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A basic systems question concerns the concept of closure, meaning autonomomy(closed) in the sense of describing the (sub)system as fully consistent withinitself. Alternatively, the system may be nonautonomous (open) meaning itreceives influence from an outside coupling subsystem. Information flow, andrelated causation inference, are tenant on this simple concept. We take theperspective of Weiner-Granger causality, descriptive of a subsystem forecastquality dependence on considering states of another subsystem. Here we developa new direct analytic discussion, rather than a data oriented approach. Thatis, we refer to the underlying Frobenius-Perron transfer operator thatmoderates evolution of densities of ensembles of orbits, and two alternativeforms of the restricted Frobenius-Perron (FP) operator, interpreted as ifeither closed (determinstic FP) or not closed (the unaccounted outsideinfluence seems stochastic and correspondingly the stochastic FP operator).From this follows contrasting the kernels of the variants of the operators, asif densities in their own rights. However, the corresponding differentialentropy to compare by Kulback-Leibler divergence, as one would when leading totransfer entropy, becomes ill-defined. Instead we build our ForecastabilityQuality Metric (FQM) upon the "symmetrized" variant known as Jensen-Shanondivergence, and also we are able to point out several useful resultingproperties that result. We illustrate the FQM by a simple coupled chaoticsystem. For now, this analysis is a new theoretical direction, but we describedata oriented directions for the future.

Open or Closed? Information Flow Decided by Transfer Operators and Forecastability Quality Metric