Causality Analysis: Identifying the Leading Element in a Coupled Dynamical System

Abstract

Physical systems with time-varying internal couplings are abundant in nature. While the full governing equations of these systems are typically unknown due to insufficient understanding of their internal mechanisms, there is often interest in determining the leading element. Here, the leading element is defined as the sub-system with the largest coupling coefficient averaged over a selected time span. Previously, the Convergent Cross Mapping (CCM) method has been employed to determine causality and dominant component in weakly coupled systems with constant coupling coefficients. In this study, CCM is applied to a pair of coupled Lorenz systems with time-varying coupling coefficients, exhibiting switching between dominant sub-systems in different periods. Four sets of numerical experiments are carried out. The first three cases consist of different coupling coefficient schemes: I) Periodic–constant, II) Normal, and III) Mixed Normal/Non-normal. In case IV, numerical experiment of cases II and III are repeated with imposed temporal uncertainties as well as additive normal noise. Our results show that, through detecting directional interactions, CCM identifies the leading sub-system in all cases except when the average coupling coefficients are approximately equal, i.e., when the dominant sub-system is not well defined.

Empirical dynamic modeling for beginners

“…daily or annual) reveal different dynamics, because the behavior of a dynamical system is scale dependent (Hsieh and Ohman 2006). In addition, the time series data needs to be stationary, as required by all time series analyses (Box et al 1994). …”

Causal dependences between the coupled ocean–atmosphere dynamics over the tropical Pacific, the North Pacific and the North Atlantic

“…It is also known that the Walker circulation is considerably affected by the upper-layer ocean temperature, and vice versa (Philander, 1990). Let us therefore consider for now the mean ocean potential temperature in the Nino3.4 region, known to have strong correlation with the variability in the North Atlantic region (Brönnimann, 2007). For the characterization of the Walker circulation, the zonal winds at 500 and 200 hPa over the same domain are chosen.…”

“…In particular, an important question that has attracted a lot of attention in the past decades is to know whether the tropical Pacific system forces the dynamics of the climate system in the extratropics. In this context, important teleconnections are found between events like El Niño or La Niña and the temperature and precipitation patterns all over the world, e.g., Fraedrich and Müller (1992), Brönnimann (2007) and Lau (2016). The origin of these teleconnections for the North Atlantic and North Pacific is currently explained through the concept of the atmospheric bridge that allows for the transfer of information from one basin to another, e.g., Sardeshmukh and Hoskins (1988), Alexander et al (2002), Yu and Lin (2016) and Lau (2016).…”

“…A powerful method has been recently proposed by Sugihara et al (2012), known as convergent cross mapping (CCM), which is a method suitable for nonlinear deterministic dynamical systems as it is based on analogs of the current state. This method has also been tested with success when nonlinear dynamical systems are affected by noise (Mønster et al, 2017) and in coupled dynamical systems in order to identify the leading element of the coupling (BozorgMagham et al, 2015). It has also been recently used to disentangle the link between galactic cosmic rays and the variations of the global temperature (Tsonis et al, 2015;Luo et al, 2015;Ye et al, 2015) or between environmental drivers and influenza (Deyle et al, 2016).…”

Inferring Causality from Noisy Time Series Data - A Test of Convergent Cross-Mapping

“…CCM has already been used in a wide range of different fields for different kinds of data (McBride et al, 2015;BozorgMagham et al, 2015;McCracken and Weigel, 2014), and it has been noted (McCracken and Weigel, 2014) that CCM results are not always consistent with theoretical intuitions. McCracken and Weigel (2014) extended CCM to pairwise asymmetric inference (PAI), which they demonstrated to give results that are in better accordance with the intuitively expected outcomes for several physical systems.…”