Generate part of peak with only initial part of peak

Question

If I have the following data:

data={{24.998, 3.01329}, {25.487, 3.1036}, {25.977, 3.18242}, {26.473, 
  3.2167}, {26.983, 3.13354}, {27.495, 3.03633}, {28.006, 
  2.95134}, {28.515, 2.88278}, {29.02, 2.8459}, {29.524, 
  2.81361}, {30.028, 2.78335}, {30.532, 2.75411}, {31.035, 
  2.73131}, {31.537, 2.71316}, {32.039, 2.69919}, {32.541, 
  2.6876}, {33.042, 2.67938}, {33.543, 2.67225}, {34.044, 
  2.66644}, {34.545, 2.66139}, {35.045, 2.65809}, {35.546, 
  2.65551}, {36.046, 2.65338}, {36.546, 2.65153}, {37.047, 
  2.65029}, {37.547, 2.6494}, {38.047, 2.64884}, {38.547, 
  2.64841}, {39.047, 2.64811}, {39.548, 2.64781}, {40.048, 
  2.64757}, {40.548, 2.64746}, {41.048, 2.6475}, {41.548, 
  2.64755}, {42.048, 2.64767}, {42.548, 2.64788}, {43.048, 
  2.64815}, {43.549, 2.6484}, {44.049, 2.64867}, {44.549, 
  2.64896}, {45.049, 2.64932}, {45.549, 2.64966}, {46.049, 
  2.65001}, {46.549, 2.65038}, {47.049, 2.65079}, {47.549, 
  2.65118}, {48.049, 2.65157}, {48.549, 2.65199}, {49.049, 
  2.65244}, {49.549, 2.65288}, {50.049, 2.65331}, {50.549, 
  2.65374}, {51.05, 2.65422}, {51.55, 2.65468}, {52.05, 
  2.65513}, {52.55, 2.65559}, {53.05, 2.65605}, {53.55, 
  2.65652}, {54.05, 2.657}, {54.55, 2.65748}, {55.05, 
  2.65792}, {55.55, 2.65836}, {56.05, 2.6588}, {56.55, 
  2.65925}, {57.05, 2.65961}, {57.55, 2.65999}, {58.05, 
  2.66038}, {58.55, 2.66079}, {59.05, 2.66121}, {59.551, 
  2.66165}, {60.051, 2.6621}, {60.551, 2.66256}, {61.051, 
  2.66311}, {61.551, 2.66363}, {62.051, 2.66415}, {62.551, 
  2.66466}, {63.051, 2.66521}, {63.551, 2.66574}, {64.051, 
  2.66627}, {64.551, 2.66681}, {65.051, 2.66733}, {65.551, 
  2.66788}, {66.051, 2.66842}, {66.551, 2.66894}, {67.051, 
  2.66947}, {67.551, 2.66998}, {68.051, 2.67049}, {68.551, 
  2.67099}, {69.051, 2.67149}, {69.551, 2.67201}, {70.051, 
  2.67255}, {70.551, 2.6731}, {71.051, 2.67372}, {71.551, 
  2.67432}, {72.051, 2.67492}, {72.551, 2.67555}, {73.052, 
  2.67622}, {73.552, 2.67684}, {74.052, 2.67745}, {74.552, 
  2.67805}, {75.052, 2.67864}, {75.552, 2.67923}, {76.052, 
  2.67982}, {76.552, 2.68041}, {77.052, 2.68101}, {77.552, 
  2.68161}, {78.052, 2.68222}, {78.552, 2.68284}, {79.052, 
  2.68349}, {79.552, 2.68413}, {80.052, 2.68477}, {80.552, 
  2.6854}, {81.052, 2.68606}, {81.552, 2.68671}, {82.052, 
  2.68737}, {82.552, 2.68802}, {83.052, 2.68861}, {83.552, 
  2.68918}, {84.052, 2.68975}, {84.552, 2.69027}, {85.052, 
  2.69063}, {85.552, 2.69108}, {86.052, 2.69153}, {86.552, 
  2.69199}, {87.052, 2.69245}, {87.552, 2.69292}, {88.052, 
  2.69341}, {88.552, 2.6939}, {89.052, 2.69442}, {89.552, 
  2.69493}, {90.052, 2.69546}, {90.552, 2.69599}, {91.052, 
  2.69655}, {91.552, 2.69711}, {92.052, 2.69766}, {92.552, 
  2.69823}, {93.052, 2.6988}, {93.552, 2.69935}, {94.052, 
  2.69989}, {94.552, 2.70045}, {95.052, 2.70099}, {95.552, 
  2.70157}, {96.052, 2.70216}, {96.552, 2.70274}, {97.052, 
  2.70336}, {97.552, 2.70396}, {98.052, 2.70455}, {98.552, 
  2.70515}, {99.052, 2.70576}, {99.552, 2.70637}, {100.052, 
  2.70698}, {100.552, 2.70758}, {101.052, 2.70822}, {101.552, 
  2.70886}, {102.052, 2.7095}, {102.552, 2.71015}, {103.052, 
  2.71082}, {103.552, 2.71149}, {104.052, 2.71215}, {104.552, 
  2.71282}, {105.052, 2.7135}, {105.552, 2.71419}, {106.052, 
  2.71487}, {106.552, 2.71556}, {107.052, 2.71623}, {107.552, 
  2.71689}, {108.052, 2.71755}, {108.552, 2.7182}, {109.052, 
  2.71881}, {109.552, 2.71943}, {110.052, 2.72004}, {110.552, 
  2.72066}, {111.052, 2.72125}, {111.552, 2.72184}, {112.052, 
  2.72245}, {112.552, 2.72307}, {113.052, 2.72369}, {113.552, 
  2.7243}, {114.052, 2.7249}, {114.552, 2.7255}, {115.051, 
  2.7261}, {115.551, 2.72669}, {116.051, 2.72727}, {116.551, 
  2.72784}, {117.051, 2.72838}, {117.551, 2.72893}, {118.051, 
  2.72948}, {118.551, 2.73001}, {119.051, 2.73053}, {119.551, 
  2.73106}, {120.051, 2.73157}, {120.551, 2.73209}, {121.051, 
  2.73259}, {121.551, 2.73308}, {122.051, 2.73357}, {122.551, 
  2.73405}, {123.051, 2.73451}, {123.551, 2.73498}, {124.051, 
  2.73546}, {124.551, 2.73593}, {125.051, 2.73642}, {125.551, 
  2.7369}, {126.051, 2.73737}, {126.551, 2.73783}, {127.051, 
  2.73826}, {127.551, 2.73873}, {128.051, 2.73907}, {128.551, 
  2.73939}, {129.052, 2.7384}, {129.552, 2.73619}, {130.052, 
  2.73579}, {130.552, 2.73656}, {131.052, 2.73776}, {131.552, 
  2.73884}, {132.052, 2.73986}, {132.552, 2.74085}, {133.052, 
  2.7418}, {133.551, 2.74274}, {134.051, 2.74361}, {134.551, 
  2.74444}, {135.051, 2.74503}, {135.551, 2.7457}, {136.051, 
  2.74637}, {136.551, 2.74702}, {137.051, 2.74762}, {137.551, 
  2.74826}, {138.051, 2.74894}, {138.551, 2.74962}, {139.051, 
  2.75039}, {139.551, 2.75116}, {140.051, 2.75195}, {140.551, 
  2.75276}, {141.05, 2.75371}, {141.55, 2.75462}, {142.05, 
  2.75555}, {142.55, 2.75655}, {143.05, 2.75773}, {143.55, 
  2.75892}, {144.05, 2.76018}, {144.549, 2.76152}, {145.049, 
  2.76299}, {145.549, 2.76453}, {146.049, 2.76618}, {146.548, 
  2.76791}, {147.048, 2.7701}, {147.547, 2.7726}, {148.047, 
  2.77549}, {148.546, 2.77866}, {149.046, 2.78203}, {149.545, 
  2.78568}, {150.044, 2.78979}, {150.543, 2.79424}, {151.042, 
  2.79957}, {151.541, 2.80519}, {152.04, 2.81119}, {152.539, 
  2.81739}, {153.037, 2.82423}, {153.536, 2.83135}, {154.034, 
  2.83883}, {154.533, 2.84651}, {155.031, 2.85469}, {155.529, 
  2.86309}, {156.027, 2.87178}, {156.526, 2.88067}, {157.024, 
  2.88982}, {157.522, 2.89892}, {158.02, 2.90806}, {158.518, 
  2.91733}, {159.016, 2.92689}, {159.514, 2.93649}, {160.012, 
  2.94622}, {160.509, 2.95616}, {161.007, 2.96665}, {161.505, 
  2.97774}, {162.002, 2.98925}, {162.5, 3.0011}, {162.997, 
  3.01387}, {163.494, 3.02748}, {163.991, 3.04203}, {164.487, 
  3.05723}, {164.984, 3.07422}, {165.48, 3.09228}, {165.975, 
  3.11176}, {166.471, 3.13235}, {166.966, 3.15638}, {167.46, 
  3.18247}, {167.953, 3.21148}, {168.446, 3.2427}, {168.938, 
  3.28076}, {169.428, 3.32355}, {169.917, 3.37256}, {170.405, 
  3.42621}, {170.89, 3.494}, {171.373, 3.57118}, {171.852, 
  3.66097}, {172.33, 3.76013}, {172.802, 3.88615}, {173.27, 
  4.02805}, {173.733, 4.18957}, {174.193, 4.36525}, {174.646, 
  4.57463}, {175.095, 4.7979}, {175.541, 5.03393}, {175.986, 
  5.277}, {176.428, 5.52958}, {176.869, 5.78474}, {177.31, 
  6.04046}, {177.752, 6.295}, {178.194, 6.54687}, {178.636, 
  6.79524}, {179.08, 7.03736}, {179.524, 7.27554}, {179.978, 
  7.47131}, {180.441, 7.62793}, {180.923, 7.70132}, {181.424, 
  7.68992}, {181.974, 7.46629}, {182.54, 7.17707}, {183.117, 
  6.83616}, {183.699, 6.47479}}

Which plotted like ListLinePlot[data,PlotRange -> {{50, 250}, All}] gives (without the red line):

How can I generate the red line in the figure that "completes the peak" following more and less a linear line from the part of the peak that is visible?. Also how to also generate the baseline after the peak ends?. YOU CAN ASSUME GAUSSIAN BEHAVIOR OF THE PEAK

If you can determine what functional shape the peak should have (Gaussian, Lorentzian, etc), then use curve fitting to fit an appropriate function representing that peak shape to your data, then plot the resulting fitted peak. However, this would make me a bit uncomfortable: it feels rather too close to fabricating the missing data. — MarcoB, Jan 13 at 18:50
@John what you posted doesn't provide the blue part of that peak — b3m2a1, Jan 13 at 19:18
@b3m2a1 It does provide that portion of the peak but one needs to add in PlotRange -> All to see it. But then it's worse in that there's a bump around 25 in the horizontal variable that I assume will be conveniently ignored. John: as MarcoB states you really need to have some physical meaning/justification. Many of your questions seem to be data manipulations without any justified underlying models. That can get one into trouble if important decisions are made from the results. — JimB, Jan 13 at 19:31
@JimB Mathematica very kindly clipped the data when I copy-pasted so that it dropped anything beyond 150 -_- — b3m2a1, Jan 13 at 19:39

Anton Antonov · Accepted Answer · 2021-01-14 01:09:58Z

This can be done in several ways. Instead of using Gaussians I am using B-splines below. (But the process can be done with Gaussians too.)

Reflect and apply QRMon

Get the QRMon package:

Import["https://raw.githubusercontent.com/antononcube/MathematicaForPrediction/master/MonadicProgramming/MonadicQuantileRegression.m"]

Sort the data and get the portion of interest:

data1 = SortBy[data, First][[50 ;; -1]];

Find the maximum y-point:

pos = Position[data1[[All, 2]], Max[data1[[All, 2]]]][[1, 1]]

(*266*)

Get the data part up to the y-maximum:

data2 = data1[[1 ;; pos]];

Reflect the “focus” data around y-maximum x-position:

data3 = Join[data2, Transpose[{data2[[-1, 1]] + Accumulate[Reverse@Differences[data2[[All, 1]]]], Reverse[Most@data2[[All, 2]]]}]];
Dimensions[data3]
ListPlot[{data3, data1[[pos ;; -1]]}, PlotLegends -> {"Reflected data", "Un-reflected data part"}, PlotStyle -> {Automatic, {PointSize[0.01], Red}}, PlotRange -> All, PlotTheme -> "Detailed", ImageSize -> Large]

(*{531, 2}*)

Remark: From the plot above we see that there is no reason to add the un-reflected data part to the derived reflected data.

Do Quantile Regression fit:

lsKnots = Sort@Join[Range @@ Append[{0.98, 1.1}*MinMax[data3[[All, 1]]], 20], Range[data2[[-1, 1]] - 20, data2[[-1, 1]] + 20, 4]];

qrObj = 
   QRMonUnit[data3]⟹
    QRMonSetRegressionFunctionsPlotOptions[{PlotStyle -> Red}]⟹
    QRMonQuantileRegression[lsKnots, 0.5]⟹
    QRMonPlot[GridLines -> {lsKnots, None}, GridLinesStyle -> Directive[{Thin, Dashed}]]⟹
    QRMonErrorPlots[GridLines -> {lsKnots, None}, GridLinesStyle -> Directive[{Thin, Dashed}]];

Get the regression function:

qFunc = (qrObj⟹QRMonTakeRegressionFunctions)[0.5];
Simplify[qFunc[x]]

Plot the regression function and the “focus” data:

Show[ListLinePlot[{#, qFunc[#]} & /@ data3[[All, 1]], PlotRange -> All, PlotLegends -> {"Fitted"}, PlotTheme -> "Detailed"], ListPlot[data1, PlotLegends -> {"Original"}, PlotStyle -> Red], ImageSize -> Large]

Here are the (relative) residuals:

Block[{lsRes = Abs[(#[[2]] - qFunc[#[[1]]])/#[[2]]] & /@ data1},
 Row[{ResourceFunction["RecordsSummary"][lsRes], Spacer[3], 
   ListPlot[lsRes, PlotTheme -> "Detailed", ImageSize -> Medium]}]
 ]

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Generate part of peak with only initial part of peak

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Reflect and apply QRMon

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Generate part of peak with only initial part of peak

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