FACULTY OF ENGINEERING AND ARCHITECTURE

Demonstration of the robustness of geodesicleast squares regression for scaling of the

LH power threshold in tokamaks

Geert Verdoolaege1,2, Aqsa Shabbir1,3 and Grégoire Hornung1

1Department of Applied Physics, Ghent University, Ghent, Belgium2Laboratory for Plasma Physics, Royal Military Academy (LPP–ERM/KMS),

Brussels, Belgium3Max-Planck-Institut für Plasmaphysik, D-85748 Garching, Germany

1st IAEA TM on Fusion Data Processing, Validation and AnalysisNice, June 1–3, 2015

Overview

1 Methodology for scaling laws

2 Geodesic least squares regression (GLS)

3 Numerical experiments

4 Power threshold scaling

Overview

1 Methodology for scaling laws

2 Geodesic least squares regression (GLS)

3 Numerical experiments

4 Power threshold scaling

Scaling laws

Scaling laws in fusion science:

Evaluate theoretical predictions

Estimate parametric dependencies

Extrapolate to future devices

Terminology:

Scaling law: scale to larger sizes, magnetic fields, etc.

Often power law: y = b0xb11 xb2

2 . . . xbpp

Regression analysis: probabilistic/statistical framework forestimation with confidence intervals

Scaling law estimation ⊂ regression analysis ⊂ parameterestimation

Applications in biology, geology, astronomy, . . .

A matter of methodology

A matter of methodology

A matter of methodology

Physical system with uncertaintyTheory and modeling→ physicsUncertainty→ probability→ machine learningExamples:

Parameter estimation in physical modelsClassification using physically meaningful quantitiesPhysics-based prior information. . .

Challenges for fusion scaling laws

Large (non-Gaussian?) stochastic uncertainties (noise)Systematic measurement uncertaintiesUncertainty on response (y ) and predictor (xj ) variablesUncertainty on regression model (nonlinear?)Near-collinearity of predictor variablesAtypical observations (outliers)Heterogeneous data and error barsLogarithmic transformation in power laws:

ln(y) = ln(b0) + b1 ln(x1) + b2 ln(x2) + . . .+ bp ln(xp)

The minimum distance approach

Need robust regression considering all uncertainties

Parameter estimation→ distance minimization:expected↔ measured:

Ordinary least squaresMaximum likelihood (ML) / maximum a posteriori (MAP):

1√2πσ

exp{−1

2[y − f (x , θ)]2

σ2

}

Measurement→ probability distribution

Minimum distance estimation: Hellinger divergence,Kullback-Leibler divergence, . . .

Firm mathematical basis: information geometry=⇒ regression on probabilistic manifolds

Overview

1 Methodology for scaling laws

2 Geodesic least squares regression (GLS)

3 Numerical experiments

4 Power threshold scaling

Information geometry

The Gaussian probability space

Estimation through distance minimization

Ordinary least squares (OLS) vs. geodesic least squares (GLS)

Linear regression

1√2π(σ2

y +∑m

j=1 βj2σ2

x ,j

) exp

−12

[y −

(β0 +

∑mj=1 βj xij

)]2

σ2y +

∑mj=1 βj

2σ2

x ,j

1√2π σobs

exp[−1

2(y − yi)

2

σobs 2

]Rao GD

To be estimated: σobs, β0, β1, . . . , βmiid data: minimize sum of squared GDs

=⇒ geodesic least squares (GLS) regressionMahalanobis distance = Rao geodesic distance for Gaussianswith same variance:

GD[N (µ1, σ

2)||N (µ2, σ2)]

=|µ1 − µ2|

σC. Atkinson and A.F.S. Mitchell, Rao’s Distance Measure, Sankhya: The Indian Journal of Statistics 43,

pp. 345—365, 1981

Overview

1 Methodology for scaling laws

2 Geodesic least squares regression (GLS)

3 Numerical experiments

4 Power threshold scaling

Experiment 1: effect of outliers (1)

Some data:ξi ∈ [0,50]

ηi = bξi , b = 3i = 1, . . . ,10

xi = ξi + εx

yi = ηi + εy

εx ∼ N(

0, σ2x

)εy ∼ N

(0, σ2

y

) σx = 0.5σy = 2.0

σmod =√σ2

y + b2σ2x = 2.5

One outlier: yj → 2yj , j ∈ {8,9,10}

Replicate 100 times

Experiment 1: estimates

Estimates of b:

Original GLS OLS MAP TLS ROB

b = 3.003.031 3.528 3.696 4.61 2.992± 0.035 ± 0.038 ± 0.049 ± 0.11 ± 0.041

GLS = geodesic least squares (proposed)

OLS = ordinary least squares

MAP = maximum a posteriori

TLS = total least squares (errors-in-variables)

ROB = robust method (iteratively reweighted leastsquares)

GLS ≈ ROB

Estimate of σobs: 5.43± 0.24

σobs > σmod =⇒ σobs captures extra variance due to outlier

Experiment 1: regression lines

Experiment 1: interpretation

Experiment 2: Pthr synthetic linear

Pthr = β0 + β1ne + β2Bt + β3S

Pthr: L-H power threshold (MW)ne: line-averaged electron density (1020 m−3)Bt: toroidal magnetic field (T )S: plasma surface area (m2)

ITPA power threshold database (IAEA02: J. Snipes et al., IAEA FEC 2002, CT/P-04)

Data + error bars from 7 tokamaks→ > 600 entriesApproximate pmod as N (µmod, σ

2mod) via Gaussian error

propagation, where

µmod = β0 + β1ne + β2Bt + β3S

σ2mod = β2

1σ2ne

+ β22σ

2Bt

+ β23σ

2S

Experiment 2: setup

Pthr = β0 + β1ne + β2Bt + β3S

Synthetic data sets:β0: 1,1.1, . . . ,20

β1, β2, β3: 0.1,0.2, . . . ,2

Percentage errors:

Pthr: 15%

ne: 4%

Bt: 1%

S: 3%

10 trials per parameter set

Experiment 2: results

Percentage error on parameter estimates

Experiment 3: Pthr synthetic linear with outliers

Pthr = β0 + β1ne + β2Bt + β3S

Synthetic data sets:β0: 1,1.1, . . . ,20

β1, β2, β3: 0.1,0.2, . . . ,2

Percentage errors:

Pthr: 15%ne: 4%Bt: 1%S: 3%

10 outliers per parameter set: Pthr → F × Pthr, F ∼ N (1.5,1.02)

10 trials per parameter set

Experiment 3: results

Percentage error on parameter estimates

Experiment 4: Pthr synthetic log-linear

Pthr = β0n β1e B β2

t S β3

=⇒ ln Pthr ≈ lnβ0 + β1 ln ne + β2 ln Bt + β3 ln S

Synthetic data sets:β0: 1,1.1, . . . ,20

β1, β2, β3: 0.1,0.2, . . . ,2

Percentage errors:

Pthr: 15%ne: 20%Bt: 5%S: 15%

10 trials per parameter set

Experiment 4: results

Percentage error on parameter estimates

Overview

1 Methodology for scaling laws

2 Geodesic least squares regression (GLS)

3 Numerical experiments

4 Power threshold scaling

Power threshold scaling: log-linear

Parameter estimation and prediction of Pthr for ITER(ne = 0.5× 1020 m−3 or 1.0× 1020 m−3):

Method b0 b1 b2 b3 Pthr,0.5 (MW) Pthr,1.0 (MW)

MAP 0.059 0.73 0.71 0.92 48 80GLS 0.065 0.93 0.64 1.02 62 117

GLS predictions higher than MAP

Power threshold scaling: nonlinear

Approximate pmod as N (µmod, σ2mod) via Gaussian error

propagation, where

µmod = b0n b1e B b2

t S b3

σ2mod = σ2

Pthr+ µ2

mod

[b2

1

(σne

ne

)2

+ b22

(σBt

Bt

)2

+ b23

(σS

S

)2]

Estimates and predictions:

Method b0 b1 b2 b3 Pthr,0.5 (MW) Pthr,1.0 (MW)

MAP 0.051 0.85 0.70 1.00 62 111GLS 0.048 0.96 0.59 1.05 64 124

GLS similar for linear and non-linear

MAP now agrees with GLS

Conclusion and future work

Geodesic least squares regression: flexible and robust

Easy to use, fast optimization

Works for linear and nonlinear relations and any distribution model

Revisit established scaling laws, contribute to new regressionanalyses

To be implemented in publicly accessible software package

Geodesic distance minimization:

Parameter estimation

Model validation