Transforming variable density on log scale with R

Question

I want to plot the density of variable whose range is the following:

 Min.   :-1214813.0  
 1st Qu.:       1.0  
 Median :      40.0  
 Mean   :     303.2  
 3rd Qu.:     166.0  
 Max.   : 1623990.0

The linear plot of the density results in a tall column in range [0,1000], with two very long tails towards positive infinity and negative infinity. Hence, I'd like to transform the variable to a log scale, so that I can see what's going on around the mean. For example, I'm thinking of something like:

log_values = c( -log10(-values[values<0]), log10(values[values>0]))

which results in:

Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
-6.085   0.699   1.708   1.286   2.272   6.211 

The main problem with this is the fact that it doesn't include the 0 values. Of course, I can shift all the values away from 0with values[values>=0]+1, but this would introduce some distortion in the data.

What would be an accepted and scientifically solid way of transforming this variable to the log scale?

Answer 1

What you have is essentially what @James suggests. This is problematic for values in (-1,1), especially those close to the origin:

x <- seq(-2, 2, by=.01)
plot(x, sign(x)*log10(abs(x)), pch='.')

Something like this may help:

y <- c(-log10(-x[x<(-1)])-1, x[x >= -1 & x <= 1], log10(x[x>1])+1)

plot(x, y, pch='.')

This is continuous. One can force C^1 by using the interval (-1/log(10), 1/log(10)), which is found by solving d/dx log10(x) = 1 :

z <- c( -log10(-x[x<(-1/log(10))]) - 1/log(10)+log10(1/log(10)),
         x[x >= -1/log(10) & x <= 1/log(10)],
         log10(x[x>1/log(10)]) + 1/log(10)-log10(1/log(10))
       )
plot(x, z, pch='.')