Find elementary function that fits the data

Question

I have the following data, i.e. a list of points:

data = {{2.4304509797414493`, 
6.729260872150955`}, {2.415783210264611`, 
6.565682138556903`}, {2.4285129031030896`, 
6.360986237044105`}, {2.430562993364792`, 
6.217242963838571`}, {2.433939194759898`, 
6.000904926489934`}, {2.439798201319317`, 
6.02056869053287`}, {2.4474754678928474`, 
5.933316971617399`}, {2.44262947185293`, 
5.7946089613451`}, {2.4363589182216328`, 
5.612020534793792`}, {2.4537323675072766`, 
5.497306106602562`}, {2.4569372953446074`, 
5.386651172652806`}, {2.4627461665071477`, 
5.030299171261363`}, {2.4589758499134926`, 
4.900876752643637`}, {2.4691747688901677`, 
4.76514655355153`}, {2.4588829440844933`, 
4.599178719008849`}, {2.4601447061957553`, 
4.681674971543597`}, {2.4620538137371892`, 
4.518731147262657`}, {2.471519575080883`, 
4.4887156063235185`}, {2.4833105962333133`, 
4.494324100614184`}, {2.482217736826656`, 
4.35346623855313`}, {2.4825618294063685`, 
4.220142948899261`}, {2.493987372063851`, 
4.18094620015567`}, {2.477097925380988`, 
4.15283186569182`}, {2.4938992549400973`, 
4.099098326478774`}, {2.497698067673524`, 
3.9815629377573485`}, {2.5041350123222705`, 
3.931932456990258`}, {2.4980391049582384`, 
3.936625697225489`}, {2.5070860043626246`, 
3.88517474618616`}, {2.5066365099981907`, 
3.8705817019895843`}, {2.506883708850762`, 
3.9637220751508466`}, {2.5173883296281137`, 
3.965338924501667`}, {2.5203532671356474`, 
4.04876386601876`}, {2.5231902325699935`, 
4.137223671828706`}, {2.5139949793472285`, 
4.273691499948905`}, {2.5192956267860365`, 
4.3969119471631695`}, {2.5226645695039496`, 
4.439285086259388`}, {2.5383526343046885`, 
4.490749717622663`}, {2.538466860171095`, 
4.643216483003511`}, {2.52984888069124`, 
4.779386383817692`}, {2.5420446701865655`, 
4.936902296968895`}, {2.5448069142094867`, 
5.074772262845685`}, {2.544654259360489`, 
5.207127340974729`}, {2.5408247460793305`, 
5.3047205272804785`}, {2.5560437914870566`, 
5.460443294447049`}, {2.544579158127718`, 
5.6194936731428635`}, {2.556227507593902`, 
5.7812272087171666`}, {2.5508346564146875`, 
5.894824642390141`}, {2.562190730269429`, 
6.114712245973943`}, {2.561307773622258`, 
6.296337290605692`}, {2.573188403870403`, 
6.441990229058111`}, {2.5787706742188594`, 
6.623612891925203`}, {2.5690015557201415`, 
6.655875936289631`}, {2.5829752008946647`, 
6.703907351725054`}, {2.582218719686103`, 
6.840545179753442`}, {2.575039020572967`, 
7.001443162050621`}, {2.5822008893060344`, 
7.002078088653101`}, {2.5812284248390114`, 
7.139306108382944`}, {2.5949647014605293`, 
7.216437578232519`}, {2.589591061915744`, 
7.295474466371815`}, {2.594889555173271`, 
7.244851925938778`}, {2.605589523411269`, 
7.324236749153339`}, {2.608225581855298`, 
7.385366871530149`}, {2.6027259153992657`, 
7.36212645925139`}, {2.6090029464931024`, 
7.401851778950361`}, {2.6218295646925274`, 
7.401182407283057`}, {2.6164487196673223`, 
7.3458824743549265`}, {2.6094479040052883`, 
7.199953825056011`}, {2.6211773871206083`, 
7.167729106856195`}, {2.63264512243632`, 
7.022429199828518`}, {2.6237439115525603`, 
7.045145997613185`}, {2.6324766606723915`, 
7.00066515585204`}, {2.63750228225642`, 
6.93478482095497`}, {2.634738772909267`, 
6.893690921643136`}, {2.644149541353371`, 
6.864219856346318`}, {2.6424065997768897`, 
6.83027937856814`}, {2.6501323010954705`, 
6.628512930884991`}, {2.6452089091628226`, 
6.582296163486687`}, {2.6583652683920556`, 
6.43362970520658`}, {2.646267706399761`, 
6.429593966187606`}, {2.656578594618857`, 
6.345478121237732`}, {2.6562868790451057`, 
6.1310366943292856`}, {2.6586865341515735`, 
6.1068907215880746`}, {2.6622125779486487`, 
5.886500137006599`}, {2.6665120396783655`, 
5.812510321428766`}, {2.673863355938594`, 
5.786625409410042`}, {2.667217252357143`, 
5.58886857056124`}, {2.673860783217352`, 
5.601011200079617`}, {2.678930403264612`, 
5.485468961983902`}, {2.68608290551096`, 
5.429617344097778`}, {2.67958293086715`, 
5.458867686446765`}, {2.6923527272114702`, 
5.422414013774456`}, {2.6961279322800076`, 
5.347402233570434`}, {2.6934341401860045`, 
5.219114253066407`}, {2.699106735299276`, 
5.16906153861877`}, {2.697715518005347`, 
5.228710625224765`}, {2.699208698684565`, 
5.16361871787311`}, {2.7100394653088937`, 
5.17386428163147`}, {2.7209106937110836`, 5.093924058102383`}};

How do I obtain an explicit and elementary function that "best" fits such data? It should also be a combination of functions, let's say: Cos, Sin, Log, Times, Plus.

One idea is to have Mathematica search a great number of functions and minimise the residuals.

In the documentation I found this:

fit = FindFormula[data, x, 100000, {"Complexity", 
"Error","Score"},TimeConstraint -> 100, PerformanceGoal -> "Quality", 
SpecificityGoal -> 1, TargetFunctions -> {Cos, Log, Times, Plus}];

So I fixed a tolerance value

TOL = 0.5;

and tried to plot the fitting funtions in increasing order of "accuracy", relying on the "error".

errors = Select[fit[[All, 2, 2]],# < TOL &];
POS = Flatten[
Position[errors,(Sort@errors)[[#]]]&/@Range@(Length@errors), 2];
bestfits = fit[[POS,1]];
plotdata = ListPlot[data];

GraphicsRow[Show[{plotdata,Plot[bestfits[[#]], {x, 1, 1000}]},
ImageSize -> Medium]& /@POS ]

But it does not even look like a fit.

---

Moreover, is there a way to work with parameters?

I might be interested in obtaining a function (of x) a particular form, like

a Log[x] + b Cos[c x] Sin[x/d] - e.

Answer 1

There are several ways. Try this:

Clear[model, ff];
model = a*Cos[b*x + c] + d*Cos[e*x + g] + h;
ff = FindFit[data, model, {a, b, c, d, e, g, h}, x]

Show[{
  ListPlot[data],
  Plot[model /. ff, {x, 2.3, 2.8}, PlotStyle -> Red]
  }]

(* {a -> 0.719, b -> 36., c -> -80.8, d -> -1.13, e -> 22.9, g -> -44.4, h -> 5.77} *)

Have fun!