Reshape interpolated data

Question

I want to reshape/smooth the following data:

gamma={
{0., 0.235269}, {0.02, 0.229659}, {0.04, 0.224064}, {0.06,0.218485}, {0.08, 0.212923}, {0.1, 0.207379}, {0.12,0.201855}, {0.14,0.196353}, {0.16, 0.190874}, {0.18,0.185421}, {0.2, 0.179995}, {0.22,0.1746}, {0.24, 0.169238}, {0.26,0.163913}, {0.28, 0.158627}, {0.3,0.153384}, {0.32,0.14819}, {0.34, 0.143048}, {0.36, 0.137965}, {0.38,0.132947}, {0.4, 0.127999}, {0.42, 0.123131}, {0.44,0.118352}, {0.46,0.11367}, {0.48, 0.109098}, {0.5,0.104649}, {0.52, 0.100336}, {0.54, 0.0961758}, {0.56,0.0921875}, {0.58, 0.0883914}, {0.6, 0.0848107},{0.62,0.0814711}, {0.64, 0.0784008}, {0.66, 0.0756304}, {0.68,0.0731923}, {0.7, 0.0711201}, {0.72, 0.0694465}, {0.74,0.0682012}, {0.76, 0.0674068}, {0.78, 0.067074}, {0.8,0.0671932}, {0.82, 0.0677263}, {0.84, 0.0685938},{0.86,0.0696635}, {0.88, 0.0707382}, {0.9, 0.0715837}, {0.92,0.0723992}, {0.94, 0.0732458}, {0.96, 0.0740598}, {0.98,0.0747586}, {1., 0.0752395}, {1.02, 0.0753813}, {1.04,0.0752247}, {1.06, 0.0748585}, {1.08, 0.0743271}, {1.1,0.0736906}, {1.12, 0.073026}, {1.14, 0.0724292}, {1.16,0.0721029}, {1.18, 0.0721423}, {1.2, 0.0724895}, {1.22,0.0730663}, {1.24, 0.0737686}, {1.26, 0.0744597}, {1.28,0.0748476}, {1.3, 0.0746472}, {1.32, 0.0738905}, {1.34,0.0726507}, {1.36, 0.0710396}, {1.38, 0.0692057}, {1.4,0.0675227}, {1.42, 0.0666539}, {1.44, 0.0665287}, {1.46,0.0670245}, {1.48, 0.0680135}, {1.5, 0.0693562}, {1.52,0.0708603}, {1.54, 0.0720217}, {1.56, 0.0727481}, {1.58,0.0730951}, {1.6, 0.0731381}, {1.62, 0.0729729}, {1.64,0.0727181}, {1.66, 0.0725961}, {1.68, 0.0726691}, {1.7,0.07291}, {1.72, 0.0732853}, {1.74, 0.0737549}, {1.76,0.0742721}, {1.78, 0.0748587}, {1.8, 0.0756793}, {1.82,0.0769077}, {1.84, 0.0787294}, {1.86, 0.0813377}, {1.88,0.0849278}, {1.9, 0.0896083}, {1.92, 0.0951742}, {1.94,0.101348}, {1.96, 0.107816}, {1.98, 0.114231}, {2., 0.12021}, {2.02,0.125595}, {2.04, 0.131118}, {2.06, 0.137176}, {2.08,0.144141}, {2.1, 0.15242}, {2.12, 0.162474}, {2.14,0.174727}, {2.16, 0.188301}, {2.18, 0.202597}, {2.2,0.21724}, {2.22, 0.231765}, {2.24, 0.245597}, {2.26,0.25801}, {2.28, 0.268195}, {2.3,0.276589}, {2.32, 0.28373}, {2.34,0.289746}, {2.36, 0.294389}, {2.38, 0.297231}, {2.4,0.304172}, {2.42, 0.320876}, {2.44, 0.347966}, {2.46,  0.387543}, {2.48, 0.444091}, {2.5, 0.52661}, {2.52,0.641105}, {2.54,0.779722}, {2.56, 0.947065}, {2.58,1.14903}, {2.6, 1.39341}, {2.62, 1.69239}, {2.64, 2.0517}, {2.66,2.4213}, {2.68, 2.76526}, {2.7, 3.06049}, {2.72,3.28789}, {2.74,3.40201}, {2.76, 3.34047}, {2.78, 3.29255}, {2.8, 3.35736}, {2.82,3.57565}, {2.84, 4.02724}, {2.86, 4.9107}, {2.88, 6.8093}, {2.9,10.3673}, {2.92, 16.4}, {2.94, 25.5976}, {2.96, 38.4815}, {2.98,62.8303}  
}

which, when plotted, look like:

I want to get rid of the bumps, oscillations, e.g. at $\sim 2.5-2.8~eV$. I used

ExponentialMovingAverage[MovingMap[Mean, gamma, {0.16,Right,Automatic},None], 0.065]

which was somewhat okay, but doesn't work satisfying anymore after the Kernel was quit. Afterwards I want to use FindFormula:

FindFormula[gamma, x, 3,PerformanceGoal -> "Quality", SpecificityGoal -> Infinity]

which gave me previously

  0.07415952155402779` + 2.253522774452971` Log[x]^15  

Combined in one plot, where the blue line is the found formula:

Since I'm only interested in energies below $2.6~eV$ this was acceptable but I know this can be a achieved way better. But as a newbie in Mathematica my capabilities are very limited

Answer 1

I would always check simple log / log-log spaces for your data. They look very nicely more like cubic in log-space:

ListLogPlot[gamma, PlotRange -> All, PlotTheme -> "Detailed"]

So transform to that log space and try fitting that:

gammaLog = Transpose[MapAt[Log, Transpose[gamma], 2]];
fitLog = FindFormula[gammaLog, PerformanceGoal -> "Quality"];
fitLog[x]
(*-1.3239779106223315`-1.8181372529411417` x+0.36137302563406115` x^3*)

The guess and fit are pretty good:

Plot[fitLog[x], {x, 0, 3}, Epilog -> {Red, Point[gammaLog]}, 
PlotTheme -> "Detailed"]

So going back into original space looks good too:

Plot[Exp[fitLog[x]], {x, 0, 3}, Epilog -> {Red, Point[gamma]}, 
 PlotTheme -> "Detailed", PlotRange -> All]

If you need values under some threshold just cut off unneeded stuff from dataset so your fit is not affected by those points.

Answer 2

You might get the kind of fit you want by simply dropping all the data points with greater than 2.6 eV.

gamma = 
  {{0., 0.235269}, {0.02, 0.229659}, {0.04, 0.224064}, {0.06, 0.218485}, 
   {0.08, 0.212923}, {0.1, 0.207379}, {0.12, 0.201855}, {0.14, 0.196353}, 
   {0.16, 0.190874}, {0.18, 0.185421}, {0.2, 0.179995}, {0.22, 0.1746}, 
   {0.24, 0.169238}, {0.26, 0.163913}, {0.28, 0.158627}, {0.3, 0.153384}, 
   {0.32, 0.14819}, {0.34, 0.143048}, {0.36, 0.137965}, {0.38, 0.132947}, 
   {0.4, 0.127999}, {0.42, 0.123131}, {0.44, 0.118352}, {0.46, 0.11367}, 
   {0.48, 0.109098}, {0.5, 0.104649}, {0.52, 0.100336}, {0.54, 0.0961758}, 
   {0.56, 0.0921875}, {0.58, 0.0883914}, {0.6, 0.0848107}, {0.62, 0.0814711}, 
   {0.64, 0.0784008}, {0.66, 0.0756304}, {0.68, 0.0731923}, {0.7, 0.0711201}, 
   {0.72, 0.0694465}, {0.74, 0.0682012}, {0.76, 0.0674068}, {0.78, 0.067074}, 
   {0.8, 0.0671932}, {0.82, 0.0677263}, {0.84, 0.0685938}, {0.86, 0.0696635}, 
   {0.88, 0.0707382}, {0.9, 0.0715837}, {0.92, 0.0723992}, {0.94, 0.0732458}, 
   {0.96, 0.0740598}, {0.98, 0.0747586}, {1., 0.0752395}, {1.02, 0.0753813}, 
   {1.04, 0.0752247}, {1.06, 0.0748585}, {1.08, 0.0743271}, {1.1, 0.0736906}, 
   {1.12, 0.073026}, {1.14, 0.0724292}, {1.16, 0.0721029}, {1.18, 0.0721423}, 
   {1.2, 0.0724895}, {1.22, 0.0730663}, {1.24, 0.0737686}, {1.26, 0.0744597}, 
   {1.28, 0.0748476}, {1.3, 0.0746472}, {1.32, 0.0738905}, {1.34, 0.0726507}, 
   {1.36, 0.0710396}, {1.38, 0.0692057}, {1.4, 0.0675227}, {1.42, 0.0666539}, 
   {1.44, 0.0665287}, {1.46, 0.0670245}, {1.48, 0.0680135}, {1.5, 0.0693562}, 
   {1.52, 0.0708603}, {1.54, 0.0720217}, {1.56, 0.0727481}, {1.58, 0.0730951}, 
   {1.6, 0.0731381}, {1.62, 0.0729729}, {1.64, 0.0727181}, {1.66, 0.0725961}, 
   {1.68, 0.0726691}, {1.7, 0.07291}, {1.72, 0.0732853}, {1.74, 0.0737549}, 
   {1.76, 0.0742721}, {1.78, 0.0748587}, {1.8, 0.0756793}, {1.82, 0.0769077}, 
   {1.84, 0.0787294}, {1.86, 0.0813377}, {1.88, 0.0849278}, {1.9, 0.0896083}, 
   {1.92, 0.0951742}, {1.94, 0.101348}, {1.96, 0.107816}, {1.98, 0.114231}, 
   {2., 0.12021}, {2.02, 0.125595}, {2.04, 0.131118}, {2.06, 0.137176}, 
   {2.08, 0.144141}, {2.1, 0.15242}, {2.12, 0.162474}, {2.14, 0.174727}, 
   {2.16, 0.188301}, {2.18, 0.202597}, {2.2, 0.21724}, {2.22, 0.231765}, 
   {2.24, 0.245597}, {2.26, 0.25801}, {2.28, 0.268195}, {2.3, 0.276589}, 
   {2.32, 0.28373}, {2.34, 0.289746}, {2.36, 0.294389}, {2.38, 0.297231}, 
   {2.4, 0.304172}, {2.42, 0.320876}, {2.44, 0.347966}, {2.46, 0.387543}, 
   {2.48, 0.444091}, {2.5, 0.52661}, {2.52, 0.641105}, {2.54, 0.779722}, 
   {2.56, 0.947065}, {2.58, 1.14903}, {2.6, 1.39341}};

formulas = 
  FindFormula[gamma, x, 5, 
    PerformanceGoal -> "Quality", 
    SpecificityGoal -> .9, 
    TargetFunctions -> {Plus, Times, Exp, Power}]

{0.106594 + 4.55534 x (4.64585 + x)^(4.42054 (-2.86045 + x)), 
 0.106625 + 4.56433 x (4.67849 + x)^(4.41374 (-2.86045 + x)), 
 0.106854 + 1.85994 x (4.67849 + x)^(4.41374 (-2.75768 + x)), 
 0.105148 + 1.72022 x (3.56921 + x)^(4.6042 (-2.75768 + x)), 
 0.109194 + 4.54899 x (4.67849 + x)^(4.41374 (-2.86045 + x))}

These are all very similar, so let's try the first one.

f[x_] = formulas[[1]];
p1 = ListLinePlot[gamma, PlotRange -> All, PlotStyle -> Blue];
p2 = Plot[f[x], {x, 0, 2.6}, PlotRange -> {0, All}, PlotStyle -> Red];
Show[p1, p2]