4

I was experimenting with different ways of understanding numbers in the range [0, 1], especially the extreme values, and came up with the function $\log{(x)}(1 − x)-\log{(1 − x)}x$. It has the nice property that it approaches logarithmic behavior near either extreme, so you could use it to compare percentages (or fractions) in a general way. Making the input closer to 0 or 1 by a factor of $e$ changes the output by 1, asymptotically.

Using $\log_{10}$ instead yields the popular use of "5 nines" to means 99.999% -- f(0.99999) is roughly 5. f(0.50) = 0, f(0.75) = 0.42. It produces fairly intuitive results: 90% is "about one step better" than 50%, and 99% has the same relationship to 90%.

So, question: Is this a function that is already named in the literature? Or perhaps it might be a specific case of a more general named formula? It's pretty simple, but I wasn't able to find any instances of it using a LaTeX-specific search engine.

Tim has moved to Codidact
  • 141
  • 3
    The logit function $f(x)=\log(x/(1-x))$ comes to mind. It converts a probability $x\in [0,1]$ to a log-odds $f(x)\in\mathbb{R}\cup{\pm\infty}$. In fact log-odds come up a lot in probability and statistics. The application is almost exactly as you suggested. It's cool that you've come up with something very similar. But what is your motivation for the extra factors of $(1-x)$ and $x$ compared to the logit function? – Benjamin Wang Nov 27 '22 at 01:28
  • 1
    Also related is the binary entropy function. – Misha Lavrov Nov 27 '22 at 01:33
  • See also https://en.m.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equations. If you set some parameters to zero and some as variable functions, you get an equation of the type in the question. – vvg Nov 27 '22 at 01:37
  • 1
    Ah! The logit function does the same thing, perfect! Simpler, with an almost identical shape, so I'd be satisfied just using that one and not seeing if there's a name for the one I came up with. The structure of mine is just a linear interpolation between $\log(x)$ and $-\log(1-x)$. Nothing more principled than that. :-) – Tim has moved to Codidact Nov 28 '22 at 02:42

0 Answers0