7 Asia-Pacific Conference on Plasma Physics, 12-17 Nov, 2023 at Port Messe Nagoya
thMeasuring permutation entropy and statistical complexity in plasma
T. Onchi1
RIAM, Kyushu University
e-mail (speaker): onchi@triam.kyushu-u.ac.jp
1

As a method based on probability information theory, theanalysis involving permutation entropy and statisticalcomplexity is becoming widely used in scientificresearches [1]. These parameters are useful to categorize
time-series data. Complexity is a measure of theinterplay between the information stored in the systemand its disequilibrium. To evaluate probabilitydistributions, the permutation entropy is applied to timeseries. Plenty of research have gone into developing thismeasure, and those have presented distinction between
stochasticity and chaos using scatter plots of the entropy
versus complexity, often referred to as the complexity–
entropy causality plane (or C–H plane). Recently, theanalytical method has spread to plasma physics
researches [2, 3].
Considering a time-series data as a set of tuples 𝑥" ∈{𝑥% , … , 𝑥( } , each tuple is sorted into a set ofpermutations 𝑦" ∈ {𝑦% , … , 𝑦( } . 𝑦" is one of thepermutations of 𝑥" . There are K permutations, where K
is a factorial of the embedding dimension, d!. To
determine the probability associated with each 𝑦" , thefrequency of each permutation is counted in a time
window. The probability for 𝑦" is given as 𝑝, =⋕ /𝑦, 0⁄𝑁 denoting the frequency occurrence of 𝑦, in N
tuples. Therefore, the probability distribution ispresented as 𝑝, ∈ 𝑃 = {𝑝% , … , 𝑝4 } so that Shannon’s
information entropy is given as 𝑆 = − ∑4,:% 𝑝, ln 𝑝, . The
normalized entropy can be written as 𝐻< = 𝑆(𝑃)⁄𝑆?@A ,which is called permutation entropy. Here, 𝑆?@A =𝑆[𝑃C ] = ln 𝐾 with 𝑃C = {1⁄𝐾 , 1⁄𝐾 , … , 1⁄𝐾} . Then,Jensen-Shannon statistical complexity is given as
𝐶H< (𝑃) = 𝑄H [𝑃, 𝑃C ]𝐻< (𝑃) , where 𝑄H [𝑃, 𝑃C ] is thedisequilibrium between the probabilities.
The analysis with ( 𝐻< , 𝐶H< ) was applied to thetime-series in the reversed field pinch (RFP) plasma [4].
The data was obtained in an RFP device, RELAX. The
typical discharge duration is about 2-3 ms. Fig.1 presentsa C-H plane scatter diagram for soft X-ray (SXR)
emissions and peripheral magnetic fluctuations. Theseparameters were measured in the conditions with highand low electron density. The number of data points toobtain a probability distribution is 151, and d = 4 isselected. Comparing SXR emission features, 𝐻<JK and𝐶<JK of the high-density regime are clearly lower thanthose of the low-density regime. This indicates that 𝐻<JKand 𝐶<JK have some dependences on the parameters ofthe plasma like electron density. Magnetic fluctuation inthese two regimes also behave differently. Fluctuations at
high-density condition approach the central area of theC–H plane but those at low density remain on thefractional Brownian motion curve around 𝐻<JK ≈ 0.9.
The statistical complexities of the SXR emissions andthe magnetic fluctuations also depend on the relationshipbetween reversal and pinch parameter in the RFPexperiment. Similar analyses can be applied totime-series data in plasma experiments or even innumerical simulation data.
In addition to the analysis of time-series data, twodimensional images are also characterized by 𝐻< and𝐶H< . Considering an image as a set of two-dimensional
data array, permutations can be formed for tuples
consisting of, for example, 2 × 3 pixels. This analysis
gives quantitative measures of complexity of the images.
As a test of this method, numerically generated images
are analyzed.
Plasma provides structures with complexity throughmultiple physical phenomena. Even if the mechanism isunknown, two-dimensional structures in plasma can beevaluated by the analysis with 𝐻< and 𝐶H< .
References[1] O.A. Rosso et al., Phys. Rev. Lett. 99 154102 (2007)[2] W. Gekelman et al., Plasma Phys. Control. Fusion 55064002 (2014)[3] J. E. Maggs and G.J. Morales, Plasma Phys. Control.
Fusion 55 850015 (2013)[4] T. Onchi et al., Phys. Scr. 92 055601 (2017)

Figure 1. Scatter plots in the C-H plane for SXRemissions and peripheral magnetic fluctuations [4].