LogSt {DescTools} | R Documentation |

## 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 |

`threshold` |
constant |

`mult` |
a tuning constant affecting the transformation of small values, see |

### 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 <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.

### See Also

### 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*version 0.99.54 Index]