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提取信息，并通过分析数据来了解自然世界和人类社会，从而为问题和决策提供信息。数据通常表现为所谓的时间序列，也称为随机过程，即一段时间内可测量量的演变。从一个时期到下一个时期，这些时间序列的变化是相当可预测的，但在涵盖许多时期的较长时间间隔内则不那么可预测。这些随机过程的各种类型在本质上有微妙的差异。例如，金融股票或指数的价值，或者全球年平均气温，都是累积的，随着时间的推移而随机累积。但股票或商品期货的日收益率，或者连续大西洋飓风的类别（强度），则具有不同的性质，通常表现出从一天或一个事件到下一个事件的很大程度的独立性，在调整趋势和季节性之后，随着时间的推移具有平稳性。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？气候科学家会谈论后者对前者的重要归因，如果这种关系在统计上是显著的。我们发现，当时间序列基本上是静止的时候，普通的统计工具可以很好地测量归因，但是对于积累性更强的时间序列，当实际上不存在归因时，同样的工具可能错误地指向一个强的归因。这种不正确的归因现象，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。它被称为尤尔的"无意义的相关性"，以荣誉的名字命名，尤尔是著名的英国统计学家，他在1926年首次根据经验描述了这种可能性。我们的工作是第一个量化这种相关性如何作为一个数学对象，累积时间序列，并为固定的时间序列。作为该奖项工作的结果，我们将为科学家提供时间序列相关性的证明正确的工具，这将帮助他们非常精确地测量自然和社会现象，例如上述现象，是否在统计上相关，或者它们是否更有可能相互独立。该项目还将为研究生提供研究培训机会。 作为开发统计推断工具时一个广为接受的方向，该奖项的工作将研究统计测试的属性，这些测试可以检测数据流是否可能不是独立的。研究的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测研究，而不是重复实验，因为单一时间序列往往是任何特定环境或经济变量的唯一类型的数据。对于平稳随机过程，我们将通过使用涉及精确分布理论和正态近似的计算，通过随机分析得到经验相关性波动的精确估计。这些结果将直接导致提出原则性的统计方法来区分依赖和独立的随机过程对。接下来，我们将研究高度非平稳路径的领域，包括随机游动和布朗运动，经验相关性的渐近性如何强烈偏离正态性，以及如何将这些信息转换为上述应用以区分依赖性和独立性。我们的大部分工作将利用高斯向量的经典方差和协方差对象的分布特性，作为随机分析的技术方面。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

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.