Exact function that generated the data












1














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,
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-15.45666273835037`}, {22, -15.164813964524406`}, {23,
-14.880338498176549`}, {24, -14.603067012321297`}, {25,
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-12.40469267417549`}, {34, -12.190531855974797`}, {35,
-11.9818349673951`}, {36, -11.77845745250421`}, {37,
-11.580257353223834`}, {38, -11.387095361836874`}, {39,
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-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,
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-1.832825441675692`}, {292, -1.8265755709541789`}, {293,
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NonlinearModelFit[data,
c0 + c1*x^c2 + c3*x^c4, {c0, c1, c2, c3, c4}, x]









share|improve this question


















  • 2




    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
    4 hours ago








  • 3




    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
    4 hours ago






  • 1




    @JimB I think you should turn your comment into an answer.
    – Anton Antonov
    23 mins ago










  • @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
    19 mins ago










  • Yeah, but I learned something from your method! Thanks for posting it. I'd have made it an answer.
    – MikeY
    7 mins ago


















1














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,
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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,
-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]









share|improve this question


















  • 2




    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
    4 hours ago








  • 3




    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
    4 hours ago






  • 1




    @JimB I think you should turn your comment into an answer.
    – Anton Antonov
    23 mins ago










  • @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
    19 mins ago










  • Yeah, but I learned something from your method! Thanks for posting it. I'd have made it an answer.
    – MikeY
    7 mins ago
















1












1








1







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,
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-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]









share|improve this question













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






fitting






share|improve this question













share|improve this question











share|improve this question




share|improve this question










asked 5 hours ago









Alex

132




132








  • 2




    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
    4 hours ago








  • 3




    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
    4 hours ago






  • 1




    @JimB I think you should turn your comment into an answer.
    – Anton Antonov
    23 mins ago










  • @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
    19 mins ago










  • Yeah, but I learned something from your method! Thanks for posting it. I'd have made it an answer.
    – MikeY
    7 mins ago
















  • 2




    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
    4 hours ago








  • 3




    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
    4 hours ago






  • 1




    @JimB I think you should turn your comment into an answer.
    – Anton Antonov
    23 mins ago










  • @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
    19 mins ago










  • Yeah, but I learned something from your method! Thanks for posting it. I'd have made it an answer.
    – MikeY
    7 mins ago










2




2




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
4 hours ago






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
4 hours ago






3




3




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
4 hours ago




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
4 hours ago




1




1




@JimB I think you should turn your comment into an answer.
– Anton Antonov
23 mins ago




@JimB I think you should turn your comment into an answer.
– Anton Antonov
23 mins ago












@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
19 mins ago




@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
19 mins ago












Yeah, but I learned something from your method! Thanks for posting it. I'd have made it an answer.
– MikeY
7 mins ago






Yeah, but I learned something from your method! Thanks for posting it. I'd have made it an answer.
– MikeY
7 mins ago












2 Answers
2






active

oldest

votes


















4














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]]


enter image description here






share|improve this answer





























    0














    It also resembles the error function:



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


    enter image description here






    share|improve this answer





















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      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      4














      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]]


      enter image description here






      share|improve this answer


























        4














        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]]


        enter image description here






        share|improve this answer
























          4












          4








          4






          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]]


          enter image description here






          share|improve this answer












          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]]


          enter image description here







          share|improve this answer












          share|improve this answer



          share|improve this answer










          answered 1 hour ago









          MikeY

          2,317411




          2,317411























              0














              It also resembles the error function:



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


              enter image description here






              share|improve this answer


























                0














                It also resembles the error function:



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


                enter image description here






                share|improve this answer
























                  0












                  0








                  0






                  It also resembles the error function:



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


                  enter image description here






                  share|improve this answer












                  It also resembles the error function:



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


                  enter image description here







                  share|improve this answer












                  share|improve this answer



                  share|improve this answer










                  answered 26 mins ago









                  David Keith

                  946213




                  946213






























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