Residual Analysis for the qualification of Equilibria

1. Residual Analysis for the qualification of Equilibria 3 2 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 5 7

2. 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)

3. Statistical estimator: c • 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:

4. c 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

5. The correlation tests method 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

6. The correlation tests method 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

7. New model validation method 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:

8. Implementation of the correlations for equilibrium 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

9. 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

10. EFIT: EFIT version: EFIT-J Pressure constraints Polarimeter constraints (ch 3, 5, 7) P’ and FF’ equal to 0 at the separatrix EFIT

11. Residuals 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

12. Residuals: monomodal and bimodal pdf Monomodal / Multimodal error shapes  No clear tendency

13. Example of global correlations je,e ju,e Outside the 95% confidence interval  Problems in the reconstructions

14. Overview of the database 60% outside outside 80% Many points outside the 95% confidence interval Failings in the model

15. Example of local correlations 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)

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

17. 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 ELMy phase ELMs free

18. 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

19. Utility function: the Z-test • 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%.

20. Z-test for the ELMy 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

21. 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

22. Relation between residuals in the ELMy and ELM-free phases versus time constant of the coils Fast and slow coils Difference 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

23. 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 Monomodal 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

24. 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.

25. Example: pendulum 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) 25

26. Example: pendulum 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)