Box-Cox or Power transforms are a family of functions for transforming data. These functions are defined for positive data values only and are used because they can often can make very skewed datasets more Normally distributed. They can also provide so-called variance stabilization. The various forms of the transforms utilize a single parameter, λ, which is either specified by the user or is computed from the data using some form of optimization procedure on the assumption that the transformed sample is to be Normally distributed. Although the transform can be useful in a wide range of situations it may not be effective for distributions with significant outliers and heavy tails. Optimization for λ may be performed by evaluating the standard deviation, s, of the transformed distribution for a range of values for λ and selecting the λ-value that minimizes s. More generally, maximum likelihood procedures are used to estimate λ, often over an initial test range (e.g. [-3,3]) and in some cases in conjunction with graphical display of the log likelihood. In some instances an understanding of the nature of the problem and data can assist in selecting the appropriate value for λ. For example, it may be appropriate to consider only integer values. Crawley (2007, p336-8, [CRA1]) cites the example of the timber dataset included in the MASS packages with the R environment. There is a clear relationship between the volume of timber in a tree and its height and girth. With raw data the relationship would suggest a cubic/cube root relationship (λ=1/3) whereas with prior transformation of height and girth to logarithmic scales, the optimized λ=-0.08 suggesting a more appropriate value of λ=0 be used, which reduces the model to log(volume)=f(log(height),log(girth)).
Box-Cox transforms (see Box and Cox, 1964 [BOX1] for more details and worked examples) are used in both general statistical analysis and in the analysis of control charts and time series data. In its simplest form the power transform is of the form:
where c is an optional constant. Note that this form embraces the standard log transforms and the basic form of the Freeman-Tukey square root transform. The basic version of the Box-Cox transform, which is the form widely used in statistical software packages, is:
For λ=0 the transform would be infinite but it is defined as ln(x) to avoid this. For λ=1 the transform reduces to z=x-1 and for λ=-1 it reduced to z=1-1/x. A graph of this transform for a range of positive λ values is provided at the end of this topic. The inverse or back-transform is:
A generalized version of the Box-Cox transform, which incorporates a data transpose or shift parameter α and the inclusion of the Geometric Mean (GM) of the data values in the denominator is:
In this form the original data values are shifted by the factor α and adjusted by the GM component which ensures constant scaling as λ varies. An example of the application of the Box-Cox transformation applied to Radon data is provided in the Graphics section (probability plots).
Box-Cox transforms: λ = [0.5,1,2,4] and x=[0,2]