# Consistent Quantification of Complex Dynamics via a Novel Statistical Complexity Measure

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## Abstract

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## 1. Introduction

## 2. Methods

#### 2.1. Complexity Change upon Perturbation

#### 2.2. Datasets

`histdata.com`for the currency pairs GBP-EUR, GBP-USD, GBP-CHF, and GBP-JYP. We averaged bit and ask prices to obtain one time series.

## 3. Results

#### 3.1. Synthetic Data Set for Coupled Logistic Maps

#### 3.2. Complexity within HIVP

#### 3.3. Foreign Exchange Rates under Perturbation

`histdata.com`and thus have market data roughly from half a year before and half a year after the referendum (the Brexit referendum was held on 23 June 2016). We binned the rates, following standard procedure, into basis points (bps).

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Coupled map: (

**A**) complexity and correlation progression; (

**B**) (Logarithmic) heatmap of the contingency tables/two-dimensional histograms $p(X,Y)$ for varying $\xi $. For these $\xi $-values, $\Delta C(X||Y)$ reached its (local) maxima/minimum.

**Figure 2.**$\Delta C$ values for the HIVP positions. In orange, peaks with $\Delta C\ge 0.1\xb7max\left(\Delta C\right)$ are highlighted, whereas peaks with $\Delta C\le -0.1\xb7max\left(\Delta C\right)$ are shown in blue. Almost all peak positions have been reported to be influenced by protease inhibitors.

**Figure 3.**Histograms of $\Delta C$ for the currency pairs under investigation obtained for resampled data.

**Table 1.**Summary of positions mutated by the four most prominent protease inhibitors in the HIVP dataset.

**Bold**positions show a $\left|\Delta C\right|\ge 0.1\xb7max\Delta C$. Underlined positions represent “major” HIVP mutations which are detected to mutate first in presence of a drug.

Drug | Number of Sequences | Affected Positions |
---|---|---|

Indinavir | 3753 | 10, 20, 24, 32, 36,
46, 54, 71, 73, 76,
82, 84, 89 |

Lopinavir | 955 | 10, 20, 24, 32,
33, 46, 47, 50, 53, 54, 63,
71, 73, 76, 82, 84,
90 |

Nelfinavir | 3178 | 10, 30, 36, 46,
71, 77, 82, 84, 88,
90 |

Saquinavir | 2526 | 10, 24, 48, 54,
62, 71, 73, 77, 82, 84,
90 |

**Table 2.**Our complexity measure for various exchange rate distributions. Here, ${p}_{\mathrm{a}}$, ${p}_{\mathrm{b}}$, and ${p}_{\mathrm{u}}$ are the distributions of the exchange rates before and after the Brexit referendum and the uniform distribution, respectively. Clearly, $\Delta C$ is always negative with respect to the uniform distribution ${p}_{\mathrm{u}}$, as the entropy of ${p}_{\mathrm{u}}$ is maximal; thus, $\Delta C$ can only decrease. Note, however, the amount of decrease differs widely. To assess the significance, we performed a permutation test and calculated the Z-score for $\Delta C\left({p}_{\mathrm{a}}\left|\right|{p}_{\mathrm{b}}\right)$ (see main text for details).

Currency Pair | $\mathbf{\Delta}\mathit{C}\left({\mathit{p}}_{\mathbf{a}}\left|\right|{\mathit{p}}_{\mathbf{b}}\right)$ | $\mathbf{\Delta}\mathit{C}\left({\mathit{p}}_{\mathbf{b}}\left|\right|{\mathit{p}}_{\mathbf{u}}\right)$ | $\mathbf{\Delta}\mathit{C}\left({\mathit{p}}_{\mathbf{a}}\left|\right|{\mathit{p}}_{\mathbf{u}}\right)$ | Z |
---|---|---|---|---|

GBP-EUR | 0.207 | −0.455 | −0.60 | $252\xb7{10}^{3}$ |

GBP-USD | 0.449 | −0.387 | −0.821 | $468\xb7{10}^{3}$ |

GBP-CHF | −0.0247 | −0.493 | −0.482 | $-23\xb7{10}^{3}$ |

GBP-JPY | −0.115 | −0.503 | −0.398 | $-12\xb7{10}^{3}$ |

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**MDPI and ACS Style**

Keul, F.; Hamacher, K. Consistent Quantification of Complex Dynamics via a Novel Statistical Complexity Measure. *Entropy* **2022**, *24*, 505.
https://doi.org/10.3390/e24040505

**AMA Style**

Keul F, Hamacher K. Consistent Quantification of Complex Dynamics via a Novel Statistical Complexity Measure. *Entropy*. 2022; 24(4):505.
https://doi.org/10.3390/e24040505

**Chicago/Turabian Style**

Keul, Frank, and Kay Hamacher. 2022. "Consistent Quantification of Complex Dynamics via a Novel Statistical Complexity Measure" *Entropy* 24, no. 4: 505.
https://doi.org/10.3390/e24040505