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Residual Analysis for the qualification of Equilibria

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Residual Analysis for the qualification of Equilibria

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1

4 University of Rome

“Tor Vergata”

A.Murari1, D.Mazon2, J.Vega3, P.Gaudio4, M.Gelfusa4, E.Peluso4, F.Maviglia5, M. Falschette6

6 École Centrale de Nantes

44000 Nantes, France

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7

Question: how to choose the value of the weighting parameter K1=Wfar/Wcoils?

Statistical Assessment of the Magnetic Reconstructions

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 c2 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 estimator: Kc

- For each probe (i) the following variable ci has been computed:

while for each shot the average over all n coils is:

having the following statistical error:

c K is not fully adequate

In case of nonlinear systems, indicators of the c2 type are not fully satisfactory. They take into account only the amplitude of the residuals.

y

The time evolution of the residuals can also provide very interesting information about the quality of the models.

t

The correlation tests method K

Hypothesis: the noise is random and additive

Consequence: 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

Residualcorrelations

The correlation tests method K

Theory: 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/√n

Advanced correlations for nonlinear systems

Autocorrelations

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 (e residuals, u inputs, y outputs):

where q is the number of the dependent variables and r is the number of the independent variables.

If the non linear model is an adequate representation of the system, in the ideal case, should be:

Implementation of the correlations for equilibrium K

Implementation of the correlations for the case of EFIT:

EFIT

Inputs:

Pickup coils

Faraday measurements

Outputs:

Pickup coils

Faraday chords reconstructed by EFIT

u

y

e

Residuals:

Differencebetween measurements & EFIT for pickup coils and Faraday

In our case the analysis consists of assessing the quality of the equilibrium reconstructions of EFIT by analysing the distribution function of the residuals

The correlation tests method K

- 2 different points of view:
- Global: all data are computed for all coils
- Local: data are computed independently for individual coils

EFIT: K

EFIT version:

EFIT-J

Pressure constraints

Polarimeter constraints (ch 3, 5, 7)

P’ and FF’ equal to 0 at the separatrix

EFIT

Residuals K

Residuals: differences between the experimental values and the model estimates or predictions

Residuals are often presented as histograms: x axis the value of the residual, y axis the number of occurrences of that value

Residuals in the case of the equilibrium (EFIT) and the magnetic measurements for two coils

Monomodal error type

Multimodal error type

Residuals: monomodal and bimodal pdf K

Monomodal / Multimodal error shapes No clear tendency

Example of global correlations K

je,e

ju,e

Outside the 95% confidence interval Problems in the reconstructions

Overview of the database K

60%

outside

outside

80%

Many points outside the 95% confidence interval Failings in the model

Example of local correlations K

je,e

ju,e

NO clear trend: the pattern changes from shot to shot and even during the same discharge (more than 120 shots analysed)

Comparison of ELM-free and ELMy phases K

- 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: Da channel; Bottom: ELMs in the EHTR channel.

Residual distributions: visual analysis K

- 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

ELMy phase

ELMs free

Summary of the shots analysed K

- 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

Utility function: the Z-test K

- In order to check if two physical quantities, two measurements etc are different, the Z-test is normally used (c1,2 are the averagesof thec 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%.

Z-test for the KELMy and ELM free periods

- The c variable has been computed separately for the ELMy (cEy ) phase and for the ELM free one (cFEy ):

Results: the c during ELMs is always higher than in ELM-free periods

Details of this application by M.Gelfusa, A.Murari et al to be submitted to NIMA

Results: the c 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

ELMs effects on the equilibrium. Three main causes: K

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

Relation between residuals in the ELMy and ELM-free phases versus time constant of the coils

Fast and slow coilsDifference Dm of the residual means between the ELMy and ELM –free phases versus the difference t between the rise time of the signals of the pick-up coils and the divertor currents.

Fast coils reconstructed more poorly

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 edgeMonomodal residual pdf

Freeing p’ and ff’ improves the situation in ELM-free periods but does not have a major effects for the reconstructions during ELMs

Bimodal residual pdf

Question: how to choose the value of the weighting parameter K1=Wfar/Wcoils?

Statistical Assessment of the Magnetic Reconstructions: Summary

Summary:

The approach based not only on the c2 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 c2 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.

Example: pendulum K

Example: non linearpendulum

Nonlinear pendulum plus 10% of Gaussian noise. Black curve: exact solution Red curve: exact solution plus noise

y’’ + g∙y’ + a∙sin(y) = b∙sin(p∙t)

Residual for Accurate model Good correlations

(inside the 95% confidence interval)

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Example: pendulum K

Example: non linearpendulum

Error added on parameter a Poor correlations

(outside the 95% confidence interval)

y’’ + g∙y’ + a∙sin(y) = b∙sin(p∙t)

Details of application to equilibrium in the paper A.Murari et al Nucl. Fusion 51 (2011) 053012 (18pp)

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