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Scaling law: Difference between revisions
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The typical scaling law expression for a (dependent) variable ''y'' as a function of some (independent) system variables ''x<sub>1</sub>'', ''x<sub>2</sub>'',... is: | The typical scaling law expression for a (dependent) variable ''y'' as a function of some (independent) system variables ''x<sub>1</sub>'', ''x<sub>2</sub>'',... is: | ||
:<math>y = \alpha_0 x_1^{\alpha_1} x_2^{\alpha_1} ...</math> | :<math>y = e^{\alpha_0} x_1^{\alpha_1} x_2^{\alpha_1} ...</math> | ||
Here, the α<sub>i</sub> are the scaling parameters. | Here, the α<sub>i</sub> are the scaling parameters. | ||
By taking the logarithm of this expression, it becomes linear and simple (multivariate) linear regression tools can be used. | By taking the logarithm of this expression, it becomes linear and simple (multivariate) linear regression tools can be used to determine the parameters from a set of data. | ||
However, a proper analysis requires: | However, a proper analysis requires: | ||
* using ''dimensionless'' variables (easily achieved by normalizing all quantities appropriately) | * using ''dimensionless'' variables (easily achieved by normalizing all quantities appropriately) | ||