DescTools (version 0.99.42)

LogSt: Started Logarithmic Transformation and Its Inverse

Description

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.

Usage

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.

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").

Details

In order to avoid $log(x)=-\mathrm{\infty }$ 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

$$g(x)=\{\begin{array}{ll}lo{g}_{10}(x)& \text{for }x\ge c\\ lo{g}_{10}(c)-\frac{c-x}{c\cdot log(10)}& \text{for }x<c\end{array}$$

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 $lo{g}_{10}$ 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))
}

References

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

See Also

log, log10

Examples

# NOT RUN {
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
# }