4.4 Power Transformations

A transformation that is similar to the logarithmic one is the Box-Cox transformation.

\[ z = \frac{1}{\lambda} (x^\lambda - 1) \]

where \(\lambda\) is a positive real number that plays the role of a power parameter.

Interestingly, the Box-Cox transformation approximates to the log-transformation as the power parameter \(\lambda\) tends to 0. The division by \(\lambda\) has a reason: it keeps the scale of the original variable from collapsing. For instance, if you take the 10th roots \(x^{0.1}\) of a variable, you should see that all the values are close to 1, so dividing by 1/10 multiplies the values by 10, which is almost a logarithmic scale. In summary, Box-Cox transformations provide a way of making data more symmetric, which could be very helpful in regression analysis tools.

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