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Functional Depth and Quantiles: Limit Theory, Comparisons and Applications

函数深度和分位数：极限理论、比较和应用

基本信息

批准号：
1208962
负责人：
Joel Zinn
金额：
$ 10万
依托单位：
Texas A&M University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2012
资助国家：
美国
起止时间：
2012-09-01 至 2015-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1208962&HistoricalAwards=false
关键词：
Functional Depth Quantiles Limit Theory

项目摘要

This investigator proposes to study functional depth and functional quantiles using techniques developed for Empirical Process theory. Some recent techniques developed by this investigator and his colleagues allow one to obtain central limit theorems for quantile processes formed from functional data, and even can be applied to introduce new methods in the study of finite dimensional data, obtaining interesting variations on already existing examples of data depth. Moreover, within the context of functional depth, this has uncovered some unforeseen problems. One such problem is that for natural processes, such as Brownian motion, and for some natural definitions of depth, one might have depth which is identically zero. Together with colleagues, this investigator has introduced a certain type of smoothing which allows one to eliminate this problem in many special cases. However, it is still necessary to develop a general, realistic, and usable approach for a wide variety of circumstances. So, in addition to developing an asymptotic theory for functional data, this project proposes to develop a coherent methodology for smoothing/modifying the data to allow for such analyses.The contemporary statistician must deal with data that appears in many quite different forms. Much energy has been spent on one-dimensional data, and researchers have achieved a great deal of success. While there are still many important questions in this area, the analysis of finite-dimensional data has also become a vibrant and important area for research. One critical difference between one-dimensional data and finite dimensional data is the lack of an obvious ordering of the data in dimensions greater than one. To alleviate this problem, and better analyze (finite) multidimensional data, the concept of data depth has been introduced and studied by many authors. There are a variety of examples of such depth, each with its own set of good properties. One can choose a particular data depth to analyze a given data problem, depending on which properties are the most important. One can also compare various forms of depth on the same set of data to find contrasts that otherwise may not be apparent. As this area has matured, so has the ability of the researcher to better fit the type of depth to the problem at hand. Even more recently, due to improved computing tools, real time monitoring of many processes is available and consequently, there is a growing need to analyze such data. This is often referred to as functional data. In mathematical parlance, this is considered infinite dimensional data. Data of this type occur, for example, in medicine, neuroscience, chemometrics, signal transmission, stock markets and meteorology. A robust methodology is important to successfully handle the resulting problems, and as is to be expected, this requires methods beyond those used to study the finite dimensional situation. It is not surprising to a researcher in statistics or mathematics, that like many other infinite dimensional problems, infinite dimensional depth is fraught with differences and difficulties not found in the finite dimensional setting. The techniques proposed by this investigator to study functional data partially rely on a theory in which this investigator has made many contributions, and which is, now, quite well developed. These techniques have already allowed this investigator and his colleagues to uncover some of the difficulties that must be overcome.

本研究者建议使用经验过程理论开发的技术来研究功能深度和功能分位数。最近的一些技术开发的这个调查员和他的同事们允许一个获得中心极限定理分位数过程形成的功能数据，甚至可以应用于引入新的方法在研究有限维数据，获得有趣的变化已经存在的例子的数据深度。此外，在功能深度的背景下，这揭示了一些不可预见的问题。一个这样的问题是，对于自然过程，如布朗运动，以及对于深度的一些自然定义，可能有深度是相同的零。与同事们一起，这位研究人员介绍了某种类型的平滑，允许在许多特殊情况下消除这个问题。然而，仍然有必要为各种各样的情况开发一种通用的、现实的和可用的方法。因此，除了为函数数据开发渐近理论外，本项目还提出开发一种连贯的方法来平滑/修改数据，以允许进行此类分析。当代统计学家必须处理以许多不同形式出现的数据。人们在一维数据上花费了大量的精力，研究人员已经取得了很大的成功。虽然在这一领域仍然有许多重要的问题，但有限维数据的分析也已成为一个充满活力的重要研究领域。一维数据和有限维数据之间的一个关键区别是在大于一维的维度中缺乏数据的明显排序。为了缓解这个问题，更好地分析（有限）多维数据，数据深度的概念已经被许多作者引入和研究。有很多这样的深度的例子，每一个都有自己的一套好的属性。人们可以选择特定的数据深度来分析给定的数据问题，这取决于哪些属性是最重要的。人们还可以比较同一组数据的各种形式的深度，以找到否则可能不明显的对比。随着这一领域的成熟，研究人员也有能力更好地适应手头问题的深度类型。甚至最近，由于改进的计算工具，许多过程的真实的时间监控是可用的，并且因此，存在对分析这样的数据的增长的需要。这通常被称为功能数据。在数学术语中，这被认为是无限维数据。这种类型的数据出现在例如医学、神经科学、化学计量学、信号传输、股票市场和气象学中。一个强大的方法是很重要的，成功地处理由此产生的问题，正如预期的那样，这需要超越那些用于研究有限维情况的方法。对于统计学或数学研究人员来说，这并不奇怪，就像许多其他无限维问题一样，无限维深度充满了有限维设置中没有的差异和困难。本研究员提出的研究功能数据的技术部分依赖于一个理论，在这个理论中，本研究员做出了许多贡献，并且现在已经发展得很好。这些技术已经使这位研究人员和他的同事发现了一些必须克服的困难。