Probability distributions for analog-to-target distances

Abstract: Some properties of chaotic dynamical systems can be probed through features of recurrences, also called analogs. In practice, analogs are nearest neighbours of the state of a system, taken from a large database called the catalog. Analogs have been used in many atmospheric applications including forecasts, downscaling, predictability estimation, and attribution of extreme events. The distances of the analogs to the target state condition the performances of analog applications. These distances can be viewed as random variables, and their probability distributions can be related to the catalog size and properties of the system at stake. A few studies have focused on the first moments of return time statistics for the best analog, fixing an objective of maximum distance from this analog to the target state. However, for practical use and to reduce estimation variance, applications usually require not just one, but many analogs. In this paper, we evaluate from a theoretical standpoint and with numerical experiments the probability distributions of the $K$-best analog-to-target distances. We show that dimensionality plays a role on the size of the catalog needed to find good analogs, and also on the relative means and variances of the $K$-best analogs. Our results are based on recently developed tools from dynamical systems theory. These findings are illustrated with numerical simulations of a well-known chaotic dynamical system and on 10m-wind reanalysis data in north-west France. A practical application of our derivations for the purpose of objective-based dimension reduction is shown using the same reanalysis data.