# Combining multiple peak fits to one

I'd like to combine three separate fits to one. As you can see in the picture below, there's an underground I manually fitted with the function

k = Divide[Log[10/35], 2000]
UndergroundFit[c_] := 37*Exp(-k*c).


The two peaks were fitted with a Lorentz-curve model like this:

LorentzModel = Divide[2*A, 4*(c - p)^2 + FWHM^2];
LorentzFitPeak1 = NonlinearModelFit[ListPeak1,
Divide[2*A, 4*(c - p)^2 + FWHM^2] + 30, {{A, 100}, p, FWHM}, c];
LorentzFitPeak2 = NonlinearModelFit[ListPeak2,
Divide[2*A, 4*(c - p)^2 + FWHM^2] + 16, {{A, 60}, p, FWHM}, c];


ListPeak1 and ListPeak2 are subsets of the main set of data:

CalibCountsPairs= {{1,0},{2,0},{3,0},{4,0},{5,0},{6,0},{7,0},{8,0},{9,0},{10,0},{11,0},{12,0},{13,0},{14,0},{15,0},{16,0},{17,0},{18,0},{19,0},{20,0},{21,0},{22,0},{23,0},{24,0},{25,0},{26,0},{27,0},{28,0},{29,0},{30,0},{31,0},{32,0},{33,0},{34,0},{35,0},{36,0},{37,0},{38,0},{39,0},{40,0},{41,0},{42,44},{43,46},{44,47},{45,42},{46,41},{47,45},{48,42},{49,46},{50,48},{51,54},{52,40},{53,45},{54,49},{55,49},{56,39},{57,36},{58,29},{59,49},{60,46},{61,41},{62,43},{63,29},{64,37},{65,31},{66,35},{67,40},{68,29},{69,37},{70,28},{71,33},{72,32},{73,35},{74,42},{75,25},{76,37},{77,36},{78,37},{79,41},{80,38},{81,43},{82,36},{83,48},{84,26},{85,31},{86,48},{87,41},{88,36},{89,41},{90,34},{91,26},{92,34},{93,32},{94,29},{95,38},{96,26},{97,38},{98,26},{99,32},{100,39},{101,44},{102,32},{103,38},{104,24},{105,37},{106,50},{107,34},{108,31},{109,30},{110,26},{111,34},{112,37},{113,32},{114,39},{115,40},{116,28},{117,37},{118,27},{119,43},{120,35},{121,29},{122,32},{123,37},{124,32},{125,48},{126,29},{127,26},{128,38},{129,25},{130,35},{131,24},{132,28},{133,44},{134,28},{135,29},{136,31},{137,27},{138,40},{139,35},{140,26},{141,31},{142,36},{143,34},{144,34},{145,29},{146,39},{147,30},{148,36},{149,27},{150,35},{151,23},{152,32},{153,25},{154,37},{155,35},{156,41},{157,33},{158,36},{159,30},{160,31},{161,35},{162,31},{163,30},{164,27},{165,33},{166,22},{167,39},{168,39},{169,29},{170,33},{171,36},{172,35},{173,30},{174,31},{175,32},{176,38},{177,28},{178,25},{179,39},{180,35},{181,36},{182,30},{183,20},{184,30},{185,28},{186,30},{187,27},{188,43},{189,22},{190,35},{191,40},{192,34},{193,32},{194,30},{195,34},{196,31},{197,28},{198,30},{199,31},{200,39},{201,42},{202,37},{203,32},{204,42},{205,34},{206,33},{207,25},{208,31},{209,34},{210,43},{211,30},{212,35},{213,32},{214,29},{215,29},{216,30},{217,31},{218,28},{219,27},{220,37},{221,31},{222,23},{223,27},{224,29},{225,33},{226,29},{227,32},{228,32},{229,35},{230,29},{231,31},{232,33},{233,31},{234,34},{235,21},{236,33},{237,29},{238,37},{239,40},{240,26},{241,40},{242,39},{243,34},{244,39},{245,29},{246,29},{247,33},{248,23},{249,29},{250,31},{251,30},{252,34},{253,31},{254,26},{255,26},{256,35},{257,34},{258,25},{259,35},{260,37},{261,32},{262,35},{263,33},{264,29},{265,31},{266,30},{267,32},{268,32},{269,27},{270,31},{271,38},{272,33},{273,28},{274,36},{275,25},{276,32},{277,35},{278,33},{279,33},{280,40},{281,37},{282,36},{283,30},{284,28},{285,33},{286,33},{287,33},{288,36},{289,31},{290,46},{291,43},{292,31},{293,37},{294,33},{295,41},{296,37},{297,34},{298,48},{299,45},{300,40},{301,48},{302,38},{303,51},{304,48},{305,48},{306,37},{307,63},{308,40},{309,44},{310,55},{311,50},{312,45},{313,58},{314,56},{315,42},{316,57},{317,67},{318,44},{319,59},{320,65},{321,72},{322,70},{323,68},{324,69},{325,61},{326,68},{327,60},{328,64},{329,83},{330,63},{331,70},{332,66},{333,61},{334,70},{335,83},{336,74},{337,66},{338,63},{339,73},{340,73},{341,54},{342,94},{343,86},{344,82},{345,87},{346,94},{347,81},{348,65},{349,89},{350,92},{351,85},{352,86},{353,86},{354,86},{355,98},{356,97},{357,79},{358,97},{359,111},{360,97},{361,96},{362,90},{363,85},{364,90},{365,98},{366,99},{367,104},{368,96},{369,79},{370,87},{371,101},{372,92},{373,103},{374,85},{375,97},{376,72},{377,92},{378,101},{379,100},{380,102},{381,86},{382,91},{383,102},{384,95},{385,110},{386,97},{387,105},{388,91},{389,98},{390,92},{391,96},{392,75},{393,94},{394,81},{395,63},{396,76},{397,89},{398,73},{399,91},{400,70},{401,76},{402,67},{403,76},{404,85},{405,71},{406,84},{407,77},{408,85},{409,69},{410,60},{411,73},{412,78},{413,78},{414,66},{415,80},{416,79},{417,67},{418,54},{419,77},{420,71},{421,64},{422,58},{423,60},{424,54},{425,55},{426,53},{427,69},{428,75},{429,60},{430,60},{431,50},{432,50},{433,77},{434,60},{435,51},{436,55},{437,54},{438,57},{439,58},{440,55},{441,58},{442,61},{443,53},{444,44},{445,54},{446,52},{447,57},{448,40},{449,54},{450,39},{451,60},{452,51},{453,47},{454,47},{455,55},{456,48},{457,47},{458,55},{459,29},{460,41},{461,43},{462,44},{463,32},{464,46},{465,40},{466,46},{467,41},{468,45},{469,40},{470,43},{471,37},{472,47},{473,43},{474,47},{475,41},{476,44},{477,34},{478,46},{479,37},{480,32},{481,29},{482,45},{483,37},{484,38},{485,40},{486,29},{487,33},{488,41},{489,31},{490,38},{491,33},{492,46},{493,37},{494,38},{495,29},{496,34},{497,37},{498,37},{499,32},{500,33},{501,32},{502,38},{503,41},{504,34},{505,29},{506,29},{507,35},{508,21},{509,38},{510,40},{511,30},{512,39},{513,26},{514,28},{515,13},{516,24},{517,31},{518,31},{519,29},{520,32},{521,46},{522,22},{523,36},{524,29},{525,36},{526,26},{527,22},{528,24},{529,31},{530,22},{531,27},{532,25},{533,23},{534,39},{535,24},{536,31},{537,35},{538,26},{539,29},{540,22},{541,25},{542,42},{543,32},{544,29},{545,30},{546,32},{547,27},{548,31},{549,28},{550,22},{551,30},{552,28},{553,20},{554,26},{555,24},{556,32},{557,22},{558,33},{559,30},{560,24},{561,33},{562,33},{563,20},{564,33},{565,20},{566,24},{567,15},{568,18},{569,36},{570,30},{571,21}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Note that I purposefully cut off data so that I wouldn't exceed the character limit for posts with

ListPeak1 = CalibCounts2Pairs[[300;;500]];
ListPeak2 = CalibCounts2Pairs[[900;;1200]];


There's two main problems I'd like to solve:

1. Combining the three fits to one so that the Lorentz fits themselves take the underground into account
2. As can be seen in the figure above, the two Lorentz fits don't quite reach the upper most data points

My approach is to just define a new fit function:

 AltPeakFit[c_]=UndergroundFit[c] -
LorentzFitIronPeak1[c] + LorentzFitPeak2[c] + 420


Adding 420 is neccessary to shift the new function up (looking at the single functions, I don't understand why this is neccesary, but adding 420 helps the result look almost as desired).

The new figure with all the fit functions looks like this:

As one can see, the second peak is quite well traced by the new fit function AltPeakFit[c_]. I don't understand why the proposed procedure doesn't work for peak1 as well and how I can fix it.

Any help would be greatly appreciated.

• Hi, I wouldn't worry about the fits not reaching the uppermost points, that is to be expected from a fit. Commented Jul 25, 2023 at 13:48
• CalibCounts2Pairs is not defined. UndergroundFit has non-executable defintion. Commented Jul 25, 2023 at 14:16
• @banone Unfortunately my lab supervisor doesn't see it that way :') Commented Jul 25, 2023 at 14:17

You can combine all three functions into one, and perform one fit. To obtain meaningful results, you have to provide good initial values for the parameters.

peak1 = 2 A1/(4*(c - p1)^2 + FWHM1^2);
peak2 = 2 A2/(4*(c - p2)^2 + FWHM2^2);
background = 37 Exp[(Log[10/35]/2000) c];
fit = NonlinearModelFit[CalibCountsPairs,
peak1 + peak2 + background, {{p1, 300}, {p2, 1000}, {FWHM1, 80}, {FWHM2, 80},
{A1, 100}, {A2, 10}}, c]

Show[Plot[fit[c], {c, 0, 2500}, PlotRange -> All], ListPlot[CalibCountsPairs]]


Note that I fixed the parameters for the background (but you could let them be fitted). Also, it is important that you know what you are actually trying to achieve with such fits and are aware of potential (statistical) pitfalls.