I’m working on a project and once again a topic has come up that I struggle with: a good way to express fractions between 0 and 1 where typical values are either very close to 0 or 1. A good example is the neutral fraction of a species in a stellar atmosphere, or the transmission of the atmosphere as a function of wavelength. In both cases, things are usually very close to 0 or 1, and rarely spend much time between, so trying to plot it is frustrating: you basically get loooong stretches where the number is pinned to 1 or 0, even though there is usually some well known value not quite at those numbers.
One solution is to use logarithms. Engineers do this when they talk about the number of “nines”: if you only have one dud in 1,000,000, then your fraction of good items is 0.999999, or “six nines”. If you waste only one unit of energy per 100, you have “two nines.” Mathematically, the number of nines is -log(1-x), where x is your number (for x = 0.999999, 1-x = 0.000001 = 10^-6, and -log(10^-6) = 6.
But this fails at the other end, since near zero you want to express things in terms of “zeros”: if you transmit only 1% (0.01) of flux through the atmosphere you want to express this as -log(x) (so, two zeros).
I like to think about this function:
log(x) – log(1-x)
It basically stretches out the the y-axis logarithmically in both directions. For values of x near zero, the second term is negligible (so small as to not matter), and the function returns a value close to -6 for “six zeros”). For values near 1 the first term is small, so the function returns almost exactly 6 for “six nines”. For exactly 50% (a perfect balance) it returns 0.
Does anyone know if this function already has a name? Wolfram alpha and Google don’t seem to recognize it. It is a simple map of (0,1) onto the real axis, but there are other ways to do that I don’t think I’ve seen this particular one before (and certainly not with base 10 logs).
Any suggestions? The “zeros and nines” function? “Nine-oh”?
Update: In the comments below Mike Bottom names that function: “logit” from probability theory. My use is strictly base 10 and in a different context, but as I suspected the function is well known and named. John Johnson correctly asserts that, like all useful math and interesting physics, it was probably discovered by a Russian in the 50’s, but Wikipedia only knows about the earlier coinage by Joseph Berkson in 1944.
this function is called the “logit” function. it is the inverse of the logistic function. you are using it in base 10, but it’s usually used in base e. it behaves a lot like the inverse of the CDF of the normal distribution (the “probit” function), which you might consider using instead if you’re doing probabilistic bayesian stuff, because the probit simplifies those sorts of calculations more than the logit.
-mike bottom
I’ve taken to calling it a Wright Function and then tell people it was actually discovered by a Russian graduate student back in 1954. -JohnJohn