Image-Based Methods to Investigate Synchronization between Time Series Relevant for Plasma Fusion Diagnostics

Abstract

Advanced time series analysis and causality detection techniques have been successfully applied to the assessment of synchronization experiments in tokamaks, such as Edge Localized Modes (ELMs) and sawtooth pacing. Lag synchronization is a typical strategy for fusion plasma instability control by pace-making techniques. The major difficulty, in evaluating the efficiency of the pacing methods, is the coexistence of the causal effects with the periodic or quasi-periodic nature of the plasma instabilities. In the present work, a set of methods based on the image representation of time series, are investigated as tools for evaluating the efficiency of the pace-making techniques. The main options rely on the Gramian Angular Field (GAF), the Markov Transition Field (MTF), previously used for time series classification, and the Chaos Game Representation (CGR), employed for the visualization of large collections of long time series. The paper proposes an original variation of the Markov Transition Matrix, defined for a couple of time series. Additionally, a recently proposed method, based on the mapping of time series as cross-visibility networks and their representation as images, is included in this study. The performances of the method are evaluated on synthetic data and applied to JET measurements.

Keywords: Gramian angular field; Markov transition field; chaos game representation; complex networks; entropy; pacing experiments; sawteeth; tokamaks.

Figure 1

An example of the time series obtained in sawteeth pacing with Ion Cyclotron Radiofrequency Heating (ICRH) modulation in the JET-ILW L mode discharge #89826. The ICRH power time series is presented in the top plot while the central electron temperature, influenced by the sawteeth, is shown in the bottom plot. 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. The sampling frequency in the sawteeth time series is 10⁻⁶ s.

Figure 2

Illustration of the construction of a new time series by the intercalated merge of a pair of time series. The graphs have been obtained using a Rössler coupled system and two different values of the coupling parameter. The driver is the blue curve and the target the green one. In the case of the left plot the coupling is zero, while in the right plot the coupling coefficient is set to C = 0.5.

Figure 3

The evolution of the coupling measures with the coupling strength for the Rössler system: Structural Similarity Index (SSIM) calculated for Gramian Angular Field (GAF) images (top-left), SSIM calculated for Markov Transition Field (MTF) images (top-right), absolute value of the $W_{cross}$ image entropy (middle-left), SSIM calculated for the CGRS images (middle-right), SNSEV entropy for the CGRS graphs (bottom-left) and absolute value of the WAM image entropy (bottom-right). The absolute values of the entropies (SNSEV, $E_{cross}$ and $E_{WAM}$) are normalized to their maximum value.

Figure 4

GAF image of the driver time series $x_2$ (left) and of the driven time series $y_2$ for $C=0$ (middle) and $C=2$ (left).

Figure 5

MTF image of the driver time series $x_2$ (top) and of the driven time series $y_2$ for $C=0,0.65,1.35$ (bottom, from left to right).

Figure 6

The evolution of the $W_{cross}$ images with the coupling parameter. Top row show the images for $C=0$ (left) and $C=0.5$ (right) and the bottom row shows the values for $C=1.0$ (left), $C=1.5$ (middle) and $C=2$ (right).

Figure 7

CGRS graph for the driver time series $x_2$ (left) and for the driven time series $y_2$ for $C=0$ (middle) and C = 2 (right). The graph has been generated using an alphabet with 10 symbols so all the points of the graph are inside the decagon drawn in red.

Figure 8

WAM images for $C=0$ (top-left), $C=0.5$ (top-right) and $C=1$ (bottom).

Figure 9

The time-lagged evolution of the coupling measures for the JET discharge #89826: SSIM calculated for GAF images (top-left), SSIM calculated for MTF images (top-right), absolute value of the CMM $W_{cross}$ image entropy (middle-left), SNSEV entropy for the CGRS graphs (middle-right) and the absolute values of the WAM image entropy (bottom-left). The absolute values of the entropies (SNSEV, $E_{cross}$ and $E_{WAM}$) are normalized to their maximum value. The Gaussian fit of the peaks is reported in red. For the case of SSIM-MTF an alternative fit (reported in blue) has been performed using a 4-th order polynomial and all the points of the plot.

Figure 10

The evolution of the $W_{cross}$ images for different time lag values. The top row shows the images for $t=0\mathrm{ms}$ (left) and $t=0.025$ ms (middle) and $t=0.052$ ms. The bottom row shows the values for $t=0.075$ ms (left), $t=0.100$ ms (middle) and t = 0.120 ms (right). The image for which the absolute value of W_cross entropy reaches its maximum is marked by a red border.