Identifying causal networks is important for effective policy and management recommendations on climate, epidemiology, financial regulation, and much else. We introduce a method, based on nonlinear state space reconstruction, that can distinguish causality from correlation. It extends to nonseparable weakly connected dynamic systems (cases not covered by the current Granger causality paradigm). The approach is illustrated both by simple models (where, in contrast to the real world, we know the underlying equations/relations and so can check the validity of our method) and by application to real ecological systems, including the controversial sardine-anchovy-temperature problem.

The heart of the scientific enterprise is a rational effort to understand the causes behind the phenomena we observe. In large-scale complex dynamical systems such as the Earth system, real experiments are rarely feasible. However, a rapidly increasing amount of observational and simulated data opens up the use of novel data-driven causal methods beyond the commonly adopted correlation techniques. Here, we give an overview of causal inference frameworks and identify promising generic application cases common in Earth system sciences and beyond. We discuss challenges and initiate the benchmark platform causeme.net to close the gap between method users and developers.

Ecological theory suggests that large-scale patterns such as community stability can be influenced by changes in interspecific interactions that arise from the behavioural and/or physiological responses of individual species varying over time. Although this theory has experimental support, evidence from natural ecosystems is lacking owing to the challenges of tracking rapid changes in interspecific interactions (known to occur on timescales much shorter than a generation time) and then identifying the effect of such changes on large-scale community dynamics. Here, using tools for analysing nonlinear time series and a 12-year-long dataset of fortnightly collected observations on a natural marine fish community in Maizuru Bay, Japan, we show that short-term changes in interaction networks influence overall community dynamics. Among the 15 dominant species, we identify 14 interspecific interactions to construct a dynamic interaction network. We show that the strengths, and even types, of interactions change with time; we also develop a time-varying stability measure based on local Lyapunov stability for attractor dynamics in non-equilibrium nonlinear systems. We use this dynamic stability measure to examine the link between the time-varying interaction network and community stability. We find seasonal patterns in dynamic stability for this fish community that broadly support expectations of current ecological theory. Specifically, the dominance of weak interactions and higher species diversity during summer months are associated with higher dynamic stability and smaller population fluctuations. We suggest that interspecific interactions, community network structure and community stability are dynamic properties, and that linking fluctuating interaction networks to community-level dynamic properties is key to understanding the maintenance of ecological communities in nature.

An important problem across many scientific fields is the identification of causal effects from observational data alone. Recent methods (convergent cross mapping, CCM) have made substantial progress on this problem by applying the idea of nonlinear attractor reconstruction to time series data. Here, we expand upon the technique of CCM by explicitly considering time lags. Applying this extended method to representative examples (model simulations, a laboratory predator-prey experiment, temperature and greenhouse gas reconstructions from the Vostok ice core, and long-term ecological time series collected in the Southern California Bight), we demonstrate the ability to identify different time-delayed interactions, distinguish between synchrony induced by strong unidirectional-forcing and true bidirectional causality, and resolve transitive causal chains.

Evidence shows that species interactions are not constant but change as the ecosystem shifts to new states. Although controlled experiments and model investigations demonstrate how nonlinear interactions can arise in principle, empirical tools to track and predict them in nature are lacking. Here we present a practical method, using available time-series data, to measure and forecast changing interactions in real systems, and identify the underlying mechanisms. The method is illustrated with model data from a marine mesocosm experiment and limnologic field data from Sparkling Lake, WI, USA. From simple to complex, these examples demonstrate the feasibility of quantifying, predicting and understanding state-dependent, nonlinear interactions as they occur in situ and in real time-a requirement for managing resources in a nonlinear, non-equilibrium world.

In temperate countries, influenza outbreaks are well correlated to seasonal changes in temperature and absolute humidity. However, tropical countries have much weaker annual climate cycles, and outbreaks show less seasonality and are more difficult to explain with environmental correlations. Here, we use convergent cross mapping, a robust test for causality that does not require correlation, to test alternative hypotheses about the global environmental drivers of influenza outbreaks from country-level epidemic time series. By moving beyond correlation, we show that despite the apparent differences in outbreak patterns between temperate and tropical countries, absolute humidity and, to a lesser extent, temperature drive influenza outbreaks globally. We also find a hypothesized U-shaped relationship between absolute humidity and influenza that is predicted by theory and experiment, but hitherto has not been documented at the population level. The balance between positive and negative effects of absolute humidity appears to be mediated by temperature, and the analysis reveals a key threshold around 75°F. The results indicate a unified explanation for environmental drivers of influenza that applies globally.epidemiology | empirical dynamic modeling | nonlinear dynamics | state-dependence | physical-biological coupling A diverse group of drivers and mechanisms has been put forward to explain the wintertime occurrence of seasonal influenza outbreaks. Laboratory experiments show that relative humidity controls droplet size and aerosol transmission rates (1). Experiments with mammalian models showed that viral shedding by hosts increases at low temperature (2). Strong laboratory evidence has emerged that absolute humidity has a controlling effect on airborne influenza transmission (3).Nevertheless, questions remain as to how these potential causal agents are expressed at the population level as epidemic control variables. At the population level, environmental factors covary, multiple mechanisms can coact, and infection dynamics are influenced by many other important processes (4), such as human crowding, rapid viral evolution, and international travel patterns. Perhaps not surprisingly, statistical analyses of population level data have produced contradictory results. Although correlations between influenza incidence and both temperature and absolute humidity are easy to find in temperate countries (5) and individual US states (6), such associations are weak or altogether absent in data from tropical countries (5).Here we use new methods appropriate for disease dynamics to identify the causal drivers of influenza acting at the population level. Using time series data across countries and latitudes, we find that absolute humidity drives influenza across latitudes, and that this effect is modulated by temperature. At low temperatures, absolute humidity negatively affects influenza incidence (drier conditions improve survival of the influenza virus when it is cold), but at high temperatures, absolute humidity positively a...

Recent developments in complex systems analysis have led to new techniques for detecting causal relationships using relatively short time series, on the order of 30 sequential observations. Although many ecological observation series are even shorter, perhaps fewer than ten sequential observations, these shorter time series are often highly replicated in space (i.e., plot replication). Here, we combine the existing techniques of convergent cross mapping (CCM) and dewdrop regression to build a novel test of causal relations that leverages spatial replication, which we call multispatial CCM. Using examples from simulated and real-world ecological data, we test the ability of multispatial CCM to detect causal relationships between processes. We find that multispatial CCM successfully detects causal relationships with as few as five sequential observations, even in the presence of process noise and observation error. Our results suggest that this technique may constitute a useful test for causality in systems where experiments are difficult to perform and long time series are not available. This new technique is available in the multispatialCCM package for the R programming language.

Takens' theorem (1981) shows how lagged variables of a single time series can be used as proxy variables to reconstruct an attractor for an underlying dynamic process. State space reconstruction (SSR) from single time series has been a powerful approach for the analysis of the complex, non-linear systems that appear ubiquitous in the natural and human world. The main shortcoming of these methods is the phenomenological nature of attractor reconstructions. Moreover, applied studies show that these single time series reconstructions can often be improved ad hoc by including multiple dynamically coupled time series in the reconstructions, to provide a more mechanistic model. Here we provide three analytical proofs that add to the growing literature to generalize Takens' work and that demonstrate how multiple time series can be used in attractor reconstructions. These expanded results (Takens' theorem is a special case) apply to a wide variety of natural systems having parallel time series observations for variables believed to be related to the same dynamic manifold. The potential information leverage provided by multiple embeddings created from different combinations of variables (and their lags) can pave the way for new applied techniques to exploit the time-limited, but parallel observations of natural systems, such as coupled ecological systems, geophysical systems, and financial systems. This paper aims to justify and help open this potential growth area for SSR applications in the natural sciences.