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I have a set of data

  {{0.01`, 0.048210580093612655`}, {0.02`, 
0.04823954678734667`}, {0.03`, 0.04821555471430313`}, {0.04`, 
0.04811564106936836`}, {0.05`, 0.04815182711185681`}, {0.06`, 
0.04800244159194423`}, {0.07`, 0.047987407067919395`}, {0.08`, 
0.04785766284608522`}, {0.09`, 0.047771957661935774`}, {0.1`, 
0.04772401202975343`}, {0.11`, 0.0476528836847016`}, {0.12`, 
0.04757077818998488`}, {0.13`, 0.04752808223718771`}, {0.14`, 
0.04743123828481375`}, {0.15`, 0.047472879479558236`}, {0.16`, 
0.04753933712481332`}, {0.17`, 0.0475047868063725`}, {0.18`, 
0.04757767540247086`}, {0.19`, 0.04759783119112234`}, {0.2`, 
0.04756174676025805`}, {0.21`, 0.04762931320390573`}}

And I'm trying to find a model to plot its best fit, so far, I have managed to come across with two seed, one with function $a\exp[-b (x - c)^8]$ and it produces enter image description here

Another one is $a\exp[-b (x - c)^6] + d \sin(\omega x + \phi)$ , and it produces enter image description here

The second one wasn't that bad but I feel it surely can be improved. Can anyone suggests me a better seed for this model?
I haven't done much this sort of work before, I appreciate any advice!

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    You need to give some background about how the data is generated. Often how the data is created suggests or dictates a particular model. Otherwise you could come up with any model and force it to fit. – 2017-01-06
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    @IanMiller. The plot represents energy with respect to a parameter value, the parameter is from a six order action term for serving density matrix in Feynman's path integral formalism, this parameter has its limit $(0.01, 0.21)$. The data points are not sequential, so neither the energy should, in which case, if we can come up with a model and force it to fit isn't a bad idea perhaps? – 2017-01-06
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    @GavyLittlewolf I don't think you gave enough information about what the model should be. It would be a decent fit even with a straight line or a second order polynomial if you want a minimum. The second form is probably an overkill. You have 22 points and you fit 6 parameters. – 2017-01-06
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    My point was you could perfectly fit it with a polynomial if you used terms up to $x^{20}$ but thats probably not appropriate. Does the generator of the data suggest a particular model that it should fit? Also what is your aim for the model? If the parameter has a limit (0.01,0.21) are you going to use it to interpolate only? – 2017-01-06

1 Answers 1

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Well, I did a bunch of tries, but I forgot the most basic one. A 6-order polynomial function is probably our best bet, and in fact the result is satisfying. So, with the seed $ax^6+bx^5+cx^4+dx^3+ex^2+fx+g$

and we get 

   {a -> -9.30484, b -> -19.5562, c -> 9.04087, d -> -0.72346, e -> -0.073357, f -> 0.00258817, g -> 0.0482028}

Here's the eventual plot

enter image description here

Thanks to anyone who intended to help!

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    What was your criteria for best? – 2017-01-06
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    A degree four polynomial looks (to my eye) very similar to a degree six polynomial. – 2017-01-06
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    @Ian . Yeah, I did realise that I should have given a deeper thought about the criteria for the best fit before the post. But at this stage, this works well enough for finding the optimised parameter value. Thanks for the help! – 2017-01-06