Exact function that generated the data

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, \
-10.492171978179679}, {44, -10.326466513462087}, {45, \
-10.164901217751611}, {46, -10.00735908041173}, {47, \
-9.853725920778135}, {48, -9.703890378719906}, {49, \
-9.557743900241988}, {50, -9.415180718431747}, {51, \
-9.27609783005945}, {52, -9.140394968148861}, {53, \
-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, \
-7.844699319567687}, {64, -7.742680759794512}, {65, \
-7.642969851469594}, {66, -7.545500222986023}, {67, \
-7.450207545755878}, {68, -7.357029480628}, {69, \
-7.26590562448199}, {70, -7.176777457127898}, {71, \
-7.089588288633837}, {72, -7.00428320718695}, {73, \
-6.920809027583852}, {74, -6.839114240434034}, {75, \
-6.759148962153092}, {76, -6.680864885807705}, {77, \
-6.604215232869001}, {78, -6.529154705921911}, {79, \
-6.455639442369452}, {80, -6.383626969162678}, {81, \
-6.31307615858577}, {82, -6.243947185110054}, {83, \
-6.176201483335542}, {84, -6.109801707026194}, {85, \
-6.04471168924599}, {86, -5.980896403591716}, {87, \
-5.918321926523271}, {88, -5.856955400784149}, {89, \
-5.796764999899467}, {90, -5.737719893744034}, {91, \
-5.67979021516316}, {92, -5.622947027629922}, {93, \
-5.567162293924735}, {94, -5.51240884581518}, {95, \
-5.4586603547111325}, {96, -5.405891303287587}, {97, \
-5.354076958038671}, {98, -5.303193342744227}, {99, \
-5.253217212836056}, {100, -5.204126030621797}, {101, \
-5.155897941359824}, {102, -5.108511750155478}, {103, \
-5.061946899645364}, {104, -5.016183448466045}, {105, \
-4.971202050471683}, {106, -4.926983934661999}, {107, \
-4.883510885836728}, {108, -4.84076522592182}, {109, \
-4.798729795945647}, {110, -4.757387938669721}, {111, \
-4.716723481825754}, {112, -4.67672072193916}, {113, \
-4.637364408757703}, {114, -4.59863973019463}, {115, \
-4.560532297842467}, {116, -4.523028132982823}, {117, \
-4.486113653103491}, {118, -4.449775658895453}, {119, \
-4.41400132171649}, {120, -4.378778171492242}, {121, \
-4.344094085051662}, {122, -4.309937274899812}, {123, \
-4.276296278348539}, {124, -4.243159947070432}, {125, \
-4.210517437006852}, {126, -4.178358198625626}, {127, \
-4.146671967559926}, {128, -4.1154487555198624}, {129, \
-4.0846788415867845}, {130, -4.054352763762313}, {131, \
-4.0244613108513585}, {132, -3.9949955146174574}, {133, \
-3.96594664218276}, {134, -3.9373061887660405}, {135, \
-3.909065870596173}, {136, -3.8812176180973057}, {137, \
-3.8537535693628917}, {138, -3.8266660637358667}, {139, \
-3.7999476358072934}, {140, -3.7735910093074665}, {141, \
-3.74758909160381}, {142, -3.7219349680465865}, {143, \
-3.6966218966629736}, {144, -3.6716433030615776}, {145, \
-3.6469927753955544}, {146, -3.6226640595680135}, {147, \
-3.5986510546565684}, {148, -3.574947808325067}, {149, \
-3.5515485125124444}, {150, -3.5284474993200767}, {151, \
-3.5056392368240044}, {152, -3.483118325276561}, {153, \
-3.460879493260421}, {154, -3.438917593975345}, {155, \
-3.4172276017590093}, {156, -3.3958046086882554}, {157, \
-3.374643821160908}, {158, -3.353740556736291}, {159, \
-3.3330902410178322}, {160, -3.312688404715038}, {161, \
-3.2925306805915473}, {162, -3.272612800745575}, {163, \
-3.2529305938873545}, {164, -3.233479982647259}, {165, \
-3.214256981045697}, {166, -3.1952576919922233}, {167, \
-3.1764783049446503}, {168, -3.1579150935109284}, {169, \
-3.139564413239762}, {170, -3.121422699346016}, {171, \
-3.103486464815515}, {172, -3.085752298105903}, {173, \
-3.06821686127576}, {174, -3.050876888100025}, {175, \
-3.0337291820666468}, {176, -3.0167706147250413}, {177, \
-2.9999981237621083}, {178, -2.983408711517164}, {179, \
-2.966999443043029}, {180, -2.950767444701468}, {181, \
-2.934709902599512}, {182, -2.9188240610407234}, {183, \
-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, \
-2.5952641239009546}, {206, -2.582801049737661}, {207, \
-2.5704560622673993}, {208, -2.558227508614336}, {209, \
-2.5461137664044258}, {210, -2.534113242995652}, {211, \
-2.522224374603854}, {212, -2.5104456257717658}, {213, \
-2.498775488706279}, {214, -2.4872124825245163}, {215, \
-2.4757551535422944}, {216, -2.464402073172508}, {217, \
-2.453151838443181}, {218, -2.442003071243755}, {219, \
-2.4309544177318334}, {220, -2.4200045476642322}, {221, \
-2.409152153992214}, {222, -2.3983959524956675}, {223, \
-2.387734681289511}, {224, -2.377167099889028}, {225, \
-2.366691989346202}, {226, -2.3563081515904245}, {227, \
-2.3460144087642822}, {228, -2.3358096032830167}, {229, \
-2.325692596783091}, {230, -2.315662270438909}, {231, \
-2.3057175233907956}, {232, -2.29585727442902}, {233, \
-2.286080459414958}, {234, -2.2763860317434053}, {235, \
-2.266772962762401}, {236, -2.2572402399963534}, {237, \
-2.247786868076797}, {238, -2.2384118676807003}, {239, \
-2.229114275276284}, {240, -2.219893143305838}, {241, \
-2.2107475390725484}, {242, -2.201676544892208}, {243, \
-2.1926792581970433}, {244, -2.1837547901839267}, {245, \
-2.174902266691395}, {246, -2.1661208267976306}, {247, \
-2.157409624059163}, {248, -2.1487678244320083}, {249, \
-2.140194607212623}, {250, -2.1316891648369265}, {251, \
-2.1232507019591473}, {252, -2.1148784350248993}, {253, \
-2.106571593566107}, {254, -2.098329418416463}, {255, \
-2.090151161998165}, {256, -2.0820360882444153}, {257, \
-2.073983472006926}, {258, -2.065992599822153}, {259, \
-2.058062768049216}, {260, -2.050193284216243}, {261, \
-2.0423834658368696}, {262, -2.0346326410997926}, {263, \
-2.0269401485288645}, {264, -2.0193053338702636}, {265, \
-2.0117275563473562}, {266, -2.004206182315287}, {267, \
-1.9967405874795818}, {268, -1.9893301568484185}, {269, \
-1.9819742855282303}, {270, -1.9746723747402435}, {271, \
-1.9674238375778639}, {272, -1.9602280932974574}, {273, \
-1.9530845707790225}, {274, -1.9459927058478763}, {275, \
-1.9389519432101352}, {276, -1.931961735476371}, {277, \
-1.925021542799568}, {278, -1.9181308327120814}, {279, \
-1.9112890808085006}, {280, -1.9044957695265645}, {281, \
-1.8977503886127203}, {282, -1.891052435105641}, {283, \
-1.884401412885268}, {284, -1.8777968326794983}, {285, \
-1.8712382123452354}, {286, -1.8647250755056284}, {287, \
-1.8582569532551345}, {288, -1.8518333819478199}, {289, \
-1.8454539057598962}, {290, -1.8391180735418549}, {291, \
-1.832825441675692}, {292, -1.8265755709541789}, {293, \
-1.820368029301432}, {294, -1.814202389691782}, {295, \
-1.8080782314221209}, {296, -1.8019951386958164}, {297, \
-1.795952701852902}, {298, -1.789950516054215}, {299, \
-1.7839881824124155}, {300, -1.7780653067123846}}

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

• Where did you get this list of 300 numbers? Why do you need "the exact function"? Given any finite collection of numbers there is an exact polynomial interpolation function. What form do you expect for the function? There is nothing specific to Mathematica here that I can see. – Somos Jan 6 at 21:55
• ff = FindFormula[data, x]; Show[ListPlot[data], Plot[ff, {x, 0, 300}, PlotStyle -> Red], ImageSize -> Large] will reproduce the data pretty well but I find it hard to believe that you'll be successful to find the "exact" formula used to generate the data. – JimB Jan 6 at 22:09
• @JimB I think you should turn your comment into an answer. – Anton Antonov Jan 7 at 2:13
• @AntonAntonov But I already feel dirty enough even using FindFormula in a comment. Plus, @MikeY's formula uses far fewer parameters and results in a much better fit. – JimB Jan 7 at 2:17
• Yeah, but I learned something from your method! Thanks for posting it. I'd have made it an answer. – MikeY Jan 7 at 2:29

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];


$$\frac{-2.10241 x^2-1735.16 x-43612.1}{x^{2.25431}+116.08 x+1843.92}$$

 nlf["AdjustedRSquared"]
nlf["FitResiduals"] // MinMax


0.999999

{-0.0134303, 0.014954}

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


The first part of the answer uses FindFormula and the results are compared with the results of the second part that uses Quantile Regression with B-splines. The two approaches produce very similar formulas (piecewise polynomials.) The errors with Quantile Regression are much smaller.

(The first part of this answer is a comment made by @JimB, who because of some purity considerations, also implied here, refuses to make it an answer.)

FindFormula

ff = FindFormula[data, x];
Show[ListPlot[data], Plot[ff, {x, 0, 300}, PlotStyle -> Red], ImageSize -> Large]


ff


Through[{Min, Mean, Max}[Abs[((ff /. x -> #[[1]]) - #[[2]])/#[[2]]] & /@ data]]
(* {3.43479*10^-7, 0.00344725, 1.} *)


Quantile regression

Load the QRMon package:

Import["https://raw.githubusercontent.com/antononcube/MathematicaForPrediction/master/MonadicProgramming/MonadicQuantileRegression.m"]


First how the formulas found with QRMon package look like:

qFunc = (QRMonUnit[data] ⟹ QRMonQuantileRegression[2, 0.5, InterpolationOrder -> 5] ⟹ QRMonTakeRegressionFunctions)[0.5];
qFunc[x] // PiecewiseExpand


Here is a bulk computation with max absolute relative errors for different combinations of B-spline basis number of knots and order:

aErrors = Association@Flatten@
Table[
{nknots, norder} ->
QRMonUnit[data]⟹
QRMonQuantileRegression[nknots, 0.5, InterpolationOrder -> norder]⟹
QRMonErrors⟹
(QRMonUnit[First[Values[#1]][[All, 2]], #2] &)⟹
QRMonTakeValue,
{nknots, 3, 12, 2}, {norder, 1, 5}];

GridTableForm[
SortBy[Flatten@*List @@@ Normal[Max /@ Abs@aErrors], Last],
TableHeadings -> {"number\nof knots", "interpolation\norder",
"max absolute\nrelative error"}]


• Sorry, I can only give you a +1. – JimB Jan 7 at 15:19
• @JimB Thanks! I mostly posted this answer in order to proclaim the similarities of the two approaches FindFormula and Quantile Regression... – Anton Antonov Jan 7 at 15:26
• And you (and I) might have greatly exaggerated my level of purity. – JimB Jan 7 at 15:27

It also resembles the error function:

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