9th ITPA Confinement Database and Modelling Topical Physics Group meeting in St. Petersburg 1/13 A maximum likelihood analysis of the L-H transition DB Darren McDonald

9th ITPA Confinement Database and Modelling Topical Physics Group meeting in St. Petersburg 2/13 Is L-H scaling sensitive to error models + if so, is the appropriate one used? OLS fits are appropriate when 1.Errors in P >> than in other parameters 2.Relative errors same for all experiments 3.Logs of variables ≈ Normally distributed All are violated to some extent Use Maximum-Likelihood to test impact Introduction

9th ITPA Confinement Database and Modelling Topical Physics Group meeting in St. Petersburg 3/13 Maximum-Likelihood method Soln is one which makes data most likely For Likelihood is Problem is now Non-Linear, but has been solved by MINUIT package. Take IAE04R dataset.

9th ITPA Confinement Database and Modelling Topical Physics Group meeting in St. Petersburg 4/13 1.Errors in P >> than in other parameter 2.Relative errors same for all experiments 3.Logs of variables ≈ Normally distributed OLS model - assumptions

9th ITPA Confinement Database and Modelling Topical Physics Group meeting in St. Petersburg 5/13 OLS model - fits M-L model + i), ii) and iii) agrees with OLS Now relax assumptions in turn Statistical model c cScS cBcB cncn 1. OLS M-L with i), ii) and iii) EVOR EVOR with mean errors M-L with ii) and iii) only M-L with iii) only OLS adjusted for log bias

9th ITPA Confinement Database and Modelling Topical Physics Group meeting in St. Petersburg 6/13 1.Errors in P >> than in other parameter Relax to include all errors 2.Relative errors same for all experiments 3.Logs of variables ≈ Normally distributed EVOR model - assumptions

9th ITPA Confinement Database and Modelling Topical Physics Group meeting in St. Petersburg 7/13 EVOR model - fits M-L model + ii) and iii) agrees with EVOR Two methods for averaging errors ≈ same answer Differ from OLS  OLS biases result Statistical model c cScS cBcB cncn 1. OLS M-L with i), ii) and iii) EVOR EVOR with mean errors M-L with ii) and iii) only M-L with iii) only OLS adjusted for log bias

9th ITPA Confinement Database and Modelling Topical Physics Group meeting in St. Petersburg 8/13 1.Errors in P >> than in other parameter Relax to include all errors 2.Relative errors same for all experiments Relax to allow machine-machine variation 3.Logs of variables ≈ Normally distributed Log M-L model - assumptions

9th ITPA Confinement Database and Modelling Topical Physics Group meeting in St. Petersburg 9/13 Log M-L model - fits M-L model iii) only differs from OLS and EVOR  assumption ii) biases results Are we sure about tokamak error estimates? Easy to extend to point-point variation Statistical model c cScS cBcB cncn 1. OLS M-L with i), ii) and iii) EVOR EVOR with mean errors M-L with ii) and iii) only M-L with iii) only OLS adjusted for log bias

9th ITPA Confinement Database and Modelling Topical Physics Group meeting in St. Petersburg 10/13 1.Errors in P >> than in other parameter Relax to include all errors 2.Relative errors same for all experiments Relax to allow machine-machine variation 3.Logs of variables ≈ Normally distributed Relax by using real variables M-L model - assumptions

9th ITPA Confinement Database and Modelling Topical Physics Group meeting in St. Petersburg 11/13 M-L model - fits M-L model differs again  skewing of logs influences results Attempt to correct this in OLS method (7) failed Are we sure real errors are Normally distributed? Statistical model c cScS cBcB cncn 1. OLS M-L with i), ii) and iii) EVOR EVOR with mean errors M-L with ii) and iii) only M-L with iii) only OLS adjusted for log bias M-L,Errors on P only M-L

9th ITPA Confinement Database and Modelling Topical Physics Group meeting in St. Petersburg 12/13 Consistency, errors and ITER All models differ by more than their errors M-L gives lowest χ 2 N for model, but still >>1  model still has missing features  must improve before confidence can be placed in this method ITER prediction highest for M-L Statistical model c cScS cBcB cncn χ2Nχ2N P ITER OLS7.7 ± ± ± ± EVOR7.5 ± ± ± ± M-L6.0 ± ± ± ±

9th ITPA Confinement Database and Modelling Topical Physics Group meeting in St. Petersburg 13/13 Conclusion M-L method shown consistent with OLS and EVOR where assumptions are the same All three assumptions looked at biased scaling χ 2 N >> 1  model has missing features  must have refine error model to use method ITER prediction higher for M-L Prudent estimates may come from average of a set of error models