Nonregular asymptotics under dependence and inference on change points in graphical networks

图网络中变化点的依赖和推理下的非正则渐近

• 批准号: 1308890
• 负责人: Moulinath Banerjee
• 金额: $ 11.5万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-09-01 至 2016-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1308890&HistoricalAwards=false
• 关键词: Nonregular; asymptotics; under; dependence; inference

The proposed project will make two main intellectual contributions: (i) the development of non-regular asymptotic theory for a class of problems involving dependent data (time series models, for example), (ii) inference on change-points for time-varying graphical networks. Non-regular problems, of increasing importance in modern statistics, are those where natural estimators are highly non-linear in the data and deriving their asymptotic properties requires the application of sophisticated tools (like modern empirical process methods), in contrast to the asymptotic linearization techniques that work for estimators arising in `regular' parametric and semiparametric problems. Shape-restricted inference for time series data, for which not much is yet known, and which has important implications for some pressing problems (as in the monotone trends observed in global warming and environmental pollution) will be one important focus in the proposed study of non-regular methods. Another goal is to develop a unified theoretical framework for the study of M-estimators (i.e. estimators obtained by minimizing/maximizing a random criterion function) for finite dimensional parameters in the dependent data setting, which will provide a generic approach (a paradigm as well as a set of tools) to the study of a variety of problems that have been, hitherto, solved on a case-by-case basis and other similar problems that arise in important applications in economics and biology. As far as (ii) is concerned, the problem of time-varying graphs, either observed or unobserved, whose structures undergo sudden massive changes at certain points in time, is of prime importance in a variety of examples, ranging from biology and engineering to social sciences and economics. Rigorous inferential procedures for determining such `change-points' in time -- regime changes -- in a variety of network models (like Markov transitioning random graphs, Markov random fields), that are interesting both from a mathematical perspective and in that they provide useful models for many observed phenomena, will be developed. The proposed research program is motivated by problems arising in a variety of fields, ranging from climatology and environmental studies to economics and genomics. More specifically, part of the proposed project deals with making intelligent predictions on increasing or decreasing `trend' functions, e.g. the GDP output of a rapidly developing country, global temperature trends over time, by taking advantage of their pre-known monotone shape. The proposed techniques are novel and expected to enjoy considerable benefits over existing statistical procedures. A related goal is to develop new theoretical tools for addressing a wealth of statistical procedures that share some key common features and are of considerable importance in problems arising in economics. The second part of the project is focused on investigating sudden `regime' changes in mechanisms called `networks' which describe interactions among a collection of entities: for example, genetic networks which capture how genes interact with each other and with proteins to regulate bodily functions, social networks where a group of people interact on sites like Facebook or Twitter and exchange information in the process. Drastic changes in network behavior typically represent the onset of a critical event, say a disease in the gene network setting, or socio-economic upheaval in the social network setting, and it is therefore important to identify them. Inter-disciplinary collaborations with biologists, economists and environmental scientists will be pursued actively to enhance the impact of the proposed research.

拟议的项目将做出两个主要的智力贡献：（i）针对涉及相关数据（例如时间序列模型）的一类问题开发非正则渐近理论，（ii）时变变化点的推断图形网络。在现代统计学中越来越重要的非常规问题是那些自然估计量在数据中高度非线性的问题，并且与渐近线性化相反，推导其渐近性质需要应用复杂的工具（如现代经验过程方法）适用于“常规”参数和半参数问题中出现的估计器的技术。时间序列数据的形状限制推理目前还知之甚少，但它对一些紧迫问题（如在全球变暖和环境污染中观察到的单调趋势）具有重要意义，将是拟议研究的一个重要焦点非常规方法。另一个目标是开发一个统一的理论框架，用于研究相关数据设置中有限维参数的 M 估计量（即通过最小化/最大化随机准则函数获得的估计量），这将提供一种通用方法（也是一种范式）作为一套工具）来研究迄今为止已在个案基础上解决的各种问题以及在经济学和生物学的重要应用中出现的其他类似问题。 就（ii）而言，时变图的问题，无论是观察到的还是未观察到的，其结构在某些时间点经历突然的巨大变化，在各种例子中都是最重要的，从生物学和工程学到社会科学和经济学。 在各种网络模型（如马尔可夫过渡随机图、马尔可夫随机场）中确定此类“时间变化点”（政权变化）的严格推理程序，从数学角度来看都很有趣，因为它们提供了将开发许多观察到的现象的有用模型。拟议的研究计划是由气候学和环境研究到经济学和基因组学等各个领域出现的问题推动的。更具体地说，拟议项目的一部分涉及对“趋势”函数的增加或减少进行智能预测，例如快速发展国家的 GDP 产出、全球温度随时间变化的趋势，利用其先前已知的单调形状。所提出的技术是新颖的，预计将比现有的统计程序带来更大的好处。一个相关的目标是开发新的理论工具来解决大量统计程序，这些程序具有一些关键的共同特征，并且对于经济学中出现的问题具有相当重要的意义。该项目的第二部分重点研究称为“网络”的机制中突然的“制度”变化，这些机制描述了一组实体之间的相互作用：例如，遗传网络捕获基因如何相互作用以及如何与蛋白质相互作用以调节身体功能社交网络，一群人在 Facebook 或 Twitter 等网站上进行互动，并在此过程中交换信息。网络行为的剧烈变化通常代表重大事件的发生，例如基因网络环境中的疾病，或社交网络环境中的社会经济剧变，因此识别它们非常重要。将积极寻求与生物学家、经济学家和环境科学家的跨学科合作，以增强拟议研究的影响。