7th Asia-Pacific Conference on Plasma Physics, 12-17 Nov, 2023 at Port Messe Nagoya

An intrinsically probabilistic approach to analyzing stochasticity
and uncertainty in fusion plasmas using information geometry
G. Verdoolaege
Department of Applied Physics, Ghent University
e-mail (speaker): geert.verdoolaege@ugent.be
Various phenomena and processes in plasmas are of astrongly stochastic nature or are affected by considerable
uncertainty. Some examples of stochastic phenomena areplasma instabilities and turbulent fluctuations, whilemeasurement of key plasma parameters occurs withinstruments that can be quite complex and that may
introduce considerable uncertainty on the estimatedquantities of interest. In such cases, a probabilistic
description of the corresponding data often provides
significant added value. In this work, we leverage thepowerful methods of information geometry (IG).
In IG, a parametric family of probability densities isinterpreted as a Riemannian differentiable manifold [1].
Each point on the manifold corresponds to a specificprobability density function (PDF) within the family, andthe family parameters represent a coordinate system onthe manifold. The Fisher information (covariance of thescore) provides a unique metric tensor. This allows amathematically well-founded distance measure between
PDFs: the Rao geodesic distance (GD). In turn, similarity
measurement between PDFs in fusion science is essential
for quantifying changes in stochastic processes ormeasurement characteristics under varying plasma
conditions. Furthermore, similarity forms the basis ofmany pattern recognition methods, such as classification
and regression.
In this work, we first illustrate our methods by a studyof waiting time distributions of a class of MHD
instabilities occurring in high-confinement (H-mode)
tokamak plasmas, called edge-localized modes (ELMs).
The variability of the distribution shape across ELM
types is highlighted and modeled with Gaussian andWeibull distributions [2]. Next, a nearest neighboralgorithm based on the GD between the PDF models isshown to successfully identify ELM types [3].
(a)

(b)

(c)

We then proceed to regression analysis based on theRao GD. This has led to geodesic least squares (GLS) asa new regression technique operating in spaces ofprobability distributions [4]. Owing to its solid
foundations rooted in IG, GLS is robust against outliers
and model misspecification, yet it is easily implemented
without expert statistics knowledge. We demonstrate therobustness properties of GLS using regression of ELM
size (energy drop) vs. waiting time [5]. Additional
interpretation is gained from visualizations using several
models for the manifold of Gaussian probabilitydistributions. We then concentrate on the application ofGLS in a recent update of the key scaling law for theglobal energy confinement time 𝜏E,th in H-mode
tokamak plasmas (Figure 1) [6]. GLS is shown tocompare favorably to other robust regression methods,
including minimum distance estimation techniques basedon alternative similarity measures.
With the applications showed in this work, we aim todemonstrate the suitability of the powerful framework ofIG for quantifying stochasticity in the complex andchallenging environment posed by fusion plasmas.
References[1] S.-I. Amari and H. Nagaoka, “Methods ofInformation Geometry”, AMS, New York, NY (2000)[2] G. Verdoolaege et al, Proc. 45th EPS Conference on
Plasma Physics, P2.1078 (2018)[3] A. Shabbir et al, Rev. Sci. Instrum. 87, 11D404(2016)[4] G. Verdoolaege, Entropy, 17, 4602 (2015)[5] G. Verdoolaege et al, AIP Conf. Proc. 1853, 100002(2016)[6] G. Verdoolaege et al, Nucl. Fusion 61, 076006 (2021)

Figure 1. Visualization ofregression results obtained withGLS from the ITPA20-IL global
H-mode confinement scaling [6],showing(a)confinement
enhancement factor H20 vs.
Greenwald fraction 𝑓GW (notethe outliers from PBX-M), (b) aprojection of the data indicatingthe gap in power 𝑃l,th andmachine size 𝑅 , and (c) thepseudosphere model of the
manifold of Gaussian distributions, where the generally higher
standard deviation 𝜎 of the observed distributions, compared tothe measured 𝜏E,th is a sign of additional sources of uncertainty.