Power transform

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

${\displaystyle y_{i}^{(\lambda )}={\begin{cases}{\dfrac {y_{i}^{\lambda }-1}{\lambda (\operatorname {GM} (y))^{\lambda -1}}},&{\text{if }}\lambda \neq 0\\[12pt]\operatorname {GM} (y)\ln {y_{i}},&{\text{if }}\lambda =0\end{cases}}}$

where

${\displaystyle \operatorname {GM} (y)=\left(\prod _{i=1}^{n}y_{i}\right)^{\frac {1}{n}}={\sqrt[{n}]{y_{1}y_{2}\cdots y_{n}}}\,}$

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 ${\displaystyle y_{i}^{\lambda }=\exp({\lambda \ln(y_{i})})=1+\lambda \ln(y_{i})+O((\lambda \ln(y_{i}))^{2})}$ - using Taylor series. Then ${\displaystyle {\dfrac {y_{i}^{\lambda }-1}{\lambda }}=\ln(y_{i})+O(\lambda )}$, and everything but ${\displaystyle \ln(y_{i})}$ becomes negligible for ${\displaystyle \lambda }$ sufficiently small.

The inclusion of the (λ − 1)th power of the geometric mean in the denominator simplifies the scientific interpretation of any equation involving ${\displaystyle y_{i}^{(\lambda )}}$, because the units of measurement do not change as λ changes.

Box and Cox (1964) introduced the geometric mean into this transformation by first including the Jacobian of rescaled power transformation

${\displaystyle {\frac {y^{\lambda }-1}{\lambda }}.}$

with the likelihood. This Jacobian is as follows:

${\displaystyle J(\lambda ;y_{1},\ldots ,y_{n})=\prod _{i=1}^{n}|dy_{i}^{(\lambda )}/dy|=\prod _{i=1}^{n}y_{i}^{\lambda -1}=\operatorname {GM} (y)^{n(\lambda -1)}}$

This allows the normal log likelihood at its maximum to be written as follows:

${\displaystyle {\begin{aligned}\log({\mathcal {L}}({\hat {\mu }},{\hat {\sigma }}))&=(-n/2)(\log(2\pi {\hat {\sigma }}^{2})+1)+n(\lambda -1)\log(\operatorname {GM} (y))\\[5pt]&=(-n/2)(\log(2\pi {\hat {\sigma }}^{2}/\operatorname {GM} (y)^{2(\lambda -1)})+1).\end{aligned}}}$

From here, absorbing ${\displaystyle \operatorname {GM} (y)^{2(\lambda -1)}}$ into the expression for ${\displaystyle {\hat {\sigma }}^{2}}$ produces an expression that establishes that minimizing the sum of squares of residuals from ${\displaystyle y_{i}^{(\lambda )}}$is equivalent to maximizing the sum of the normal log likelihood of deviations from ${\displaystyle (y^{\lambda }-1)/\lambda }$ and the log of the Jacobian of the transformation.

The value at Y = 1 for any λ is 0, and the derivative with respect to Y there is 1 for any λ. Sometimes Y is a version of some other variable scaled to give Y = 1 at some sort of average value.

The transformation is a power transformation, but done in such a way as to make it continuous with the parameter λ at λ = 0. It has proved popular in regression analysis, including econometrics.

Box and Cox also proposed a more general form of the transformation that incorporates a shift parameter.

${\displaystyle \tau (y_{i};\lambda ,\alpha )={\begin{cases}{\dfrac {(y_{i}+\alpha )^{\lambda }-1}{\lambda (\operatorname {GM} (y+\alpha ))^{\lambda -1}}}&{\text{if }}\lambda \neq 0,\\\\\operatorname {GM} (y+\alpha )\ln(y_{i}+\alpha )&{\text{if }}\lambda =0,\end{cases}}}$

which holds if y_i + α > 0 for all i. If τ(Y, λ, α) follows a truncated normal distribution, then Y is said to follow a Box–Cox distribution.

Bickel and Doksum eliminated the need to use a truncated distribution by extending the range of the transformation to all y, as follows:

${\displaystyle \tau (y_{i};\lambda ,\alpha )={\begin{cases}{\dfrac {\operatorname {sgn} (y_{i}+\alpha )|y_{i}+\alpha |^{\lambda }-1}{\lambda (\operatorname {GM} (y+\alpha ))^{\lambda -1}}}&{\text{if }}\lambda \neq 0,\\\\\operatorname {GM} (y+\alpha )\operatorname {sgn} (y+\alpha )\ln(y_{i}+\alpha )&{\text{if }}\lambda =0,\end{cases}}}$

where sgn(.) is the sign function. This change in definition has little practical import as long as ${\displaystyle \alpha }$ is less than ${\displaystyle \operatorname {min} (y_{i})}$, which it usually is.[5]

Bickel and Doksum also proved that the parameter estimates are consistent and asymptotically normal under appropriate regularity conditions, though the standard Cramér–Rao lower bound can substantially underestimate the variance when parameter values are small relative to the noise variance.[5] However, this problem of underestimating the variance may not be a substantive problem in many applications.[6][7]

The one-parameter Box–Cox transformations are defined as

${\displaystyle y_{i}^{(\lambda )}={\begin{cases}{\dfrac {y_{i}^{\lambda }-1}{\lambda }}&{\text{if }}\lambda \neq 0,\\\ln y_{i}&{\text{if }}\lambda =0,\end{cases}}}$

and the two-parameter Box–Cox transformations as

${\displaystyle y_{i}^{({\boldsymbol {\lambda }})}={\begin{cases}{\dfrac {(y_{i}+\lambda _{2})^{\lambda _{1}}-1}{\lambda _{1}}}&{\text{if }}\lambda _{1}\neq 0,\\\ln(y_{i}+\lambda _{2})&{\text{if }}\lambda _{1}=0,\end{cases}}}$

as described in the original article.[8][9] Moreover, the first transformations hold for ${\displaystyle y_{i}>0}$, and the second for ${\displaystyle y_{i}>-\lambda _{2}}$.[8]

The parameter ${\displaystyle \lambda }$ is estimated using the profile likelihood function and using goodness-of-fit tests.[10]

The Yeo–Johnson transformation[15] allows also for zero and negative values of ${\displaystyle y}$. ${\displaystyle \lambda }$ can be any real number, where ${\displaystyle \lambda =1}$ produces the identity transformation. The transformation law reads:

${\displaystyle y_{i}^{(\lambda )}={\begin{cases}((y_{i}+1)^{\lambda }-1)/\lambda &{\text{if }}\lambda \neq 0,y\geq 0\\[4pt]\ln(y_{i}+1)&{\text{if }}\lambda =0,y\geq 0\\[4pt]-((-y_{i}+1)^{(2-\lambda )}-1)/(2-\lambda )&{\text{if }}\lambda \neq 2,y<0\\[4pt]-\ln(-y_{i}+1)&{\text{if }}\lambda =2,y<0\end{cases}}}$

• Box, George E. P.; Cox, D. R. (1964). "An analysis of transformations". Journal of the Royal Statistical Society, Series B. 26 (2): 211–252. JSTOR 2984418. MR 0192611.
• Carroll, R. J.; Ruppert, D. (1981). "On prediction and the power transformation family" (PDF). Biometrika. 68 (3): 609–615. doi:10.1093/biomet/68.3.609.
• DeGroot, M. H. (1987). "A Conversation with George Box". Statistical Science. 2 (3): 239–258. doi:10.1214/ss/1177013223.
• Handelsman, D. J. (2002). "Optimal Power Transformations for Analysis of Sperm Concentration and Other Semen Variables". Journal of Andrology. 23 (5).
• Gluzman, S.; Yukalov, V. I. (2006). "Self-similar power transforms in extrapolation problems". Journal of Mathematical Chemistry. 39 (1): 47–56. arXiv:cond-mat/0606104. Bibcode:2006cond.mat..6104G. doi:10.1007/s10910-005-9003-7. S2CID 118965098.
• Howarth, R. J.; Earle, S. A. M. (1979). "Application of a generalized power transformation to geochemical data". Journal of the International Association for Mathematical Geology. 11 (1): 45–62. doi:10.1007/BF01043245. S2CID 121582755.

• Nishii, R. (2001) [1994], "Box–Cox transformation", Encyclopedia of Mathematics, EMS Press (fixed link)
• Sanford Weisberg, Yeo-Johnson Power Transformations