1. A.Murari 1 (24) Frascati 27 th March 2012 Residual Analysis for the qualification of Equilibria A.Murari 1, D.Mazon 2, J.Vega 3, P.Gaudio 4, M.Gelfusa 4, E.Peluso 4, F.Maviglia 5, M. Falschette 6 1 3 2 4 University of Rome “Tor Vergata” 7 5 6 École Centrale de Nantes 44000 Nantes, France

2. A.Murari 2 (24) Frascati 27 th March 2012 Question: how to choose the value of the weighting parameter K 1 =W far /W coils ? 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

3. A.Murari 3 (24) Frascati 27 th 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:

4. A.Murari 4 (24) Frascati 27 th 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

5. A.Murari 5 (24) Frascati 27 th March 2012 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

6. A.Murari 6 (24) Frascati 27 th March 2012 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. A.Murari 7 (24) Frascati 27 th 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): If the non linear model is an adequate representation of the system, in the ideal case, should be: New model validation method

8. A.Murari 8 (24) Frascati 27 th March 2012 EFIT  y u Inputs: Pickup coils Faraday measurements Outputs: Pickup coils Faraday 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

9. A.Murari 9 (24) Frascati 27 th 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

10. A.Murari 10 (24) Frascati 27 th 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

11. A.Murari 11 (24) Frascati 27 th March 2012 Residuals Monomodal error typeMultimodal 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

12. A.Murari 12 (24) Frascati 27 th March 2012 Monomodal / Multimodal error shapes  No clear tendency Residuals: monomodal and bimodal pdf

13. A.Murari 13 (24) Frascati 27 th March 2012 Example of global correlations ,,  u,  Outside the 95% confidence interval  Problems in the reconstructions

14. A.Murari 14 (24) Frascati 27 th March 2012 60% outsi de 80% Many points outside the 95% confidence interval  Failings in the model Overview of the database

15. A.Murari 15 (24) Frascati 27 th 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)

16. A.Murari 16 (24) Frascati 27 th 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.

17. A.Murari 17 (24) Frascati 27 th 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

18. A.Murari 18 (24) Frascati 27 th 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

19. A.Murari 19 (24) Frascati 27 th 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%.

20. A.Murari 20 (24) Frascati 27 th 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

21. A.Murari 21 (24) Frascati 27 th 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

22. A.Murari 22 (24) Frascati 27 th 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

23. A.Murari 23 (24) Frascati 27 th 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 freeELMy phase 00 1719 01 1419 10 1719 11 817 ELMs freeELMy phase 00 3730 01 3930 10 3730 11 4534

24. A.Murari 24 (24) Frascati 27 th March 2012 Question: how to choose the value of the weighting parameter K 1 =W far /W coils ? 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

25. A.Murari 25 (24) Frascati 27 th March 2012 Example: pendulum 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

26. A.Murari 26 (24) Frascati 27 th March 2012 Example: pendulum 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)