Box–Cox transformation

In statistics, a power transform is a family of functions that are applied to create a monotonic transformation of data using power functions. This is a useful data transformation technique used to stabilize variance, make the data more normal distribution-like, improve the validity of measures of association such as the Pearson correlation between variables and for other data stabilization procedures.

The power transformation is defined as a continuously varying function, with respect to the power parameter λ, in a piece-wise function form that makes it continuous at the point of singularity (λ = 0). For data vectors (y₁,..., y_n) in which each y_i > 0, the power transform is

where

is the geometric mean of the observations y₁, ..., y_n. The case for ${\displaystyle \lambda =0}$ is the limit as ${\displaystyle \lambda }$ approaches 0. To see this, note that ${\displaystyle y_{i}^{\lambda }}$ = ${\displaystyle \operatorname {exp} ({\lambda \operatorname {log} (y_{i})})}$ = ${\displaystyle 1+\lambda \operatorname {log} (y_{i})+O((\lambda \operatorname {log} (y_{i}))^{2})}$. Then ${\displaystyle {\dfrac {y_{i}^{\lambda }-1}{\lambda }}}$ = ${\displaystyle \operatorname {log} (y_{i})+O(\lambda )}$, and everything but ${\displaystyle \operatorname {log} (y_{i})}$ becomes negligible for ${\displaystyle \lambda }$ sufficiently small.

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