a.murari 1 (24) frascati 27 th march 2012 residual analysis for the qualification of equilibria...
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A.Murari 1 (24) Frascati 27th March 2012
Residual Analysis for the Residual Analysis for the qualification of Equilibriaqualification of Equilibria
A.MurariA.Murari11, D.Mazon, D.Mazon22, J.Vega, J.Vega33, P.Gaudio, P.Gaudio44, , M.GelfusaM.Gelfusa44, E.Peluso, E.Peluso44, F.Maviglia, F.Maviglia55, M. , M. FalschetteFalschette66
1
32
4 University of Rome
“Tor Vergata”
7
5
6 École Centrale de Nantes44000 Nantes, France
A.Murari 2 (24) Frascati 27th March 2012
Question: how to choose the value of the weighting parameter K1=Wfar/Wcoils?
Goal of the analysis:
Identify a statistically sound methodology to determine the quality of the equilibrium
reconstructions.
The approach is based not only on the 2 but also on high order correlations of the residuals, which have been proved to be adequate for nonlinear
systems.
Statistical method from Billing and Zu (1995)
Statistical Assessment of the Magnetic Reconstructions
A.Murari 3 (24) Frascati 27th March 2012
Statistical estimator:
• For each probe (i) the following variable i has been computed:
while for each shot the average over all n coils is:
having the following statistical error:
A.Murari 4 (24) Frascati 27th March 2012
is not fully adequate
In case of nonlinear systems, indicators of the 2 type are not fully satisfactory. They take into account only the amplitude of the residuals.
The time evolution of the residuals can also
provide very interesting information about the quality of the models.
t
y
A.Murari 5 (24) Frascati 27th March 2012
The correlation tests method Hypothesis: the noise is random and additiveConsequence: the residuals of a perfect model should be randomly distributed
The model with the distribution of the residuals closer to a random one is preferred
Cost function: correlation functions of the following type
A.Murari 6 (24) Frascati 27th March 2012
The correlation tests methodTheory: for an infinite series of random number the autocorrelations should be zero
With finite samples the autocorrelations will not exactly be zero
Anderson, Bartlett and Quenouille showed in the 40s that the autocorrelation coefficients of white noise data can be approximated by a normal curve with mean zero and standard error 1/√n wher n is the number of samples
95% confidence level can be calculated 1.96 1/√nAdvanced correlations for nonlinear systems
Autocorrelations
A.Murari 7 (24) Frascati 27th March 2012
where q is the number of the dependent variables and r is the number of the independent variables.
A complete and adequate set of tests for a nonlinear, MIMO system is provided by the higher order correlations between the residual and input and output vectors given by the following relations ( residuals, u inputs, y outputs):
tE)(
tE)(
ttt q22
1 ...
ttyttyt qq ...11
tutut r22
1 ...
If the non linear model is an adequate representation of the system, in the ideal case, should be:
otherwise
k
0
0
,)( ,)( 0
New model validation method
A.Murari 8 (24) Frascati 27th March 2012
EFIT
yu
Inputs:Pickup coils
Faraday measurements
Outputs:Pickup coilsFaraday chords reconstructed by EFIT
Residuals:Difference between measurements & EFIT
for pickup coils and Faraday
Implementation of the correlations for equilibrium
Implementation of the correlations for the case of EFIT:
In our case the analysis consists of assessing the quality of the equilibrium reconstructions of EFIT by analysing the distribution function of the residuals
A.Murari 9 (24) Frascati 27th March 2012
The correlation tests method• 2 different points of view:– Global: all data are computed for all coils– Local: data are computed independently for
individual coils
A.Murari 10 (24) Frascati 27th March 2012
EFIT:
EFIT
EFIT version:
EFIT-J
Pressure constraints
Polarimeter constraints (ch 3, 5, 7)
P’ and FF’ equal to 0 at the separatrix
A.Murari 11 (24) Frascati 27th March 2012
Residuals
-6 -5 -4 -3 -2 -1 0 1 2
x 10-3
0
100
200
300
400
500
600Coil n°21 [TP203 ] - Error distribution
Error (T)
Poin
ts n
um
ber
-3 -2 -1 0 1 2 3 4 5 6
x 10-3
0
100
200
300
400
500
600Coil n°40 [TN211 ] - Error distribution
Error (T)
Poin
ts n
um
ber
Monomodal error type Multimodal error type
Residuals: differences between the experimental values and the model estimates or predictions
Residuals in the case of the equilibrium (EFIT) and the magnetic measurements for two coils
Residuals are often presented as histograms: x axis the value of the residual, y axis the number of occurrences of
that value
A.Murari 12 (24) Frascati 27th March 2012
Monomodal / Multimodal error shapes No clear tendency
Residuals: monomodal and bimodal pdf
A.Murari 13 (24) Frascati 27th March 2012
Example of global correlations
42 44 46 48 50 52 54 56 58 60-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Linear auto-correlation , (residuals & residuals) - Pulse 68671
Delay (s)
Corr
ela
tion
42 44 46 48 50 52 54 56 58 60-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
Linear cross-correlation u, (measurements & residuals) - Pulse 68671
Delay (s)
Corr
ela
tion
, u,
Outside the 95% confidence interval Problems in the reconstructions
A.Murari 14 (24) Frascati 27th March 2012
60%outsi
de
outside
80%
Many points outside the 95% confidence interval Failings in the model
Overview of the database
A.Murari 15 (24) Frascati 27th March 2012
Example of local correlations
u,,
NO clear trend: the pattern changes from shot to shot and even during the same discharge (more than
120 shots analysed)
A.Murari 16 (24) Frascati 27th March 2012
Comparison of ELM-free and ELMy phases
• Hypothesis: ELMs are of the the causes of the multimodal distribution
• Comparison of the EFIT quality during ELMs and during ELM-free periods
Figure: Shot 75412. Top: D channel; Bottom: ELMs in the EHTR channel.
A.Murari 17 (24) Frascati 27th March 2012
Residual distributions: visual analysis
• The residual distribution function of the pick-up coils shows typically a multimodal shape. The ELMs typically account for one of the peaks
Total Residual distribution
ELMs free ELMy phase
A.Murari 18 (24) Frascati 27th March 2012
Summary of the shots analysed
• Details of the shots analysed:
• Abut 350 type I ELMs studied. Results are consistent not only for the shots but also for the individual coils so the statistical basis is considered sufficient of the shots
A.Murari 19 (24) Frascati 27th March 2012
Utility function: the Z-test
• In order to check if two physical quantities, two measurements
etc are different, the Z-test is normally used ( 1,2 are the
averages of the in the ELMy and ELM-free phases):
• If the Z variable is higher than 1.96, the two quantities are statistically different with a
confidence exceeding 95%.
FEyEy
FEyEyZ22
A.Murari 20 (24) Frascati 27th March 2012
Z-test for the ELMy and ELM free periods
• The variable has been computed separately for the ELMy (Ey ) phase and for the ELM free one (FEy ):
Results: the for ELMy and ELM-free periods are different with a confidence well in excess of 95%.Not an academic exercise: in statistics quantity is
quality
Details of this application by
M.Gelfusa, A.Murari et al to be submitted
to NIMA
Results: the during ELMs is always higher than in ELM-free periods
A.Murari 21 (24) Frascati 27th March 2012
- ELMs effects on the equilibrium. Three main causes:- a) EFIT hypotheses not valid: equilibrium, toroidal symmetry,
current at the boundary etc- b) Coils: delays, eddy current in metallic structures etc. - c) Not optimal constraints in EFIT
ELMs
- A specific dry shot in which the currents in the divertor coils have been modulated is being used to assess the time response of the coils
A.Murari 22 (24) Frascati 27th March 2012
Relation between residuals in the ELMy and ELM-free phases versus time constant of the coils
Fast and slow coils
Difference m of the residual means between the ELMy and ELM –free phases versus the difference between the rise time of the signals of the pick-up coils and the divertor currents.
Fast coils reconstructed more poorly
A.Murari 23 (24) Frascati 27th March 2012
The constraints of p’ and ff’ to go to zero at the separatrix have been relaxed.
11 both zero 00 both parameters free
Constraints at the edge
Freeing p’ and ff’ improves the situation in ELM-free periods but does not have a major effects for the reconstructions during ELMs
Monomodal residual pdf
Bimodal residual pdf
ELMs free ELMy phase
00 17 19
01 14 19
10 17 19
11 8 17
ELMs free ELMy phase
00 37 30
01 39 30
10 37 30
11 45 34
A.Murari 24 (24) Frascati 27th March 2012
Question: how to choose the value of the weighting parameter K1=Wfar/Wcoils?
Summary:
The approach based not only on the 2 but also on high order correlations of the residuals increases
the confidence in the results
However, no principle method has been found yet to determine the relative importance of the 2 and
the high order correlations.
The application to the investigation of the ELMs seems to indicate that the main issue resides in the limited physics in EFIT more than in the coils
or the constraints.
Statistical Assessment of the Magnetic Reconstructions:
Summary
A.Murari 25 (24) Frascati 27th March 2012
Example: pendulum
0 5 10 15 20 25 30 35 40-2.5
-2
-1.5
-1
-0.5
0
Time
Solu
tion
Accurate model
100 200 300 400 500 600 700 800 900 1000
-0.1
-0.05
0
0.05
0.1
Linear auto-correlation , (residuals & residuals)
Delay (s)
Corr
ela
tion
0 100 200 300 400 500 600 700 800 900 1000-0.1
-0.05
0
0.05
0.1
Linear cross-correlation u, (inputs & residuals)
Delay (s)
Corr
ela
tion
Residual for Accurate model Good correlations(inside the 95% confidence interval)
y’’ + ∙y’ + a sin(∙ y) = b sin(∙ ∙t)
25
Nonlinear pendulum plus 10% of Gaussian noise. Black curve: exact solution Red curve: exact solution plus noise
A.Murari 26 (24) Frascati 27th March 2012
Example: pendulum
0 5 10 15 20 25 30 35 40-2.5
-2
-1.5
-1
-0.5
0Wrong model
Time
Solu
tion
0 100 200 300 400 500 600 700 800 900 1000-0.5
0
0.5
1
Linear auto-correlation , (residuals & residuals)
Delay (s)
Corr
ela
tion
0 100 200 300 400 500 600 700 800 900 1000-0.5
0
0.5
Linear cross-correlation u, (inputs & residuals)
Delay (s)
Corr
ela
tion
Error added on parameter a Poor correlations (outside the 95% confidence interval)
y’’ + ∙y’ + a sin(∙ y) = b sin(∙ ∙t)
Details of application to equilibrium in the paper A.Murari et al Nucl. Fusion 51 (2011) 053012 (18pp)