Started Logarithmic Transformation and Its Inverse

Transforms the data by a log transformation, modifying small and zero observations such that the transformation is linear for $x <= threshold$ and logarithmic for x > threshold. So the transformation yields finite values and is continuously differentiable.

LogSt(x, base = 10, calib = x, threshold = NULL, mult = 1)

LogStInv(x, base = NULL, threshold = NULL)

Arguments

x: a vector or matrix of data, which is to be transformed
base: a positive or complex number: the base with respect to which logarithms are computed. Defaults to 10. Use=exp(1) for natural log.
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.

Details

In order to avoid $log(x) = -\infty$ 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 $c = \frac{q_1^{1+r}}{q_3^r}$ 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 by default because they can be backtransformed relatively easily in mind.

A generalized log (see: Rocke 2003) can be calculated in order to stabilize the variance as:

function (x, a) {
 return(log((x + sqrt(x^2 + a^2)) / 2))
}

Value

the transformed data. The value $c$ used for the transformation and needed for inverse transformation is returned as attr(.,"threshold") and the used base as attr(.,"base").

Author

Werner A. Stahel, ETH Zurich
slight modifications Andri Signorell <andri@signorell.net>

References

Rocke, D M, Durbin B (2003): Approximate variance-stabilizing transformations for gene-expression microarray data, Bioinformatics. 22;19(8):966-72.

Examples

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
#>   [1]  4.440892e-16  0.000000e+00  0.000000e+00  0.000000e+00  0.000000e+00
#>   [6]  0.000000e+00 -1.776357e-15  0.000000e+00  0.000000e+00  0.000000e+00
#>  [11]  0.000000e+00  0.000000e+00  8.881784e-16  0.000000e+00  0.000000e+00
#>  [16]  5.551115e-17  8.881784e-16  0.000000e+00 -4.440892e-16  0.000000e+00
#>  [21]  0.000000e+00 -8.881784e-16  0.000000e+00  0.000000e+00  0.000000e+00
#>  [26]  0.000000e+00  0.000000e+00  0.000000e+00  0.000000e+00  4.440892e-16
#>  [31]  0.000000e+00  0.000000e+00  0.000000e+00  0.000000e+00  0.000000e+00
#>  [36]  0.000000e+00  0.000000e+00  8.881784e-16  0.000000e+00  0.000000e+00
#>  [41]  0.000000e+00  0.000000e+00  0.000000e+00  0.000000e+00  0.000000e+00
#>  [46]  0.000000e+00  0.000000e+00  0.000000e+00  0.000000e+00  8.881784e-16
#>  [51]  0.000000e+00  0.000000e+00  8.881784e-16  0.000000e+00  0.000000e+00
#>  [56]  0.000000e+00  0.000000e+00  0.000000e+00  0.000000e+00  0.000000e+00
#>  [61]  0.000000e+00  0.000000e+00 -4.440892e-16  0.000000e+00  0.000000e+00
#>  [66]  0.000000e+00  0.000000e+00  0.000000e+00  0.000000e+00  0.000000e+00
#>  [71]  8.881784e-16  0.000000e+00 -4.440892e-16  0.000000e+00  0.000000e+00
#>  [76]  0.000000e+00 -8.881784e-16  1.387779e-17  0.000000e+00 -4.440892e-16
#>  [81]  0.000000e+00 -1.387779e-17  0.000000e+00  0.000000e+00  0.000000e+00
#>  [86]  0.000000e+00  0.000000e+00  0.000000e+00  0.000000e+00 -1.776357e-15
#>  [91]  0.000000e+00  0.000000e+00  0.000000e+00  0.000000e+00  0.000000e+00
#>  [96]  0.000000e+00  0.000000e+00  4.440892e-16  0.000000e+00  0.000000e+00