{
"cells": [
{
"cell_type": "markdown",
"id": "df56a0ed",
"metadata": {},
"source": [
"# rf704_amplitudefit\n",
"Special pdf's: using a pdf defined by a sum of real-valued amplitude components\n",
"\n",
"\n",
"\n",
"\n",
"**Author:** Clemens Lange, Wouter Verkerke (C++ version) \n",
"This notebook tutorial was automatically generated with ROOTBOOK-izer from the macro found in the ROOT repository on Tuesday, May 19, 2026 at 08:34 PM."
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "fee3c732",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:34:35.320479Z",
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"shell.execute_reply": "2026-05-19T20:34:36.284637Z"
}
},
"outputs": [],
"source": [
"import ROOT"
]
},
{
"cell_type": "markdown",
"id": "72cde419",
"metadata": {},
"source": [
"Setup 2D amplitude functions\n",
"-------------------------------------------------------"
]
},
{
"cell_type": "markdown",
"id": "a38e42d0",
"metadata": {},
"source": [
"Observables"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "972d1765",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:34:36.293713Z",
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"iopub.status.idle": "2026-05-19T20:34:36.536314Z",
"shell.execute_reply": "2026-05-19T20:34:36.535782Z"
}
},
"outputs": [],
"source": [
"t = ROOT.RooRealVar(\"t\", \"time\", -1.0, 15.0)\n",
"cosa = ROOT.RooRealVar(\"cosa\", \"cos(alpha)\", -1.0, 1.0)\n",
"\n",
"tau = ROOT.RooRealVar(\"tau\", \"#tau\", 1.5)\n",
"deltaGamma = ROOT.RooRealVar(\"deltaGamma\", \"deltaGamma\", 0.3)\n",
"coshG = ROOT.RooFormulaVar(\"coshGBasis\", \"exp(-@0/ @1)*cosh(@0*@2/2)\", [t, tau, deltaGamma])\n",
"sinhG = ROOT.RooFormulaVar(\"sinhGBasis\", \"exp(-@0/ @1)*sinh(@0*@2/2)\", [t, tau, deltaGamma])"
]
},
{
"cell_type": "markdown",
"id": "7ffcca56",
"metadata": {},
"source": [
"Construct polynomial amplitudes in cos(a)"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "69ad56d5",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:34:36.539919Z",
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"shell.execute_reply": "2026-05-19T20:34:36.665135Z"
}
},
"outputs": [],
"source": [
"poly1 = ROOT.RooPolyVar(\"poly1\", \"poly1\", cosa, [0.5, 0.2, 0.2], 0)\n",
"poly2 = ROOT.RooPolyVar(\"poly2\", \"poly2\", cosa, [1.0, -0.2, 3.0], 0)"
]
},
{
"cell_type": "markdown",
"id": "485eb08a",
"metadata": {},
"source": [
"Construct 2D amplitude as uncorrelated product of amp(t)*amp(cosa)"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "782f0b04",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:34:36.667433Z",
"iopub.status.busy": "2026-05-19T20:34:36.667307Z",
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"shell.execute_reply": "2026-05-19T20:34:36.777728Z"
}
},
"outputs": [],
"source": [
"ampl1 = ROOT.RooProduct(\"ampl1\", \"amplitude 1\", [poly1, coshG])\n",
"ampl2 = ROOT.RooProduct(\"ampl2\", \"amplitude 2\", [poly2, sinhG])"
]
},
{
"cell_type": "markdown",
"id": "22db19de",
"metadata": {},
"source": [
"Construct amplitude sum pdf\n",
"-----------------------------------------------------"
]
},
{
"cell_type": "markdown",
"id": "23107e84",
"metadata": {},
"source": [
"Amplitude strengths"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "9ee16f77",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:34:36.780189Z",
"iopub.status.busy": "2026-05-19T20:34:36.780030Z",
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"shell.execute_reply": "2026-05-19T20:34:36.889759Z"
}
},
"outputs": [],
"source": [
"f1 = ROOT.RooRealVar(\"f1\", \"f1\", 1, 0, 2)\n",
"f2 = ROOT.RooRealVar(\"f2\", \"f2\", 0.5, 0, 2)"
]
},
{
"cell_type": "markdown",
"id": "2bc52d40",
"metadata": {},
"source": [
"Construct pdf"
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "6036ec3a",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:34:36.892056Z",
"iopub.status.busy": "2026-05-19T20:34:36.891928Z",
"iopub.status.idle": "2026-05-19T20:34:37.009513Z",
"shell.execute_reply": "2026-05-19T20:34:37.008952Z"
}
},
"outputs": [],
"source": [
"pdf = ROOT.RooRealSumPdf(\"pdf\", \"pdf\", [ampl1, ampl2], [f1, f2])"
]
},
{
"cell_type": "markdown",
"id": "3a84f5b5",
"metadata": {},
"source": [
"Generate some toy data from pdf"
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "ec9090b8",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:34:37.011239Z",
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"shell.execute_reply": "2026-05-19T20:34:37.209801Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[#1] INFO:NumericIntegration -- RooRealIntegral::init(coshGBasis_Int[t]) using numeric integrator RooIntegrator1D to calculate Int(t)\n",
"[#1] INFO:NumericIntegration -- RooRealIntegral::init(sinhGBasis_Int[t]) using numeric integrator RooIntegrator1D to calculate Int(t)\n",
"[#1] INFO:NumericIntegration -- RooRealIntegral::init(coshGBasis_Int[t]) using numeric integrator RooIntegrator1D to calculate Int(t)\n",
"[#1] INFO:NumericIntegration -- RooRealIntegral::init(sinhGBasis_Int[t]) using numeric integrator RooIntegrator1D to calculate Int(t)\n"
]
}
],
"source": [
"data = pdf.generate({t, cosa}, 10000)"
]
},
{
"cell_type": "markdown",
"id": "577c774c",
"metadata": {},
"source": [
"Fit pdf to toy data with only amplitude strength floating"
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "4420efa9",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:34:37.212027Z",
"iopub.status.busy": "2026-05-19T20:34:37.211897Z",
"iopub.status.idle": "2026-05-19T20:34:37.405691Z",
"shell.execute_reply": "2026-05-19T20:34:37.405138Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[#1] INFO:Fitting -- RooAbsPdf::fitTo(pdf_over_pdf_Int[cosa,t]) fixing normalization set for coefficient determination to observables in data\n",
"[#1] INFO:Fitting -- using generic CPU library compiled with no vectorizations\n",
"[#1] INFO:Fitting -- Creation of NLL object took 934.074 μs\n",
"[#1] INFO:Fitting -- RooAddition::defaultErrorLevel(nll_pdf_over_pdf_Int[cosa,t]_pdfData) Summation contains a RooNLLVar, using its error level\n",
"[#1] INFO:Minimization -- [fitFCN] No discrete parameters, performing continuous minimization only\n",
"[#1] INFO:NumericIntegration -- RooRealIntegral::init(coshGBasis_Int[t]) using numeric integrator RooIntegrator1D to calculate Int(t)\n",
"[#1] INFO:NumericIntegration -- RooRealIntegral::init(sinhGBasis_Int[t]) using numeric integrator RooIntegrator1D to calculate Int(t)\n"
]
},
{
"data": {
"text/plain": [
""
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"pdf.fitTo(data, PrintLevel=-1)"
]
},
{
"cell_type": "markdown",
"id": "472041a4",
"metadata": {},
"source": [
"Plot amplitude sum pdf\n",
"-------------------------------------------"
]
},
{
"cell_type": "markdown",
"id": "3c9affa1",
"metadata": {},
"source": [
"Make 2D plots of amplitudes"
]
},
{
"cell_type": "code",
"execution_count": 9,
"id": "222a2f4e",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:34:37.407739Z",
"iopub.status.busy": "2026-05-19T20:34:37.407576Z",
"iopub.status.idle": "2026-05-19T20:34:37.566894Z",
"shell.execute_reply": "2026-05-19T20:34:37.566345Z"
}
},
"outputs": [],
"source": [
"hh_cos = ampl1.createHistogram(\"hh_cos\", t, Binning=50, YVar=dict(var=cosa, Binning=50))\n",
"hh_sin = ampl2.createHistogram(\"hh_sin\", t, Binning=50, YVar=dict(var=cosa, Binning=50))\n",
"hh_cos.SetLineColor(\"kBlue\")\n",
"hh_sin.SetLineColor(\"kRed\")"
]
},
{
"cell_type": "markdown",
"id": "20d761cb",
"metadata": {},
"source": [
"Make projection on t, data, and its components\n",
"Note component projections may be larger than sum because amplitudes can\n",
"be negative"
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "3017216d",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:34:37.568634Z",
"iopub.status.busy": "2026-05-19T20:34:37.568496Z",
"iopub.status.idle": "2026-05-19T20:34:37.743988Z",
"shell.execute_reply": "2026-05-19T20:34:37.743455Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[#1] INFO:Plotting -- RooAbsReal::plotOn(pdf) plot on t integrates over variables (cosa)\n",
"[#1] INFO:NumericIntegration -- RooRealIntegral::init(coshGBasis_Int[t]) using numeric integrator RooIntegrator1D to calculate Int(t)\n",
"[#1] INFO:NumericIntegration -- RooRealIntegral::init(sinhGBasis_Int[t]) using numeric integrator RooIntegrator1D to calculate Int(t)\n"
]
},
{
"data": {
"text/plain": [
""
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"[#1] INFO:Plotting -- RooAbsPdf::plotOn(pdf) directly selected PDF components: (ampl1)\n",
"[#1] INFO:Plotting -- RooAbsPdf::plotOn(pdf) indirectly selected PDF components: (poly1,coshGBasis)\n",
"[#1] INFO:Plotting -- RooAbsReal::plotOn(pdf) plot on t integrates over variables (cosa)\n",
"[#1] INFO:NumericIntegration -- RooRealIntegral::init(coshGBasis_Int[t]) using numeric integrator RooIntegrator1D to calculate Int(t)\n",
"[#1] INFO:NumericIntegration -- RooRealIntegral::init(sinhGBasis_Int[t]) using numeric integrator RooIntegrator1D to calculate Int(t)\n",
"[#1] INFO:Plotting -- RooAbsPdf::plotOn(pdf) directly selected PDF components: (ampl2)\n",
"[#1] INFO:Plotting -- RooAbsPdf::plotOn(pdf) indirectly selected PDF components: (poly2,sinhGBasis)\n",
"[#1] INFO:Plotting -- RooAbsReal::plotOn(pdf) plot on t integrates over variables (cosa)\n",
"[#1] INFO:NumericIntegration -- RooRealIntegral::init(coshGBasis_Int[t]) using numeric integrator RooIntegrator1D to calculate Int(t)\n",
"[#1] INFO:NumericIntegration -- RooRealIntegral::init(sinhGBasis_Int[t]) using numeric integrator RooIntegrator1D to calculate Int(t)\n"
]
}
],
"source": [
"frame1 = t.frame()\n",
"data.plotOn(frame1)\n",
"pdf.plotOn(frame1)\n",
"pdf.plotOn(frame1, Components=ampl1, LineStyle=\"--\")\n",
"pdf.plotOn(frame1, Components=ampl2, LineStyle=\"--\", LineColor=\"r\")"
]
},
{
"cell_type": "markdown",
"id": "af4996db",
"metadata": {},
"source": [
"Make projection on cosa, data, and its components\n",
"Note that components projection may be larger than sum because\n",
"amplitudes can be negative"
]
},
{
"cell_type": "code",
"execution_count": 11,
"id": "5928298c",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:34:37.745735Z",
"iopub.status.busy": "2026-05-19T20:34:37.745579Z",
"iopub.status.idle": "2026-05-19T20:34:38.034408Z",
"shell.execute_reply": "2026-05-19T20:34:38.033853Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[#1] INFO:Plotting -- RooAbsReal::plotOn(pdf) plot on cosa integrates over variables (t)\n",
"[#1] INFO:NumericIntegration -- RooRealIntegral::init(coshGBasis_Int[t]) using numeric integrator RooIntegrator1D to calculate Int(t)\n",
"[#1] INFO:NumericIntegration -- RooRealIntegral::init(sinhGBasis_Int[t]) using numeric integrator RooIntegrator1D to calculate Int(t)\n",
"[#1] INFO:NumericIntegration -- RooRealIntegral::init(coshGBasis_Int[t]) using numeric integrator RooIntegrator1D to calculate Int(t)\n",
"[#1] INFO:NumericIntegration -- RooRealIntegral::init(sinhGBasis_Int[t]) using numeric integrator RooIntegrator1D to calculate Int(t)\n",
"[#1] INFO:Plotting -- RooAbsPdf::plotOn(pdf) directly selected PDF components: (ampl1)\n",
"[#1] INFO:Plotting -- RooAbsPdf::plotOn(pdf) indirectly selected PDF components: (poly1,coshGBasis)\n",
"[#1] INFO:Plotting -- RooAbsReal::plotOn(pdf) plot on cosa integrates over variables (t)\n",
"[#1] INFO:NumericIntegration -- RooRealIntegral::init(coshGBasis_Int[t]) using numeric integrator RooIntegrator1D to calculate Int(t)\n",
"[#1] INFO:NumericIntegration -- RooRealIntegral::init(sinhGBasis_Int[t]) using numeric integrator RooIntegrator1D to calculate Int(t)\n",
"[#1] INFO:NumericIntegration -- RooRealIntegral::init(coshGBasis_Int[t]) using numeric integrator RooIntegrator1D to calculate Int(t)\n",
"[#1] INFO:Plotting -- RooAbsPdf::plotOn(pdf) directly selected PDF components: (ampl2)\n",
"[#1] INFO:Plotting -- RooAbsPdf::plotOn(pdf) indirectly selected PDF components: (poly2,sinhGBasis)\n",
"[#1] INFO:Plotting -- RooAbsReal::plotOn(pdf) plot on cosa integrates over variables (t)\n",
"[#1] INFO:NumericIntegration -- RooRealIntegral::init(coshGBasis_Int[t]) using numeric integrator RooIntegrator1D to calculate Int(t)\n",
"[#1] INFO:NumericIntegration -- RooRealIntegral::init(sinhGBasis_Int[t]) using numeric integrator RooIntegrator1D to calculate Int(t)\n",
"[#1] INFO:NumericIntegration -- RooRealIntegral::init(sinhGBasis_Int[t]) using numeric integrator RooIntegrator1D to calculate Int(t)\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"Info in : png file rf704_amplitudefit.png has been created\n"
]
}
],
"source": [
"frame2 = cosa.frame()\n",
"data.plotOn(frame2)\n",
"pdf.plotOn(frame2)\n",
"pdf.plotOn(frame2, Components=ampl1, LineStyle=\"--\")\n",
"pdf.plotOn(frame2, Components=ampl2, LineStyle=\"--\", LineColor=\"r\")\n",
"\n",
"c = ROOT.TCanvas(\"rf704_amplitudefit\", \"rf704_amplitudefit\", 800, 800)\n",
"c.Divide(2, 2)\n",
"c.cd(1)\n",
"ROOT.gPad.SetLeftMargin(0.15)\n",
"frame1.GetYaxis().SetTitleOffset(1.4)\n",
"frame1.Draw()\n",
"c.cd(2)\n",
"ROOT.gPad.SetLeftMargin(0.15)\n",
"frame2.GetYaxis().SetTitleOffset(1.4)\n",
"frame2.Draw()\n",
"c.cd(3)\n",
"ROOT.gPad.SetLeftMargin(0.20)\n",
"hh_cos.GetZaxis().SetTitleOffset(2.3)\n",
"hh_cos.Draw(\"surf\")\n",
"c.cd(4)\n",
"ROOT.gPad.SetLeftMargin(0.20)\n",
"hh_sin.GetZaxis().SetTitleOffset(2.3)\n",
"hh_sin.Draw(\"surf\")\n",
"\n",
"c.SaveAs(\"rf704_amplitudefit.png\")"
]
},
{
"cell_type": "markdown",
"id": "a4fbdfff",
"metadata": {},
"source": [
"Draw all canvases "
]
},
{
"cell_type": "code",
"execution_count": 12,
"id": "27344253",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:34:38.035916Z",
"iopub.status.busy": "2026-05-19T20:34:38.035785Z",
"iopub.status.idle": "2026-05-19T20:34:38.231022Z",
"shell.execute_reply": "2026-05-19T20:34:38.230539Z"
}
},
"outputs": [
{
"data": {
"text/html": [
"\n",
"\n",
"\n",
"
\n",
"\n",
"\n",
"\n"
],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"from ROOT import gROOT \n",
"gROOT.GetListOfCanvases().Draw()"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
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"file_extension": ".py",
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