A Detailed Analysis of Geodesic Least Squares Regression and Its Application to Edge-Localized Modes in Fusion Plasmas

Transcription

1 A Detailed Analysis of Geodesic Least Squares Regression and Its Application to Edge-Localized Modes in Fusion Plasmas Geert Verdoolaege1,2, Aqsa Shabbir1,3 and JET Contributors EUROfusion Consortium, JET, Culham Science Centre, Abingdon, OX14 3DB, UK 1 Department of Applied Physics, Ghent University, Ghent, Belgium 2 Laboratory for Plasma Physics, Royal Military Academy (LPP ERM/KMS), 3 Brussels, Belgium Max Planck Institute for Plasma Physics, D Garching, Germany See the Appendix of F. Romanelli et al., Proceedings of the 25th IAEA Fusion Energy Conference 2014, Saint Petersburg, Russia MaxEnt 2016,

2 Overview 1 Motivation 2 Gaussian information geometry 3 Geodesic least squares regression (GLS) 4 Regression for edge-localized modes 2

3 Overview 1 Motivation 2 Gaussian information geometry 3 Geodesic least squares regression (GLS) 4 Regression for edge-localized modes 3

4 Challenges in regression analysis Data uncertainty: measurement error, fluctuations,... Model uncertainty: missing variables, linear vs. nonlinear, Gaussian vs. non-gaussian,... Heterogeneous data and error bars Uncertainty on response (y) and predictor (x j ) variables Atypical observations (outliers) Near-collinearity of predictor variables Data transformations, e.g. ln(y) = ln(β 0 ) + β 1 ln(x 1 ) + β 2 ln(x 2 ) β p ln(x p ) 4

5 Least squares and maximum a posteriori Workhorse: ordinary least squares (OLS) Need flexible and robust regression (And preferably simple) Parameter estimation distance minimization: Expected measured: Ordinary least squares Michigan circa 1890s. Detroit Publishing Company. Maximum likelihood (ML) / maximum a posteriori (MAP): { 1 exp 1 [y f (x, θ)] 2 } 2πσ 2 σ 2 5

6 The minimum distance approach Minimum distance estimation (Wolfowitz, 1952): Which distribution does the model predict? vs. Which distribution do you observe? Gaussian case: different means and standard deviations Kullback-Leibler divergence, Hellinger divergence (Beran, 1977),... Observed distribution: kernel density estimate 6

7 Number distribution Model observations by distributions More flexibility, more information 7

8 Overview 1 Motivation 2 Gaussian information geometry 3 Geodesic least squares regression (GLS) 4 Regression for edge-localized modes 8

9 Information geometry Family of probability distributions differentiable manifold Parameters = coordinates Metric tensor: Fisher information matrix Parametric probability model: p (x θ) = [ 2 ] g µν (θ) = E θ µ ln p (x θ), µ, ν = 1,..., m θν θ = m-dimensional parameter vector Line element: Minimum-length curve: geodesic Rao geodesic distance (GD) ds 2 = g µν dθ µ dθ ν 9

10 Information geometry scheme 10

11 The Gaussian manifold PDF: p(x µ, σ) = 1 ] (x µ)2 exp [ 2πσ 2σ 2 Line element: ds 2 = dµ2 σ 2 + 2dσ2 σ 2 Hyperbolic geometry: Poincaré half-plane model, pseudosphere,... Analytic geodesic distance 11

12 The pseudosphere (tractroid) Original Compressed 12

13 Geodesic intuition 13

14 Centroid of a cluster Cluster of Gaussian distributions 20 points per distribution 14

15 Overview 1 Motivation 2 Gaussian information geometry 3 Geodesic least squares regression (GLS) 4 Regression for edge-localized modes 15

16 GD minimization (1) 1 ( 2π σy 2 + ) exp p j=1 β 2 j σ x,j [ ( y β 0 + p )] 2 j=1 β j x ij σy 2 + p j=1 β 2 j σ 2 x,j Rao geodesic distance (GD) 1 2π σobs [ exp 1 2 (y y i ) 2 ] σ obs 2 16

17 GD minimization (2) Minimize GD between modeled (p mod ) and observed (p obs ) distributions To be estimated: σ obs, β 0, β 1,..., β p iid data: minimize sum of squared GDs = geodesic least squares (GLS) regression Regression on probabilistic manifold Gaussian: limit equal standard deviation Mahalanobis distance 17

18 Numerical experiment: L-H power threshold Input power threshold for transition to high confinement (L H) Log-linear model: P thr = β 0 n e βn B β B t S β S = ln P thr ln β 0 + β n ln n e + β B ln B t + β S ln S P thr : L-H power threshold (MW) n e: central line-averaged electron density (10 20 m 3 ) B t: toroidal magnetic field (T ) S: plasma surface area (m 2 ) ITPA Power Threshold Database: 2002 version (J. Snipes et al., IAEA FEC 2002, CT/P-04) Data + error bars from 7 tokamaks: > 600 entries 18

19 Synthetic regression models p mod N (µ mod, σ 2 mod ): β 0 : 1, 1.1,..., 20 β 1, β 2, β 3 : 0.1, 0.2,..., 2 Percentage errors: µ mod = ln β 0 + β n ln n e + β B ln B t + β S ln S σ 2 mod = β 2 nσ 2 ln n e + β 2 B σ2 ln B t + β 2 S σ2 ln S P thr : 15% B t : 5% n e : 20% S : 15% 10 trials per parameter set 19

20 Experimental results Percentage error on parameter estimates 20

21 Overview 1 Motivation 2 Gaussian information geometry 3 Geodesic least squares regression (GLS) 4 Regression for edge-localized modes 21

22 Edge-localized modes (ELMs) Repetitive instabilities in plasma edge Magnetohydrodynamic origin MAST, Culham Centre for Fusion Energy, UK 22

23 Analogy 1: Solar flares 23

24 Analogy 2: Cooking pot 24

25 Importance of ELMs Confinement loss Potential damaging effects Impurity outflux ELM control/mitigation Energy (frequency) 1 25

26 Data extraction: waiting times 32 recent JET discharges Waiting time: time before ELM burst 26

27 Data extraction: energies Energy carried from the plasma by an ELM 27

28 Average waiting times and energies 28

29 Error bars on averages Standard deviation / n error bars 29

30 Regression on averages E ELM = β 0 + β 1 t ELM, σ E,obs µ E,obs 30

31 Regression results on pseudosphere 31

32 Projected regression results Multidimensional scaling: 32

33 Correlations in individual plasmas A. Shabbir et al., EPS

34 Individual waiting times and energies 34

35 Error bars for individual ELMs Standard deviation error bars 35

36 Regression on individual measurements E ELM = β 0 + β 1 t ELM 36

37 Average vs. collective trend Average Method β 0 (MJ) β 1 (MJ/s) OLS GLS Individual Method β 0 (MJ) β 1 (MJ/s) OLS GLS

38 Conclusions and future work Geodesic least squares regression: flexible and robust Simple and fast Consistent results Geometrical intuition Applicable to general regression analyses 38