压下差异”的特性,并且与关于自动递减平均移动模型空间(线性过程)的重新标准化小组地图的固定点相对应。这解释了为什么衍生物重建和局部实时推断方法的区别性办法不能用于对二等或更高顺序的相偏差方程式进行时间序列分析,即使相应的集成计划对于数字模拟来说可能是可以接受的。