{
"cells": [
{
"cell_type": "markdown",
"id": "cab3f50e",
"metadata": {},
"source": [
"# rf308_normintegration2d\n",
"Multidimensional models: normalization and integration of pdfs, construction of\n",
"cumulative distribution functions from pdfs in two dimensions\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 08:30 PM."
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "0992c586",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:30:54.949582Z",
"iopub.status.busy": "2026-05-19T20:30:54.949467Z",
"iopub.status.idle": "2026-05-19T20:30:55.902581Z",
"shell.execute_reply": "2026-05-19T20:30:55.901849Z"
}
},
"outputs": [],
"source": [
"import ROOT"
]
},
{
"cell_type": "markdown",
"id": "73d599db",
"metadata": {},
"source": [
"Set up model\n",
"---------------------"
]
},
{
"cell_type": "markdown",
"id": "a2f8fb3b",
"metadata": {},
"source": [
"Create observables x,y"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "401c3979",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:30:55.904363Z",
"iopub.status.busy": "2026-05-19T20:30:55.904231Z",
"iopub.status.idle": "2026-05-19T20:30:56.061671Z",
"shell.execute_reply": "2026-05-19T20:30:56.060992Z"
}
},
"outputs": [],
"source": [
"x = ROOT.RooRealVar(\"x\", \"x\", -10, 10)\n",
"y = ROOT.RooRealVar(\"y\", \"y\", -10, 10)"
]
},
{
"cell_type": "markdown",
"id": "75e3b63e",
"metadata": {},
"source": [
"Create pdf gaussx(x,-2,3), gaussy(y,2,2)"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "0d0de452",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:30:56.074490Z",
"iopub.status.busy": "2026-05-19T20:30:56.074351Z",
"iopub.status.idle": "2026-05-19T20:30:56.201127Z",
"shell.execute_reply": "2026-05-19T20:30:56.200448Z"
}
},
"outputs": [],
"source": [
"gx = ROOT.RooGaussian(\"gx\", \"gx\", x, -2.0, 3.0)\n",
"gy = ROOT.RooGaussian(\"gy\", \"gy\", y, +2.0, 2.0)"
]
},
{
"cell_type": "markdown",
"id": "53482bb7",
"metadata": {},
"source": [
"gxy = gx(x)*gy(y)"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "b7f83fc4",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:30:56.203219Z",
"iopub.status.busy": "2026-05-19T20:30:56.203095Z",
"iopub.status.idle": "2026-05-19T20:30:56.381330Z",
"shell.execute_reply": "2026-05-19T20:30:56.380643Z"
}
},
"outputs": [],
"source": [
"gxy = ROOT.RooProdPdf(\"gxy\", \"gxy\", [gx, gy])"
]
},
{
"cell_type": "markdown",
"id": "ed4ed652",
"metadata": {},
"source": [
"Retrieve raw & normalized values of RooFit pdfs\n",
"--------------------------------------------------------------------------------------------------"
]
},
{
"cell_type": "markdown",
"id": "a99c332c",
"metadata": {},
"source": [
"Return 'raw' unnormalized value of gx"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "a722a223",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:30:56.383522Z",
"iopub.status.busy": "2026-05-19T20:30:56.383395Z",
"iopub.status.idle": "2026-05-19T20:30:56.506925Z",
"shell.execute_reply": "2026-05-19T20:30:56.506261Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"gxy = 0.4856717852477124\n"
]
}
],
"source": [
"print(\"gxy = \", gxy.getVal())"
]
},
{
"cell_type": "markdown",
"id": "94414777",
"metadata": {},
"source": [
"Return value of gxy normalized over x _and_ y in range [-10,10]"
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "895eb5ff",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:30:56.508655Z",
"iopub.status.busy": "2026-05-19T20:30:56.508514Z",
"iopub.status.idle": "2026-05-19T20:30:56.630125Z",
"shell.execute_reply": "2026-05-19T20:30:56.629424Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"gx_Norm[x,y] = 0.012933200957206766\n"
]
}
],
"source": [
"nset_xy = ROOT.RooArgSet(x, y)\n",
"print(\"gx_Norm[x,y] = \", gxy.getVal(nset_xy))"
]
},
{
"cell_type": "markdown",
"id": "ec726200",
"metadata": {},
"source": [
"Create object representing integral over gx\n",
"which is used to calculate gx_Norm[x,y] == gx / gx_Int[x,y]"
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "20cf3b2b",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:30:56.632035Z",
"iopub.status.busy": "2026-05-19T20:30:56.631913Z",
"iopub.status.idle": "2026-05-19T20:30:56.752558Z",
"shell.execute_reply": "2026-05-19T20:30:56.751909Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"gx_Int[x,y] = 37.552326516436096\n"
]
}
],
"source": [
"x_and_y = {x, y}\n",
"igxy = gxy.createIntegral(x_and_y)\n",
"print(\"gx_Int[x,y] = \", igxy.getVal())"
]
},
{
"cell_type": "markdown",
"id": "4a81abcd",
"metadata": {},
"source": [
"NB: it is also possible to do the following"
]
},
{
"cell_type": "markdown",
"id": "f96baaec",
"metadata": {},
"source": [
"Return value of gxy normalized over x in range [-10,10] (i.e. treating y\n",
"as parameter)"
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "ce16a86e",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:30:56.765480Z",
"iopub.status.busy": "2026-05-19T20:30:56.765331Z",
"iopub.status.idle": "2026-05-19T20:30:56.869043Z",
"shell.execute_reply": "2026-05-19T20:30:56.868358Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"gx_Norm[x] = 0.1068955044839622\n"
]
}
],
"source": [
"nset_x = ROOT.RooArgSet(x)\n",
"print(\"gx_Norm[x] = \", gxy.getVal(nset_x))"
]
},
{
"cell_type": "markdown",
"id": "f66e75d3",
"metadata": {},
"source": [
"Return value of gxy normalized over y in range [-10,10] (i.e. treating x\n",
"as parameter)"
]
},
{
"cell_type": "code",
"execution_count": 9,
"id": "ee889ba3",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:30:56.879388Z",
"iopub.status.busy": "2026-05-19T20:30:56.879255Z",
"iopub.status.idle": "2026-05-19T20:30:56.985171Z",
"shell.execute_reply": "2026-05-19T20:30:56.984591Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"gx_Norm[y] = 0.12098919425696865\n"
]
}
],
"source": [
"nset_y = ROOT.RooArgSet(y)\n",
"print(\"gx_Norm[y] = \", gxy.getVal(nset_y))"
]
},
{
"cell_type": "markdown",
"id": "92572583",
"metadata": {},
"source": [
"Integrate normalized pdf over subrange\n",
"----------------------------------------------------------------------------"
]
},
{
"cell_type": "markdown",
"id": "502ba548",
"metadata": {},
"source": [
"Define a range named \"signal\" in x from -5,5"
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "57d36bb3",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:30:56.987156Z",
"iopub.status.busy": "2026-05-19T20:30:56.987035Z",
"iopub.status.idle": "2026-05-19T20:30:57.096408Z",
"shell.execute_reply": "2026-05-19T20:30:57.095695Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[#1] INFO:Eval -- RooRealVar::setRange(x) new range named 'signal' created with bounds [-5,5]\n",
"[#1] INFO:Eval -- RooRealVar::setRange(y) new range named 'signal' created with bounds [-3,3]\n"
]
}
],
"source": [
"x.setRange(\"signal\", -5, 5)\n",
"y.setRange(\"signal\", -3, 3)"
]
},
{
"cell_type": "markdown",
"id": "82e62aeb",
"metadata": {},
"source": [
"Create an integral of gxy_Norm[x,y] over x and y in range \"signal\"\n",
"ROOT.This is the fraction of of pdf gxy_Norm[x,y] which is in the\n",
"range named \"signal\""
]
},
{
"cell_type": "code",
"execution_count": 11,
"id": "825747ed",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:30:57.098140Z",
"iopub.status.busy": "2026-05-19T20:30:57.098018Z",
"iopub.status.idle": "2026-05-19T20:30:57.236131Z",
"shell.execute_reply": "2026-05-19T20:30:57.235442Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"gx_Int[x,y|signal]_Norm[x,y] = 0.5720351351990985\n"
]
}
],
"source": [
"igxy_sig = gxy.createIntegral(x_and_y, NormSet=x_and_y, Range=\"signal\")\n",
"print(\"gx_Int[x,y|signal]_Norm[x,y] = \", igxy_sig.getVal())"
]
},
{
"cell_type": "markdown",
"id": "2a63188d",
"metadata": {},
"source": [
"Construct cumulative distribution function from pdf\n",
"-----------------------------------------------------------------------------------------------------"
]
},
{
"cell_type": "markdown",
"id": "73d26e64",
"metadata": {},
"source": [
"Create the cumulative distribution function of gx\n",
"i.e. calculate Int[-10,x] gx(x') dx'"
]
},
{
"cell_type": "code",
"execution_count": 12,
"id": "d21a6470",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:30:57.237974Z",
"iopub.status.busy": "2026-05-19T20:30:57.237854Z",
"iopub.status.idle": "2026-05-19T20:30:57.348159Z",
"shell.execute_reply": "2026-05-19T20:30:57.347474Z"
}
},
"outputs": [],
"source": [
"gxy_cdf = gxy.createCdf({x, y})"
]
},
{
"cell_type": "markdown",
"id": "16fdcf42",
"metadata": {},
"source": [
"Plot cdf of gx versus x"
]
},
{
"cell_type": "code",
"execution_count": 13,
"id": "3d512374",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:30:57.349955Z",
"iopub.status.busy": "2026-05-19T20:30:57.349832Z",
"iopub.status.idle": "2026-05-19T20:30:57.607146Z",
"shell.execute_reply": "2026-05-19T20:30:57.606524Z"
}
},
"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": [
"Info in : png file rf308_normintegration2d.png has been created\n"
]
}
],
"source": [
"hh_cdf = gxy_cdf.createHistogram(\"hh_cdf\", x, Binning=40, YVar=dict(var=y, Binning=40))\n",
"hh_cdf.SetLineColor(\"kBlue\")\n",
"\n",
"c = ROOT.TCanvas(\"rf308_normintegration2d\", \"rf308_normintegration2d\", 600, 600)\n",
"ROOT.gPad.SetLeftMargin(0.15)\n",
"hh_cdf.GetZaxis().SetTitleOffset(1.8)\n",
"hh_cdf.Draw(\"surf\")\n",
"\n",
"c.SaveAs(\"rf308_normintegration2d.png\")"
]
},
{
"cell_type": "markdown",
"id": "afa1e1bc",
"metadata": {},
"source": [
"Draw all canvases "
]
},
{
"cell_type": "code",
"execution_count": 14,
"id": "052c994f",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:30:57.609295Z",
"iopub.status.busy": "2026-05-19T20:30:57.609172Z",
"iopub.status.idle": "2026-05-19T20:30:57.787677Z",
"shell.execute_reply": "2026-05-19T20:30:57.787124Z"
}
},
"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",
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"nbformat": 4,
"nbformat_minor": 5
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