Coupling of statistics and math models identifies power tail jump patterns in time series
Coupling of statistics and math models identifies power tail jump patterns in time series lead image
In a 1999 article, climatologist Peter Ditlevsen gave strong evidence of an alpha-stable noise component in ice-core records with fast fluctuations of the last glacial period. His method was based on a heuristic study of the tails of the jump distributions. Recently, researchers from universities in Colombia, Germany, Ukraine and Canada have amplified Ditlevsen’s focus and derived a rigorous minimal distance procedure for this and more general time series, which they report in Chaos: An Interdisciplinary Journal of Nonlinear Science.
Co-author and mathematician Michael Hoegele says that such time series often occur in large aggregations with threshold effects, and tend to have flat tails (having many outliers in the distribution). Good models which produce such effects are Lévy diffusions with non-Gaussian polynomial tails which are discontinuous, characterized by jumps in the noise.
For their procedure, Hoegele and colleagues brought in the notion of coupling distance which measures the Wasserstein distance between larger and larger parts of the increments. Their device is strong enough to distinguish such models even on path space only in terms of the distributions of individual jumps while being sufficiently weak to allow for an easy implementation and fast rates of convergence confirmed in systematic simulation studies.
The researchers applied their method successfully to Ditlevsen’s data confirming his results on power-tails in the data, however with different tail exponents. They also tested recent precipitation data of the south Pacific rain-off patterns and detected a power-tail component in them.
Hoegele says one innovation of their work was to provide a rigorous mathematical tool that elucidates these noisy Lévy tails in data for accurate modelling, and may serve as an exploration tool for non-specialists for many focuses.
Source: “How close are time series to power tail Lévy diffusions?,” by Jan M. Gairing, Michael A. Hoegele, Tania Kosenkova, and Adam Monahan, Chaos: An Interdisciplinary Journal of Nonlinear Science (2017). The article can be accessed at https://doi.org/10.1063/1.4986496 .
