# Correlation Dimension Detects Causal Links in Coupled Dynamical Systems

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. State Space Reconstruction

#### 2.2. Correlation Dimension Estimation

#### 2.3. Causality Detection Based on Correlation Dimension

#### 2.3.1. $X\to Y$ or $Y\to X$

#### 2.3.2. X and Y Are Independent

#### 2.3.3. Uncoupled X and Y with a Hidden Common Driver

#### 2.3.4. $X\leftrightarrow Y$

## 3. Results

#### 3.1. $X\to Y$

#### 3.2. X and Y Are Independent

#### 3.3. Uncoupled X and Y with a Hidden Common Driver

#### 3.4. $X\leftrightarrow Y$

## 4. Discussion

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Interaction graph for unidirectional coupling of two Hénon systems described by Equation (1) (on the left) and the connections detected by causal analysis in reconstructed state spaces (on the right).

**Figure 2.**Four examples of detectable types of causal relations between time series. Each row contains plots of 30 points of investigated time series x and y, two-dimensional projections of the reconstructed state portraits of systems X and Y, and the plateaus of the correlation exponents used to estimate ${D}_{2}\left(X\right)$, ${D}_{2}\left(Y\right)$, and ${D}_{2}\left(\left[XY\right]\right)$.

**Figure 3.**Estimates of ${D}_{2}\left(X\right)$ (red), ${D}_{2}\left(Y\right)$ (blue) and ${D}_{2}\left(\left[XY\right]\right)$ (green) of state portraits reconstructed from time series ${x}_{1}$ and ${y}_{1}$ generated by Equation (1) for 21 different values of coupling C. The plus signs are for the sums ${D}_{2}\left(X\right)+{D}_{2}\left(Y\right)$.

**Figure 4.**Interaction graph for two independent systems X and Y with a common driver Z described by Equation (2).

**Figure 5.**Estimates of ${D}_{2}\left(X\right)$ (red), ${D}_{2}\left(Y\right)$ (blue) and ${D}_{2}\left(\left[XY\right]\right)$ (green) of state portraits reconstructed from time series ${x}_{1}$ and ${y}_{1}$ generated by Equation (2) for different driving strength C of the hidden common driver. The plus signs are for the sums ${D}_{2}\left(X\right)+{D}_{2}\left(Y\right)$.

**Table 1.**Rules for deriving causal relationships between systems X and Y based on dimensions ${D}_{2}\left(X\right)$, ${D}_{2}\left(Y\right)$ and ${D}_{2}\left(\left[XY\right]\right)$.

Causal Relation | Relations between Correlation Dimensions |
---|---|

$X\to Y$ | ${D}_{2}\left(\left[XY\right]\right)={D}_{2}\left(Y\right)>{D}_{2}\left(X\right)$ |

$Y\to X$ | ${D}_{2}\left(\left[XY\right]\right)={D}_{2}\left(X\right)>{D}_{2}\left(Y\right)$ |

X independent of Y | ${D}_{2}\left(\left[XY\right]\right)={D}_{2}\left(X\right)+{D}_{2}\left(Y\right)$ |

X and Y uncoupled, with a common driver | ${D}_{2}\left(\left[XY\right]\right)<{D}_{2}\left(X\right)+{D}_{2}\left(Y\right)$, ${D}_{2}\left(X\right)<{D}_{2}\left(\left[XY\right]\right)>{D}_{2}\left(Y\right)$ |

$X\leftrightarrow Y$ | ${D}_{2}\left(\left[XY\right]\right)={D}_{2}\left(X\right)={D}_{2}\left(Y\right)$ |

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Krakovská, A.
Correlation Dimension Detects Causal Links in Coupled Dynamical Systems. *Entropy* **2019**, *21*, 818.
https://doi.org/10.3390/e21090818

**AMA Style**

Krakovská A.
Correlation Dimension Detects Causal Links in Coupled Dynamical Systems. *Entropy*. 2019; 21(9):818.
https://doi.org/10.3390/e21090818

**Chicago/Turabian Style**

Krakovská, Anna.
2019. "Correlation Dimension Detects Causal Links in Coupled Dynamical Systems" *Entropy* 21, no. 9: 818.
https://doi.org/10.3390/e21090818