{ "cells": [ { "cell_type": "markdown", "id": "e6cf992b", "metadata": {}, "source": [ "# rf201_composite\n", "Addition and convolution: composite pdf with signal and background component\n", "\n", "```\n", "pdf = f_bkg * bkg(x,a0,a1) + (1-fbkg) * (f_sig1 * sig1(x,m,s1 + (1-f_sig1) * sig2(x,m,s2)))\n", "```\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 08:29 PM." ] }, { "cell_type": "code", "execution_count": 1, "id": "ecc55586", "metadata": { "collapsed": false, "execution": { "iopub.execute_input": "2026-05-19T20:29:35.755709Z", "iopub.status.busy": "2026-05-19T20:29:35.755547Z", "iopub.status.idle": "2026-05-19T20:29:36.697884Z", "shell.execute_reply": "2026-05-19T20:29:36.697479Z" } }, "outputs": [], "source": [ "import ROOT" ] }, { "cell_type": "markdown", "id": "150bd8b8", "metadata": {}, "source": [ "Setup component pdfs\n", "---------------------------------------" ] }, { "cell_type": "markdown", "id": "07c3f370", "metadata": {}, "source": [ "Declare observable x" ] }, { "cell_type": "code", "execution_count": 2, "id": "b52a6a9d", "metadata": { "collapsed": false, "execution": { "iopub.execute_input": "2026-05-19T20:29:36.700198Z", "iopub.status.busy": "2026-05-19T20:29:36.700066Z", "iopub.status.idle": "2026-05-19T20:29:36.860490Z", "shell.execute_reply": "2026-05-19T20:29:36.859711Z" } }, "outputs": [], "source": [ "x = ROOT.RooRealVar(\"x\", \"x\", 0, 10)" ] }, { "cell_type": "markdown", "id": "72440edb", "metadata": {}, "source": [ "Create two Gaussian PDFs g1(x,mean1,sigma) anf g2(x,mean2,sigma) and\n", "their parameters" ] }, { "cell_type": "code", "execution_count": 3, "id": "35aa2ba9", "metadata": { "collapsed": false, "execution": { "iopub.execute_input": "2026-05-19T20:29:36.862447Z", "iopub.status.busy": "2026-05-19T20:29:36.862317Z", "iopub.status.idle": "2026-05-19T20:29:36.990106Z", "shell.execute_reply": "2026-05-19T20:29:36.989411Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[#0] WARNING:InputArguments -- The parameter 'sigma1' with range [-inf, inf] of the RooGaussian 'sig1' exceeds the safe range of (0, inf). Advise to limit its range.\n", "[#0] WARNING:InputArguments -- The parameter 'sigma2' with range [-inf, inf] of the RooGaussian 'sig2' exceeds the safe range of (0, inf). Advise to limit its range.\n" ] } ], "source": [ "mean = ROOT.RooRealVar(\"mean\", \"mean of gaussians\", 5)\n", "sigma1 = ROOT.RooRealVar(\"sigma1\", \"width of gaussians\", 0.5)\n", "sigma2 = ROOT.RooRealVar(\"sigma2\", \"width of gaussians\", 1)\n", "\n", "sig1 = ROOT.RooGaussian(\"sig1\", \"Signal component 1\", x, mean, sigma1)\n", "sig2 = ROOT.RooGaussian(\"sig2\", \"Signal component 2\", x, mean, sigma2)" ] }, { "cell_type": "markdown", "id": "924e19b4", "metadata": {}, "source": [ "Build Chebychev polynomial pdf" ] }, { "cell_type": "code", "execution_count": 4, "id": "0427c922", "metadata": { "collapsed": false, "execution": { "iopub.execute_input": "2026-05-19T20:29:36.991637Z", "iopub.status.busy": "2026-05-19T20:29:36.991488Z", "iopub.status.idle": "2026-05-19T20:29:37.173992Z", "shell.execute_reply": "2026-05-19T20:29:37.173321Z" } }, "outputs": [], "source": [ "a0 = ROOT.RooRealVar(\"a0\", \"a0\", 0.5, 0.0, 1.0)\n", "a1 = ROOT.RooRealVar(\"a1\", \"a1\", -0.2, 0.0, 1.0)\n", "bkg = ROOT.RooChebychev(\"bkg\", \"Background\", x, [a0, a1])" ] }, { "cell_type": "markdown", "id": "387e201a", "metadata": {}, "source": [ "Method 1 - Two RooAddPdfs\n", "------------------------------------------\n", "Add signal components" ] }, { "cell_type": "markdown", "id": "0040907c", "metadata": {}, "source": [ "Sum the signal components into a composite signal pdf" ] }, { "cell_type": "code", "execution_count": 5, "id": "86f67f23", "metadata": { "collapsed": false, "execution": { "iopub.execute_input": "2026-05-19T20:29:37.176218Z", "iopub.status.busy": "2026-05-19T20:29:37.176091Z", "iopub.status.idle": "2026-05-19T20:29:37.288674Z", "shell.execute_reply": "2026-05-19T20:29:37.288019Z" } }, "outputs": [], "source": [ "sig1frac = ROOT.RooRealVar(\"sig1frac\", \"fraction of component 1 in signal\", 0.8, 0.0, 1.0)\n", "sig = ROOT.RooAddPdf(\"sig\", \"Signal\", [sig1, sig2], [sig1frac])" ] }, { "cell_type": "markdown", "id": "49921a36", "metadata": {}, "source": [ "Add signal and background\n", "------------------------------------------------" ] }, { "cell_type": "markdown", "id": "477cc4cf", "metadata": {}, "source": [ "Sum the composite signal and background" ] }, { "cell_type": "code", "execution_count": 6, "id": "73d97a97", "metadata": { "collapsed": false, "execution": { "iopub.execute_input": "2026-05-19T20:29:37.290741Z", "iopub.status.busy": "2026-05-19T20:29:37.290591Z", "iopub.status.idle": "2026-05-19T20:29:37.394293Z", "shell.execute_reply": "2026-05-19T20:29:37.393641Z" } }, "outputs": [], "source": [ "bkgfrac = ROOT.RooRealVar(\"bkgfrac\", \"fraction of background\", 0.5, 0.0, 1.0)\n", "model = ROOT.RooAddPdf(\"model\", \"g1+g2+a\", [bkg, sig], [bkgfrac])" ] }, { "cell_type": "markdown", "id": "530dfcfe", "metadata": {}, "source": [ "Sample, fit and plot model\n", "---------------------------------------------------" ] }, { "cell_type": "markdown", "id": "78b389b1", "metadata": {}, "source": [ "Generate a data sample of 1000 events in x from model" ] }, { "cell_type": "code", "execution_count": 7, "id": "9a184ee3", "metadata": { "collapsed": false, "execution": { "iopub.execute_input": "2026-05-19T20:29:37.396294Z", "iopub.status.busy": "2026-05-19T20:29:37.396165Z", "iopub.status.idle": "2026-05-19T20:29:37.541956Z", "shell.execute_reply": "2026-05-19T20:29:37.541297Z" } }, "outputs": [], "source": [ "data = model.generate({x}, 1000)" ] }, { "cell_type": "markdown", "id": "c76a5a1a", "metadata": {}, "source": [ "Fit model to data" ] }, { "cell_type": "code", "execution_count": 8, "id": "b69288b1", "metadata": { "collapsed": false, "execution": { "iopub.execute_input": "2026-05-19T20:29:37.544099Z", "iopub.status.busy": "2026-05-19T20:29:37.543976Z", "iopub.status.idle": "2026-05-19T20:29:37.734083Z", "shell.execute_reply": "2026-05-19T20:29:37.733446Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[#1] INFO:Fitting -- RooAbsPdf::fitTo(model) 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 828.944 μs\n", "[#1] INFO:Fitting -- RooAddition::defaultErrorLevel(nll_model_modelData) Summation contains a RooNLLVar, using its error level\n", "[#1] INFO:Minimization -- [fitFCN] No discrete parameters, performing continuous minimization only\n" ] }, { "data": { "text/plain": [ "" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "model.fitTo(data, PrintLevel=-1)" ] }, { "cell_type": "markdown", "id": "cb815db5", "metadata": {}, "source": [ "Plot data and PDF overlaid" ] }, { "cell_type": "code", "execution_count": 9, "id": "041f6c6f", "metadata": { "collapsed": false, "execution": { "iopub.execute_input": "2026-05-19T20:29:37.736271Z", "iopub.status.busy": "2026-05-19T20:29:37.736144Z", "iopub.status.idle": "2026-05-19T20:29:37.912410Z", "shell.execute_reply": "2026-05-19T20:29:37.911776Z" } }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "xframe = x.frame(Title=\"Example of composite pdf=(sig1+sig2)+bkg\")\n", "data.plotOn(xframe)\n", "model.plotOn(xframe)" ] }, { "cell_type": "markdown", "id": "b6c033d9", "metadata": {}, "source": [ "Overlay the background component of model with a dashed line" ] }, { "cell_type": "code", "execution_count": 10, "id": "63cfd450", "metadata": { "collapsed": false, "execution": { "iopub.execute_input": "2026-05-19T20:29:37.914586Z", "iopub.status.busy": "2026-05-19T20:29:37.914463Z", "iopub.status.idle": "2026-05-19T20:29:38.032751Z", "shell.execute_reply": "2026-05-19T20:29:38.032101Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[#1] INFO:Plotting -- RooAbsPdf::plotOn(model) directly selected PDF components: (bkg)\n", "[#1] INFO:Plotting -- RooAbsPdf::plotOn(model) indirectly selected PDF components: ()\n" ] }, { "data": { "text/plain": [ "" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "model.plotOn(xframe, Components={bkg}, LineStyle=\"--\")" ] }, { "cell_type": "markdown", "id": "8a5786ae", "metadata": {}, "source": [ "Overlay the background+sig2 components of model with a dotted line" ] }, { "cell_type": "code", "execution_count": 11, "id": "6330f3f9", "metadata": { "collapsed": false, "execution": { "iopub.execute_input": "2026-05-19T20:29:38.034971Z", "iopub.status.busy": "2026-05-19T20:29:38.034811Z", "iopub.status.idle": "2026-05-19T20:29:38.157195Z", "shell.execute_reply": "2026-05-19T20:29:38.156433Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[#1] INFO:Plotting -- RooAbsPdf::plotOn(model) directly selected PDF components: (bkg,sig2)\n", "[#1] INFO:Plotting -- RooAbsPdf::plotOn(model) indirectly selected PDF components: (sig)\n" ] }, { "data": { "text/plain": [ "" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "model.plotOn(xframe, Components={bkg, sig2}, LineStyle=\":\")" ] }, { "cell_type": "markdown", "id": "3a0b25cc", "metadata": {}, "source": [ "Print structure of composite pdf" ] }, { "cell_type": "code", "execution_count": 12, "id": "20c323d7", "metadata": { "collapsed": false, "execution": { "iopub.execute_input": "2026-05-19T20:29:38.158888Z", "iopub.status.busy": "2026-05-19T20:29:38.158762Z", "iopub.status.idle": "2026-05-19T20:29:38.267932Z", "shell.execute_reply": "2026-05-19T20:29:38.267316Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "0x5639e795a9a0 RooAddPdf::model = 0.886326/1 [Auto,Clean] \n", " 0x5639e780f3f0/V- RooChebychev::bkg = 0.734412 [Auto,Dirty] \n", " 0x5639e1cba8f0/V- RooRealVar::x = 5\n", " 0x5639e735c370/V- RooRealVar::a0 = 0.506755 +/- 0.0795919\n", " 0x5639e709eba0/V- RooRealVar::a1 = 0.265588 +/- 0.133931\n", " 0x5639e76ecab0/V- RooRealVar::bkgfrac = 0.428008 +/- 0.0356013\n", " 0x5639e7954100/V- RooAddPdf::sig = 1/1 [Auto,Clean] \n", " 0x5639e73f4bc0/V- RooGaussian::sig1 = 1 [Auto,Dirty] \n", " 0x5639e1cba8f0/V- RooRealVar::x = 5\n", " 0x5639e7310280/V- RooRealVar::mean = 5\n", " 0x5639e736a9c0/V- RooRealVar::sigma1 = 0.5\n", " 0x5639e74331d0/V- RooRealVar::sig1frac = 0.641992 +/- 0.0969095\n", " 0x5639e744c7e0/V- RooGaussian::sig2 = 1 [Auto,Dirty] \n", " 0x5639e1cba8f0/V- RooRealVar::x = 5\n", " 0x5639e7310280/V- RooRealVar::mean = 5\n", " 0x5639e730caf0/V- RooRealVar::sigma2 = 1\n" ] } ], "source": [ "model.Print(\"t\")" ] }, { "cell_type": "markdown", "id": "90efa4e8", "metadata": {}, "source": [ "Method 2 - One RooAddPdf with recursive fractions\n", "---------------------------------------------------" ] }, { "cell_type": "markdown", "id": "e7e06666", "metadata": {}, "source": [ "Construct sum of models on one go using recursive fraction interpretations\n", "\n", " model2 = bkg + (sig1 + sig2)" ] }, { "cell_type": "code", "execution_count": 13, "id": "8289a5f9", "metadata": { "collapsed": false, "execution": { "iopub.execute_input": "2026-05-19T20:29:38.269535Z", "iopub.status.busy": "2026-05-19T20:29:38.269336Z", "iopub.status.idle": "2026-05-19T20:29:38.373025Z", "shell.execute_reply": "2026-05-19T20:29:38.372416Z" } }, "outputs": [], "source": [ "model2 = ROOT.RooAddPdf(\"model\", \"g1+g2+a\", [bkg, sig1, sig2], [bkgfrac, sig1frac], True)" ] }, { "cell_type": "markdown", "id": "3783727d", "metadata": {}, "source": [ "NB: Each coefficient is interpreted as the fraction of the\n", "left-hand component of the i-th recursive sum, i.e.\n", "\n", " sum4 = A + ( B + ( C + D) with fraction fA, and fC expands to\n", "\n", " sum4 = fA*A + (1-fA)*(fB*B + (1-fB)*(fC*C + (1-fC)*D))" ] }, { "cell_type": "markdown", "id": "8f385d63", "metadata": {}, "source": [ "Plot recursive addition model\n", "---------------------------------------------------------" ] }, { "cell_type": "code", "execution_count": 14, "id": "d1ee2766", "metadata": { "collapsed": false, "execution": { "iopub.execute_input": "2026-05-19T20:29:38.375145Z", "iopub.status.busy": "2026-05-19T20:29:38.375022Z", "iopub.status.idle": "2026-05-19T20:29:38.492075Z", "shell.execute_reply": "2026-05-19T20:29:38.491463Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[#1] INFO:Plotting -- RooAbsPdf::plotOn(model) directly selected PDF components: (bkg,sig2)\n", "[#1] INFO:Plotting -- RooAbsPdf::plotOn(model) indirectly selected PDF components: ()\n", "0x5639e8033be0 RooAddPdf::model = 0.886326/1 [Auto,Clean] \n", " 0x5639e780f3f0/V- RooChebychev::bkg = 0.734412 [Auto,Dirty] \n", " 0x5639e1cba8f0/V- RooRealVar::x = 5\n", " 0x5639e735c370/V- RooRealVar::a0 = 0.506755 +/- 0.0795919\n", " 0x5639e709eba0/V- RooRealVar::a1 = 0.265588 +/- 0.133931\n", " 0x5639e76ecab0/V- RooRealVar::bkgfrac = 0.428008 +/- 0.0356013\n", " 0x5639e73f4bc0/V- RooGaussian::sig1 = 1 [Auto,Dirty] \n", " 0x5639e1cba8f0/V- RooRealVar::x = 5\n", " 0x5639e7310280/V- RooRealVar::mean = 5\n", " 0x5639e736a9c0/V- RooRealVar::sigma1 = 0.5\n", " 0x5639e8208ac0/V- RooRecursiveFraction::model_recursive_fraction_sig1_2 = 0.367214 [Auto,Clean] \n", " 0x5639e74331d0/V- RooRealVar::sig1frac = 0.641992 +/- 0.0969095\n", " 0x5639e76ecab0/V- RooRealVar::bkgfrac = 0.428008 +/- 0.0356013\n", " 0x5639e744c7e0/V- RooGaussian::sig2 = 1 [Auto,Dirty] \n", " 0x5639e1cba8f0/V- RooRealVar::x = 5\n", " 0x5639e7310280/V- RooRealVar::mean = 5\n", " 0x5639e730caf0/V- RooRealVar::sigma2 = 1\n", " 0x5639e7faca90/V- RooRecursiveFraction::model_recursive_fraction_sig2_3 = 0.204778 [Auto,Clean] \n", " 0x5639e8197a60/V- RooConstVar::1 = 1\n", " 0x5639e74331d0/V- RooRealVar::sig1frac = 0.641992 +/- 0.0969095\n", " 0x5639e76ecab0/V- RooRealVar::bkgfrac = 0.428008 +/- 0.0356013\n" ] } ], "source": [ "model2.plotOn(xframe, LineColor=\"r\", LineStyle=\"--\")\n", "model2.plotOn(xframe, Components={bkg, sig2}, LineColor=\"r\", LineStyle=\"--\")\n", "model2.Print(\"t\")" ] }, { "cell_type": "markdown", "id": "a3c9f573", "metadata": {}, "source": [ "Draw the frame on the canvas" ] }, { "cell_type": "code", "execution_count": 15, "id": "c65d6435", "metadata": { "collapsed": false, "execution": { "iopub.execute_input": "2026-05-19T20:29:38.493749Z", "iopub.status.busy": "2026-05-19T20:29:38.493623Z", "iopub.status.idle": "2026-05-19T20:29:38.705387Z", "shell.execute_reply": "2026-05-19T20:29:38.704662Z" } }, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "Info in : png file rf201_composite.png has been created\n" ] } ], "source": [ "c = ROOT.TCanvas(\"rf201_composite\", \"rf201_composite\", 600, 600)\n", "ROOT.gPad.SetLeftMargin(0.15)\n", "xframe.GetYaxis().SetTitleOffset(1.4)\n", "xframe.Draw()\n", "\n", "c.SaveAs(\"rf201_composite.png\")" ] }, { "cell_type": "markdown", "id": "b76335af", "metadata": {}, "source": [ "Draw all canvases " ] }, { "cell_type": "code", "execution_count": 16, "id": "7c952af4", "metadata": { "collapsed": false, "execution": { "iopub.execute_input": "2026-05-19T20:29:38.707164Z", "iopub.status.busy": "2026-05-19T20:29:38.707042Z", "iopub.status.idle": "2026-05-19T20:29:38.888018Z", "shell.execute_reply": "2026-05-19T20:29:38.887408Z" } }, "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 }