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I have a dataset whose mathematical model is unknown. There is my code.

ClearAll["Global`*"]
data = {{0., 0.25}, {0.05, 
    0.04535957501876631}, {0.1, -0.22959683050251156}, \
{0.15000000000000002, -0.5158795465693689}, {0.2, \
-0.7301957913348458}, {0.25, -0.8011017169956666}, \
{0.30000000000000004, -0.7044608640550588}, {0.35000000000000003, \
-0.4765378605500501}, {0.4, -0.196100747567105}, {0.45, 
    0.05366775689601985}, {0.5, 0.2206242256461488}, {0.55, 
    0.29732043449572876}, {0.6000000000000001, 
    0.3095987116186817}, {0.65, 
    0.29278966180408206}, {0.7000000000000001, 
    0.27284518841568056}, {0.75, 0.261387359850386}, {0.8, 
    0.2606941881651699}, {0.8500000000000001, 
    0.26817927575775835}, {0.9, 
    0.27446733372325716}, {0.9500000000000001, 
    0.2592721592749264}, {1., 0.1947001957678513}, {1.05, 
    0.06047018239905877}, {1.1, -0.1362654725629964}, \
{1.1500000000000001, -0.3517586997981886}, {1.2000000000000002, \
-0.5183389114682548}, {1.25, -0.5755346214961556}, {1.3, \
-0.5019115834325736}, {1.35, -0.3272715823202753}, \
{1.4000000000000001, -0.11688696144090496}, {1.4500000000000002, 
    0.06223484823497412}, {1.5, 0.17182231969774298}, {1.55, 
    0.21215161276646213}, {1.6, 
    0.2092238920099766}, {1.6500000000000001, 
    0.1923165533808785}, {1.7000000000000002, 
    0.17854999898792742}, {1.75, 0.1720188782322812}, {1.8, 
    0.17190219696043543}, {1.85, 
    0.17794131796485643}, {1.9000000000000001, 
    0.18708794903860115}, {1.9500000000000002, 
    0.18618443013013833}, {2., 
    0.15163266492815844}, {2.0500000000000003, 
    0.062023864530717204}, {2.1, -0.08252831016642331}, {2.15, \
-0.24937887284517374}, {2.2, -0.38250999453025925}, {2.25, \
-0.42977193986190576}, {2.3000000000000003, -0.3719090193656952}, \
{2.35, -0.23403979566383987}, {2.4000000000000004, \
-0.07150507862117503}, {2.45, 0.060544004366313964}, {2.5, 
    0.1338153571297475}, {2.5500000000000003, 
    0.15377710551268767}, {2.6, 
    0.1458096898465104}, {2.6500000000000004, 
    0.1338788539262672}, {2.7, 0.12782685785681855}, {2.75, 
    0.12571190340963798}, {2.8000000000000003, 
    0.12435717478341927}, {2.85, 
    0.12558951152535677}, {2.9000000000000004, 
    0.13173895990209336}, {2.95, 0.13577668352986438}, {3., 
    0.11809163818525376}, {3.0500000000000003, 
    0.05703848372174477}, {3.1, -0.052171141264986065}, \
{3.1500000000000004, -0.18507407808261417}, {3.2, \
-0.2944160290738003}, {3.25, -0.3343926651223669}, \
{3.3000000000000003, -0.28748988516911944}, {3.35, \
-0.17539126742962546}, {3.4000000000000004, -0.04605407250026007}, \
{3.45, 0.054154817787421754}, {3.5, 
    0.10421550491962703}, {3.5500000000000003, 
    0.11313996787827663}, {3.6, 
    0.10537370919096235}, {3.6500000000000004, 
    0.10038340729687355}, {3.7, 0.10210038914116974}, {3.75, 
    0.10377607675326449}, {3.8000000000000003, 
    0.10043392741012536}, {3.85, 
    0.09569030102719618}, {3.9000000000000004, 
    0.09633736364909878}, {3.95, 0.10046944586840713}, {4., 
    0.09196986029286065}, {4.05, 
    0.04939817146023659}, {4.1000000000000005, -0.035500038827273483},
    {4.15, -0.14435142990428168}, {4.2, -0.23653772606398482}, {4.25, \
-0.2711141722776035}, {4.3, -0.23195026722976075}, \
{4.3500000000000005, -0.13818319486254826}, {4.4, \
-0.03223287701492064}, {4.45, 0.0461023748153309}, {4.5, 
    0.08116311683958738}, {4.55, 
    0.08441822812800531}, {4.6000000000000005, 
    0.0793112218398028}, {4.65, 0.08158512653257002}, {4.7, 
    0.09042061843787946}, {4.75, 
    0.09526155242756693}, {4.800000000000001, 
    0.08975097883615274}, {4.8500000000000005, 
    0.07899922694726703}, {4.9, 0.07343943043119187}, {4.95, 
    0.07536815287115564}, {5., 
    0.07162619921504758}, {5.050000000000001, 
    0.04116873379851412}, {5.1000000000000005, \
-0.026744747659383464}, {5.15, -0.11831249917611593}, {5.2, \
-0.1979697358294582}, {5.25, -0.22850646569942168}, \
{5.300000000000001, -0.19488626358397199}, {5.3500000000000005, \
-0.11434128511531431}, {5.4, -0.025108144357342724}, {5.45, 
    0.03796511041395856}, {5.5, 
    0.06320989895118646}, {5.550000000000001, 
    0.06382472357812302}, {5.6000000000000005, 
    0.06230664342404383}, {5.65, 0.07136540577634531}, {5.7, 
    0.08639258817951433}, {5.75, 
    0.09382786497802414}, {5.800000000000001, 
    0.08625209616707691}, {5.8500000000000005, 
    0.06999994420319763}, {5.9, 0.058440733553212644}, {5.95, 
    0.05727250056847864}, {6., 
    0.05578254003710749}, {6.050000000000001, 
    0.033377263014135045}, {6.1000000000000005, \
-0.022490638702671296}, {6.15, -0.10147596664039486}, {6.2, \
-0.1718800607438274}, {6.25, -0.19937131359947818}, \
{6.300000000000001, -0.1697753918045676}, {6.3500000000000005, \
-0.09888818685910143}, {6.4, -0.02176669865153779}, {6.45, 
    0.03049600690836037}, {6.5, 
    0.04922791880104855}, {6.550000000000001, 
    0.04886308340261533}, {6.6000000000000005, 
    0.05106071357932963}, {6.65, 0.06608738435836632}, {6.7, 
    0.08632956423403415}, {6.75, 
    0.09586370917900694}, {6.800000000000001, 
    0.08645125158199114}, {6.8500000000000005, 
    0.0654172599800155}, {6.9, 0.048478784946397874}, {6.95, 
    0.044061035698346705}, {7., 
    0.04344348586261131}, {7.050000000000001, 
    0.026470804650265044}, {7.1000000000000005, \
-0.020732994511128804}, {7.15, -0.09045170867633029}, {7.2, \
-0.1539554421089161}, {7.25, -0.1791359267816674}, \
{7.300000000000001, -0.15249630720578145}, {7.3500000000000005, \
-0.08874264467440107}, {7.4, -0.02050317425559245}, {7.45, 
    0.023992594506665153}, {7.5, 
    0.038338741711232055}, {7.550000000000001, 
    0.03786393558494788}, {7.6000000000000005, 
    0.04351376744916997}, {7.65, 0.0636030295959461}, {7.7, 
    0.08814496194819264}, {7.75, 
    0.09936125416148656}, {7.800000000000001, 
    0.08837990298510912}, {7.8500000000000005, 
    0.0633196836002487}, {7.9, 0.0417630670498372}, {7.95, 
    0.034306558906331726}, {8., 0.033833820809153196}, {8.05, 
    0.020583500915716844}, {8.1, -0.020307573530223846}, {8.15, \
-0.08313247231935975}, {8.200000000000001, -0.1414477941963944}, \
{8.25, -0.16486568052709186}, {8.3, -0.14042058394731663}, {8.35, \
-0.0819872328343009}, {8.4, -0.020331161370387024}, \
{8.450000000000001, 0.018511367496210886}, {8.5, 
    0.029858242066679915}, {8.55, 0.029693848662359766}, {8.6, 
    0.038370946791177404}, {8.65, 
    0.06265453561103906}, {8.700000000000001, 
    0.09068965988664124}, {8.75, 0.10324484849113105}, {8.8, 
    0.09095765005921527}, {8.85, 0.06257855183652253}, {8.9, 
    0.03716526320884847}, {8.950000000000001, 
    0.027034029615809366}, {9., 0.026349806140466097}, {9.05, 
    0.015690013868017962}, {9.1, -0.02054847679666216}, {9.15, \
-0.07820039478789328}, {9.200000000000001, -0.13258754658167485}, \
{9.25, -0.1546551796088675}, {9.3, -0.13185376644945565}, {9.35, \
-0.07742116319072777}, {9.4, -0.020689713175164805}, \
{9.450000000000001, 0.013990045870832553}, {9.5, 
    0.02325362230266585}, {9.55, 
    0.02357119989594019}, {9.600000000000001, 
    0.034811360556287134}, {9.65, 
    0.06251408316536598}, {9.700000000000001, 
    0.09335737679307088}, {9.75, 0.10697279056057489}, {9.8, 
    0.09361765371778913}, {9.850000000000001, 
    0.06254266362460573}, {9.9, 
    0.03396806149215307}, {9.950000000000001, 
    0.021566854697117256}, {10., 0.020521249655974707}};
lp = ListPlot[data, PlotStyle -> {PointSize[0.01]}, 
   DisplayFunction -> Identity];
Show[lp, DisplayFunction -> $DisplayFunction, PlotRange -> Full]

enter image description here

By a very long manual selection of functions, I was able to establish that this curve is well described by the following function.

     Plot[0.25` E^(-0.245` t) - 0.48` E^(-0.47` t) Sin[2 \[Pi] t] - 
  0.51` E^(-0.253` t) Sin[2 \[Pi] t]^2 - 0.126` Sin[2 \[Pi] t]^3, {t, 
  0, 10}]

Very similar, isn't it?

Very similar, isn't it?

I decided to try using the FindFormula command.

But the result that I got does not make me happy. It turns out that this curve is described by a set of not the most complex functions, and FindFormula cannot determine this.

fit = FindFormula[data, t, 5, TargetFunctions -> {Exp, Sin}]

Out[548]= {-0.00731088, Sin[21.096^(-23. t)], Sin[18.2321^(-15 t)], 
 Sin[10.8084^(-7 t)], Sin[Sin[t]]}

Show[ListPlot[data], Plot[fits, {x, 0, 10}, PlotRange -> All]]

How to choose the model structure for NonlinearFitModel?

I would be grateful for any help.

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  • $\begingroup$ What exactly would you like our help with? FindFormula is not going to help here; your data is far too complicated. If your model is truly unknown (which seems a problem by itself), then the model parameters have no meaning to you, so you could use any functional form to fit it. At that point you might as well use an interpolation, if what you need is to reproduce the shape of the function faithfully. $\endgroup$ – MarcoB Jun 25 '20 at 15:08
  • $\begingroup$ I need not only to get a form, but also have a fairly accurate analytical description. $\endgroup$ – dtn Jun 25 '20 at 15:12
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I'm afraid that what you are asking is impossible. Let me explain:

If you do not have a mathematical model restricting the set of possible functions in some way, but only require the function to be e.g. continuous you have a few problems:

  1. You optimize over an infinite-dimensional space which is generally a hard task.
  2. You only have a finite amount of information (=data points) to pick an element from an infinite-dimensional space. Thus, your optimal solution will not be unique (as @MarcoB said you could interpolate the points, but you can also find infinitely many other functions that go perfectly through these points.) So which of the functions do you choose?
  3. Most continuous functions cannot easily be described using mathematical formulas. Therefore you have many solutions but you can not write them down.
  4. These solutions most certainly do not give you any useful insight into what is going on in your problem, because all the optimal solutions are too different from each other.

Therefore you should restrict the set of functions to consider only functions of a certain form like in the answer of @Bob Hanlon https://mathematica.stackexchange.com/a/224707/67019.

Even a choice of functions like this often not enough to solve all the problems above (e.g. get a unique solution). Therefore one often chooses to regularize the parameters (i.e. penalizing weird parameter values). However, then you have to decide how exactly you want to penalize the parameters. And there is also no perfect answer to this task.

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  • $\begingroup$ Thanks for these clarifications. That's true. $\endgroup$ – dtn Jun 25 '20 at 16:01
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ClearAll["Global`*"]

To find the best fit, generalize your model and use NonlinearModelFit

model = {a E^(-b t) - c E^(-d t) Sin[2 π t] - 
    f E^(-g t) Sin[2 π t]^2 - h Sin[2 π t]^3, b > 0, d > 0, g > 0};

(nlm = NonlinearModelFit[data, model, {a, b, c, d, f, g, h}, t]) // Normal

(* 0.25 E^(-0.25 t) - 0.5 E^(-0.5 t) Sin[2 π t] - 
 0.5 E^(-0.25 t) Sin[2 π t]^2 - 0.125 Sin[2 π t]^3 *)

Plot[nlm[t], {t, 0, 10}, Epilog -> {Red,
   AbsolutePointSize[3], Point[data]},
 PlotRange -> 1.05*MinMax[data[[All, 2]]]]

enter image description here

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  • $\begingroup$ This works well when the structure of the mathematical model is known. What if the structure of the mathematical model is not known? However, thank you for your reply. $\endgroup$ – dtn Jun 25 '20 at 15:35
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    $\begingroup$ @dtn - If you don't have a model use func = Interpolation[data]; Plot[func[t], {t, 0, 10}], adjust InterpolationOrder to taste. $\endgroup$ – Bob Hanlon Jun 25 '20 at 15:42
  • $\begingroup$ Does interpolation in Mathematics provide an analytical description for interpolating functions? $\endgroup$ – dtn Jun 25 '20 at 15:45
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    $\begingroup$ If you need a "fairly accurate analytical description" then you will need to analyze the process that is producing the data and develop a model based on an understanding of that process. $\endgroup$ – Bob Hanlon Jun 25 '20 at 15:59

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