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

BozorgMagham AE, Motesharrei S, Penny SG, Kalnay E - PLoS ONE (2015)

Bottom Line: Four sets of numerical experiments are carried out.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.

View Article: PubMed Central - PubMed

Affiliation: Department of Atmospheric and Oceanic Science (AOSC), University of Maryland, College Park, MD, 20742, USA.

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.

No MeSH data available.


Related in: MedlinePlus

Case I: periodic–constant coupling coefficients.(a)  at L = 500 over the range of μ and η given by Eq (10) and for random initial condition. (b) Difference between the CCM coefficients, , at L = 500 over the specified range of μ and η.
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pone.0131226.g003: Case I: periodic–constant coupling coefficients.(a) at L = 500 over the range of μ and η given by Eq (10) and for random initial condition. (b) Difference between the CCM coefficients, , at L = 500 over the specified range of μ and η.

Mentions: Fig 3a shows at L = 500 for different values of μ and η and random initial conditions. We observe similar patterns in and (not shown here) such as vertical and horizontal bands. For example, we see a blue horizontal band near μ = 1 in Fig 3a, showing that the influence of 𝔏Y on 𝔏x is small. Therefore, (the recovered phase space of 𝔏Y from 𝔏X), is poorly correlated to My (the reconstructed phase space of 𝔏Y), hence the small values for on the band. Another observed pattern is the high values of ρ in the regions with the strongest coupling between the two sub–systems (high μ and η), showing the large mutual influence between the two systems.


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

BozorgMagham AE, Motesharrei S, Penny SG, Kalnay E - PLoS ONE (2015)

Case I: periodic–constant coupling coefficients.(a)  at L = 500 over the range of μ and η given by Eq (10) and for random initial condition. (b) Difference between the CCM coefficients, , at L = 500 over the specified range of μ and η.
© Copyright Policy
Related In: Results  -  Collection

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getmorefigures.php?uid=PMC4488350&req=5

pone.0131226.g003: Case I: periodic–constant coupling coefficients.(a) at L = 500 over the range of μ and η given by Eq (10) and for random initial condition. (b) Difference between the CCM coefficients, , at L = 500 over the specified range of μ and η.
Mentions: Fig 3a shows at L = 500 for different values of μ and η and random initial conditions. We observe similar patterns in and (not shown here) such as vertical and horizontal bands. For example, we see a blue horizontal band near μ = 1 in Fig 3a, showing that the influence of 𝔏Y on 𝔏x is small. Therefore, (the recovered phase space of 𝔏Y from 𝔏X), is poorly correlated to My (the reconstructed phase space of 𝔏Y), hence the small values for on the band. Another observed pattern is the high values of ρ in the regions with the strongest coupling between the two sub–systems (high μ and η), showing the large mutual influence between the two systems.

Bottom Line: Four sets of numerical experiments are carried out.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.

View Article: PubMed Central - PubMed

Affiliation: Department of Atmospheric and Oceanic Science (AOSC), University of Maryland, College Park, MD, 20742, USA.

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.

No MeSH data available.


Related in: MedlinePlus