Transfer entropy computation using the Perron-Frobenius operator.

We propose a method for computing the transfer entropy between time series using Ulam's approximation of the Perron-Frobenius (transfer) operator associated with the map generating the dynamics. Our method differs from standard transfer entropy estimators in that the invariant measure is estimated not directly from the data points, but from the invariant distribution of the transfer operator approximated from the data points. For sparse time series and low embedding dimension, the transfer operator is approximated using a triangulation of the attractor, whereas for data-rich time series or higher embedding dimension, we use a faster grid approach. We compare the performance of our methods with existing estimators such as the k nearest neighbors method and kernel density estimation method, using coupled instances of well known chaotic systems: coupled logistic maps and a coupled Rössler-Lorenz system. We find that our estimators are robust against moderate levels of noise. For sparse time series with less than 100 observations and low embedding dimension, our triangulation estimator shows improved ability to detect coupling directionality, relative to standard transfer entropy estimators.

Forcing of late Pleistocene ice volume by spatially variable summer energy.

Changes in Earth's orbit set the pace of glacial cycles, but the role of spatial variability in the insolation forcing of global ice volume remains unknown. Here, we leverage the intrinsic dynamical information in empirical records to show that ice volume responded to summer energy at high northern latitudes, as predicted by Milankovitch theory. However, the external forcing of ice volume encompasses insolation signals with a wide range of orbital frequency content, and cannot be fully accounted for by a unique time series. Southern mid-latitude insolation forcing coincides with the position of the subtropical front and the westerlies, which have been implicated in Quaternary climate changes. Dominant forcing modes at northern mid-latitudes are anti-phased with the canonical Milankovitch forcing, consistent with ice volume sensitivity to latitudinal insolation gradients.

Common species link global ecosystems to climate change: dynamical evidence in the planktonic fossil record.

Common species shape the world around us, and changes in their commonness signify large-scale shifts in ecosystem structure and function. However, our understanding of long-term ecosystem response to environmental forcing in the deep past is centred on species richness, neglecting the disproportional impact of common species. Here, we use common and widespread species of planktonic foraminifera in deep-sea sediments to track changes in observed global occupancy (proportion of sampled sites at which a species is present and observed) through the turbulent climatic history of the last 65 Myr. Our approach is sensitive to relative changes in global abundance of the species set and robust to factors that bias richness estimators. Using three independent methods for detecting causality, we show that the observed global occupancy of planktonic foraminifera has been dynamically coupled to past oceanographic changes captured in deep-ocean temperature reconstructions. The causal inference does not imply a direct mechanism, but is consistent with an indirect, time-delayed causal linkage. Given the strong quantitative evidence that a dynamical coupling exists, we hypothesize that mixotrophy (symbiont hosting) may be an ecological factor linking the global abundance of planktonic foraminifera to long-term climate changes via the relative extent of oligotrophic oceans.