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 &alpha;<sub>i</sub> are the scaling parameters.
Here, the &alpha;<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)

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