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Scaling law: Difference between revisions
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== General method == | == General method == | ||
The typical scaling law expression for a (dependent) variable ''y'' as a function of some (positive and independent) system variables ''x<sub>1</sub>'', ''x<sub>2</sub>'',... is: | The typical scaling law expression for a (dependent or response) variable ''y'' as a function of some (positive and independent) system (or predictor) variables ''x<sub>1</sub>'', ''x<sub>2</sub>'',... is: | ||
:<math>y = e^{\alpha_0} x_1^{\alpha_1} x_2^{\alpha_2} ...</math> | :<math>y = e^{\alpha_0} x_1^{\alpha_1} x_2^{\alpha_2} ...</math> | ||
Here, the α<sub>i</sub> are the scaling parameters. | Here, the α<sub>i</sub> are the scaling parameters. | ||
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However, a proper analysis requires: | However, a proper analysis requires: | ||
* using ''dimensionless'' variables (easily achieved by normalizing all quantities appropriately)<ref>T.C. Luce, C.C. Petty, and J.G. Cordey,''Application of dimensionless parameter scaling techniques to the design and interpretation of magnetic fusion experiments'', [[doi:10.1088/0741-3335/50/4/043001|Plasma Phys. Control. Fusion, '''50''', 4 (2008) 043001]]</ref> | * using ''dimensionless'' variables (easily achieved by normalizing all quantities appropriately)<ref>T.C. Luce, C.C. Petty, and J.G. Cordey,''Application of dimensionless parameter scaling techniques to the design and interpretation of magnetic fusion experiments'', [[doi:10.1088/0741-3335/50/4/043001|Plasma Phys. Control. Fusion, '''50''', 4 (2008) 043001]]</ref> | ||
* guaranteeing the (linear) ''statistical independence'' of the independent variables (applying, e.g., [[:Wikipedia:Principal component analysis|Principal Component Analysis]]) | * guaranteeing the (linear) ''statistical independence'' of the independent variables (applying, e.g., [[:Wikipedia:Principal component analysis|Principal Component Analysis]]), or at least taking mutual linear correlations into account <ref>S. M. Kaye et al, ''The role of aspect ratio and beta in H-mode confinement scalings, [[doi:10.1088/0741-3335/48/5A/S44|Plasma Phys. Control. Fusion '''48''' (2006) A429]]</ref> | ||
* considering errors in the predictor variables and the propagation of these errors in order to define confidence intervals for the resulting predictions | |||
== Dimensionless parameters == | == Dimensionless parameters == | ||