Applications of stochastic analysis to statistical inference for stationary and non-stationary Gaussian processes

科学家和工程师希望通过分析数据来为问题和决策提供依据，从而提取信息并发展对自然世界和人类社会的理解。数据通常表现为所谓的时间序列，也称为随机过程，即可测量量在一段时间内的演变。这些时间序列的变化可以从一个时期到下一个时期相当可预测，但在跨越许多时期的较长时间间隔上就不那么可预测了。不同类型的随机过程在性质上有细微的差别。例如，金融股或指数的价值，或全球年平均气温，都是累积的，随着时间的推移随机累积。但股票或大宗商品期货的每日回报，或连续发生的大西洋飓风的类别(强度)具有不同的性质，通常表现出一天或一天到另一天的极大独立性，在根据趋势和季节性进行调整后，具有随时间推移的平稳性。一个至关重要的问题是，这些时间序列中的一些是如何相互关联的。例如，全球平均气温是否与大西洋飓风活动密切相关？如果这种关系在统计学上具有重要意义，气候科学家会谈论将后者归因于前者。我们发现，当时间序列在很大程度上是平稳的时，普通的统计工具可以很好地衡量归因，但对于更具累积性的时间序列，相同的工具可能会错误地指出实际上不存在的强烈归因。这种不正确的归因现象使用所谓的相关系数来衡量，在科学论文中发生的频率比人们希望的要高。它被称为尤勒的“无稽之谈”，以纪念著名的英国统计学家，他在1926年首次经验性地描述了这种可能性。我们的工作首次准确地量化了这种相关性如何表现为一个数学对象，对于累积时间序列和平稳时间序列。作为该奖项工作的结果，我们将为科学家提供用于时间序列相关性的明显正确的工具，这将帮助他们非常精确地测量自然和社会现象，如上文所述，是统计上相关的，还是它们更有可能相互独立。该项目还将为研究生提供研究培训机会。在开发统计推断工具时，这是一个被广泛接受的方向，该奖项的工作将研究统计测试的性质，这些测试检测数据流是否可能不独立。研究对象是时间序列或随机过程的路径对，以及任何这样的路径对的经验皮尔逊型相关统计量。特别是，这项工作将适用于观察性研究，而不是重复实验，因为对于任何给定的环境或经济变量，单一时间序列往往是唯一类型的数据。对于平稳随机过程，我们将通过使用精确分布理论和随机分析的正态近似计算，得到经验相关的波动的精确估计。这些结果将直接导致提出区分随机过程对的相依和独立的原则性统计方法。接下来，我们将研究高度非平稳路径的领域，包括随机行走和布朗运动，经验关联的渐近如何强烈偏离正态分布，以及如何将这些信息转换为前述应用来区分依赖和独立。我们的大部分工作将利用经典的高斯方差和协方差对象的分布特性，作为随机分析的一个技术方面。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Scientists and engineers want to extract information and develop an understanding of the natural world and human society by analyzing data to inform questions and decisions. The data often presents itself as the so-called time series, also known as stochastic processes, evolutions of measurable quantities over periods of time. The variations of these time series can be rather predictable from one period to the next, but less so over longer time intervals covering many periods. There are subtle differences in the nature of various types of these stochastic processes. For instance, the value of a financial stock or index, or the yearly global mean temperature, are buildups, accumulating stochastically over time. But daily returns on stocks or commodities futures, or the category (intensity) of successive Atlantic hurricanes, are of a different nature, typically showing a great deal of independence from one day or one event to the next, featuring a property of stationarity over time after adjusting for trends and seasonality. A critically important question is how some of these time series relate to each other. For instance, are global mean temperatures closely tied to Atlantic hurricane activity? Climate scientists would talk about significant attribution of the latter to the former if the relation is statistically significant. We have discovered that ordinary statistical tools work well to measure attribution when time series are largely stationary, but that the same tools can incorrectly point to a strong attribution when none actually exists, for time series, which are more accumulative. This incorrect attribution phenomenon, measured using a so-called correlation coefficient, occurs more frequently in scientific papers than one would hope. It is known as Yule's "nonsense correlation" in honor of the famed British statistician who first described the possibility empirically in 1926. Our work is the first to quantify exactly how this correlation can behave as a mathematical object, for accumulative time series, and for stationary time series. As a consequence of this award's work, we will provide scientists with demonstrably correct tools for correlations of time series, which will help them measure with great precision whether natural and societal phenomena, such as those described above, are statistically related, or whether they are more likely to be independent of each other. The project will also provide research training opportunities for graduate students. As is a well-accepted direction when developing tools for statistical inference, this award's work will study the properties of statistical tests which detect whether data streams are likely not to be independent. The objects of study are pairs of paths of times series or stochastic processes, and the empirical Pearson-type correlation statistic for any such pair. In particular, the work will apply to observational studies, rather than repeated experiments, since single time series are often the only type of data for any given environmental or economic variable. For stationary stochastic processes, we will derive precise estimates of the empirical correlation's fluctuations, by using calculations involving both exact distribution theory and normal approximations via stochastic analysis. These results will lead directly to proposing principled statistical methods for distinguishing between dependent and independent of pairs of stochastic processes. Next, we will investigate the realm of highly non-stationary paths, including random walks and Brownian motion, how the asymptotics for the empirical correlations deviate strongly from normality, and how to convert this information to the aforementioned application to distinguish between dependence and independence. Much of our work will draw on the distributional properties of classical variance and covariance objects for Gaussian vectors, as a technical aspect of stochastic analysis.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.