scholarly journals Entropy Analysis of Soccer Dynamics

Entropy ◽  
2019 ◽  
Vol 21 (2) ◽  
pp. 187 ◽  
Author(s):  
António Lopes ◽  
J. Tenreiro Machado
Keyword(s):  
Complex System ◽  
Mutual Information ◽  
Fractional Calculus ◽  
Multidimensional Scaling ◽  
Discrete Time ◽  
Entropy Analysis ◽  
Relative Positioning ◽  
Direct Interpretation ◽  
Jensen Shannon Divergence

This paper adopts the information and fractional calculus tools for studying the dynamics of a national soccer league. A soccer league season is treated as a complex system (CS) with a state observable at discrete time instants, that is, at the time of rounds. The CS state, consisting of the goals scored by the teams, is processed by means of different tools, namely entropy, mutual information and Jensen–Shannon divergence. The CS behavior is visualized in 3-D maps generated by multidimensional scaling. The points on the maps represent rounds and their relative positioning allows for a direct interpretation of the results.

scholarly journals Ranking the scientific output of researchers in fractional calculus

2019 ◽  
Vol 22 (1) ◽  
pp. 11-26 ◽  
Author(s):  
J. A. Tenreiro Machado ◽  
António M. Lopes
Keyword(s):  
Probability Distribution ◽  
Fractional Calculus ◽  
Multidimensional Scaling ◽  
Gamma Distribution ◽  
Scientific Output ◽  
Second Phase ◽  
Scaling Technique ◽  
3 Dimensional ◽  
Distribution Parameters ◽  
Canberra Distance

Abstract This paper analyses the citation profiles (CP) of 130 researchers in fractional calculus. In a first phase, the Canberra distance is used to measure the similarities between the researchers’ CP, and the multidimensional scaling technique (MDS) is adopted for processing and visualizing the information. In a second phase, the gamma probability distribution is used to fit the normalized CP and the gamma parameters are used to characterize the researchers. The MDS results and the gamma distribution parameters are represented graphically in 2- and 3-dimensional locus depicting the relative positions of the researchers.

Discrete Fractional Diffusion Equation of Chaotic Order

2016 ◽  
Vol 26 (01) ◽  
pp. 1650013 ◽  
Author(s):  
Guo-Cheng Wu ◽  
Dumitru Baleanu ◽  
He-Ping Xie ◽  
Sheng-Da Zeng
Keyword(s):  
Porous Media ◽  
Fractional Calculus ◽  
Diffusion Equation ◽  
Discrete Time ◽  
Chaotic Map ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Variable Order ◽  
Diffusion Modeling ◽  
Discrete Fractional Calculus

Discrete fractional calculus is suggested in diffusion modeling in porous media. A variable-order fractional diffusion equation is proposed on discrete time scales. A function of the variable order is constructed by a chaotic map. The model shows some new random behaviors in comparison with other variable-order cases.

scholarly journals Fractional Dynamics in Soccer Leagues

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 356 ◽  
Author(s):  
António M. Lopes ◽  
Jose A. Tenreiro Machado
Keyword(s):  
Mathematical Modeling ◽  
Fractional Calculus ◽  
Real World ◽  
Complex Dynamics ◽  
Statistical Information ◽  
Fractional Dynamics ◽  
Entropy Analysis ◽  
Real World Data ◽  
Proposed Model ◽  
Soccer Teams

This paper addresses the dynamics of four European soccer teams over the season 2018–2019. The modeling perspective adopts the concepts of fractional calculus and power law. The proposed model embeds implicitly details such as the behavior of players and coaches, strategical and tactical maneuvers during the matches, errors of referees and a multitude of other effects. The scale of observation focuses the teams’ behavior at each round. Two approaches are considered, namely the evaluation of the team progress along the league by a variety of heuristic models fitting real-world data, and the analysis of statistical information by means of entropy. The best models are also adopted for predicting the future results and their performance compared with the real outcome. The computational and mathematical modeling lead to results that are analyzed and interpreted in the light of fractional dynamics. The emergence of patterns both with the heuristic modeling and the entropy analysis highlight similarities in different national leagues and point towards some underlying complex dynamics.

A new look into the discrete-time fractional calculus: transform and linear systems

2013 ◽  
Vol 46 (1) ◽  
pp. 635-640 ◽  
Author(s):  
Manuel D. Ortigueira ◽  
Fernando J.V. Coito ◽  
Juan J. Trujillo
Keyword(s):  
Fractional Calculus ◽  
Linear Systems ◽  
Discrete Time ◽  
New Look

scholarly journals A Review of Sample and Hold Systems and Design of a New Fractional Algorithm

2020 ◽  
Vol 10 (20) ◽  
pp. 7360
Author(s):  
Manuel Duarte Ortigueira ◽  
José Tenreiro Machado
Keyword(s):  
Fractional Calculus ◽  
Multidimensional Scaling ◽  
Digital Systems ◽  
Superior Performance ◽  
The Novel ◽  
Analog To Digital ◽  
Exponential Order ◽  
First Order ◽  
Sample And Hold ◽  
Novel Strategy

Digital systems require sample and hold (S&H) systems to perform the conversion from analog to digital and vice versa. Besides the standard zero and first order holds, we find in the literature other versions, namely the fractional and exponential order holds, involving parameters that can be tuned to produce a superior performance. This paper reviews the fundamental concepts associated with the S&H and proposes a new fractional version. The systems are modeled both in the time and Laplace domains. The new S&H stemming from fractional calculus generalizes these devices. The different S&H systems are compared in the frequency domain and their relationships visualized by means of hierarchical clustering and multidimensional scaling representations. The novel strategy allows a better understanding of the possibilities and limitations of S&H systems.

Nonextensive Diffusion Entropy Analysis and Teen Birth Phenomena

2004 ◽  
Author(s):  
N. Scafetta ◽  
P. Grigolin
Keyword(s):  
Complex System ◽  
Time Series ◽  
Entropy Analysis ◽  
Fractal Structures ◽  
Mathematical Methods ◽  
Teen Birth ◽  
Working Definition ◽  
Diffusion Entropy ◽  
Wavelet Multiresolution Analysis ◽  
Definition Of

A complex process is often a balance between nonscaling and scaling components. We show how the nonextensive Tsallis g-entropy indicator may be interpreted as a measure of the nonscaling condition in time series. This is done by applying the nonextensive entropy formalism to the diffusion entropy analysis (DEA). We apply the analysis to the study of the teen birth phenomenon. We find that the number of unmarried teen births is strongly influenced by social processes that induce an anomalous memory in the data. This memory is related to the strength of the nonscaling component of the signal and is more intense than that in the married teen birth time series. By using a wavelet multiresolution analysis, we attempt to provide a social interpretation of this effect…. One of the most exciting and rapidly developing areas of modern research is the quantitative study of "complexity." Complexity has special interdisciplinary impacts in the fields of physics, mathematics, information science, biology, sociology, and medicine. No definition of a complex system has been universally embraced, so here we adopt the working definition, "an arrangement of parts so intricate as to be hard to understand or deal with." Therefore, the main goal of the science of complexity is to develop mathematical methods in order to discriminate among the fundamental microscopic and macroscopic constituents of a complex system and to describe their interrelations in a concise way. Experiments usually yield results in the form of time series for physical observables. Typically, these time series contain both a slow regular variation, usually called a "signal," and a rapid erratic fluctuation, usually called "noise." Historically, the techniques applied to processing such time series have been based on equilibrium statistical mechanics and, therefore, they are not applicable to phenomena far from equilibrium. Among the fluctuating phenomena, a particularly important place is occupied by those phenomena characterized by some type of self-similar or scaling-fractal structures [4]. In this chapter we show that the nonextensive Tsallis g-entropy indicator may be interpreted as a measure of the strength of the nonscaling component of a time series.

The Robustness and Mutual Information Entropy of Random Modular Boolean Networks

2014 ◽  
Vol 989-994 ◽  
pp. 4417-4420 ◽  
Author(s):  
Nan Zhao ◽  
Bing Hui Guo ◽  
Fan Chao Meng
Keyword(s):  
Mutual Information ◽  
Regulatory Networks ◽  
Boolean Networks ◽  
Genetic Regulatory Networks ◽  
Information Propagation ◽  
Entropy Analysis ◽  
Random Boolean Networks ◽  
Basic Model ◽  
Different Types

Random Boolean networks have been proposed as a basic model of genetic regulatory networks for more than four decades. Attractors have been considered as the best way to represent the long-term behaviors of random Boolean networks. Most studies on attractors are made with random topologies. However, the real regulatory networks have been found to be modular or more complex topologies. In this work, we extend classical robustness and entropy analysis of random Boolean networks to random modular Boolean networks. We firstly focus on the robustness of the attractor to perturbations with different parameters. Then, we investigate and calculate how the amount of information propagated between the nodes when on an attractor, as quantified by the average pairwise mutual information. The results can be used to study the capability of genetic information propagation of different types of genetic regulatory networks.

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