Abstract
Information flow or information transfer the widely applicable general physics notion can be rigorously derived from first principles, rather than axiomatically proposed as an ansatz. Its logical association with causality is firmly rooted in the dynamical system that lies beneath. The principle of nil causality that reads, an event is not causal to another if the evolution of the latter is independent of the former, which transfer entropy analysis and Granger causality test fail to verify in many situations, turns out to be a proven theorem here. Established in this study are the information flows among the components of time-discrete mappings and time-continuous dynamical systems, both deterministic and stochastic. They have been obtained explicitly in closed form, and put to applications with the benchmark systems such as the Kaplan-Yorke map, Rössler system, baker transformation, Hénon map, and stochastic potential flow. Besides unraveling the causal relations as expected from the respective systems, some of the applications show that the information flow structure underlying a complex trajectory pattern could be tractable. For linear systems, the resulting remarkably concise formula asserts analytically that causation implies correlation, while correlation does not imply causation, providing a mathematical basis for the long-standing philosophical debate over causation versus correlation.
Highlights
Information flow, or information transfer as it may be referred to in the literature, has been realized as a fundamental notion in general physics
Though literally one may associate it with communication, its importance lies far beyond in that it implies causation [1,2,3,4,5], uncertainty propagation [6], predictability transfer [7,8,9], etc. It is the recognition of its causality association that has attracted enormous interest from a wide variety of disciplines, e.g., neuroscience [10,11,12,13,14,15,16], finance [17,18], climate science [19,20], turbulence research [21,22], network dynamics [23,24,25,26], and dynamical systems the field of synchronization [27,28,29,30,31,32,33]. This recognition has been further substantiated by the finding that transfer entropy [5] and Granger causality [34] are equivalent for Gaussian variables [35]
= F2(x1,x2,t), following section, the extension relies on an assumption that (1) is, again, axiomatically proposed. This makes the resulting formalism not one fully derived from first principles, and as (2) we realize later on, it does not work for multidimensional stochastic systems
Summary
Information flow, or information transfer as it may be referred to in the literature, has been realized as a fundamental notion in general physics. Though literally one may associate it with communication, its importance lies far beyond in that it implies causation [1,2,3,4,5], uncertainty propagation [6], predictability transfer [7,8,9], etc It is the recognition of its causality association that has attracted enormous interest from a wide variety of disciplines, e.g., neuroscience [10,11,12,13,14,15,16], finance [17,18], climate science [19,20], turbulence research [21,22], network dynamics [23,24,25,26], and dynamical systems the field of synchronization [27,28,29,30,31,32,33]. Aside from the failure in recovering the many preset one-way causalities, evidence has
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