Speaker: Dr. Frederi Viens
Title: Asymptotic distribution of Yule’s nonsense correlation with a view towards environmental statistics
Abstract: The empirical correlation for two related sequences of data of length n is defined classically via Pearson’s correlation statistic. When the data is i.i.d. with two moments, it is known to converge to the correlation coefficient of the pair of random variables behind the two data series, with normal fluctuations, as n tends to infinity. This property remains if the two sequences of data have some level of sequential correlation and are stationary. However, it fails under stronger memory conditions, and when the sequences are sufficiently non-stationary. 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 nonsense correlation for random walks. We will present an explicit expression for the variance of Yule’s nonsense correlation when the random walks are Gaussian, as well as a surprisingly rapid rate of convergence of that correlation to its limiting 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. We believe they are non-Gaussian; we will try to explain why, and what this might mean for statistical testing of attribution in environmental sciences.
Day & Time: Friday, September 15, 2023 at 3:00pm
Location: Lambton 9-118
Counts toward seminar attendance for MSc and PhD students in Math & Stats.