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Statistical inference of causal interactions and synchronization between dynamical phenomena evolving on different temporal scales is of vital importance for better understanding and prediction of natural complex systems such as the Earth’s climate. This article introduces and applies information theory diagnostics to phase and amplitude time series of different oscillatory components of observed data that characterizes El Nino/Southern Oscillation. A suite of significant interactions between processes operating on different time scales is detected and shown to be important for emergence of extreme events. The mechanisms of these nonlinear interactions are further studied in conceptual low-order and state-of-the-art dynamical, as well as statistical climate models. Observed and simulated interactions exhibit substantial discrepancies, whose understanding may be the key to an improved prediction of ENSO. Moreover, the statistical framework applied here is suitable for inference of cross-scale interactions in human brain dynamics and other complex systems. Strong El Nino and La Nina events arise from the interaction and synchronization between El Nino Southern Oscillation (ENSO) cycles that operate on different time scales. The warm, El Nino, and cold, La Nina, phases of ENSO are irregular and occur every 2 to 7 years, governed by the interaction between the annual, biannual and interannual cycles. Milan Palus, from the Czech Academy of Sciences in Prague, and colleagues apply statistical diagnostics from information theory to ENSO data to detect the causal interactions between these three variability modes that lead to extreme El Nino/La Nina event. They find a particularly important role for the biannual cycle in these extreme events. The authors suggest that this statistical framework could also be used for inferring cross-scale interactions in neuronal networks in the human brain and other complex systems.

In this comparative study, six causality detection methods were compared, namely, the Granger vector autoregressive test, the extended Granger test, the kernel version of the Granger test, the conditional mutual information (transfer entropy), the evaluation of cross mappings between state spaces, and an assessment of predictability improvement due to the use of mixed predictions. Seven test data sets were analyzed: linear coupling of autoregressive models, a unidirectional connection of two H\'enon systems, a unidirectional connection of chaotic systems of R\"ossler and Lorenz type and of two different R\"ossler systems, an example of bidirectionally connected two-species systems, a fishery model as an example of two correlated observables without a causal relationship, and an example of mediated causality. We tested not only $20\phantom{\rule{0.16em}{0ex}}000$ points long clean time series but also noisy and short variants of the data. The standard and the extended Granger tests worked only for the autoregressive models. The remaining methods were more successful with the more complex test examples, although they differed considerably in their capability to reveal the presence and the direction of coupling and to distinguish causality from mere correlation.

Air temperature variability on different time scales exhibits recurring patterns and quasi-oscillatory phenomena. Climate oscillations with the period about 7–8 years have been observed in many instrumental records in Europe. Although these oscillations are weak if considering their amplitude, they might have nonnegligible influence on temperature variability on shorter time scales due to cross-scale interactions recently observed by Palus (2014). In order to quantify the cross-scale influence, we propose a simple conditional mean approach which estimates the effect of the cycle with the period close to 8 years on the amplitude of the annual cycle in surface air temperature (SAT) in the range 0.7–1.4°C and the effect on the overall variability of the SAT anomalies (SATA) leads to the changes 1.5–1.7°C in the annual SATA means. The strongest effect in the winter SATA means reaches 4–5°C in central European station and reanalysis data.

Nonparametric detection of coupling delay in unidirectionally and bidirectionally coupled nonlinear dynamical systems is examined. Both continuous and discrete-time systems are considered. Two methods of detection are assessed-the method based on conditional mutual information-the CMI method (also known as the transfer entropy method) and the method of convergent cross mapping-the CCM method. Computer simulations show that neither method is generally reliable in the detection of coupling delays. For continuous-time chaotic systems, the CMI method appears to be more sensitive and applicable in a broader range of coupling parameters than the CCM method. In the case of tested discrete-time dynamical systems, the CCM method has been found to be more sensitive, while the CMI method required much stronger coupling strength in order to bring correct results. However, when studied systems contain a strong oscillatory component in their dynamics, results of both methods become ambiguous. The presented study suggests that results of the tested algorithms should be interpreted with utmost care and the nonparametric detection of coupling delay, in general, is a problem not yet solved.

Complex systems are commonly characterized by the properties of their graph representation. Dynamical complex systems are then typically represented by a graph of temporal dependencies between time series of state variables of their subunits. It has been shown recently that graphs constructed in this way tend to have relatively clustered structure, potentially leading to spurious detection of small-world properties even in the case of systems with no or randomly distributed true interactions. However, the strength of this bias depends heavily on a range of parameters and its relevance for real-world data has not yet been established. In this work, we assess the relevance of the bias using two examples of multivariate time series recorded in natural complex systems. The first is the time series of local brain activity as measured by functional magnetic resonance imaging in resting healthy human subjects, and the second is the time series of average monthly surface air temperature coming from a large reanalysis of climatological data over the period 1948-2012. In both cases, the clustering in the thresholded correlation graph is substantially higher compared with a realization of a density-matched random graph, while the shortest paths are relatively short, showing thus distinguishing features of small-world structure. However, comparable or even stronger small-world properties were reproduced in correlation graphs of model processes with randomly scrambled interconnections. This suggests that the small-world properties of the correlation matrices of these real-world systems indeed do not reflect genuinely the properties of the underlying interaction structure, but rather result from the inherent properties of correlation matrix.

A directed climate network is constructed by Granger causality analysis of air temperature time series from a regular grid covering the whole Earth. Using winner-takes-all network thresholding approach, a structure of a smooth information flow is revealed, hidden to previous studies. The relevance of this observation is confirmed by comparison with the air mass transfer defined by the wind field. Their close relation illustrates that although the information transferred due to the causal influence is not a physical quantity, the information transfer is tied to the transfer of mass and energy.