Oftentimes I wish to express one time series (say) relative to a second one. The two obvious candidate approaches are to take their difference or their ratio. Whereas the difference is symmetric with respect to which variable is subtracted from which (save for sign of the result), ratios are not, however symmetry can be restored by taking the log of the ratio.

When would one prefer a difference over a log-ratio? I have a hunch the answer may have to do with scaling, although I cannot see how. I also wonder if there is an L1 versus L2 norm distinction buried deep in here somewhere. Those are just 'feelings' that could be completely misplaced though.

Relatedly, why do I so frequently see the ratio of a difference over one of the variables (i.e. [X-Y]/Y) computed? What is the advantage of this added complexity? I am specifically interested in use-cases where there is no clear 'reference' among the two variables, meaning the choice of Y here would be arbitrary, which is the same problem suffered by simple (non-log) ratios in this circumstance.

At present I find myself often making two arbitrary decisions which nonetheless determine the resulting composite time-series: whether to go the difference or the ratio route, and then (unless taking the log of a ratio and not using [X-Y]/Y) which variable should be the 'reference' against which the other is described.


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