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Generate part of peak with only initial part of peak

Mathematica Asked on March 16, 2021

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):

enter image description here

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

list manipulation

One Answer

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}*)

enter image description here

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}]];

enter image description here

enter image description here

Get the regression function:

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

enter image description here

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]

enter image description here

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]}]
 ]

enter image description here

Answered by Anton Antonov on March 16, 2021

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