# Power transform

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Power transform

In statistics, the power transform is a family of transformations that map data from one space to another using power functions. This is a useful data (pre)processing technique used to reduce data variation, make the data more normal distribution-like, improve the correlation between variables and for other data stabilization procedures. The Box-Cox transformation is one particular way of parameterising a power transform that has advantageous properties.

Definition

The power transformation is defined as a continuously varying function, with respect to the power parameter "&lambda;", in a piece-wise function form that makes it continuous at the point of singularity ("&lambda;" = 0). For data vectors ("y"1,..., "y""n") in which each "y""i" > 0, the power transform is

:

where

: $operatorname\left\{GM\right\}\left(y\right) = \left(y_1cdots y_n\right)^\left\{1/n\right\} ,$

is the geometric mean of the observations "y"1, ..., "y""n".

The inclusion of the ("&lambda;" − 1)th power of the geometric mean in the denominator implies that the units of measurement do not change as "&lambda;" changes. That makes it possible to compare sums of squares of residuals and choose the value of "&lambda;" that minimizes that sum.

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 which incorporates a shift parameter.

:

If &tau;("Y", &lambda;, α) follows a normal distribution, then "Y" is said to follow a Box-Cox distribution.

Use of the power transform

* Power transforms are ubiquitously used in various fields. For example, [http://portal.acm.org/citation.cfm?id=1172964.1173292&coll=&dl=acm&CFID=15151515&CFTOKEN=6184618 multi-resolution and wavelet analysis] , statistical data analysis, [http://www.andrologyjournal.org/cgi/reprint/23/5/629.pdf medical research] , [http://www.springerlink.com/content/y25q020x24602701/ modeling of physical processes] , [http://www.springerlink.com/content/mt81u60813077641/ geochemical data analysis] , [http://www.blackwell-synergy.com/doi/abs/10.1111/j.1467-9876.2005.00476.x epidemiology] and many other clinical, environmental and social research areas.

Power transform activities

The SOCR resource pages contain a number of [http://wiki.stat.ucla.edu/socr/index.php/SOCR_EduMaterials_Activities_PowerTransformFamily_Graphs hands-on interactive activities with the Power Transform] using Java applets and charts.

Example

The BUPA liver data set contains data on liver enzymes ALT and &gamma;GT. The data can be found via the classic data sets page. Suppose we are interested in using log(&gamma;GT) to predict ALT. A plot of the data appears in panel (a) of the figure. There appears to be non-constant variance, and a Box-Cox transformation might help.

The log-likelihood of the power parameter appears in panel (b). The horizontal reference line is at a distance of &chi;12/2 from the maximum and can be used to read off an approximate 95% confidence interval for &lambda;. It appears as though a value close to zero would be good, so we take logs.

Possibly, the transformation could be improved by adding a shift parameter to the log transformation. Panel (c) of the figure shows the log-likelihood. In this case, the maximum of the likelihood is close to zero suggesting that a shift parameter is not needed. The final panel shows the transformed data with a superimposed regression line.

Note that although Box-Cox transformations can make big improvements in model fit, there are some issues that the transformation cannot help with. In the current example, the data are rather heavy-tailed so that the assumption of normality is not realistic and a robust regression approach leads to a more precise model.

Econometric application

Economists often characterize production relationships by some variant of the Box-Cox transformation.

Consider a common representation of production "Q" as dependent on services provided by a capital stock "K" and by labor hours "N":

:$au\left(Q\right)=alpha au\left(K\right)+ \left(1-alpha\right) au\left(N\right).,$

Solving for "Q" by inverting the Box-Cox transformation we find

:

which is known as the "constant elasticity of substitution (CES)" production function.

The CES production function is a homogeneous function of degree one.

When "&lambda;" = 1, this produces the linear production function:

: $Q=alpha K + \left(1-alpha\right)N.,$

When "λ" → 0 this produces the famous Cobb-Douglas production function:

: $Q=K^alpha N^\left\{1-alpha\right\}.,$

Activities and demonstrations

The SOCR resource pages contain a number of [http://wiki.stat.ucla.edu/socr/index.php/SOCR_EduMaterials_Activities_PowerTransformFamily_Graphs hands-on interactive activities] demonstrating the Box-Cox (Power) Transformation using Java applets and charts. These directly illustrate the effects of this transform on Qq plots, X-Y scatterplots, time-series plots and histograms.

References

*
* Carroll, RJ and Ruppert, D. [http://wiki.stat.ucla.edu/socr/uploads/b/b8/PowerTransformFamily_Biometrica609.pdf On prediction and the power transformation family] . Biometrika 68: 609&ndash;615.
*
* Handelsman, DJ. Optimal Power Transformations for Analysis of Sperm Concentration and Other Semen Variables. Journal of Andrology, Vol. 23, No. 5, September/October 2002.
* Gluzman, S and Yukalov, VI. Self-similar power transforms in extrapolation problems. Journal of Mathematical Chemistry, Volume 39, Number 1 / January, 2006, DOI 10.1007/s10910-005-9003-7, 47&ndash;56.
* Howarth, RJ and Earle, SAM. Application of a generalized power transformation to geochemical data Journal Mathematical Geology, Volume 11, Number 1 / February, 1979, DOI 10.1007/BF01043245, pages 45&ndash;62.
* Peters, JL Rushton, L, Sutton, AJ, Jones, DR, Abrams, KR, Mugglestone, MA. (2005) Bayesian methods for the cross-design synthesis of epidemiological and toxicological evidence. Journal of the Royal Statistical Society: Series C (Applied Statistics) 54 (1), 159–172, doi:10.1111/j.1467-9876.2005.00476.x

* [http://wiki.stat.ucla.edu/socr/index.php/SOCR_EduMaterials_Activities_PowerTransformFamily_Graphs SOCR Power Transform Activities and Applets]
* [http://www.stat.uconn.edu/~studentjournal/index_files/pengfi_s05.pdf Box-Cox Transformation: An Overview, Pengfei Li]

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