Guess the function! January 15, 2009Posted by dorigo in mathematics, personal, physics, science.
Tags: function, Z boson
I have a problem today -actually I’ve fiddled with it for a couple of days now. So, since it does not involve particles (at least not directly), I figured I’d bounce it off the mathematically inclined among you: maybe I get an answer before I can figure my problem out by myself!
The problem is simple: find a functional form that can be a good fit, with suitable parameters, to the following graph:
(This is a residual of a Z lineshape fit to a relativistic Breit-Wigner function by the way, but you need not bother with these unnecessary details).
As you can see, we have a negative asymptote and a positive asymptote that have different values, and a central wiggling which has different “width” for the negative and positive component. I have been trying several combinations like , where g(x) is a gaussian and h(x) some kind of “warping factor” with a different slope in the negative and positive side (with respect to x=91)… But I am getting nowhere. I am sure there is somebody out there that has a good advice, so please shoot!
UPDATE: Marius suggests a function in the comments thread below. I thank him for his input, but as is, the function does not work well: see the best fit below (parameters in the upper right legend are A,B,C,D as in the function suggested by Marius):
Maybe with suitable modifications this might work, though. Hmmmm…
UPDATE: Using the hint by Marius that the addition of another arctangent could account for the different height of the two asymptotes, I have cooked up a better fit:
This is better, but I am really not satisfied. The function has 11 degrees of freedom -which is not too troublesome since there are 300 points in the graph to fit anyway; but the function is UGLY:
Any further idea on how to improve it ?
Hmmm, and I should add that having 11 parameters is a curse for me, because what I am going to do after I have a reasonable functional form is to study the parameters as a function of Z rapidity (which modifies the original graph), and parameterize those 11 dependencies… I already have a headache!
UPDATE: Lubos makes a very good attempt with a simple ratio of polynomials in the comments thread, offering (he even offers some eyeballed parameters). Nice try, but the problem is that the function seems to be very irregular. If one fits the center region, Lubos’ function obtains a good fit (see upper plot below); if one tries to extend the fit further out on the tails, however, the fit rapidly worsens (lower plot).
Despite the shortcomings, I think I will investigate some ways to fix the function offered by Lubos -it has the potential of describing with few parameters the whole shape, once tweaked a bit…
UPDATE: Lubos tried to mend himself the function he proposed above, by adding a hyperbolic tangent. The function fits better the whole range, but it still fails to catch the subtleties of the slopes… Here is a fit using his suggested parameters:
I think I will remove the hyperbolic tangent and work on some warping of the polynomial…
UPDATE: warping the x values above 91 GeV from Lubos’ polynomial with a function seems to work. The result is below:
The fit is not extremely precise, but these are residuals from a Breit-Wigner, so I guess that the multiplication of this function by the original shape will give a more than adequate parametrization, for my goals. Next up is obtaining 50 different fits like the one above, one per each interval in Z rapidity from 0 to 5.0, and parametrizing each of the seven parameter of the fits…