Rule of thumb for using logarithmic scale When I am given a variable, I usually decide whether to take its logarithm based on gut feeling. Usually I base it on its distribution - if it has long tail (like: salaries, GDP, ...) I use logarithms.
However, when I need to preprocess a large number of variables, I use ad hoc techniques. With some tweaking I can arrive at "desired" results, but without a good argumentation.
Is there a common or widely accepted way to decide whether to scale a (single) variable with log (or, say, square root)?
Of course, for more refined techniques I need scaling related used method, the meaning of particular parameters or their relations. But e.g. for deciding whether to use log scale in a plot - distribution of a single variable should suffice.
Requirements:


*

*It should be relatively method-agnostic (I can do further rescaling, if needed).

*It should based only on the distribution of values (not e.g. semantics of data).

*It should be a sensible rule for choosing scales in plots.


I know:


*

*In linear regression, when is it appropriate to use the log of an independent variable instead of the actual values? - Stats.SE

*When are Log scales appropriate? - Stats.SE

*Box-Cox and making variance uniform (but it is for 2+ variables)

 A: As a rule of thumb, try to make the data fit a (standard) normal distribution, a uniform distribution or any other distribution where the values are more or less “evenly” distributed.
As a measurement, one thing that you could aim for is to maximize the distribution’s entropy for a fixed variance.
So, if your data is approximately log-normal distributed, taking its logarithm would probably be a good idea since afterwards it would be approximately normal distributed.
Another way to determine how to preprocess the data would be to transform it to a distribution in which an additive perturbation of a certain size would be equally significant no matter what the value that was being perturbed was. For example, if a 5 % raise in salary can be said to be equally significant no matter how much money you earn, you should probably logarithmize the data since that would make an additive perturbation equally significant for all values.
A: (So to be kosher and not mix the question with an answer.)
Right now I am using scale which minimized the following ratio:
$$\frac{\sqrt[4]{\langle (x - \bar{x})^4 \rangle}}{\sqrt{\langle (x - \bar{x})^2 \rangle}}$$
That is, after normalizing a variable (i.e. mean 0 and variance 1) I am looking to have the 4th moment as low as possible (so to penalize too long-tailed, or otherwise disperse, distributions).
For me it works (but I am not sure if it's only my using it; and if there are any easy pitfalls).
