In statistics, a power transform is a family of functions applied to create a monotonic transformation of data using power functions. It is a 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. Power transforms are used in multiple fields, including multi-resolution and wavelet analysis, statistical data analysis, medical research, modeling of physical processes, geochemical data analysis, epidemiology and many other clinical, environmental and social research areas.
Definition The power transformation is defined as a continuous function of power parameter λ, typically given in piece-wise form that makes it continuous at the point of singularity (λ = 0). For data vectors (y1,..., yn) in which each yi > 0, the power transform is
y i ( λ ) = { y i λ − 1 λ ( GM ( y ) ) λ − 1 , if λ ≠ 0 GM ( y ) ln y i , if λ = 0 {\displaystyle y_{i}^{(\lambda )}={\begin{cases}{\dfrac {y_{i}^{\lambda }-1}{\lambda (\operatorname {GM} (y))^{\lambda -1}}},&{\text{if }}\lambda \neq 0\\[12pt]\operatorname {GM} (y)\ln {y_{i}},&{\text{if }}\lambda =0\end{cases}}}
where
GM ( y ) = ( ∏ i = 1 n y i ) 1 n = y 1 y 2 ⋯ y n n {\displaystyle \operatorname {GM} (y)=\left(\prod _{i=1}^{n}y_{i}\right)^{\frac {1}{n}}={\sqrt[{n}]{y_{1}y_{2}\cdots y_{n}}}\,}
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