1. 1Statistical studies ITPA L-mode DBITPA Lausanne meeting 2007 Association Euratom-CEA ITPA L-Mode Database Statistical studies for the scaling of confinement C. Gaborit, D. Elbèze, F. Imbeaux

2. 2Statistical studies ITPA L-mode DBITPA Lausanne meeting 2007 Association Euratom-CEA Motivations  Standard OLS regression hypothesis are not fulfilled  Error on regressors is NOT much smaller than the error on the confinement time  Error on regressors may be correlated (e.g. a and A) and may also be correlated with the error on the confinement time (e.g. P)  Different regression methods yield different results in terms of dimensionless parameter scalings (in particular  and  dependence)  physical dependence of the confinement time? vs. engineer parameters vs. physical parameters  A statistical study has been done on the L-mode database version 3.1 as an example for a better understanding of the database.

3. 3Statistical studies ITPA L-mode DBITPA Lausanne meeting 2007 Association Euratom-CEA Motivations For the ITPA L-mode database version 3.1 Method  q OLS on engineer parameters-1.54-1.830.33-3.78 OLS on engineer parameters with W instead Pl0.39-2.25-0.34-0.51 OLS on dimensionless parameters0.30-2.08-0.34-0.31 EIV-0.78-2.070.082.24

4. 4Statistical studies ITPA L-mode DBITPA Lausanne meeting 2007 Association Euratom-CEA Variable correlations Significant correlations between regressors :  Problems of multi-colinearity (on a, R, S, Pl and I)

5. 5Statistical studies ITPA L-mode DBITPA Lausanne meeting 2007 Association Euratom-CEA Variable correlations +Error  is the official "error standard set" as used for H mode DB publications on EIV +Errors on some variables is significant with respect to the variance  Instability of the regression vector aBIMnPlRAW Error  0.0290.0150.0130.0840.050.1420.0130.0470.141 STD  0.5860.4790.9890.2660.6011.0710.4791.1881.880  /  (%) 4.93.21.331.68.313.32.74.07.5

6. 6Statistical studies ITPA L-mode DBITPA Lausanne meeting 2007 Association Euratom-CEA Multi-colinearity How to remove the colinearity ? Find a good compromise between the explanatory capability of the variable (accuracy of the fit) and its independence with respect to the other ones.  1 st solution: Remove the variables a and A from the regression  Loss of information, not justified  2 nd solution: Principal component analysis on the regressors  The goal is to eliminate the principal components whose inertia is too small.  An OLS is done on the principal components whose variance is regarded as sufficient and with a significant influence on the accuracy of the fit

7. 7Statistical studies ITPA L-mode DBITPA Lausanne meeting 2007 Association Euratom-CEA PCA More than 95% of the inertia is contained in the first 3 axes with R 2 close to 0.91 Problem for the unicity of the exponents Prediction is reliable if the predicted point is close to the axes of strong intertia Inertia (%)R 2 partialR 2 on OLS PC174.540.690 PC215.930.0310.721 PC34.860.1850.906 PC42.260.0120.918 PC51.090.0270.945 PC60.950.0070.952 PC70.310.0120.964 PC80.060.0050.969 Principal component analysis (PCA) on the regressors

8. 8Statistical studies ITPA L-mode DBITPA Lausanne meeting 2007 Association Euratom-CEA PCA OLS regressions done using different numbers of PC The coefficients vary significantly but the accuracy of the fit is similar. PC 1-8 PC 1-6 PC 1-3 PCaBIMNPlRS 1 - 3 0.16-0.04 0.6 9 0.32 0.1 2 -0.49-0.060.58 1 - 6 0.380.42 0.4 5 0.04 0.4 8 -0.790.460.72 1 - 8 -0.830.12 0.8 3 0.21 0.4 6 -0.761.550.63 EIV -2.160.16 0.7 3 0.38 0.4 3 -0.661.961.13

9. 9Statistical studies ITPA L-mode DBITPA Lausanne meeting 2007 Association Euratom-CEA Predictions Despite of the multi-colinearity and the errors on measurement, predictions using a regression of the database can be done. But, the prediction will be reliable when the point has large coordinate along the axes of strong inertia and small coordinate along the axes of small inertia. τ predict = 2.46 (OLS) τ predict = 3.22 (EIV) Coor d Angle Inertia (%) PC 1 4.77974.54 PC 2 0.408515.93 PC 3 -0.52964.86 PC 4 -0.10912.26 PC 5 0.10891.09 PC 6 -0.26930.95 PC 7 0.29870.31 PC 8 -0.19920.06 ITER a 2 m S 22 m 2 B 5.3 T I 15 MA M 2.5 n 5.10 19 m -3 Pl 28 MW R 6.2 m Coord: coordinate of the ITER point on PCA axe Angle: angle (°) with the ITER point and the PCA axe For the ITER point, criteria seem respected, prediction can be done… but which ones ?!

10. 10Statistical studies ITPA L-mode DBITPA Lausanne meeting 2007 Association Euratom-CEA Predictions Test accuracy of the OLS fit on random entries of the DB : good accuracy is obtained on all machines but NSTX (strong aspect ratio machine). For the ITER L-mode point, the predicted confinement time varies by a factor of 3 ! The contribution of the components of small intertia has a significant impact on the predicted confinement time  prediction for this particular point is not reliable !  (s) ITERASDEXD3DJETJT60NSTXTFTRTS Measure d 0.0340.0590.4830.0730.0420.1190.100 Pr ed ict ed 1 - 31.130.0300.0960.4780.0490.1520.1240.111 1 - 61.420.0340.0540.3360.0600.0490.1320.107 1 - 82.460.0340.0600.4280.0760.0340.1330.107 EIV3.220.0360.0770.4850.0900.0260.1260.100

11. 11Statistical studies ITPA L-mode DBITPA Lausanne meeting 2007 Association Euratom-CEA Conclusions Several difficulties prevent a unique determination of the regression coefficients –Multicolinearity –Significant errors on variables Strong variation (factor 3) of the predicted confinement time of an ITER L mode point depending on the number of PC used (and the regression method also). How to estimate the confidence interval of the OLS and is it compatible with the observation above ? Similar problems exist in the H mode database. Are they as critical for the prediction of ITER ?