Transformations

Box Cox power transformation

The Box-Cox power transformation is a data transformation to achieve a better normality and to stabilize the variance. In MCRA, the transformation parameter $p$ in $(y^p - 1)/p$ is determined by maximizing the log-likelihood function

$$l(p) = -\frac{n}{s} \log \left[ \frac{1}{n} \sum_{i=1}^{n} (y_{i}^{(p)} - \overline { y^{(p)}})^2 \right] + (p-1) \sum_{i=1}^n \log y_{i}$$

where $i$ indexes the $n$ observations and

$$\overline { y^{(p)}} = \frac{1}{n} \sum_{i=1}^{n} y_{i}^{(p)}$$

is the average of the $y_{i}^{(p)}$ (Box & Cox, 1964) [Box et al., 1964].