Wasserstein recurrence networks for multiscale time series pattern analysis

Time series data are often generated by systems which operate on multiple temporal scales, of which Earth's climate system is a paramount example.

Variations in global climate are recorded in paleo-environmental archives as temporal patterns across a wide range of time scales, from seasonal or decadal to multi-millennial.

In this context, recurrence analysis, where repeating patterns are identified in time series, is limited by the underlying properties of the distance function used and of the time series data themselves, especially in terms of temporal resolution and scale dependence.

In this paper, we present a novel recurrence analysis framework designed for multiscale time series data with abrupt changes and irregular temporal resolution as found in paleoclimate records.

We introduce a simple mathematical transform to use the $1-$Wasserstein distance for recurring pattern detection in time series.

The scale invariance of $1-$Wasserstein distance distributions between patterns in Brownian motion is demonstrated numerically, which provides a principled threshold choice for recurrences.

At any time scale, recurrences are defined as local minima of the distance, granted that they are below a threshold given by the probability of encountering patterns at least as similar in one-dimensional Brownian motion.

Recurrences can be further combined according to a non-overlapping condition to yield a distinct set of multiscale recurring events.

We provide examples of climatic applications from ice-rafted debris and ice core records, where detected recurrences have durations spanning over two orders of magnitude.

Our method extends recurrence analysis to more complex time series data and provides new avenues for statistical identification and analyses of recurring events at multiple temporal scales.