Wikipedia:Reference desk/Archives/Mathematics/2015 September 30

= September 30 =

Curve fitting
I need an approximation formula for a family of functions that I cannot evaluate easily. Each function $$f(x)$$ has these properties: Except for property 4, the graph of $$f$$ bears a superficial resemblance to that of a Poisson distribution.
 * 1) $$f(0) = 0$$
 * 2) $$f(x) \rightarrow 0$$, as $$x \rightarrow \infty$$
 * 3) $$f$$ has a single maximum, at $$x_{max}$$ (the value of $$x_{max}$$ varies among the functions in the family)
 * 4) between 0 and $$x_{max}$$, $$f$$ is concave (aka concave downward) (Edited: This is not exactly true: $$f$$ has a small region of convexity where $$x \in [0,y)$$, for some small value of $$y$$. It's OK if this is ignored.)
 * 5) between $$x_{max}$$ and $$\infty$$, $$f$$ is initially concave, then convex

What function families might provide a good basis for an approximation formula? Thanks. --134.242.92.2 (talk) 14:28, 30 September 2015 (UTC)
 * $$ax^be^{-cx}$$, for $$a>0,\ 00$$. $$x_{max}=b/c$$. -- Meni Rosenfeld (talk) 17:03, 30 September 2015 (UTC)


 * I'm pretty sure your expression actually gives $$x_{max} = b/c$$. Dragons flight (talk) 08:55, 1 October 2015 (UTC)
 * You're right of course, fixed. That's what happens when I try to use different notation in my calculations and the writeup... -- Meni Rosenfeld (talk) 12:49, 1 October 2015 (UTC)


 * Just to make sure I understand the description, does the graph look something like this ?

^ |         o  |     o         o |  o                o |o                        o   o->


 * StuRat (talk) 17:47, 30 September 2015 (UTC)
 * Qualitatively, yes, but I'd add that graphs of the functions are markedly skewed to the left.--134.242.92.2 (talk) 19:05, 30 September 2015 (UTC)
 * Maxwell distribution. 120.145.34.119 (talk) 05:55, 1 October 2015 (UTC)
 * This doesn't start out downward convex. -- Meni Rosenfeld (talk) 08:36, 1 October 2015 (UTC)
 * On closer examination, I found that $$f$$ actually has a small region of convexity in $$[0,y)$$ for some small $$y$$. This is kind of subtle and it's OK if it is ignored. --134.242.92.2 (talk) 22:35, 1 October 2015 (UTC)
 * In this case, b can be slightly higher than 1 in my function above. -- Meni Rosenfeld (talk) 10:39, 2 October 2015 (UTC)
 * If you want more parameters, you can use something like $$e^{a+bx+cx^2}((x+x_0)^d-x_0^d)$$. In particular, the quadratic term in the exponent (which also exists in the Maxwell distribution pdf) allows for a steeper decline once the maximum is reached. -- Meni Rosenfeld (talk) 13:31, 2 October 2015 (UTC)