what is your opinion for the best function can be fitted to these plots?

Question

I have three sets of data (x,f(x)) as follows:

setA={{0., -0.089772065541}, {0.405465108108, -0.108090184631}, \
{0.69314718056, -0.12075482495}, {1.09861228867, -0.136713310362}, \
{2.30258509299, -0.157557628096}, {3.91202300543, -0.124471537798}, \
{5.33271879327, -0.0801408424879}, {5.99146454711, -0.0626234699497}, \
{6.80239476332, -0.0451132549058}, {7.51534457118, -0.0332461226979}}

setB={{0., -2.73726021706}, {0.405465108108, -2.5635349861}, \
{0.69314718056, -2.45687874502}, {1.09861228867, -2.33885028583}, \
{2.30258509299, -2.25010185834}, {3.91202300543, -2.61123676241}, \
{5.33271879327, -3.14138743301}, {5.99146454711, -3.4084055453}, \
{6.80239476332, -3.74171459151}, {7.51534457118, -4.05221791235}}

setC={{0., -0.0513220378362}, {0.405465108108, -0.0598535159266}, \
{0.69314718056, -0.0628371387026}, {1.09861228867, -0.0622774482268}, \
{2.30258509299, -0.0396889415734}, {3.91202300543, -0.0121637513607}, \
{5.33271879327, -0.00330346154785}, {5.99146454711, \
-0.00172143838541}, {6.80239476332, -0.000717519675494}, \
{7.51534457118, -0.000325599564725}}

where their plots respectively are

I look for a function which can be fitted to these data and has small error. I tried the Morse function by Mathematica with the following form $$ a (1 - \exp[-b (x - c)])^4 + d $$ where $a$, $b$, $c$ and $d$ are fit parameters. The fitted plot seem so good as follows respectively

but they have large errors up to 370% (specially in very small data) which is not good at all. As I'm not a mathematician and don't know large category of functions I decided to expose it to discussion in this site hoping that you (who know mathematics well) can help me by suggesting an appropriate function. Thanks in advance.

Answer 1

You can fit a polynomial function with the structure

$$ s(x,t) = \sum_{k=1}^nc_k(t)x^k $$

with $c_k(t) = \sum_{j=1}^m \alpha_j t^j$ as follows in the MATHEMATICA script

grA = ListPlot[setA];
grB = ListPlot[setB];
grC = ListPlot[setC];

fA = Fit[setA, {1, x, x^2, x^3, x^4, x^5, x^6}, x];
fB = Fit[setB, {1, x, x^2, x^3, x^4, x^5, x^6}, x];
fC = Fit[setC, {1, x, x^2, x^3, x^4, x^5, x^6}, x];

coeft[pol_, k_] := D[pol, {x, k}]/k! /. {x -> 0}

Ct = Table[Fit[{{0, coeft[fA, k]}, {1/2, coeft[fB, k]}, {1, coeft[fC, k]}}, {1, t, t^2}, t], {k, 0, 6}];

fun = Sum[Ct[[k]] x^(k - 1), {k, 1, 7}];

gr0 = Plot[{fun /. {t -> 0}, fun /. {t -> 1/2}, fun /. {t -> 1}}, {x, 0, 7.6}];
Show[gr0, grA, grB, grC]

Plot3D[fun, {x, 0, 7.6}, {t, 0, 1}, PlotRange -> All]