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September 30

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Curve fitting

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I need an approximation formula for a family of functions that I cannot evaluate easily. Each function has these properties:

  1. , as
  2. has a single maximum, at (the value of varies among the functions in the family)
  3. between 0 and , is concave (aka concave downward) (Edited: This is not exactly true: has a small region of convexity where , for some small value of . It's OK if this is ignored.)
  4. between and , is initially concave, then convex

Except for property 4, the graph of 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]

, for . . -- Meni Rosenfeld (talk) 17:03, 30 September 2015 (UTC)[reply]
I'm pretty sure your expression actually gives . 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 actually has a small region of convexity in for some small . 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 . 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]