Title: Wiener chaos calculus for Yule's correlation statistic, and a strategy for attribution testing in observational time series.
Date: Friday October 6, 2023
Place and Time: Love 101, 3:05-3:55 pm
Abstract. The empirical correlation for two related sequences of data of length n is defined classically via Pearson's correlation statistic. As as n tends to infinity, this converge to the relevant correlation coefficient when both data streams come from i.i.d. measurements, or under stationarity and weak memory conditions. The fluctuations fail to be normal under longer memory conditions. When the sequences are sufficiently non-stationary, convergence will fail. Famously, the statistic is asymptotically diffuse, over the entire interval (-1,1), when the data are random walks. The asymptotic statistic is known as Yule's "nonsense correlation" in honor of the statistician G. Udny Yule who first described the phenomenon in 1926. Many decades later, there still exist vexing instances of applied scientists who draw incorrect attribution conclusions based on invalid inference about correlations of time series, in ignorance of Yule's observation. We will describe the mathematical question of understanding the asymptotics of Yule's correlation for random walks, including an explicit expression for its variance when the random walks are Gaussian, and a surprisingly rapid rate of convergence to the limiting "nonsense correlation" object. These results appeared in a paper with Philip Ernst and Dongzhou Huang, in SPA in April 2023. The question of what types of fluctuations this convergence has is open, but we will present arguments for the conjecture that the fluctuations are non-Gaussian, which is work in progress. We will explain how these fluctuation could lead to testing procedures and what this might mean for statistical attribution in environmental sciences.