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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:

$f(0)=0$
$f(x)\rightarrow 0$ , as $x\rightarrow \infty$
$f$ has a single maximum, at $x_{max}$ (the value of $x_{max}$ varies among the functions in the family)
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.)
between $x_{max}$ and $\infty$ , $f$ 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)[reply]

ax^{b}e^{-cx}

, for

a>0,\ 0<b\leq 1,\ c>0

.

x_{max}=b/c

. -- Meni Rosenfeld (talk) 17:03, 30 September 2015 (UTC)[reply]

I'm pretty sure your expression actually gives

x_{max}=b/c

. Dragons flight (talk) 08:55, 1 October 2015 (UTC)[reply]

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)[reply]

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)[reply]

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)[reply]

Maxwell distribution. 120.145.34.119 (talk) 05:55, 1 October 2015 (UTC)[reply]

This doesn't start out downward convex. -- Meni Rosenfeld (talk) 08:36, 1 October 2015 (UTC)[reply]

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)[reply]

In this case, b can be slightly higher than 1 in my function above. -- Meni Rosenfeld (talk) 10:39, 2 October 2015 (UTC)[reply]

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)[reply]