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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
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