# Dynamical Complexity of the 2015 St. Patrick’s Day Magnetic Storm at Swarm Altitudes Using Entropy Measures

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## Abstract

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## 1. Introduction

## 2. Methodology

#### 2.1. Shannon Entropy

#### 2.2. Symbolic Dynamics and Block Entropy

#### 2.3. Alternative Entropy Formulations

#### 2.4. Approximate Entropy

#### 2.5. Sample Entropy

#### 2.6. Fuzzy Entropy

- The uncertainty of an open system state can be quantified by the Boltzmann-Gibbs (B-G) entropy, which is the widest known uncertainty measure in statistical mechanics. B-G entropy cannot, however, describe nonequilibrium physical systems characterized by long-range interactions or long-term memory or being of a multi-fractal nature. Inspired by multi-fractal concepts, Tsallis [38,39] has proposed a generalization of the B-G statistics, that is, the Tsallis Entropy, ${S}_{q}$.
- Approximate entropy ($ApEn$) has been introduced by Pincus as a measure for characterizing the regularity in relatively short and potentially noisy data. More specifically, $ApEn$ examines time series for detecting the presence of similar epochs; more similar and more frequent epochs lead to lower values of $ApEn$.
- Sample entropy ($SampEn$) was proposed by Richman and Moorman as an alternative that would provide an improvement of the intrinsic bias of $ApEn$.
- Fuzzy entropy ($FuzzyEn$), like its ancestors, $ApEn$ and $SampEn$, is a “regularity statistic” that quantifies the (un)predictability of fluctuations in a time series. For the calculation of $FuzzyEn$, the similarity between vectors is defined based on fuzzy membership functions and the vectors’ shapes. $FuzzyEn$ can be considered as an upgraded alternative of $SampEn$ (and $ApEn$) for the evaluation of complexity, especially for short time series contaminated by noise.

## 3. Data and Analysis

#### 3.1. Swarm Magnetic Field Data

#### 3.2. Entropy Analysis of the Swarm B Magnetic Field Data for the St. Patrick’s 2015 Storm

## 4. Conclusions and Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Comparison between the Dst index (

**top panel**) and the pre-processed series of the total (external) magnetic field (

**bottom panel**), as measured by Swarm B, during the March 2015 storm.

**Figure 2.**Example of three segments before (

**left**) and after filtering (

**right**), for the pre-storm phase (

**top row**), the peak of the storm (

**middle row**) and after the end of event (

**bottom row**), respectively.

**Figure 3.**Entropy analysis according to Shannon formalism of the Swarm B total (external) field for the March 2015 magnetic storm.

**Figure 4.**Entropy analysis according to Tsallis formalism of the Swarm B total (external) field for the March 2015 magnetic storm.

**Figure 5.**Approximate, Sample and Fuzzy entropy analysis of the Swarm B total (external) magnetic field for the March 2015 magnetic storm.

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**MDPI and ACS Style**

Papadimitriou, C.; Balasis, G.; Boutsi, A.Z.; Daglis, I.A.; Giannakis, O.; Anastasiadis, A.; Michelis, P.D.; Consolini, G. Dynamical Complexity of the 2015 St. Patrick’s Day Magnetic Storm at Swarm Altitudes Using Entropy Measures. *Entropy* **2020**, *22*, 574.
https://doi.org/10.3390/e22050574

**AMA Style**

Papadimitriou C, Balasis G, Boutsi AZ, Daglis IA, Giannakis O, Anastasiadis A, Michelis PD, Consolini G. Dynamical Complexity of the 2015 St. Patrick’s Day Magnetic Storm at Swarm Altitudes Using Entropy Measures. *Entropy*. 2020; 22(5):574.
https://doi.org/10.3390/e22050574

**Chicago/Turabian Style**

Papadimitriou, Constantinos, Georgios Balasis, Adamantia Zoe Boutsi, Ioannis A. Daglis, Omiros Giannakis, Anastasios Anastasiadis, Paola De Michelis, and Giuseppe Consolini. 2020. "Dynamical Complexity of the 2015 St. Patrick’s Day Magnetic Storm at Swarm Altitudes Using Entropy Measures" *Entropy* 22, no. 5: 574.
https://doi.org/10.3390/e22050574