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)
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 |
mult |
a tuning constant affecting the transformation of small values, see |
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)=log_10(x), if x>c, log_10(c) - (c-x)/(c log(10)), otherwise
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 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))
}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").
Werner A. Stahel, ETH Zurich
slight modifications Andri Signorell <andri@signorell.net>
Rocke, D M, Durbin B (2003): Approximate variance-stabilizing transformations for gene-expression microarray data, Bioinformatics. 22;19(8):966-72.
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
Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.