Approximate Entropy and Sample Entropy: A Comprehensive Tutorial

Delgado-Bonal, Alfonso; Marshak, Alexander

doi:10.3390/e21060541

Cited by 443 publications

(302 citation statements)

References 41 publications

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“…These above mentioned studies use the Shannon entropy which is a classical probabilistic approach analyzing the randomness of the generating process of a series and not the randomness of a series itself. In other words, Shannon entropy is a metric to study the process of the data, and the order of the generated data does not have any influence on this metric (Chaitin, 1975; Delgado‐Bonal & Marshak, 2019).…”

Section: Introductionmentioning

confidence: 99%

Entropy and Intermittency of River Bed Elevation Fluctuations

Ranjbar

Singh

2020

JGR Earth Surface

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River beds evolve as a result of a complex interaction between strongly nonlinear processes such as near‐bed turbulence, particle‐particle interaction, and particle‐bed interaction. This interaction contributes to the initiation and evolution of extremely variable river bed elevation patterns, commonly known as bedforms that span across a range of spatiotemporal scales. In this paper, we employ a refined definition of entropy, that is, the multiscale entropy (MSE), to characterize the observed variability in the fluctuations of bed elevation time series under variable discharges obtained from a field‐scale laboratory flume. The MSE accounts for the sequence of data points in a series and quantifies the complexity and lack of information in a system. We show that the MSE of bed elevation fluctuations is higher for higher discharges. When compared with surrogates, which are the linearized series of a signal, the MSE provides across‐scale information about the underlying nonlinearity and linear correlation in a signal. The MSE difference between the original and surrogate series is due to the inherent nonlinearity that is higher for higher discharge at the smaller scales and peaks at intermediate scales. These results indicate the presence of a heterogeneous arrangement of extreme fluctuations that enhances the underlying complexity in bed elevation, rendering them less predictable at higher discharges. We further investigate the role of asymmetry of the bed elevation increments in observed complexity. Our results of asymmetry together with entropy suggest that characteristics of both small‐ and large‐scale features should be included for the accurate predictive modeling of sediment transport.

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Section: Introductionmentioning

confidence: 99%

Entropy and Intermittency of River Bed Elevation Fluctuations

Ranjbar

Singh

2020

JGR Earth Surface

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“…A description of the algorithm can be found in 24 , and a comprehensive step by step tutorial with visual examples is available in 44 . Both Approximate Entropy and Sample Entropy require the selection of two parameters for their calculations, the embedding dimension m (the size of the template being compared) and the noise filter r (points within a distance of r are considered equal).…”

Section: Approximate Entropymentioning

confidence: 99%

Analyzing changes in the complexity of climate in the last four decades using MERRA-2 radiation data

Delgado-Bonal

Marshak

Yang

et al. 2020

Sci Rep

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the energy balance of the earth is controlled by the shortwave and longwave radiation emitted to space. Changes in the thermodynamic state of the system over time affect climate and are noticeable when viewing the system as a whole. In this paper, we study the changes in the complexity of climate in the last four decades using data from the Modern-era Retrospective analysis for Research and Applications, Version 2 (MERRA-2). First, we study the complexity of the shortwave and longwave radiation fields independently using Approximate Entropy and Sample Entropy, observing that the rate of complexity change is faster for shortwave radiation. Then, we study the causality of those changes using Transfer Entropy to capture the non-linear dynamics of climate, showing that the changes are mainly driven by the variations in shortwave radiation. the observed behavior of climatic complexity could be explained by the changes in cloud amount, and we research that possibility by investigating its evolution from a complexity perspective using data from the international Satellite cloud climatology Project (ISCCP). Climate is a complex system 1 composed of many different processes operating at all spatiotemporal scales, showing non-linear relations between its variables in long time spans 2 or its acceleration in the last four decades 3. Understanding the changes in these non-linear relations is of the utmost importance for understanding current climate change and its implications for the future 4,5. Great efforts have been made in the last years to characterize those variations due to its importance for future abrupt changes in the structure of climate, being it potentially harmful to biodiversity and society 6,7. The atmosphere of the Earth can be considered as a closed non-linear thermodynamic system which basically only exchanges radiation with its environment 8. The solar radiation flux maintains the atmosphere in a non-equilibrium thermodynamics situation, generating a climate dynamics which has an influence on the temperature of the planet. The energy balance is modulated by the incoming solar radiation and the outgoing shortwave and longwave Earth's radiation. Changes in the energy balance of the atmosphere are the drivers of climate change. Nowadays, it is believed that the Earth absorbs more energy from the Sun than it is emitted to the space, creating a situation of energy imbalance 9,10. Although those differences can be monitored from the ground, the use of satellite sensors to provide measures of the fluxes of energy to and from Earth offers a more precise view of the system 11. The evolution of the energy imbalance has been investigated using satellite data and models, and it seems to be increasing in the last decades 12,13. While those changes have been argued to be due to human activities 14 , the questions of which of the parameters are changing faster and if that is affecting the dynamics of the system are still open. When reaching the Earth, part of the incoming solar radiation is reflected off clouds and the...

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“…Both Figures 13 and 14 demonstrate numerical investigation of systems (12) and (13) with respect to the entropic properties of the generated signals. The threshold r is the main parameter of the numerical algorithm, which measures and quantify similarity patterns in the data sequence of the increasing length (up to the self-comparison)-see tutorial paper [47] for a better understanding. In this picture, the rainbow color scale for ApEn quantity is utilized, see legend.…”

Section: Wideband Cpe As Part Of Chaotic Systemmentioning

confidence: 99%

Fractional-Order Chaotic Memory with Wideband Constant Phase Elements

Petržela

2020

Entropy

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This paper provides readers with three partial results that are mutually connected. Firstly, the gallery of the so-called constant phase elements (CPE) dedicated for the wideband applications is presented. CPEs are calculated for 9° (decimal orders) and 10° phase steps including ¼, ½, and ¾ orders, which are the most used mathematical orders between zero and one in practice. For each phase shift, all necessary numerical values to design fully passive RC ladder two-terminal circuits are provided. Individual CPEs are easily distinguishable because of a very high accuracy; maximal phase error is less than 1.5° in wide frequency range beginning with 3 Hz and ending with 1 MHz. Secondly, dynamics of ternary memory composed by a series connection of two resonant tunneling diodes is investigated and, consequently, a robust chaotic behavior is discovered and reported. Finally, CPEs are directly used for realization of fractional-order (FO) ternary memory as lumped chaotic oscillator. Existence of structurally stable strange attractors for different orders is proved, both by numerical analyzed and experimental measurement.

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Approximate Entropy and Sample Entropy: A Comprehensive Tutorial

Cited by 443 publications

References 41 publications

Entropy and Intermittency of River Bed Elevation Fluctuations

Entropy and Intermittency of River Bed Elevation Fluctuations

Analyzing changes in the complexity of climate in the last four decades using MERRA-2 radiation data

Fractional-Order Chaotic Memory with Wideband Constant Phase Elements

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