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Robust Bayesian linear regression for Tokamakplasma boundary estimation
Vít Škvára1,2, Jakub Urban1, Václav Šmídl2,3
1Institute of Plasma Physics, Prague, Czech Republic2Institute of Information Theory and Automation, Prague, Czech Republic
3Regional Innovation Center for Electrical Engineering, Pilsen, Czech Republic
2nd IAEA Technical Meeting on Fusion Data Processing, Validation andAnalysis / May 30, 2017
V. Škvára: Bayesian linear regression for boundary estimation / 2nd IAEA TM FDPVA / May 19, 2017 1/16
Outline
1 Toroidal harmonics equilibrium model
2 Bayesian formulation of the regression problem
3 Results evaluation
4 Conclusion
[1] Faugeras, B. et al. 2D interpolation and extrapolation of discrete magneticmeasurements with toroidal harmonics for equilibrium reconstruction in aTokamak. Plasma Physics and Controlled Fusion. 2014, 56.
V. Škvára: Bayesian linear regression for boundary estimation / 2nd IAEA TM FDPVA / May 19, 2017 2/16
Magnetic flux in a tokamak
0.3 0.4 0.5 0.6 0.7 0.8
r [m]
−0.4
−0.3
−0.2
−0.1
0.0
0.1
0.2
0.3
0.4
z [m
]
ψ
chamber wall
magnetic probe
We seek a form of magnetic flux ψ, that satisfies partial differential equation
−∆∗ψ = 0 in domain D,ψ = ψ0 on ∂D,
where D is a domain without plasma or other current source.
V. Škvára: Bayesian linear regression for boundary estimation / 2nd IAEA TM FDPVA / May 19, 2017 3/16
1 There is a known solution, given by decomposition into toroidalharmonics series:
ψ(r , z) =∞∑
n=0
unF (n, r , z)
where F (.) are known functions (containing the toroidal harmonics),un ∈ R are unknown coefficients.
2 Knowledge of measurements ψ(r , z) + selection of a finite number ofelements of the decomposition⇒ inverse problem of coefficientsdetermination.
3 With vector u, we reconstruct ψ(r , z)⇒ boundary is identified as lastclosed flux surface.
V. Škvára: Bayesian linear regression for boundary estimation / 2nd IAEA TM FDPVA / May 19, 2017 4/16
1 There is a known solution, given by decomposition into toroidalharmonics series:
ψ(r , z) =∞∑
n=0
unF (n, r , z)
where F (.) are known functions (containing the toroidal harmonics),un ∈ R are unknown coefficients.
2 Knowledge of measurements ψ(r , z) + selection of a finite number ofelements of the decomposition⇒ inverse problem of coefficientsdetermination.
3 With vector u, we reconstruct ψ(r , z)⇒ boundary is identified as lastclosed flux surface.
V. Škvára: Bayesian linear regression for boundary estimation / 2nd IAEA TM FDPVA / May 19, 2017 4/16
1 There is a known solution, given by decomposition into toroidalharmonics series:
ψ(r , z) =∞∑
n=0
unF (n, r , z)
where F (.) are known functions (containing the toroidal harmonics),un ∈ R are unknown coefficients.
2 Knowledge of measurements ψ(r , z) + selection of a finite number ofelements of the decomposition⇒ inverse problem of coefficientsdetermination.
3 With vector u, we reconstruct ψ(r , z)⇒ boundary is identified as lastclosed flux surface.
V. Škvára: Bayesian linear regression for boundary estimation / 2nd IAEA TM FDPVA / May 19, 2017 4/16
Problem formulation
The previous transforms into a linear model
Y = Xu + e,
whereY are magnetic flux measurements from coils,X is regressor matrix given by F (n, r , z),u is a vector of unknown coefficients,e is noise.
Optimal choice of the degree of decomposition (length of u) is unknown butthe result is extremely sensitive to it.
Proposed solutionConstruction of a hierarchical probability model that
automatically selects the proper decomposition modelis robust to outliers in measurements.
V. Škvára: Bayesian linear regression for boundary estimation / 2nd IAEA TM FDPVA / May 19, 2017 5/16
Problem formulation
The previous transforms into a linear model
Y = Xu + e,
whereY are magnetic flux measurements from coils,X is regressor matrix given by F (n, r , z),u is a vector of unknown coefficients,e is noise.
Optimal choice of the degree of decomposition (length of u) is unknown butthe result is extremely sensitive to it.
Proposed solutionConstruction of a hierarchical probability model that
automatically selects the proper decomposition modelis robust to outliers in measurements.
V. Škvára: Bayesian linear regression for boundary estimation / 2nd IAEA TM FDPVA / May 19, 2017 5/16
Least squares approach
The modelY = Xu + e,e ∼ N (e|0, β−1I)
leads to the likelihood function
p(Y |u, β) = N (Y |Xu, β−1I).
An estimate of u is obtained by maximizing the likelihood which results in theclosed-form solution
u = (X T X )−1X T Y .
How to extend this?
V. Škvára: Bayesian linear regression for boundary estimation / 2nd IAEA TM FDPVA / May 19, 2017 6/16
Least squares approach
The modelY = Xu + e,e ∼ N (e|0, β−1I)
leads to the likelihood function
p(Y |u, β) = N (Y |Xu, β−1I).
An estimate of u is obtained by maximizing the likelihood which results in theclosed-form solution
u = (X T X )−1X T Y .
How to extend this?
V. Škvára: Bayesian linear regression for boundary estimation / 2nd IAEA TM FDPVA / May 19, 2017 6/16
Least squares approach
The modelY = Xu + e,e ∼ N (e|0, β−1I)
leads to the likelihood function
p(Y |u, β) = N (Y |Xu, β−1I).
An estimate of u is obtained by maximizing the likelihood which results in theclosed-form solution
u = (X T X )−1X T Y .
How to extend this?
V. Škvára: Bayesian linear regression for boundary estimation / 2nd IAEA TM FDPVA / May 19, 2017 6/16
Hierarchical model 1 - LS-ARD
Now begins the construction of a hierarchical probability model. So far wehave
p(Y |u, β) = N (Y |Xu, β−1I).
Now, we add a prior distribution on the vector u:
p(u|α) = N (u|0,diag(α−1)).
Also, we add a gamma prior on both precisions parameters:
p(β) = G(β|a0,b0),
p(αi ) = G(αi |c0,d0).
Now, u, β, α are unknown. How to estimate them?
V. Škvára: Bayesian linear regression for boundary estimation / 2nd IAEA TM FDPVA / May 19, 2017 7/16
Hierarchical model 1 - LS-ARD
Now begins the construction of a hierarchical probability model. So far wehave
p(Y |u, β) = N (Y |Xu, β−1I).
Now, we add a prior distribution on the vector u:
p(u|α) = N (u|0,diag(α−1)).
Also, we add a gamma prior on both precisions parameters:
p(β) = G(β|a0,b0),
p(αi ) = G(αi |c0,d0).
Now, u, β, α are unknown. How to estimate them?
V. Škvára: Bayesian linear regression for boundary estimation / 2nd IAEA TM FDPVA / May 19, 2017 7/16
Hierarchical model 1 - LS-ARD
Now begins the construction of a hierarchical probability model. So far wehave
p(Y |u, β) = N (Y |Xu, β−1I).
Now, we add a prior distribution on the vector u:
p(u|α) = N (u|0,diag(α−1)).
Also, we add a gamma prior on both precisions parameters:
p(β) = G(β|a0,b0),
p(αi ) = G(αi |c0,d0).
Now, u, β, α are unknown. How to estimate them?
V. Škvára: Bayesian linear regression for boundary estimation / 2nd IAEA TM FDPVA / May 19, 2017 7/16
Variational Bayes
We must estimate the posterior distribution p(u, β, α|Y ) = distribution ofparameters given the data Y . For that, we use the variational Bayesframework. We approximate the true posterior with a mixture of marginals
p(u, β, α|Y ) ≈ p(u, β, α|Y ) = p(u|Y )p(β|Y )p(α|Y ).
What does this result into?
We obtain an iterative algorithm that computes hyperparameters (means andvariances) of the posterior distributions.
In effect, the iterative algorithm solves the regularized least squares problem
minu
∑i
(Yi − Xiu)2 +∑
j
αju2j
with adaptive αj .
V. Škvára: Bayesian linear regression for boundary estimation / 2nd IAEA TM FDPVA / May 19, 2017 8/16
Variational Bayes
We must estimate the posterior distribution p(u, β, α|Y ) = distribution ofparameters given the data Y . For that, we use the variational Bayesframework. We approximate the true posterior with a mixture of marginals
p(u, β, α|Y ) ≈ p(u, β, α|Y ) = p(u|Y )p(β|Y )p(α|Y ).
What does this result into?
We obtain an iterative algorithm that computes hyperparameters (means andvariances) of the posterior distributions.
In effect, the iterative algorithm solves the regularized least squares problem
minu
∑i
(Yi − Xiu)2 +∑
j
αju2j
with adaptive αj .
V. Škvára: Bayesian linear regression for boundary estimation / 2nd IAEA TM FDPVA / May 19, 2017 8/16
Sparsity of the model
Equations of the posterior distribution moments are solved iteratively.By estimating α - the precision (inverse of variance) of u ⇒ less informativeelements are forced to zero.
1 30 60
N
−0.04
−0.03
−0.02
−0.01
0.00
0.01
ui
i=1
i=4
i=6
1 30 60
N
103
106
109
αi
Figure : Convergence of elements of ui and their respective precisions αi .
V. Škvára: Bayesian linear regression for boundary estimation / 2nd IAEA TM FDPVA / May 19, 2017 9/16
Hierarchical model 2-RLS-ARD
To achieve robustness, we assume that the noise has the Student’s tdistribution
p(Y |u, ω, z) = N(
Y |Xu, (ω diag(z))−1),
p(ω) = G(ω|ρ0, κ0),
p(zi |d) = G(z|d/2,d/2),
p(d) = G(d |ψ0, φ0),
which follows from∫N (e|0, (ω diag(z))−1)G(z|d/2,d/2)dz = S(e|0, ω,d).
Again, the iterative algorithm solves the problem
minu
∑i
√zi (Yi − Xiu)2 +
∑j
αju2j
with adaptive zi , αj .
V. Škvára: Bayesian linear regression for boundary estimation / 2nd IAEA TM FDPVA / May 19, 2017 10/16
Hierarchical model 2-RLS-ARD
To achieve robustness, we assume that the noise has the Student’s tdistribution
p(Y |u, ω, z) = N(
Y |Xu, (ω diag(z))−1),
p(ω) = G(ω|ρ0, κ0),
p(zi |d) = G(z|d/2,d/2),
p(d) = G(d |ψ0, φ0),
which follows from∫N (e|0, (ω diag(z))−1)G(z|d/2,d/2)dz = S(e|0, ω,d).
Again, the iterative algorithm solves the problem
minu
∑i
√zi (Yi − Xiu)2 +
∑j
αju2j
with adaptive zi , αj .
V. Škvára: Bayesian linear regression for boundary estimation / 2nd IAEA TM FDPVA / May 19, 2017 10/16
Outlier detection
Estimate of d represents normality of the measurements, whilemeasurements with zi < 1 are regarded as outliers.
1.00 1.05 1.10 1.15 1.20 1.25
t [s]
10
12
14
16
18
20
22
24d
1.00 1.05 1.10 1.15 1.20 1.25
t [s]
0.85
0.90
0.95
1.00
1.05
1.10z
Figure : Time evolution of d and zi during a discharge.
V. Škvára: Bayesian linear regression for boundary estimation / 2nd IAEA TM FDPVA / May 19, 2017 11/16
Validation
COMPASS shot number 9339.Ground truth data simulated with the EFIT++ equilibrium reconstructioncode.The second model was updated to distinguish between flux loops andmagnetic probes.
V. Škvára: Bayesian linear regression for boundary estimation / 2nd IAEA TM FDPVA / May 19, 2017 12/16
−0.4
−0.3
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−0.1
0.0
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0.3
0.4
sim
ula
ted d
ata
r
[m
]
t = 1.02 s t = 1.13 s
limiter
EFIT++
OLS Student2 measurement point
t = 1.21 s
0.3 0.4 0.5 0.6 0.7 0.8
z [m]
−0.4
−0.3
−0.2
−0.1
0.0
0.1
0.2
0.3
0.4
real m
easu
rem
ents
r
[m
]
0.3 0.4 0.5 0.6 0.7 0.8
z [m]0.3 0.4 0.5 0.6 0.7 0.8
z [m]
Figure : Boundary shapes computed with EFIT++ (ground truth), original model (OLS)and Student’s t noise models. Enlarged circles depict "outlying" measurements.
V. Škvára: Bayesian linear regression for boundary estimation / 2nd IAEA TM FDPVA / May 19, 2017 13/16
1.00 1.05 1.10 1.15 1.20 1.250.00
0.02
0.04
0.06
0.08
0.10
0.12
MSE
simulated data
1.00 1.05 1.10 1.15 1.20 1.250.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35measured data
Student1
Student2
Gauss
OLS
1.00 1.05 1.10 1.15 1.20 1.25
t [s]
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
DH
1.00 1.05 1.10 1.15 1.20 1.25
t [s]
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
Figure : Fit MSE (upper row) and Hausdorff distance DH for different estimationmethods during a discharge.
V. Škvára: Bayesian linear regression for boundary estimation / 2nd IAEA TM FDPVA / May 19, 2017 14/16
EFIT++
It solves the Grad-Shafranov equation
jp(ψ, r) = rp′ +1µ0r
(ff ′)(ψ)
for unknown functions p′, ff ′.These are approximated with a decomposition into polynomial, spline orChebyshev basis functions.With a proper model, we may extend the base and achieve robustness.
V. Škvára: Bayesian linear regression for boundary estimation / 2nd IAEA TM FDPVA / May 19, 2017 15/16
Conclusion
We have presented a Bayesian approach to linear regression.Universal algorithm with self-tuning regularization and robustnessparameters.By modeling prior knowledge, we improved the results of boundaryidentification.Outlook - implementation of the method into EFIT++ code.
V. Škvára: Bayesian linear regression for boundary estimation / 2nd IAEA TM FDPVA / May 19, 2017 16/16