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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

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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

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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.

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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

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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 1.10 2 cScS cBcB cncn 1. OLS7.70.800.650.44 2. M-L with i), ii) and iii)7.70.800.650.44 3. EVOR7.50.850.580.56 4. EVOR with mean errors 7.50.850.580.56 5. M-L with ii) and iii) only 7.50.850.580.56 6. M-L with iii) only7.70.970.320.88 7. OLS adjusted for log bias 7.60.800.650.44

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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

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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 1.10 2 cScS cBcB cncn 1. OLS7.70.800.650.44 2. M-L with i), ii) and iii)7.70.800.650.44 3. EVOR7.50.850.580.56 4. EVOR with mean errors 7.50.850.580.56 5. M-L with ii) and iii) only 7.50.850.580.56 6. M-L with iii) only7.70.970.320.88 7. OLS adjusted for log bias 7.60.800.650.44

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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

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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 1.10 2 cScS cBcB cncn 1. OLS7.70.800.650.44 2. M-L with i), ii) and iii)7.70.800.650.44 3. EVOR7.50.850.580.56 4. EVOR with mean errors 7.50.850.580.56 5. M-L with ii) and iii) only 7.50.850.580.56 6. M-L with iii) only7.70.970.320.88 7. OLS adjusted for log bias 7.60.800.650.44

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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

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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 1.10 2 cScS cBcB cncn 1. OLS7.70.800.650.44 2. M-L with i), ii) and iii)7.70.800.650.44 3. EVOR7.50.850.580.56 4. EVOR with mean errors 7.50.850.580.56 5. M-L with ii) and iii) only 7.50.850.580.56 6. M-L with iii) only7.70.970.320.88 7. OLS adjusted for log bias 7.60.800.650.44 8. M-L,Errors on P only5.60.860.720.58 9. M-L6.00.960.450.80

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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 1.10 2 cScS cBcB cncn χ2Nχ2N P ITER OLS7.7 ± 0.30.80 ± 0.010.65 ± 0.030.44 ± 0.037.4335.6 EVOR7.5 ± 0.30.85 ± 0.020.58 ± 0.030.56 ± 0.037.0943.4 M-L6.0 ± 0.30.96 ± 0.020.45 ± 0.040.80 ± 0.056.2659.0

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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

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