Planes of Change: New Statistical Methods for Complex Non-Standard Systems

变化平面：复杂非标准系统的新统计方法

• 批准号: 1712962
• 负责人: Moulinath Banerjee
• 金额: $ 35万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1712962&HistoricalAwards=false
• 关键词: Planes; Change; New; Statistical; Methods

The project aims to develop new statistical methodologies for analysis of systems in a variety of fields, such as personalized medicine, internet traffic, and economics, in which sharp threshold effects occur. Such sharp effects are typically experienced when a system is subjected to a sudden shock (e.g., the effect of political tension on stock prices, the effect of socio-political upheaval on social-media networks, or the effect of a medical intervention on disease progression). Such sharp changes are of critical interest to practitioners in these different fields as they typically have important implications for future decision-making. Statisticians model such sharp changes in time, for example, through what are called "change-points;" when the sharp change happens due to the effect of multiple variables simultaneously, such regions are described in terms of "change-planes." This project aims to develop novel methods of identifying such change-points or change-planes in problems where massive amounts of data -- which have now become the norm given advances in storage capabilities as well as collection mechanisms -- are available, and furthermore, the number of variables on which data are recorded is also very large. The performance of such methods will be carefully analyzed using mathematical theory as well as computer-generated simulations, and the methods will also be validated on real data coming from a variety of sources. It is anticipated that the results of the research will have impact in a variety of natural science as well as social science disciplines. The overarching theme of this project is to develop methodology and inference in a class of problems in which thresholds or boundaries (in one or multiple dimensions) that induce discontinuities arise naturally, either in the statistical model or in the estimation paradigm. The problems are studied both in the setting of massive amounts of data as well as in scenarios where the number of covariates can exceed the number of observations. The boundaries considered in one-dimension are change-points, while those in multiple dimensions are hyper-planes. The studied problems present two different kinds of complexities: (a) massive amounts of available data, and/or (b) large numbers of covariates relative to number of observations. In particular: (i) A number of ideas are developed for sampling intelligently from (retrospectively observed) long time-series to determine the locations of multiple change-points via procedures that require analyzing only a vanishing fraction of the entire series (thereby providing computational benefits), yet produce estimates that match, in precision, the standard estimates that would have been obtained analyzing the entire series. This idea is extended to regression/likelihood based models with covariates in multiple dimensions where the parameters of the regression or the likelihood are different on either side of a hyper-plane in covariate space. (ii) Problems involving hyper-planes, either in the structure of the model or in the criterion function to be optimized, with high-dimensional covariates are studied and new variable selection and estimation methods are investigated. The problems under consideration here are important from the perspective of applications but difficult because the high-dimensional paradigm has to be extended to intrinsically discontinuous settings, outside the (almost) square-root-n rate. Effective solutions to these problems will advance statistical methodology for these important classes of systems.

该项目旨在开发新的统计方法，用于分析各个领域的系统，例如个性化医学，互联网流量和经济学，在其中发生了尖锐的阈值效应。当系统突然冲击时，通常会出现这种敏锐的影响（例如，政治紧张局势对股票价格的影响，社会政治动荡对社交媒体网络的影响或医疗干预对疾病进展的影响）。这种急剧的变化对于这些不同领域的从业者来说是关键的关注，因为它们通常对未来的决策具有重要意义。统计学家通过所谓的“变更点”来模拟这种时间的急剧变化；当由于多个变量同时效果而发生急剧变化时，这些区域将用“变更平面”描述。该项目旨在开发新的方法来识别此类变更点或变更平面的问题，这些问题可以提供大量数据（现在已经成为存储能力和收集机制进步的规范）的问题，而且还可以使用，此外，记录数据的变量的数量也很大。此类方法的性能将使用数学理论以及计算机生成的模拟仔细分析，并且这些方法还将在来自各种来源的真实数据上进行验证。预计研究结果将影响各种自然科学以及社会科学学科。该项目的总体主题是在一系列问题中开发方法和推断，其中阈值或边界（在一个或多个维度中）在统计模型或估计范式中自然会引起不连续性。在大量数据的设置以及协变量数量可以超过观测值数量的情况下，研究了这些问题。一维中考虑的边界是变化点，而多个维度的边界是超平面。研究的问题提出了两种不同类型的复杂性：（a）大量可用数据和/或（b）相对于观察次数的大量协变量。特别是：（i）从（回顾性观察到）长时间序列中智能采样的许多想法，以通过过程确定多个更改点的位置，这些程序需要分析整个系列中的一部分（从而提供计算效益）中消失的部分（从而提供计算益处），但要精确地估算了该系列的标准估计值，这些产品估算了整个系列的标准估计值。这个想法扩展到基于回归/可能性的模型，该模型在多个维度上具有协变量，其中回归或可能性的参数在协变量空间的超平面的两侧都不同。 （ii）在模型的结构或要优化的标准函数中涉及超平面的问题，研究了高维协变量，并研究了新的变量选择和估计方法。从应用的角度来看，这里考虑的问题很重要，但很困难，因为高维范式必须扩展到（几乎）平方 - 根 - N速率之外的本质上不连续的设置。解决这些问题的有效解决方案将推动这些重要类别类别的统计方法。