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

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.