Interpreting the Lambdas of Yeo Johnson Transformation?

Question

The following is the table of Lambda values that describe what the resulting dataset would look like after a Box Cox transformation:

What is the equivalent table for Yeo Johnson's lambda values? Cant seem to find it online.

Answer 1

A table adds little, but a picture can add a lot more to our understanding. I offer two pictures.

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Unlike the Box-Cox transformation, which applies to positive numbers, the Yeo-Johnson transformation applies to all numbers. It does so by splitting the real line at zero, shifting the positive values by $1$ and the negative values by $-1,$ and applying a Box-Cox transformation to the absolute values, negating them when the argument is negative. In effect, it sews two Box-Cox transformations together. However, they have "inverse" Box-Cox parameters. The natural origin of the Box-Cox parameters is $\lambda = 1$ and the "inverse" parameter is

$$\lambda^\prime = 2 - \lambda,$$

reflecting the parameter line around $\lambda = 1.$ The sewing is smooth (as you will see in the first plot below) because all Box-Cox transformations are by design made to agree with the identity transformation at $x = 1.$

For pictures of the Box-Cox transformations and some explanation of their construction, see https://stats.stackexchange.com/a/467525/919. These transformations are given by

$$\operatorname{BC}(x;\lambda) = \frac{x^\lambda - 1}{\lambda}$$

(which has the limiting value of $\log(x)$ when $\lambda = 0$). They can be inverted: when $y$ is the transformed value, the original $x$ is recovered by

$$\operatorname{BC}^{-1}(y;\lambda) = (1 + \lambda y)^{1/\lambda}$$

(limiting to the exponential function when $\lambda = 0$).

The Yeo-Johnson transformation is

$$\operatorname{YJ}(x;\lambda) = \left\{\begin{aligned}\operatorname{BC}(1+x,\lambda), && x \ge 0\\ -\operatorname{BC}(1-x, \lambda^\prime),&& x \lt 0.\end{aligned} \right.$$

These can all be inverted by inverting the positive and negative values separately.

The implementation in any programming language is thereby simple. In R, for instance, it is

BC <- function(x, lambda) ifelse(lambda != 0, (x^lambda - 1) / lambda, log(x))
YJ <- function(y, lambda) ifelse(y >= 0, BC(y + 1, lambda), -BC(1 - y, 2-lambda))

The graphs of $\operatorname{YJ}$ show the effects on the data for various $\lambda,$

Here's what they do to a reference (Normal) distribution (the green distribution for $\lambda = 1$ in the middle panel):

Like the Box-Cox family, these transformations make a distribution more positively skewed when $\lambda \gt 1$ and more negatively skewed when $\lambda \lt 1.$