Mathematical Sciences: Archetypal Analysis Applied to Dynamical Systems

959622642 Stone这个项目探索了原型分析在动力系统数据集上的应用。 原型分析是一种从实验数据集中提取相关特征的新的统计方法，由Cutler和Breiman开发（Technometrics 1995）。 原型表征极端数据值（位于数据集的凸船体上的那些值），并且可以被认为是主成分分析的变体，主成分分析也被动力系统社区称为适当的正交或Karhunen-Loeve（KL）分解。 本项目的目标之一是比较和对比两种方法，KL分解和原型，在其应用到几个实验动力系统，物理和数值。 拥有旅行结构的数据集提出了一个特殊的问题，希望减少他们使用KL分解。如果移动结构足够规则，则本征函数将是西内斯和余弦，因为分解从几何不变的数据产生傅立叶基。因此，在行进结构本身是要提取的特征的情况下，直接应用KL分解将是不够的。 提出者正在开发一种基于原型分析的方法，通过以客观的方式找到与移动结构一起移动的原型来分析具有平移连贯结构的数据。 提取结构本身的形状（或者可能的形状，如果结构本身变化以及平移）沿着关于结构如何跨空间域移动的信息。 最后，沿着上面提到的模式识别问题，所提出的研究的第三个目标是确定使用原型作为基础的可行性，在此基础上对动力系统的控制方程进行Galerkin投影（当它们存在时），用于建模目的。 理解、预测和建模复杂的动力系统是当今动力系统界的当务之急之一。 在这项工作中使用的技术被称为“动力系统理论”的家庭正在迅速扩大，并在许多方向，结合从数学，物理学和工程等不同领域的想法。 随着高速、大容量计算机的出现，更复杂的统计方法在分析动力系统的数据中变得实用。 其中一种方法是原型分析，这是一种新的统计技术，它将系统在给定时间的状态建模为极端状态的混合。提出者研究原型分析在动力系统中的应用。 他们正在将原型分析与其他动力系统的统计过程进行比较，并正在开发一种新的原型分析变体来处理移动波。 这项研究是跨学科的性质，涉及统计学，数学和物理学的想法。 ***

959622642 Stone This project explores the application of Archetypal analysis to data sets from dynamical systems. Archetypal analysis is a new statistical method for extracting relevant features from experimental data sets that has been developed by Cutler and Breiman (Technometrics 1995). Archetypes characterize extreme data values (those lying on the convex hull of the data set), and can be thought of as a variation of principal component analysis, which is also called the proper orthogonal or Karhunen-Loeve (KL) decomposition by the dynamical systems community. One of the goals of this project is to compare and contrast the two methods, the KL decomposition and Archetypes, in their application to several experimental dynamical systems, both physical and numerical. Data sets possessing traveling structures present a special problem to those hoping to reduce them using the KL decomposition. If the traveling structure is sufficiently regular the eigenfunctions will be sines and cosines, since the decomposition produces a Fourier basis from data that are translationally invariant. Therefore, in situations where the traveling structure itself is the feature to be extracted, a straight application the KL decomposition will not suffice. The proposers are developing a method that builds on Archetypal analysis to analyze data with translating coherent structures, by finding, in an objective manner, archetypes that move with the traveling structure. The shape of the structure itself is extracted (or possibly shapes, if the structure itself is varying as well as translating) along with information on how the structure is moving across the spatial domain. Finally, along with the pattern recognition problems mentioned above, the third goal of the proposed research is determining the feasibility of using archetypes as a basis upon which to do a Galerkin projection of the governing equations of the dynamical system (when they exist), for modeling purposes. %%% Understan ding, predicting and modeling complex dynamical systems is one of the preoccupations of the dynamical systems community today. The family of techniques dubbed "dynamical systems theory" utilized in this effort is expanding rapidly and in many directions, incorporating ideas from such diverse fields as mathematics, physics, and engineering. With the advent of high-speed, large capacity computers, more complex methods from statistics have become practical in the analysis of data from dynamical systems. One such method is archetypal analysis, a new statistical technique which models the state of the system at a given time as a mixture of extreme states. The proposers study the application of archetypal analysis to dynamical systems. They are comparing archetypal analysis with other statistical procedures for dynamical systems, and are developing a new variant of archetypal analysis to handle moving waves. The research is interdisciplinary in nature, involving ideas from statistics, mathematics, and physics. ***