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list manipulation - Estimating the full width half maximum of a peak



I have a dataset of which I'm attempting to determine the full width at half maximum. A picture of the data is given below, and the dataset itself can be found here (pastebin). enter image description here


Now, this set has a minimum, which can be found rather easily. Say we call the data yy, then by using Min[yy] we've found the minimum. To find the full width half max, it is then probably easiest to look for a point closest to 1-(1-Min[yy])*0.5 in yy. The first thing that came to mind was using Nearest[yy,1-(1-Min[yy])*0.5,1] which indeed returns a value of yyclosest to half of the FWHM. But I'm having some trouble finding the actual position of this point in the data. Ironically, using Position[yy,Nearest[yy,1-(1-Min[yy])*0.5,1]] seems to come up empty. I'm not sure why exactly this is, but maybe there is some rounding going on, or incompatible outputs. I've tried thinking of a solution to this, but it doesn't seem obvious to me.


My question is therefore if anyone seems a workaround, or an all around better idea. My plan was to find this position of the half-max point, find the corresponding xx, subtract the xx element of the position of the max, and multiply by two. I don't need a 100% perfect estimation; this should be good enough.



Answer



Here is an approach. You could also search around the midpoint. In the following data was imported using ImportString from pastebin link.


A non-linear model was developed. The starting points were 'guessed' with the assistance of manipulate.


nlm = NonlinearModelFit[N@ToExpression[data], 
  d - c/(1 + a (x - b)^2), {{a, 150000.}, {b, 7.62}, {c, 0.064}, {d, 
    1.}}, x]
Show[Plot[nlm[x], {x, 7.6, 7.65}, PlotRange -> {0.90, 1}, 

  PlotStyle -> Red], 
 ListPlot[data, PlotMarkers -> {Style[\[FilledDiamond], Black], 5}]]

enter image description here The fit (visually is not unreasonable, adjusted $R^2$ 0.999989).


Now to determine width.


bf = nlm["BestFitParameters"]
ypeak = nlm[x] /. First@Quiet[Solve[D[nlm[x], x] == 0, x]];
hmw = ypeak + ((d /. bf) - ypeak)/2;
{xr, xl} = x /. NSolve[nlm[x] == hmw, x];
xr - xl

Show[Plot[nlm[x], {x, 7.6, 7.65}, PlotRange -> {0.90, 1}, 
  PlotStyle -> Red, 
  Epilog -> {Green, PointSize[0.02], Point[{#, nlm@#} & /@ {xl, xr}]},
   GridLines -> {{xl, xr}, {d /. bf, hmw, ypeak}}, 
  GridLinesStyle -> Directive[Gray, Dashed]], 
 ListPlot[data, PlotMarkers -> {Style[\[FilledDiamond], Black], 5}]]

The x values: {7.61527, 7.61774}. The width: 0.00247168.


enter image description here


This can adpated, improved and I look forward to better answers.



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