论文标题
随机过程的组合视图：白噪声
A combinatorial view of stochastic processes: White noise
论文作者
论文摘要
白噪声是一个基本且相当广泛的随机过程，它符合许多其他过程的概念基础以及时间序列的建模。在这里，我们将新的视角推向白噪声，基于组合考虑，这有助于为建模和理论目的提供新的有趣见解。为此，我们结合了序数模式分析方法，该方法使我们能够将时间序列抽象为模式及其相关排列的顺序，并在排列上引入一个简单的功能，将它们分配到编码其不对称水平的类中。我们计算了该功能的确切概率质量函数（p.m.f.），而对称$ n $的对称组，从而为无限白噪声实现的情况提供了描述。这个p.m.f.可以通过指数族的连续概率密度（高斯）方便地近似，因此提供了自然的足够统计数据，通过序数模式提供了方便而简单的统计分析。在3D扩散中金纳米颗粒的轨道中的空间增量的实验数据中，这种分析被例证。
White noise is a fundamental and fairly well understood stochastic process that conforms the conceptual basis for many other processes, as well as for the modeling of time series. Here we push a fresh perspective toward white noise that, grounded on combinatorial considerations, contributes to give new interesting insights both for modelling and theoretical purposes. To this aim, we incorporate the ordinal pattern analysis approach which allows us to abstract a time series as a sequence of patterns and their associated permutations, and introduce a simple functional over permutations that partitions them into classes encoding their level of asymmetry. We compute the exact probability mass function (p.m.f.) of this functional over the symmetric group of degree $n$, thus providing the description for the case of an infinite white noise realization. This p.m.f. can be conveniently approximated by a continuous probability density from an exponential family, the Gaussian, hence providing natural sufficient statistics that render a convenient and simple statistical analysis through ordinal patterns. Such analysis is exemplified on experimental data for the spatial increments from tracks of gold nanoparticles in 3D diffusion.
