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A maximum likelihood analysis of the L-H transition DB

A maximum likelihood analysis of the L-H transition DB. Darren McDonald. Introduction. Is L-H scaling sensitive to error models + if so, is the appropriate one used? OLS fits are appropriate when Errors in P >> than in other parameters Relative errors same for all experiments

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A maximum likelihood analysis of the L-H transition DB

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  1. A maximum likelihood analysis of the L-H transition DB Darren McDonald

  2. Introduction • Is L-H scaling sensitive to error models + if so, is the appropriate one used? • OLS fits are appropriate when • Errors in P >> than in other parameters • Relative errors same for all experiments • Logs of variables ≈ Normally distributed • All are violated to some extent • Use Maximum-Likelihood to test impact

  3. 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.

  4. OLS model - assumptions • Errors in P >> than in other parameter • Relative errors same for all experiments • Logs of variables ≈ Normally distributed

  5. OLS model - fits • M-L model + i), ii) and iii) agrees with OLS • Now relax assumptions in turn

  6. EVOR model - assumptions • Errors in P >> than in other parameter • Relax to include all errors • Relative errors same for all experiments • Logs of variables ≈ Normally distributed

  7. 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

  8. Log M-L model - assumptions • Errors in P >> than in other parameter • Relax to include all errors • Relative errors same for all experiments • Relax to allow machine-machine variation • Logs of variables ≈ Normally distributed

  9. 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

  10. M-L model - assumptions • Errors in P >> than in other parameter • Relax to include all errors • Relative errors same for all experiments • Relax to allow machine-machine variation • Logs of variables ≈ Normally distributed • Relax by using real variables

  11. 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?

  12. Consistency, errors and ITER • All models differ by more than their errors • M-L gives lowest χ2N 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

  13. Conclusion • M-L method shown consistent with OLS and EVOR where assumptions are the same • All three assumptions looked at biased scaling • χ2N >> 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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