Tuesday, November 25, 2014

Graphs with log-scales in the positive and negative domains

Often to show data that is highly dispersed one will compress the data by graphing its log. One down-side of this is that it only works if the data is all-positive or all-negative (if you use \(-ln(-x)\)). If your data contains zero and/or points in both domains then the you have to do something else. Here is a simple extension that uses a linear function around zero to smoothly connect a log function and it's opposite. $$x= \begin{cases} \ln(x) & \text{if }x>e\\ x/e & \text{if }-e\leq x\leq e\\ -\ln(-x) & \text{if }x<-e \end{cases}$$ The function is log-linear-log ("trilog").
You can get a simple Stata utility -trilog- from here to make this transformation and create axis labels.
Another intuitive extension would be to shift the log and its opposite closer to zero, such as $$x= \begin{cases} \ln(x+1) & \text{if }x\geq0\\ -\ln(-x+1) & \text{if }x<0 \end{cases}$$ The downside of this is that no longer are equal proportional changes reflected as equal distance changes.

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