Dynamics of non-stationary processes that follow the maximum of the Rényi entropy principle

Shalymov, Dmitry S.; Fradkov, Alexander L.

doi:10.1098/rspa.2015.0324

Cited by 13 publications

(12 citation statements)

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“…Refs. [40][41][42][43]). The novel connection between information geometry and a generalised MEP opens the door for fertile explorations combining non-Euclidean methods and statistical analyses, which may lead to new insights and techniques to further deepen our understanding of complex systems.…”

Section: Introductionmentioning

confidence: 99%

A generalization of the maximum entropy principle for curved statistical manifolds

Morales,

Rosas

2021

Preprint

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The maximum entropy principle (MEP) is one of the most prominent methods to investigate and model complex systems. Despite its popularity, the standard form of the MEP can only generate Boltzmann-Gibbs distributions, which are ill-suited for many scenarios of interest. As a principled approach to extend the reach of the MEP, this paper revisits its foundations in information geometry and shows how the geometry of curved statistical manifolds naturally leads to a generalization of the MEP based on the Rényi entropy. By establishing a bridge between non-Euclidean geometry and the MEP, our proposal sets a solid foundation for the numerous applications of the Rényi entropy, and enables a range of novel methods for complex systems analysis.

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Section: Introductionmentioning

confidence: 99%

A generalization of the maximum entropy principle for curved statistical manifolds

Morales,

Rosas

2021

Preprint

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“…Indeed, a complete characterization of landscape diversity can be achieved if, instead of a single index, a parametric family of indices is used, whose members have varying sensitivities to the presence of rare and abundant elements (Jost 2007). Unlike Shannon entropy, which is single-valued, Re ´nyi entropy is useful in measuring a range of information levels such as those in A-yeh and , the dynamics of non-stationary processes (Shalymov and Fradkov 2016) and the measuring of the distribution of entropy changes over time (see, e.g., Jauregui et al 2018). In other words, Re ´nyi entropy offers a ''continuum of possible diversity measures'' (Ricotta and Avena 2003) at diverse spatiotemporal scales, becoming increasingly regulated by the commonest when a gets higher.…”

Section: Introducing Re ´Nyi Entropymentioning

confidence: 99%

The informational entropy endowed in cortical oscillations

2018

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A two-dimensional shadow may encompass more information than its corresponding three-dimensional object. Indeed, if we rotate the object, we achieve a pool of observed shadows from different angulations, gradients, shapes and variable length contours that make it possible for us to increase our available information. Starting from this simple observation, we show how informational entropies might turn out to be useful in the evaluation of scale-free dynamics in the brain. Indeed, brain activity exhibits a scale-free distribution that leads to the variations in the power law exponent typical of different functional neurophysiological states. Here we show that modifications in scaling slope are associated with variations in Rényi entropy, a generalization of Shannon informational entropy. From a three-dimensional object's perspective, by changing its orientation (standing for the cortical scale-free exponent), we detect different two-dimensional shadows from different perception angles (standing for Rényi entropy in different brain areas). We show how, starting from known values of Rényi entropy (easily detectable in brain fMRIs or EEG traces), it is feasible to calculate the scaling slope in a given moment and in a given brain area. Because changes in scale-free cortical dynamics modify brain activity, this issue points towards novel approaches to mind reading and description of the forces required for transcranial stimulation.

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“…Considering the maximization of the Shannon entropy under mean and standard deviation constrains, the Gaussian distribution is obtained. Recently, in 2016, Shalymov and Fradkov [27], applied the maximum entropy principle to analysis the dynamics of non-stationary processes that follow this principle. More, in 2016, the same problem was studied considering Tsallis statistics [28].…”

Section: Introductionmentioning

confidence: 99%