LogSt: Started Logarithmic Transformation and Its Inverse

Description

Transforms the data by a $log10$ transformation, modifying small and zero observations such that the transformation yields finite values.

Usage

LogSt(x, calib = x, threshold = NULL, mult = 1)
LogStInv(x, threshold = NULL)

Arguments

a vector or matrix of data, which is to be transformed

calib

a vector or matrix of data used to calibrate the transformation(s), i.e., to determine the constant $c$ needed

threshold

constant $c$ that determines the transformation. The inverse function LogStInv will look for an attribute named "threshold" if the argument is set to NULL.

mult

a tuning constant affecting the transformation of small values, see Details.

Value

$c$ used for the transformation and needed for inverse transformation is returned as attr(.,"threshold").

Details

In order to avoid $log(x) = -inf$ for $x=0$ in log-transformations there's often a constant added to the variable before taking the $log$. This is not always a pleasable strategy. The function LogSt handles this problem based on the following ideas:

The modification should only affect the values for "small" arguments.
What "small" is should be determined in connection with the non-zero values of the original variable, since it should behave well (be equivariant) with respect to a change in the "unit of measurement".
The function must remain monotone, and it should remain (weakly) convex.

These criteria are implemented here as follows: The shape is determined by a threshold $c$ at which - coming from above - the log function switches to a linear function with the same slope at this point.

This is obtained by

$$g(x) = \left\{\begin{array}{ll} log_{10}(x) &\textup{for }x \ge c\\ log_{10}(c) - \frac{c - x}{c \cdot log(10)} &\textup{for } x < c \end{array}\right. $$

Small values are determined by the threshold $c$. If not given by the argument threshold, it is determined by the quartiles $q_1$ and $q_3$ of the non-zero data as those smaller than $c=q_1^{1+r}/q_3^r$ where $r$ can be set by the argument mult. The rationale is, that, for lognormal data, this constant identifies 2 percent of the data as small. Beyond this limit, the transformation continues linear with the derivative of the log curve at this point.

Another idea for choosing the threshold $c$ was: median(x) / (median(x)/quantile(x, 0.25))^2.9) The function chooses $log_{10}$ rather than natural logs because they can be backtransformed relatively easily in the mind.

Examples

Run this code

dd <- c(seq(0,1,0.1), 5 * 10^rnorm(100, 0, 0.2))
dd <- sort(dd)
r.dl <- LogSt(dd)
plot(dd, r.dl, type="l")
abline(v=attr(r.dl, "threshold"), lty=2)

x <- rchisq(df=3, n=100)
# should give 0 (or at least something small):
LogStInv(LogSt(x)) - x