Qfyilyi

I have "data" points as given below (e.g., for x-value = 1, the corresponding value of y is -23.110606616537147. (I apologize, it is rather large data array.) I need to find out the exact function that generated these values. I tried to guess by assuming some functional forms like below in Nonlinearfit, but no matter what I do, I do not get a perfect match between the actual data points and the fitted model. For some similar looking data, earlier I successfully guessed a simple functional form like c0*x^c1, and it was indeed a correct one. But this one gives me a headache. Any hints would be appreciated.

 data = {{1, -23.110606616537147`}, {2, -22.634559807032698`}, {3, 

-22.169391395259122`}, {4, -21.714928417099323`}, {5, 

-21.27099702070698`}, {6, -20.837422557417913`}, {7, 

-20.414029677397547`}, {8, -20.00064242987733`}, {9, 

-19.59708436779354`}, {10, -19.20317865660647`}, {11, 

-18.818748187036604`}, {12, -18.44361569142125`}, {13, 

-18.077603863354696`}, {14, -17.72053548024153`}, {15, 

-17.37223352835917`}, {16, -17.03252132999208`}, {17, 

-16.701222672174307`}, {18, -16.37816193655099`}, {19, 

-16.06316422984783`}, {20, -15.756055514421238`}, {21, 

-15.45666273835037`}, {22, -15.164813964524406`}, {23, 

-14.880338498176549`}, {24, -14.603067012321297`}, {25, 

-14.332831670558821`}, {26, -14.069466246725915`}, {27, 

-13.81280624089262`}, {28, -13.562688991228022`}, {29, 

-13.318953781288066`}, {30, -13.081441942312981`}, {31, 

-12.849996950157491`}, {32, -12.62446451651955`}, {33, 

-12.40469267417549`}, {34, -12.190531855974797`}, {35, 

-11.9818349673951`}, {36, -11.77845745250421`}, {37, 

-11.580257353223834`}, {38, -11.387095361836874`}, {39, 

-11.198834866724152`}, {40, -11.015341991362185`}, {41, 

-10.83648562665372`}, {42, -10.662137456702512`}, {43, 

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-9.00797457083459`}, {54, -8.878741746823117`}, {55, 

-8.752604237770383`}, {56, -8.629472377884344`}, {57, 

-8.509259051052561`}, {58, -8.391879645785975`}, {59, 

-8.277252008260307`}, {60, -8.165296393723994`}, {61, 

-8.05593541652889`}, {62, -7.949093999027778`}, {63, 

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-6.176201483335542`}, {84, -6.109801707026194`}, {85, 

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-2.9999981237621083`}, {178, -2.983408711517164`}, {179, 

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-2.9031072210435833`}, {184, -2.887556738792709`}, {185, 

-2.872170024766015`}, {186, -2.8569445415004098`}, {187, 

-2.8418778032806804`}, {188, -2.826967374155622`}, {189, 

-2.812210867058904`}, {190, -2.7976059425004576`}, {191, 

-2.7831503072851684`}, {192, -2.7688417138905446`}, {193, 

-2.754677958553913`}, {194, -2.7406568810289835`}, {195, 

-2.726776362987283`}, {196, -2.713034327288908`}, {197, 

-2.6994287369175294`}, {198, -2.685957594145642`}, {199, 

-2.6726189392571844`}, {200, -2.659410850234966`}, {201, 

-2.6463314412821766`}, {202, -2.6333788625233256`}, {203, 

-2.620551298593924`}, {204, -2.607846968355005`}, {205, 

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-2.073983472006926`}, {258, -2.065992599822153`}, {259, 

-2.058062768049216`}, {260, -2.050193284216243`}, {261, 

-2.0423834658368696`}, {262, -2.0346326410997926`}, {263, 

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-2.0117275563473562`}, {266, -2.004206182315287`}, {267, 

-1.9967405874795818`}, {268, -1.9893301568484185`}, {269, 

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-1.9389519432101352`}, {276, -1.931961735476371`}, {277, 

-1.925021542799568`}, {278, -1.9181308327120814`}, {279, 

-1.9112890808085006`}, {280, -1.9044957695265645`}, {281, 

-1.8977503886127203`}, {282, -1.891052435105641`}, {283, 

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-1.820368029301432`}, {294, -1.814202389691782`}, {295, 

-1.8080782314221209`}, {296, -1.8019951386958164`}, {297, 

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-1.7839881824124155`}, {300, -1.7780653067123846`}}



    NonlinearModelFit[data, 

 c0 + c1*x^c2 + c3*x^c4, {c0, c1, c2, c3, c4}, x]

In the absence of additional information about the form, and just eyeballing the shape makes it look like a rational polynomial-ish thing, I vote for...

nlf = NonlinearModelFit[data, (c0 + c1 x + c2 x^2)/(c3 + c4 x + x^c5), {c0, c1, c2, c3, c4, c5}, x];

(-43612.1 - 1735.16 x - 2.10241 x^2)/(1843.92 + 116.08 x + x^2.25431)

 nlf["AdjustedRSquared"]

 nlf["FitResiduals"] // MinMax

0.999999

{-0.0134303, 0.014954}

 Plot[nlf[x], {x, 1, 300}, Epilog -> Point[data]]

It also resembles the error function:

fit = NonlinearModelFit[data, a Erf[(x - x0)/(Sqrt[2] s)] + y0, {a, x0, y0, s}, x]