RTG: Data-Oriented Mathematical and Statistical Sciences

分析大量不同数据的需求和愿望导致了重大挑战。这个RTG（研究培训组）计划将创建一个研究环境和相关的课程元素，将从事美国公民和永久居民学员的活动，这将促进统计，计算数学和应用谐波分析在解决面向数据的问题和欣赏的协同作用，可以体现在这些领域的想法时，通常由不同的学生小组学习，很少交叉，同时进行。亚利桑那州立大学通过培养跨学科的教师合作和课程创新来寻求持久的影响，旨在刺激严格的，综合的面向数据的数学和统计研究的长期力量。该计划的创新元素的积极传播将寻求为其他机构提供现代综合数据导向数学培训的模式，我们将启动年轻研究人员的职业生涯，他们将携带这一愿景。研究将集中在以下三个问题领域。(i)用于有效数据采集的实验闭环设计：传统的数据收集和分析方法本质上是“前馈”，例如，收集数据，使用数值算法来处理它（例如，用于压缩或以某种其它方式对其进行变换），并且统计方法最终用于推断。统计学家们早就认识到序贯实验设计的吸引力，其中样本的性质不是预先固定的，而是取决于先前观察到的样本。最近和正在进行的技术进步已经导致测量设备具有许多自由度，这些自由度使得能够通常以电子方式并且以真实的时间操纵测量的性质。在这种情况下，实验的顺序设计在节流回否则无法管理的数据洪流中具有根本的重要性。不是一直收集所有数据，而是可以使用反馈策略来基于已经观察到的内容仅获取手头任务的最重要的新数据;（ii）数据驱动的非经典数值逼近工具：传感和模型仿真中的一个中心问题是恢复函数、信号、图像的特征，或者从一组收集到的数据（现在经常是一组非常大的数据）中提取操作符。在几乎所有这些情况下，数据集构成了系统的不完整和嘈杂的描述。经典的数值方法大多使用有限的系统模型以及数据插值或近似来提取相关的模型信息和特征。推导出大量的数据中的关键功能需要新的方法，这些方法很容易适应模型的改进，并包含有关所提供数据的适当的先验或统计信息;（iii）统计推断的近似：近似理论及其数值实现中的传统方法试图从函数空间上的度量所描述的保真度测量中重建函数。我们的RTG在数据不是底层函数的直接样本的问题上具有特殊的专业知识（例如，它们可以是函数的某个变换版本的测量值），并且还有目标是重构函数的特定特征而不是函数本身的情况。RTG PI之间的早期合作已经导致使用过完备表示（例如，框架）和统计思想在这方面的研究。

The need and desire to analyze copious volumes of disparate data result in significant challenges. This RTG (Research Training Group) program will create a research environment and associated curricular elements that will engage U.S. citizen and permanent resident trainees in activities that will foster an understanding of the roles that statistics, computational mathematics, and applied harmonic analysis play in addressing data-oriented problems and appreciation of the synergies that can manifest when ideas from these areas, which are often studied by separate groups of students with little crossover, are brought to bear simultaneously. Durable impact is sought at ASU through cultivation of crosscutting faculty collaborations and curricular innovations intended to stimulate long-term strength in rigorous, integrated data-oriented mathematical and statistical research. Aggressive dissemination of innovative elements of the program will seek to provide a model for modern integrated data-oriented mathematics training for other institutions, and we will launch the careers of young researchers who will carry this vision.The research will focus on the following three problem areas. (i) Closed-loop design of experiment for efficient data acquisition: Traditional approaches to collection and analysis of data are essentially "feed-forward" in nature, for example, data are collected, numerical algorithms are used to process it (e.g., for compression or to transform it in some other way), and statistical methods are ultimately employed for inference. Statisticians have long recognized the appeal of sequential design of experiments in which the nature of a sample is not fixed in advance, rather depends on previously observed samples. Recent and ongoing technological advances have led to measurement devices possessing many degrees of freedom that enable manipulating the nature of the measurement, often electronically and in real time. In this context, sequential design of experiments takes on a fundamental importance in throttling back otherwise unmanageable data torrents. Instead of collecting all the data all the time, a feedback strategy can be used to acquire only the most important new data for the task at hand based on what has already been observed; (ii) Data driven non-classical numerical approximation tools: A central problem in sensing and model simulation is recovery of characteristic features of a function, signal, image, or operator from a set (frequently these days a very large set) of collected data. In almost all such situations, the data set constitutes an incomplete and noisy description of the system. Classical numerical methods mostly use a limited system model as well data interpolation or approximation to extract pertinent model information and features. Deducing crucial features in large volumes of data calls out for new methods that are readily adaptable to model improvements and inclusion of appropriate prior or statistical information on provided data; (iii) Approximation for statistical inference: Traditional methods in approximation theory and their numerical realizations seek to reconstruct functions from measurements with fidelity described by a metric on a function space. Our RTG has particular expertise with problems where the data are not direct samples of the underlying function (e.g., they may be measurements of some transformed version of the function) and also where the objective is to reconstruct specific features of the function rather than the function itself. Earlier collaborations among the RTG PIs has led to the use of both overcomplete representations (e.g., frames) and statistical ideas in this vein of research.