Mathematical Sciences: Archetypal Analysis Applied to Dynamical Systems

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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. ***