Editorial

3 pages, 200 KiB

Open AccessEditorial

Information Geometry, Complexity Measures and Data Analysis

by José M. Amigó and Piergiulio Tempesta

Entropy 2022, 24(12), 1797; https://0-doi-org.brum.beds.ac.uk/10.3390/e24121797 - 09 Dec 2022

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In the last several years, a new approach to information theory, called information geometry, has emerged [...] Full article

(This article belongs to the Special Issue Information Geometry, Complexity Measures and Data Analysis)

Research

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11 pages, 408 KiB

Open AccessArticle

f-Gintropy: An Entropic Distance Ranking Based on the Gini Index

by Tamás Sándor Biró, András Telcs, Máté Józsa and Zoltán Néda

Entropy 2022, 24(3), 407; https://0-doi-org.brum.beds.ac.uk/10.3390/e24030407 - 14 Mar 2022

Cited by 3 | Viewed by 1695

Abstract

We consider an entropic distance analog quantity based on the density of the Gini index in the Lorenz map, i.e., gintropy. Such a quantity might be used for pairwise mapping and ranking between various countries and regions based on income and wealth inequality. [...] Read more.

We consider an entropic distance analog quantity based on the density of the Gini index in the Lorenz map, i.e., gintropy. Such a quantity might be used for pairwise mapping and ranking between various countries and regions based on income and wealth inequality. Its generalization to f-gintropy, using a function of the income or wealth value, distinguishes between regional inequalities more sensitively than the original construction. Full article

(This article belongs to the Special Issue Information Geometry, Complexity Measures and Data Analysis)

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16 pages, 604 KiB

Open AccessArticle

Along the Lines of Nonadditive Entropies: q-Prime Numbers and q-Zeta Functions

by Ernesto P. Borges, Takeshi Kodama and Constantino Tsallis

Entropy 2022, 24(1), 60; https://0-doi-org.brum.beds.ac.uk/10.3390/e24010060 - 28 Dec 2021

Cited by 4 | Viewed by 1266

Abstract

The rich history of prime numbers includes great names such as Euclid, who first analytically studied the prime numbers and proved that there is an infinite number of them, Euler, who introduced the function [...] Read more.

The rich history of prime numbers includes great names such as Euclid, who first analytically studied the prime numbers and proved that there is an infinite number of them, Euler, who introduced the function

ζ (s) \equiv \sum_{n = 1}^{\infty} n^{- s} = \prod_{p p r i m e} \frac{1}{1 - p^{- s}}

, Gauss, who estimated the rate at which prime numbers increase, and Riemann, who extended

ζ (s)

to the complex plane z and conjectured that all nontrivial zeros are in the

R (z) = 1 / 2

axis. The nonadditive entropy

S_{q} = k \sum_{i} p_{i} {ln}_{q} (1 / p_{i}) (q \in R; S_{1} = S_{B G} \equiv - k \sum_{i} p_{i} ln p_{i}

, where BG stands for Boltzmann-Gibbs) on which nonextensive statistical mechanics is based, involves the function

{ln}_{q} z \equiv \frac{z^{1 - q} - 1}{1 - q} ({ln}_{1} z = ln z)

. It is already known that this function paves the way for the emergence of a q-generalized algebra, using q-numbers defined as

{⟨ x ⟩}_{q} \equiv e^{{ln}_{q} x}

, which recover the number x for

q = 1

. The q-prime numbers are then defined as the q-natural numbers

{⟨ n ⟩}_{q} \equiv e^{{ln}_{q} n} (n = 1, 2, 3, \dots)

, where n is a prime number

p = 2, 3, 5, 7, \dots

We show that, for any value of q, infinitely many q-prime numbers exist; for

q \leq 1

they diverge for increasing prime number, whereas they converge for

q > 1

; the standard prime numbers are recovered for

q = 1

. For

q \leq 1

, we generalize the

ζ (s)

function as follows:

ζ_{q} (s) \equiv {⟨ ζ (s) ⟩}_{q}

(

s \in R

). We show that this function appears to diverge at

s = 1 + 0

,

\forall q

. Also, we alternatively define, for

q \leq 1

,

ζ_{q}^{\sum} (s) \equiv \sum_{n = 1}^{\infty} \frac{1}{{⟨ n ⟩}_{q}^{s}} = 1 + \frac{1}{{⟨ 2 ⟩}_{q}^{s}} + \dots

and

ζ_{q}^{\prod} (s) \equiv \prod_{p p r i m e} \frac{1}{1 - {⟨ p ⟩}_{q}^{- s}} = \frac{1}{1 - {⟨ 2 ⟩}_{q}^{- s}} \frac{1}{1 - {⟨ 3 ⟩}_{q}^{- s}} \frac{1}{1 - {⟨ 5 ⟩}_{q}^{- s}} \dots

, which, for

q < 1

, generically satisfy

ζ_{q}^{\sum} (s) < ζ_{q}^{\prod} (s)

, in variance with the

q = 1

case, where of course

ζ_{1}^{\sum} (s) = ζ_{1}^{\prod} (s)

. Full article

(This article belongs to the Special Issue Information Geometry, Complexity Measures and Data Analysis)

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17 pages, 360 KiB

Open AccessArticle

The Causal Interaction between Complex Subsystems

by X. San Liang

Entropy 2022, 24(1), 3; https://0-doi-org.brum.beds.ac.uk/10.3390/e24010003 - 21 Dec 2021

Cited by 6 | Viewed by 2463

Abstract

Information flow provides a natural measure for the causal interaction between dynamical events. This study extends our previous rigorous formalism of componentwise information flow to the bulk information flow between two complex subsystems of a large-dimensional parental system. Analytical formulas have been obtained [...] Read more.

Information flow provides a natural measure for the causal interaction between dynamical events. This study extends our previous rigorous formalism of componentwise information flow to the bulk information flow between two complex subsystems of a large-dimensional parental system. Analytical formulas have been obtained in a closed form. Under a Gaussian assumption, their maximum likelihood estimators have also been obtained. These formulas have been validated using different subsystems with preset relations, and they yield causalities just as expected. On the contrary, the commonly used proxies for the characterization of subsystems, such as averages and principal components, generally do not work correctly. This study can help diagnose the emergence of patterns in complex systems and is expected to have applications in many real world problems in different disciplines such as climate science, fluid dynamics, neuroscience, financial economics, etc. Full article

(This article belongs to the Special Issue Information Geometry, Complexity Measures and Data Analysis)

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26 pages, 6259 KiB

Open AccessArticle

Application of Generalized Composite Multiscale Lempel–Ziv Complexity in Identifying Wind Turbine Gearbox Faults

by Xiaoan Yan, Daoming She, Yadong Xu and Minping Jia

Entropy 2021, 23(11), 1372; https://0-doi-org.brum.beds.ac.uk/10.3390/e23111372 - 20 Oct 2021

Cited by 17 | Viewed by 1676

Abstract

Wind turbine gearboxes operate in harsh environments; therefore, the resulting gear vibration signal has characteristics of strong nonlinearity, is non-stationary, and has a low signal-to-noise ratio, which indicates that it is difficult to identify wind turbine gearbox faults effectively by the traditional methods. [...] Read more.

Wind turbine gearboxes operate in harsh environments; therefore, the resulting gear vibration signal has characteristics of strong nonlinearity, is non-stationary, and has a low signal-to-noise ratio, which indicates that it is difficult to identify wind turbine gearbox faults effectively by the traditional methods. To solve this problem, this paper proposes a new fault diagnosis method for wind turbine gearboxes based on generalized composite multiscale Lempel–Ziv complexity (GCMLZC). Within the proposed method, an effective technique named multiscale morphological-hat convolution operator (MHCO) is firstly presented to remove the noise interference information of the original gear vibration signal. Then, the GCMLZC of the filtered signal was calculated to extract gear fault features. Finally, the extracted fault features were input into softmax classifier for automatically identifying different health conditions of wind turbine gearboxes. The effectiveness of the proposed method was validated by the experimental and engineering data analysis. The results of the analysis indicate that the proposed method can identify accurately different gear health conditions. Moreover, the identification accuracy of the proposed method is higher than that of traditional multiscale Lempel–Ziv complexity (MLZC) and several representative multiscale entropies (e.g., multiscale dispersion entropy (MDE), multiscale permutation entropy (MPE) and multiscale sample entropy (MSE)). Full article

(This article belongs to the Special Issue Information Geometry, Complexity Measures and Data Analysis)

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14 pages, 844 KiB

Open AccessArticle

On α-Limit Sets in Lorenz Maps

by Łukasz Cholewa and Piotr Oprocha

Entropy 2021, 23(9), 1153; https://0-doi-org.brum.beds.ac.uk/10.3390/e23091153 - 02 Sep 2021

Cited by 2 | Viewed by 1739

Abstract

The aim of this paper is to show that

α

-limit sets in Lorenz maps do not have to be completely invariant. This highlights unexpected dynamical behavior in these maps, showing gaps existing in the literature. Similar result is obtained for unimodal maps [...] Read more.

The aim of this paper is to show that

α

-limit sets in Lorenz maps do not have to be completely invariant. This highlights unexpected dynamical behavior in these maps, showing gaps existing in the literature. Similar result is obtained for unimodal maps on

[0, 1]

. On the basis of provided examples, we also present how the performed study on the structure of

α

-limit sets is closely connected with the calculation of the topological entropy. Full article

(This article belongs to the Special Issue Information Geometry, Complexity Measures and Data Analysis)

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7 pages, 284 KiB

Open AccessArticle

Decoherence, Anti-Decoherence, and Fisher Information

by Andres M. Kowalski and Angelo Plastino

Entropy 2021, 23(8), 1035; https://0-doi-org.brum.beds.ac.uk/10.3390/e23081035 - 12 Aug 2021

Cited by 4 | Viewed by 1575

Abstract

In this work, we study quantum decoherence as reflected by the dynamics of a system that accounts for the interaction between matter and a given field. The process is described by an important information geometry tool: Fisher’s information measure (FIM). We find that [...] Read more.

In this work, we study quantum decoherence as reflected by the dynamics of a system that accounts for the interaction between matter and a given field. The process is described by an important information geometry tool: Fisher’s information measure (FIM). We find that it appropriately describes this concept, detecting salient details of the quantum–classical changeover (qcc). A good description of the qcc report can thus be obtained; in particular, a clear insight into the role that the uncertainty principle (UP) plays in the pertinent proceedings is presented. Plotting FIM versus a system’s motion invariant related to the UP, one can also visualize how anti-decoherence takes place, as opposed to the decoherence process studied in dozens of papers. In Fisher terms, the qcc can be seen as an order (quantum)–disorder (classical, including chaos) transition. Full article

(This article belongs to the Special Issue Information Geometry, Complexity Measures and Data Analysis)

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Review

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18 pages, 732 KiB

Open AccessReview

Attention Mechanisms and Their Applications to Complex Systems

by Adrián Hernández and José M. Amigó

Entropy 2021, 23(3), 283; https://0-doi-org.brum.beds.ac.uk/10.3390/e23030283 - 26 Feb 2021

Cited by 30 | Viewed by 5991

Abstract

Deep learning models and graphics processing units have completely transformed the field of machine learning. Recurrent neural networks and long short-term memories have been successfully used to model and predict complex systems. However, these classic models do not perform sequential reasoning, a process [...] Read more.

Deep learning models and graphics processing units have completely transformed the field of machine learning. Recurrent neural networks and long short-term memories have been successfully used to model and predict complex systems. However, these classic models do not perform sequential reasoning, a process that guides a task based on perception and memory. In recent years, attention mechanisms have emerged as a promising solution to these problems. In this review, we describe the key aspects of attention mechanisms and some relevant attention techniques and point out why they are a remarkable advance in machine learning. Then, we illustrate some important applications of these techniques in the modeling of complex systems. Full article

(This article belongs to the Special Issue Information Geometry, Complexity Measures and Data Analysis)

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