# Detection of Causal Relations in Time Series Affected by Noise in Tokamaks Using Geodesic Distance on Gaussian Manifolds

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## Abstract

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## 1. Time Series and Causal Relations in Magnetic Confinement Nuclear Fusion

## 2. Recurrence Plots

**x**

_{i}even if the dimensionality of the phase space is much higher than two [8]. This is achieved by showing the times at which phase space trajectories visit approximately the same region in phase space. This visualization approach is called Recurrence Plot (RP) and is mathematically expressed by a two-dimensional squared matrix, the so called recurrence matrix:

_{i}, ε is a threshold distance, ||·|| a norm and Θ(·) is the Heaviside function. Each element indicates the recurrence of a state at time i at a different time j, equal to one if the distance between the two states is lower than ε, and equal to zero otherwise. A graphical representation of a RP is obtained by plotting the matrix in a square map, whose axes report time and black dots indicate ones and white dots zeroes.

## 3. Complex Networks

## 4. Information Geometry: Geodesic Distance on Gaussian Manifolds

_{1}Q

_{1}(pink and red in Figure 5) and P

_{2}Q

_{2}(blue and black in Figure 5). The distance between the means of the members of the two couples is the same. On the other hand, the Gaussian pdfs P

_{1}Q

_{1}have a standard deviation about an order of magnitude smaller than the other couple. The distance between the pdfs with higher standard deviation, P

_{2}Q

_{2}, is therefore significantly lower than the one of the more concentrated pdfs. Inspection of Figure 5 reveals the meaning of this fact, since there is a much larger overlap between the P

_{2}Q

_{2}pdfs and therefore it is appropriate to consider them closer than the two of the other couple.

## 5. Application of the Geodesic Distance to Synchronization Experiments in JET

## 6. Discussion, Conclusions and Future Perspectives

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Top: Dα signals identifying the occurrence of Edge Localised Modes (ELMs). Bottom: Dα signal indicating the arrival of the pellets into the plasma.

**Figure 2.**An example of sawteeth pacing with Ion Cyclotron Radiofrequency Heating (ICRH) modulation in a JET-ILW (ITER-like wall) L mode discharge (shot number 89826). The frequency of the modulation is 5 Hz (150 ms on and 50 ms off). The maximum power is 4 MW in a minority heating scheme with 4% of H in D. From top to bottom: plasma internal energy, ICRH power and central electron temperature.

**Figure 3.**Two examples of recurrence plots. (

**Left**) discharge 89,995 devoted to sawteeth pacing. The input signals to obtain the plot are the ICRH power and the central electron temperature; (

**Right**) discharge 82,439 devoted to ELM pacing with pellets. The signals used to obtain the plot are the Dα of ELMs and pellets.

**Figure 5.**Examples to illustrate how the Geodesic Distance on Gaussian Manifolds (GDGM) determines the distance between two Gaussians. The two couples of probability distribution function (pdf) in the figure have the same mean but different σ. The geodesic distance between the two with higher σ is much smaller. GD indicates the geodesic distance and EU the Euclidean distance.

**Figure 6.**The time evolution of ENTR (Equation (3)) derived from the Recurrence Plot, using the Euclidean distance for the JET discharge #89826, the experiment of IRCH pacing of sawteeth shown in Figure 2. The fit of the peak in the Entropy is reported in red.

**Figure 7.**Evolution of Recurrence Plot (RP) entropy of diagonal length for JET discharge #82439, an example of ELM pacing with pellets (

**left**) and for the discharge #90005, an example of sawteeth pacing with ICRH modulation (

**right**). The RP has been calculated using both the Euclidean distance (green) and the GDGM (blue). The fit of the peaks is reported in red.

**Figure 8.**Evolution of WCVG entropy for JET discharge #82854, whose raw signals are reported in Figure 1. The adjacency matrix has been calculated using both Euclidean distance (green) and GDGM (blue). The peak fitting is indicated in red. Two different phases of this discharge can be identified visually on the Dα plot of Figure 1; a phase of increased frequency appears in the period between 10.3 and 10.7 s.

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**MDPI and ACS Style**

Murari, A.; Craciunescu, T.; Peluso, E.; Gelfusa, M.; JET Contributors.
Detection of Causal Relations in Time Series Affected by Noise in Tokamaks Using Geodesic Distance on Gaussian Manifolds. *Entropy* **2017**, *19*, 569.
https://doi.org/10.3390/e19100569

**AMA Style**

Murari A, Craciunescu T, Peluso E, Gelfusa M, JET Contributors.
Detection of Causal Relations in Time Series Affected by Noise in Tokamaks Using Geodesic Distance on Gaussian Manifolds. *Entropy*. 2017; 19(10):569.
https://doi.org/10.3390/e19100569

**Chicago/Turabian Style**

Murari, Andrea, Teddy Craciunescu, Emmanuele Peluso, Michela Gelfusa, and JET Contributors.
2017. "Detection of Causal Relations in Time Series Affected by Noise in Tokamaks Using Geodesic Distance on Gaussian Manifolds" *Entropy* 19, no. 10: 569.
https://doi.org/10.3390/e19100569