{
"cells": [
{
"cell_type": "markdown",
"id": "5b60a08f",
"metadata": {},
"source": [
"# rf315_projectpdf\n",
"Multidimensional models: marginizalization of multi-dimensional pdfs through integration\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 08:31 PM."
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "7fb8eaa0",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:31:23.189259Z",
"iopub.status.busy": "2026-05-19T20:31:23.189147Z",
"iopub.status.idle": "2026-05-19T20:31:24.151888Z",
"shell.execute_reply": "2026-05-19T20:31:24.151364Z"
}
},
"outputs": [],
"source": [
"import ROOT"
]
},
{
"cell_type": "markdown",
"id": "c8256352",
"metadata": {},
"source": [
"Create pdf m(x,y) = gx(x|y) * g(y)\n",
"--------------------------------------------------------------"
]
},
{
"cell_type": "markdown",
"id": "fd296db3",
"metadata": {},
"source": [
"Increase default precision of numeric integration\n",
"as self exercise has high sensitivity to numeric integration precision"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "36585e52",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:31:24.153946Z",
"iopub.status.busy": "2026-05-19T20:31:24.153814Z",
"iopub.status.idle": "2026-05-19T20:31:24.334784Z",
"shell.execute_reply": "2026-05-19T20:31:24.329293Z"
}
},
"outputs": [],
"source": [
"ROOT.RooAbsPdf.defaultIntegratorConfig().setEpsRel(1e-8)\n",
"ROOT.RooAbsPdf.defaultIntegratorConfig().setEpsAbs(1e-8)"
]
},
{
"cell_type": "markdown",
"id": "acfefcec",
"metadata": {},
"source": [
"Create observables"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "2ce2fdb2",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:31:24.343880Z",
"iopub.status.busy": "2026-05-19T20:31:24.343746Z",
"iopub.status.idle": "2026-05-19T20:31:24.460476Z",
"shell.execute_reply": "2026-05-19T20:31:24.459736Z"
}
},
"outputs": [],
"source": [
"x = ROOT.RooRealVar(\"x\", \"x\", -5, 5)\n",
"y = ROOT.RooRealVar(\"y\", \"y\", -2, 2)"
]
},
{
"cell_type": "markdown",
"id": "5269c212",
"metadata": {},
"source": [
"Create function f(y) = a0 + a1*y"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "5dffbbf9",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:31:24.462372Z",
"iopub.status.busy": "2026-05-19T20:31:24.462241Z",
"iopub.status.idle": "2026-05-19T20:31:24.650036Z",
"shell.execute_reply": "2026-05-19T20:31:24.649269Z"
}
},
"outputs": [],
"source": [
"a0 = ROOT.RooRealVar(\"a0\", \"a0\", 0)\n",
"a1 = ROOT.RooRealVar(\"a1\", \"a1\", -1.5, -3, 1)\n",
"fy = ROOT.RooPolyVar(\"fy\", \"fy\", y, [a0, a1])"
]
},
{
"cell_type": "markdown",
"id": "218461ff",
"metadata": {},
"source": [
"Create gaussx(x,f(y),sx)"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "05436251",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:31:24.651765Z",
"iopub.status.busy": "2026-05-19T20:31:24.651639Z",
"iopub.status.idle": "2026-05-19T20:31:24.764813Z",
"shell.execute_reply": "2026-05-19T20:31:24.763988Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[#0] WARNING:InputArguments -- The parameter 'sigmax' with range [-inf, inf] of the RooGaussian 'gaussx' exceeds the safe range of (0, inf). Advise to limit its range.\n"
]
}
],
"source": [
"sigmax = ROOT.RooRealVar(\"sigmax\", \"width of gaussian\", 0.5)\n",
"gaussx = ROOT.RooGaussian(\"gaussx\", \"Gaussian in x with shifting mean in y\", x, fy, sigmax)"
]
},
{
"cell_type": "markdown",
"id": "ed95ffc1",
"metadata": {},
"source": [
"Create gaussy(y,0,2)"
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "f6e601b4",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:31:24.766325Z",
"iopub.status.busy": "2026-05-19T20:31:24.766202Z",
"iopub.status.idle": "2026-05-19T20:31:24.880060Z",
"shell.execute_reply": "2026-05-19T20:31:24.879645Z"
}
},
"outputs": [],
"source": [
"gaussy = ROOT.RooGaussian(\"gaussy\", \"Gaussian in y\", y, 0.0, 2.0)"
]
},
{
"cell_type": "markdown",
"id": "b47c0e3f",
"metadata": {},
"source": [
"Create gaussx(x,sx|y) * gaussy(y)"
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "5204aa0a",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:31:24.898387Z",
"iopub.status.busy": "2026-05-19T20:31:24.898249Z",
"iopub.status.idle": "2026-05-19T20:31:25.047297Z",
"shell.execute_reply": "2026-05-19T20:31:25.046525Z"
}
},
"outputs": [],
"source": [
"model = ROOT.RooProdPdf(\n",
" \"model\",\n",
" \"gaussx(x|y)*gaussy(y)\",\n",
" {gaussy},\n",
" Conditional=({gaussx}, {x}),\n",
")"
]
},
{
"cell_type": "markdown",
"id": "550bfb4d",
"metadata": {},
"source": [
"Marginalize m(x,y) to m(x)\n",
"----------------------------------------------------"
]
},
{
"cell_type": "markdown",
"id": "0d1e5fe2",
"metadata": {},
"source": [
"modelx(x) = Int model(x,y) dy"
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "9a4112d4",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:31:25.070479Z",
"iopub.status.busy": "2026-05-19T20:31:25.070336Z",
"iopub.status.idle": "2026-05-19T20:31:25.187399Z",
"shell.execute_reply": "2026-05-19T20:31:25.186745Z"
}
},
"outputs": [],
"source": [
"modelx = model.createProjection({y})"
]
},
{
"cell_type": "markdown",
"id": "d318a0e3",
"metadata": {},
"source": [
"Use marginalized pdf as regular 1D pdf\n",
"-----------------------------------------------"
]
},
{
"cell_type": "markdown",
"id": "00c5e3b2",
"metadata": {},
"source": [
"Sample 1000 events from modelx"
]
},
{
"cell_type": "code",
"execution_count": 9,
"id": "659f6916",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:31:25.189377Z",
"iopub.status.busy": "2026-05-19T20:31:25.189253Z",
"iopub.status.idle": "2026-05-19T20:31:25.313200Z",
"shell.execute_reply": "2026-05-19T20:31:25.312507Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[#1] INFO:NumericIntegration -- RooRealIntegral::init(SPECINT[gaussy_NORM[y]_X_gaussx_NORM[x]]_Int[y]) using numeric integrator RooIntegrator1D to calculate Int(y)\n"
]
}
],
"source": [
"data = modelx.generateBinned({x}, 1000)"
]
},
{
"cell_type": "markdown",
"id": "4cad1a42",
"metadata": {},
"source": [
"Fit modelx to toy data"
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "0a1f5417",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:31:25.314913Z",
"iopub.status.busy": "2026-05-19T20:31:25.314792Z",
"iopub.status.idle": "2026-05-19T20:31:25.533174Z",
"shell.execute_reply": "2026-05-19T20:31:25.532470Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[#1] INFO:Fitting -- RooAbsPdf::fitTo(model_Int[y]_Norm[x,y]_wrapped_pdf) 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 1.20203 ms\n",
"[#1] INFO:Fitting -- RooAddition::defaultErrorLevel(nll_model_Int[y]_Norm[x,y]_wrapped_pdf_genData) Summation contains a RooNLLVar, using its error level\n",
"[#0] WARNING:Minimization -- RooAbsMinimizerFcn::synchronize: WARNING: no initial error estimate available for a1: using 0.4\n",
"[#1] INFO:Minimization -- [fitFCN] No discrete parameters, performing continuous minimization only\n",
"[#1] INFO:NumericIntegration -- RooRealIntegral::init(SPECINT[gaussy_NORM[y]_X_gaussx_NORM[x]]_Int[y]) using numeric integrator RooIntegrator1D to calculate Int(y)\n",
"\n",
"prevFCN = 1900.156536 a1=-1.488, \n",
"prevFCN = 1899.96972 a1=-1.512, \n",
"prevFCN = 1900.566064 a1=-1.499, \n",
"prevFCN = 1900.127678 a1=-1.501, \n",
"prevFCN = 1900.187622 a1=-1.484, \n",
"prevFCN = 1899.958651 a1=-1.483, \n",
"prevFCN = 1899.959674 a1=-1.484, \n",
"prevFCN = 1899.958586 a1=-1.484, \n",
"prevFCN = 1899.958651 a1=-1.483, \n",
"prevFCN = 1899.959674 a1=-1.484, \n",
"prevFCN = 1899.958586 a1=-1.483, \n",
"prevFCN = 1899.958779 a1=-1.484, \n",
"prevFCN = 1899.958562 a1=-1.484, \n",
"prevFCN = 1899.958651 a1=-1.483, \n",
"prevFCN = 1899.958779 a1=-1.484, \n",
"prevFCN = 1899.958562 a1=-1.484, \n",
"prevFCN = 1899.958674 a1=-1.484, \n",
"prevFCN = 1899.95863 a1=-1.484, "
]
},
{
"data": {
"text/plain": [
""
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"modelx.fitTo(data, Verbose=True, PrintLevel=-1)"
]
},
{
"cell_type": "markdown",
"id": "58e60f84",
"metadata": {},
"source": [
"Plot modelx over data"
]
},
{
"cell_type": "code",
"execution_count": 11,
"id": "44430e67",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:31:25.535315Z",
"iopub.status.busy": "2026-05-19T20:31:25.535188Z",
"iopub.status.idle": "2026-05-19T20:31:25.700544Z",
"shell.execute_reply": "2026-05-19T20:31:25.699844Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[#1] INFO:NumericIntegration -- RooRealIntegral::init(SPECINT[gaussy_NORM[y]_X_gaussx_NORM[x]]_Int[y]) using numeric integrator RooIntegrator1D to calculate Int(y)\n"
]
},
{
"data": {
"text/plain": [
""
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"frame = x.frame(40)\n",
"data.plotOn(frame)\n",
"modelx.plotOn(frame)"
]
},
{
"cell_type": "markdown",
"id": "342f294b",
"metadata": {},
"source": [
"Make 2D histogram of model(x,y)"
]
},
{
"cell_type": "code",
"execution_count": 12,
"id": "812f980d",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:31:25.702452Z",
"iopub.status.busy": "2026-05-19T20:31:25.702327Z",
"iopub.status.idle": "2026-05-19T20:31:26.007078Z",
"shell.execute_reply": "2026-05-19T20:31:26.006371Z"
}
},
"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": [
"Info in : png file rf315_projectpdf.png has been created\n"
]
}
],
"source": [
"hh = model.createHistogram(\"x,y\")\n",
"hh.SetLineColor(\"kBlue\")\n",
"\n",
"c = ROOT.TCanvas(\"rf315_projectpdf\", \"rf315_projectpdf\", 800, 400)\n",
"c.Divide(2)\n",
"c.cd(1)\n",
"ROOT.gPad.SetLeftMargin(0.15)\n",
"frame.GetYaxis().SetTitleOffset(1.4)\n",
"frame.Draw()\n",
"c.cd(2)\n",
"ROOT.gPad.SetLeftMargin(0.20)\n",
"hh.GetZaxis().SetTitleOffset(2.5)\n",
"hh.Draw(\"surf\")\n",
"c.SaveAs(\"rf315_projectpdf.png\")"
]
},
{
"cell_type": "markdown",
"id": "429b0cc3",
"metadata": {},
"source": [
"Draw all canvases "
]
},
{
"cell_type": "code",
"execution_count": 13,
"id": "b7487ca9",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:31:26.009262Z",
"iopub.status.busy": "2026-05-19T20:31:26.009125Z",
"iopub.status.idle": "2026-05-19T20:31:26.204186Z",
"shell.execute_reply": "2026-05-19T20:31:26.203489Z"
}
},
"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": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.12.12"
}
},
"nbformat": 4,
"nbformat_minor": 5
}