# Find a best fitting curve for some data with no regular pattern

Suppose some data with no regular pattern is given, so common patterns of functions all fail to fit it. Then how can one find the best fitting curve? I believe there will be some data for which we can not find any regular pattern. One can try with the following data (luckily, we know that this sequence has a limit):

  {{1, 0.08888888888888872},{2, 0.13842641081508972},{3, 0.17153280014540232},
{4, 0.19586894934728782},{5, 0.21482500502852542},{6, 0.23017213935351592},
{7, 0.24294479920054762},{8, 0.25379708095478332},{9, 0.26316771779066082},
{10, 0.27136463872018062},{11, 0.27861184976714762},{12, 0.28507709754351612},
{13, 0.29088903838678592},{14, 0.29614834727916052},{15, 0.30093516063430942},
{16, 0.30531421254734282},{17, 0.30933847047870212},{18, 0.31305176609215322},
{19, 0.3164907360488469}}


Any help or suggestion will be appreciated!

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May be something here of use: http://mathematica.stackexchange.com/a/14232/363

data=
{{1, 0.08888888888888872},{2, 0.13842641081508972},{3, 0.17153280014540232},
{4, 0.19586894934728782},{5, 0.21482500502852542},{6, 0.23017213935351592},
{7, 0.24294479920054762},{8, 0.25379708095478332},{9, 0.26316771779066082},
{10, 0.27136463872018062},{11, 0.27861184976714762},{12, 0.28507709754351612},
{13, 0.29088903838678592},{14, 0.29614834727916052},{15, 0.30093516063430942},
{16, 0.30531421254734282},{17, 0.30933847047870212},{18, 0.31305176609215322},
{19, 0.3164907360488469}};

nlm = NonlinearModelFit[data, {a + b Log[c x], c > 0}, {a, b, c}, x];
Print@Normal@nlm;
Print@FindFit[data, {a + b Log[c x], c > 0}, {a, b, c}, x];
Show[Plot[nlm[x], {x, 1, Length@data}, PlotRange -> All],
ListPlot[data, PlotStyle -> Directive[Red, PointSize[0.02]]]]


See also, http://reference.wolfram.com/mathematica/tutorial/CurveFitting.html - particularly the section: 'Searching for general fits to data'.

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Thanks very much! As the example in the question, I can give more data and the sequence has a limit. But in more general case waht if we can not find a common pattern of function to fit the data very well? –  Eden Harder Aug 15 '13 at 10:12
I've added a link to the curve fitting tutorial. There are only a few general cases to try before one may need to resort to a polynomial fit. –  Chris Degnen Aug 15 '13 at 10:19
Thanks a lot! As the example in the question, I have try some possible cases. Since the sequence will converge, a polynomial fit will not help after all. –  Eden Harder Aug 15 '13 at 10:41