Table 1.

Summary of transformations

Transformation	Mathematical Definition f(y;θ), f-1 (x; θ)	Jacobian Jθ (y)	Parameter Bounds and Constraints
Linlog	$\begin{array}{l}f(y;\theta )=\{\begin{array}{l}(y-\theta )/\theta +\log (\theta );y\le \theta \\ \log (y);y>\theta \end{array}\\ f^{-1}(x;\theta )=\{\begin{array}{l}\theta (x-\log \theta +1);x<\log (\theta )\\ \exp (x);x\ge \log (\theta )\end{array}\end{array}$	$\begin{array}{l}1/\theta ;y\le \theta \\ 1/y;y>\theta \end{array}$	$\begin{array}{l}\theta \in [\min (y),\max (y)],\\ \theta \ge 0\end{array}$
Generalized Arcsinh	$\begin{array}{l}f(y;\mathit{\theta })=\log (a+by+\sqrt{{(a+by)}^2+1})+c\\ f^{-1}(x;\mathit{\theta })=\frac{1}{2}\left(e^{(x-c)}-e^{-(x-c)}\right)\end{array}$	$\frac{b+\frac{1}{2}(2(ba+b^{2}y){({(a+by)}^2+1)}^{-1/2})}{a+by+\sqrt{{(a+by)}^2+1}}$	$\begin{array}{l}\mathit{\theta }=\{a,b,c\};a,c\ge 0;\\ b>0\end{array}$
Biexponential	$\begin{array}{l}f(y;\mathit{\theta })=\text{no closed form}\\ f^{-1}(x;\mathit{\theta })=a{e}^{(b(x-w))}-c{e}^{(-d(x-w))}+\text{f}\end{array}$	1 = (abeb(x-w)+ cde-d(x-w))	$\begin{array}{l}\mathit{\theta }=\{a,b,c,d,\text{f},w\};a,c\in (0,1]\\ \text{f}=0,w\in ℝ,b,d\ge 0\end{array}$
Generalized Box-Cox	$\begin{array}{l}f(y;\theta )=\begin{array}{c}\frac{\mathrm{sgn}(y)|y{|}^{\theta }-1}{\theta };\theta \in ℝ\end{array}\\ f^{-1}(x;\theta )=\begin{array}{c}\mathrm{sgn}(\theta x+1)|\theta x+1{|}^{\frac{1}{\theta }};\theta \in ℝ\end{array}\end{array}$	|y|θ-1	θ ∈ ℝ

Summary of transformations for flow cytometry. The transformations examined in this study, together with their inverses, Jacobians and parameter restrictions. f(y; θ) is the transformation function typicallly applied to untransformed flow cytometry data, y, whereas f^-1(x; θ) is its inverse. For the biexponential, the transformation f(.) has no closed form and must be solved numerically. Consequently, the Jacobian of the biexponential transformation is given by the reciprocal of the Jacobian of the inverse transformation, and therefore depends directly on the transformed data, x. sgn is the signum function, also known as the sign function, which extracts the sign of a real number.