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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:

1. $f(0)=0$
2. $f(x)\to 0$, as $x\to \mathrm{\infty }$
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 $\mathrm{\infty }$, $f$ is initially concave, then convex

Except for property 4, the graph of $f$ bears a superficial resemblance to that of a Poisson distribution.

What function families might provide a good basis for an approximation formula? Thanks. --134.242.92.2 (talk) 14:28, 30 September 2015 (UTC)

$a{x}^b{e}^{-cx}$, for $a>0,0<b\le 1,c>0$. $x_{max}=b/c$. -- 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 $e^{a+bx+c{x}^2}((x+x_0{)}^d-x_0^d)$. 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)