LogSt: Started Logarithmic Transformation and Its Inverse

In DescTools: Tools for Descriptive Statistics

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.

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(s)

Werner A. Stahel, ETH Zurich
slight modifications Andri Signorell <[e-mail]>

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

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

DescTools documentation built on Sept. 26, 2024, 1:07 a.m.