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wiso/UncorrelatedCategories.ipynb

Created April 28, 2016 14:47
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UncorrelatedCategories.ipynb
{
"cells": [
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"import ROOT\n",
"from scipy import stats\n",
"import numpy as np\n",
"from IPython.display import SVG\n",
"import re\n",
"from matplotlib import pylab as plt"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Define three categories with different fraction of signal and background"
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"ksigs = ksig1, ksig2, ksig3 = 0.7, 0.2, 0.1 # signal partitioning in 3 categories\n",
"kbkgs = kbkg1, kbkg2, kbkg3 = 0.3, 0.3, 0.4 # background partitioning in 3 categories"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"<ROOT.RooRealVar object (\"kbkg3\") at 0x62dea30>"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"signal_xsec_inc = 20 # inclusive signal xsection\n",
"bkg_xsec_inc = 4000 # inclusive bkg xsection\n",
"lumi = 3 # luminosity\n",
"\n",
"# load constants into ws\n",
"ws = ROOT.RooWorkspace('ws')\n",
"ws.factory('M[750, 300, 3400]')\n",
"ws.var(\"M\").setVal(750)\n",
"ws.var(\"M\").setConstant(True)\n",
"mgg = ws.factory('mgg[200, 3500]')\n",
"ws.factory(\"lumi[%f]\" % lumi)\n",
"ws.factory(\"signal_xsec[%f]\" % (signal_xsec_inc))\n",
"ws.factory(\"bkg_xsec[%f]\" % (bkg_xsec_inc))\n",
"ws.factory(\"ksig1[%f]\" % ksig1)\n",
"ws.factory(\"ksig2[%f]\" % ksig2)\n",
"ws.factory(\"ksig3[%f]\" % ksig3)\n",
"ws.factory(\"kbkg1[%f]\" % kbkg1)\n",
"ws.factory(\"kbkg2[%f]\" % kbkg2)\n",
"ws.factory(\"kbkg3[%f]\" % kbkg3)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## $s / \\sqrt{b}$ in the whole range"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"s/sqrt(b) inclusive\n",
"0.547722557505\n",
"s/sqrt(b) per categories\n",
"[ 0.7 0.2 0.08660254]\n"
]
}
],
"source": [
"s = np.array([ksig1, ksig2, ksig3]) * signal_xsec_inc * lumi\n",
"b = np.array([kbkg1, kbkg2, kbkg3]) * bkg_xsec_inc * lumi\n",
"s_over_sqrtb = s / np.sqrt(b)\n",
"\n",
"print \"s/sqrt(b) inclusive\"\n",
"print np.sum(s) / np.sqrt(np.sum(b))\n",
"print \"s/sqrt(b) per categories\"\n",
"print s_over_sqrtb"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Create the model for one category (inclusive)\n",
"\n",
"$$\\text{Pois}(n_s+n_b|N_s+N_b)\\frac{N_s f_s + N_b f_b}{N_s + N_b}$$\n",
"\n",
"$$N_s = L\\sigma_s$$\n",
"$$N_b = L\\sigma_b$$"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"True"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bkg = ws.factory(\"RooChebychev:bkg(mgg, {a[-0.5, -10, 10], b[0.1, -10, 10]})\")\n",
"sig = ws.factory(\"RooGaussian:sig(mgg, M, sigma[50])\")\n",
"ws.factory(\"prod:Nsig(lumi, signal_xsec)\")\n",
"ws.factory(\"prod:Nbkg(lumi, bkg_xsec)\")\n",
"model = ws.factory(\"SUM:model(Nsig * sig, Nbkg * bkg)\")\n",
"ws.saveSnapshot('prefit', ws.allVars())"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
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"<title>model</title>\n",
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"<!-- model -->\n",
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"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"645\" y=\"-195.8\">model</text>\n",
"</g>\n",
"<!-- sig -->\n",
"<g class=\"node\" id=\"node2\"><title>sig</title>\n",
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"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"295\" y=\"-120.8\">RooGaussian</text>\n",
"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"295\" y=\"-105.8\">sig</text>\n",
"</g>\n",
"<!-- model&#45;&gt;sig -->\n",
"<g class=\"edge\" id=\"edge8\"><title>model-&gt;sig</title>\n",
"<path d=\"M595.225,-193.485C532.806,-177.791 425.753,-150.875 357.775,-133.784\" fill=\"none\" stroke=\"black\"/>\n",
"<polygon fill=\"black\" points=\"358.591,-130.38 348.039,-131.336 356.884,-137.168 358.591,-130.38\" stroke=\"black\"/>\n",
"</g>\n",
"<!-- Nsig -->\n",
"<g class=\"node\" id=\"node6\"><title>Nsig</title>\n",
"<ellipse cx=\"744\" cy=\"-117\" fill=\"none\" rx=\"56.9149\" ry=\"26.7407\" stroke=\"red\"/>\n",
"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"744\" y=\"-120.8\">RooProduct</text>\n",
"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"744\" y=\"-105.8\">Nsig</text>\n",
"</g>\n",
"<!-- model&#45;&gt;Nsig -->\n",
"<g class=\"edge\" id=\"edge10\"><title>model-&gt;Nsig</title>\n",
"<path d=\"M671.011,-182.879C683.061,-172.168 697.53,-159.307 710.395,-147.871\" fill=\"none\" stroke=\"black\"/>\n",
"<polygon fill=\"black\" points=\"713.014,-150.226 718.163,-140.967 708.363,-144.994 713.014,-150.226\" stroke=\"black\"/>\n",
"</g>\n",
"<!-- bkg -->\n",
"<g class=\"node\" id=\"node9\"><title>bkg</title>\n",
"<ellipse cx=\"508\" cy=\"-117\" fill=\"none\" rx=\"70.5656\" ry=\"26.7407\" stroke=\"red\"/>\n",
"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"508\" y=\"-120.8\">RooChebychev</text>\n",
"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"508\" y=\"-105.8\">bkg</text>\n",
"</g>\n",
"<!-- model&#45;&gt;bkg -->\n",
"<g class=\"edge\" id=\"edge7\"><title>model-&gt;bkg</title>\n",
"<path d=\"M612.185,-184.922C593.981,-173.228 571.111,-158.538 551.439,-145.903\" fill=\"none\" stroke=\"black\"/>\n",
"<polygon fill=\"black\" points=\"553.247,-142.904 542.942,-140.444 549.464,-148.794 553.247,-142.904\" stroke=\"black\"/>\n",
"</g>\n",
"<!-- Nbkg -->\n",
"<g class=\"node\" id=\"node12\"><title>Nbkg</title>\n",
"<ellipse cx=\"881\" cy=\"-117\" fill=\"none\" rx=\"56.9149\" ry=\"26.7407\" stroke=\"red\"/>\n",
"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"881\" y=\"-120.8\">RooProduct</text>\n",
"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"881\" y=\"-105.8\">Nbkg</text>\n",
"</g>\n",
"<!-- model&#45;&gt;Nbkg -->\n",
"<g class=\"edge\" id=\"edge9\"><title>model-&gt;Nbkg</title>\n",
"<path d=\"M688.837,-189.654C728.039,-175.036 785.607,-153.57 827.508,-137.946\" fill=\"none\" stroke=\"black\"/>\n",
"<polygon fill=\"black\" points=\"828.851,-141.181 836.998,-134.407 826.406,-134.622 828.851,-141.181\" stroke=\"black\"/>\n",
"</g>\n",
"<!-- mgg -->\n",
"<g class=\"node\" id=\"node3\"><title>mgg</title>\n",
"<ellipse cx=\"333\" cy=\"-27\" fill=\"none\" rx=\"59.096\" ry=\"26.7407\" stroke=\"blue\"/>\n",
"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"333\" y=\"-30.8\">RooRealVar</text>\n",
"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"333\" y=\"-15.8\">mgg</text>\n",
"</g>\n",
"<!-- sig&#45;&gt;mgg -->\n",
"<g class=\"edge\" id=\"edge5\"><title>sig-&gt;mgg</title>\n",
"<path d=\"M305.991,-90.5468C309.7,-81.957 313.908,-72.2129 317.88,-63.0159\" fill=\"none\" stroke=\"black\"/>\n",
"<polygon fill=\"black\" points=\"321.151,-64.2691 321.902,-53.7009 314.724,-61.494 321.151,-64.2691\" stroke=\"black\"/>\n",
"</g>\n",
"<!-- M -->\n",
"<g class=\"node\" id=\"node4\"><title>M</title>\n",
"<ellipse cx=\"59\" cy=\"-27\" fill=\"none\" rx=\"59.096\" ry=\"26.7407\" stroke=\"blue\"/>\n",
"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"59\" y=\"-30.8\">RooRealVar</text>\n",
"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"59\" y=\"-15.8\">M</text>\n",
"</g>\n",
"<!-- sig&#45;&gt;M -->\n",
"<g class=\"edge\" id=\"edge4\"><title>sig-&gt;M</title>\n",
"<path d=\"M248.546,-98.8225C215.026,-86.4111 168.698,-69.2225 128,-54 123.178,-52.1963 118.168,-50.3176 113.148,-48.432\" fill=\"none\" stroke=\"black\"/>\n",
"<polygon fill=\"black\" points=\"114.121,-45.0588 103.529,-44.8153 111.657,-51.6109 114.121,-45.0588\" stroke=\"black\"/>\n",
"</g>\n",
"<!-- sigma -->\n",
"<g class=\"node\" id=\"node5\"><title>sigma</title>\n",
"<ellipse cx=\"196\" cy=\"-27\" fill=\"none\" rx=\"59.096\" ry=\"26.7407\" stroke=\"blue\"/>\n",
"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"196\" y=\"-30.8\">RooRealVar</text>\n",
"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"196\" y=\"-15.8\">sigma</text>\n",
"</g>\n",
"<!-- sig&#45;&gt;sigma -->\n",
"<g class=\"edge\" id=\"edge6\"><title>sig-&gt;sigma</title>\n",
"<path d=\"M268.47,-92.4176C256.612,-81.8774 242.495,-69.3285 229.884,-58.1189\" fill=\"none\" stroke=\"black\"/>\n",
"<polygon fill=\"black\" points=\"232.06,-55.3701 222.26,-51.3423 227.409,-60.6019 232.06,-55.3701\" stroke=\"black\"/>\n",
"</g>\n",
"<!-- lumi -->\n",
"<g class=\"node\" id=\"node7\"><title>lumi</title>\n",
"<ellipse cx=\"881\" cy=\"-27\" fill=\"none\" rx=\"59.096\" ry=\"26.7407\" stroke=\"blue\"/>\n",
"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"881\" y=\"-30.8\">RooRealVar</text>\n",
"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"881\" y=\"-15.8\">lumi</text>\n",
"</g>\n",
"<!-- Nsig&#45;&gt;lumi -->\n",
"<g class=\"edge\" id=\"edge14\"><title>Nsig-&gt;lumi</title>\n",
"<path d=\"M776.815,-94.9218C795.433,-82.9627 818.931,-67.8691 838.897,-55.0444\" fill=\"none\" stroke=\"black\"/>\n",
"<polygon fill=\"black\" points=\"840.979,-57.8669 847.501,-49.5175 837.196,-51.9772 840.979,-57.8669\" stroke=\"black\"/>\n",
"</g>\n",
"<!-- signal_xsec -->\n",
"<g class=\"node\" id=\"node8\"><title>signal_xsec</title>\n",
"<ellipse cx=\"744\" cy=\"-27\" fill=\"none\" rx=\"59.096\" ry=\"26.7407\" stroke=\"blue\"/>\n",
"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"744\" y=\"-30.8\">RooRealVar</text>\n",
"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"744\" y=\"-15.8\">signal_xsec</text>\n",
"</g>\n",
"<!-- Nsig&#45;&gt;signal_xsec -->\n",
"<g class=\"edge\" id=\"edge13\"><title>Nsig-&gt;signal_xsec</title>\n",
"<path d=\"M744,-90.0731C744,-81.9993 744,-72.9428 744,-64.2961\" fill=\"none\" stroke=\"black\"/>\n",
"<polygon fill=\"black\" points=\"747.5,-64.0457 744,-54.0457 740.5,-64.0458 747.5,-64.0457\" stroke=\"black\"/>\n",
"</g>\n",
"<!-- bkg&#45;&gt;mgg -->\n",
"<g class=\"edge\" id=\"edge1\"><title>bkg-&gt;mgg</title>\n",
"<path d=\"M466.527,-95.1452C440.85,-82.2331 407.78,-65.6034 380.975,-52.1244\" fill=\"none\" stroke=\"black\"/>\n",
"<polygon fill=\"black\" points=\"382.284,-48.865 371.777,-47.4993 379.139,-55.1189 382.284,-48.865\" stroke=\"black\"/>\n",
"</g>\n",
"<!-- a -->\n",
"<g class=\"node\" id=\"node10\"><title>a</title>\n",
"<ellipse cx=\"607\" cy=\"-27\" fill=\"none\" rx=\"59.096\" ry=\"26.7407\" stroke=\"blue\"/>\n",
"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"607\" y=\"-30.8\">RooRealVar</text>\n",
"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"607\" y=\"-15.8\">a</text>\n",
"</g>\n",
"<!-- bkg&#45;&gt;a -->\n",
"<g class=\"edge\" id=\"edge3\"><title>bkg-&gt;a</title>\n",
"<path d=\"M535.052,-91.9536C546.81,-81.5023 560.711,-69.1454 573.145,-58.0932\" fill=\"none\" stroke=\"black\"/>\n",
"<polygon fill=\"black\" points=\"575.516,-60.6687 580.665,-51.409 570.865,-55.4368 575.516,-60.6687\" stroke=\"black\"/>\n",
"</g>\n",
"<!-- b -->\n",
"<g class=\"node\" id=\"node11\"><title>b</title>\n",
"<ellipse cx=\"470\" cy=\"-27\" fill=\"none\" rx=\"59.096\" ry=\"26.7407\" stroke=\"blue\"/>\n",
"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"470\" y=\"-30.8\">RooRealVar</text>\n",
"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"470\" y=\"-15.8\">b</text>\n",
"</g>\n",
"<!-- bkg&#45;&gt;b -->\n",
"<g class=\"edge\" id=\"edge2\"><title>bkg-&gt;b</title>\n",
"<path d=\"M496.804,-90.0731C493.1,-81.4939 488.916,-71.8051 484.974,-62.6774\" fill=\"none\" stroke=\"black\"/>\n",
"<polygon fill=\"black\" points=\"488.162,-61.2316 480.985,-53.4385 481.736,-64.0067 488.162,-61.2316\" stroke=\"black\"/>\n",
"</g>\n",
"<!-- Nbkg&#45;&gt;lumi -->\n",
"<g class=\"edge\" id=\"edge11\"><title>Nbkg-&gt;lumi</title>\n",
"<path d=\"M881,-90.0731C881,-81.9993 881,-72.9428 881,-64.2961\" fill=\"none\" stroke=\"black\"/>\n",
"<polygon fill=\"black\" points=\"884.5,-64.0457 881,-54.0457 877.5,-64.0458 884.5,-64.0457\" stroke=\"black\"/>\n",
"</g>\n",
"<!-- bkg_xsec -->\n",
"<g class=\"node\" id=\"node13\"><title>bkg_xsec</title>\n",
"<ellipse cx=\"1018\" cy=\"-27\" fill=\"none\" rx=\"59.096\" ry=\"26.7407\" stroke=\"blue\"/>\n",
"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"1018\" y=\"-30.8\">RooRealVar</text>\n",
"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"1018\" y=\"-15.8\">bkg_xsec</text>\n",
"</g>\n",
"<!-- Nbkg&#45;&gt;bkg_xsec -->\n",
"<g class=\"edge\" id=\"edge12\"><title>Nbkg-&gt;bkg_xsec</title>\n",
"<path d=\"M913.815,-94.9218C932.433,-82.9627 955.931,-67.8691 975.897,-55.0444\" fill=\"none\" stroke=\"black\"/>\n",
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"</g>\n",
"</g>\n",
"</svg>"
],
"text/plain": [
"<IPython.core.display.SVG object>"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"model.graphVizTree(\"model.dot\")\n",
"!dot -Tsvg model.dot > model.svg; rm model.dot\n",
"s = SVG(\"model.svg\")\n",
"s.data = re.sub(r'width=\"[0-9]+pt\"', r'width=\"90%\"', s.data)\n",
"s.data = re.sub(r'height=\"[0-9]+pt\"', r'height=\"\"', s.data)\n",
"s"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Expected significances"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"def compute_qvalue(ws, model, pois, data):\n",
" frame = mgg.frame()\n",
" data.plotOn(frame)\n",
" \n",
" old_values = {k: ws.var(k).getVal() for k in pois}\n",
" set_pois = ROOT.RooArgSet(*(ws.var(k) for k in pois))\n",
" # compute alternative qvalue\n",
" r_alt = model.fitTo(data, ROOT.RooFit.Save())\n",
" nll_alt = r_alt.minNll()\n",
"\n",
" model.plotOn(frame, ROOT.RooFit.LineColor(ROOT.kRed))\n",
" \n",
" for k, v in pois.iteritems():\n",
" ws.var(k).setVal(v)\n",
" ws.var(k).setConstant(True)\n",
" \n",
" r_null = model.fitTo(data, ROOT.RooFit.Save())\n",
" nll_zero = r_null.minNll()\n",
" \n",
" model.plotOn(frame, ROOT.RooFit.LineColor(ROOT.kBlue))\n",
" \n",
" for k, v in old_values.iteritems():\n",
" ws.var(k).setVal(v)\n",
" ws.var(k).setConstant(False) \n",
" \n",
" t = 2 * (nll_zero - nll_alt)\n",
" frame.Draw()\n",
" return t"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Inclusive Asimov dataset"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"12059.9172414\n"
]
}
],
"source": [
"ws.loadSnapshot('prefit')\n",
"data_inclusive = model.generate(ROOT.RooArgSet(mgg), ROOT.RooFit.ExpectedData())\n",
"print data_inclusive.sumEntries()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Inclusive expected significance\n",
"\n",
"Null hypothesis: $\\sigma = 0$\n",
"\n",
"$$q = -2\\log \\frac{L(\\sigma=0)}{L(\\hat\\sigma)}\\quad \\text{with } \\hat\\sigma \\geq 0$$\n",
"$$f(q) = \\frac{1}{2}\\delta(q) + \\frac{1}{2}\\chi^2(q)$$\n",
"$$\\text{p-value} = \\int_{q_{obs}}^\\infty f(q) dq = \\frac{1}{2}\\int_{q_{obs}}^\\infty \\chi^2(q) dq$$\n",
"$$z = 1 - \\phi^{-1}(p)$$"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"1.93095194839\n"
]
}
],
"source": [
"qvalue = compute_qvalue(ws, model, {'signal_xsec': 0}, data_inclusive)\n",
"pvalue = stats.chi2(1).sf(qvalue) / 2. # sf -> survival function\n",
"zvalue_inclusive = stats.norm().isf(pvalue) # inverse survival function\n",
"print zvalue_inclusive"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Per categories"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[2.4252775819727441, 0.70848868283284172, 0.30916495429276014]\n"
]
}
],
"source": [
"zvalues_cat = []\n",
"for icat in range(1, 3 + 1):\n",
" ws.loadSnapshot('prefit')\n",
" ws.var(\"signal_xsec\").setVal(signal_xsec_inc * ksigs[icat - 1])\n",
" ws.var(\"bkg_xsec\").setVal(bkg_xsec_inc * kbkgs[icat - 1])\n",
" data_cat = model.generate(ROOT.RooArgSet(mgg), ROOT.RooFit.ExpectedData())\n",
" qvalue = compute_qvalue(ws, model, {'signal_xsec': 0}, data_cat)\n",
" pvalue = stats.chi2(1).sf(qvalue) / 2.\n",
" zvalue_cat = stats.norm(0, 1).isf(pvalue) \n",
" zvalues_cat += [zvalue_cat]\n",
"print zvalues_cat"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The first category alone has a better expected significance than the inclusive one"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Create model with categories\n",
"### Uncorrelated background and signal yield\n",
"\n",
"$$\\sigma_{s,1}, \\sigma_{s,2}, \\sigma_{s,3}$$\n",
"$$\\sigma_{b,1}, \\sigma_{b,2}, \\sigma_{b,3}$$"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"cat = ws.factory(\"cat[cat1=1, cat2=2, cat3=3]\")\n",
"model_sim_uncorr = ws.factory(\"SIMCLONE:model_sim_uncorr(model, $SplitParam({signal_xsec, bkg_xsec}, cat))\")"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {
"collapsed": false
},
"outputs": [
{
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"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"405\" y=\"-30.8\">RooRealVar</text>\n",
"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"405\" y=\"-15.8\">signal_xsec_cat2</text>\n",
"</g>\n",
"<!-- Nsig_cat2&#45;&gt;signal_xsec_cat2 -->\n",
"<g class=\"edge\" id=\"edge24\"><title>Nsig_cat2-&gt;signal_xsec_cat2</title>\n",
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"</g>\n",
"<!-- Nbkg_cat2&#45;&gt;lumi -->\n",
"<g class=\"edge\" id=\"edge25\"><title>Nbkg_cat2-&gt;lumi</title>\n",
"<path d=\"M684.644,-96.2511C661.131,-83.4787 630.32,-66.7416 605.094,-53.0385\" fill=\"none\" stroke=\"black\"/>\n",
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"</g>\n",
"<!-- bkg_xsec_cat2 -->\n",
"<g class=\"node\" id=\"node19\"><title>bkg_xsec_cat2</title>\n",
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"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"705\" y=\"-30.8\">RooRealVar</text>\n",
"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"705\" y=\"-15.8\">bkg_xsec_cat2</text>\n",
"</g>\n",
"<!-- Nbkg_cat2&#45;&gt;bkg_xsec_cat2 -->\n",
"<g class=\"edge\" id=\"edge26\"><title>Nbkg_cat2-&gt;bkg_xsec_cat2</title>\n",
"<path d=\"M716.286,-90.0731C714.802,-81.9105 713.135,-72.7436 711.547,-64.0112\" fill=\"none\" stroke=\"black\"/>\n",
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"<g class=\"edge\" id=\"edge28\"><title>model_cat3-&gt;sig</title>\n",
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"<!-- model_cat3&#45;&gt;bkg -->\n",
"<g class=\"edge\" id=\"edge27\"><title>model_cat3-&gt;bkg</title>\n",
"<path d=\"M1126.33,-181.487C1134.19,-172.017 1143.3,-161.037 1151.71,-150.898\" fill=\"none\" stroke=\"black\"/>\n",
"<polygon fill=\"black\" points=\"1154.58,-152.924 1158.27,-142.993 1149.19,-148.455 1154.58,-152.924\" stroke=\"black\"/>\n",
"</g>\n",
"<!-- Nsig_cat3 -->\n",
"<g class=\"node\" id=\"node21\"><title>Nsig_cat3</title>\n",
"<ellipse cx=\"869\" cy=\"-117\" fill=\"none\" rx=\"56.9149\" ry=\"26.7407\" stroke=\"red\"/>\n",
"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"869\" y=\"-120.8\">RooProduct</text>\n",
"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"869\" y=\"-105.8\">Nsig_cat3</text>\n",
"</g>\n",
"<!-- model_cat3&#45;&gt;Nsig_cat3 -->\n",
"<g class=\"edge\" id=\"edge29\"><title>model_cat3-&gt;Nsig_cat3</title>\n",
"<path d=\"M1062.8,-189.087C1054.89,-186.034 1046.71,-182.9 1039,-180 1000.33,-165.447 956.526,-149.484 923.067,-137.4\" fill=\"none\" stroke=\"black\"/>\n",
"<polygon fill=\"black\" points=\"924.071,-134.041 913.476,-133.94 921.695,-140.625 924.071,-134.041\" stroke=\"black\"/>\n",
"</g>\n",
"<!-- Nbkg_cat3 -->\n",
"<g class=\"node\" id=\"node23\"><title>Nbkg_cat3</title>\n",
"<ellipse cx=\"1033\" cy=\"-117\" fill=\"none\" rx=\"56.9149\" ry=\"26.7407\" stroke=\"red\"/>\n",
"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"1033\" y=\"-120.8\">RooProduct</text>\n",
"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"1033\" y=\"-105.8\">Nbkg_cat3</text>\n",
"</g>\n",
"<!-- model_cat3&#45;&gt;Nbkg_cat3 -->\n",
"<g class=\"edge\" id=\"edge30\"><title>model_cat3-&gt;Nbkg_cat3</title>\n",
"<path d=\"M1085.67,-181.487C1077.6,-171.761 1068.21,-160.442 1059.61,-150.079\" fill=\"none\" stroke=\"black\"/>\n",
"<polygon fill=\"black\" points=\"1062.3,-147.843 1053.23,-142.381 1056.92,-152.313 1062.3,-147.843\" stroke=\"black\"/>\n",
"</g>\n",
"<!-- Nsig_cat3&#45;&gt;lumi -->\n",
"<g class=\"edge\" id=\"edge31\"><title>Nsig_cat3-&gt;lumi</title>\n",
"<path d=\"M823.234,-100.726C811.483,-97.0145 798.822,-93.1977 787,-90 716.64,-70.9674 696.777,-75.0702 627,-54 622.06,-52.5083 616.966,-50.8311 611.893,-49.0655\" fill=\"none\" stroke=\"black\"/>\n",
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"<g class=\"node\" id=\"node22\"><title>signal_xsec_cat3</title>\n",
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"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"869\" y=\"-30.8\">RooRealVar</text>\n",
"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"869\" y=\"-15.8\">signal_xsec_cat3</text>\n",
"</g>\n",
"<!-- Nsig_cat3&#45;&gt;signal_xsec_cat3 -->\n",
"<g class=\"edge\" id=\"edge32\"><title>Nsig_cat3-&gt;signal_xsec_cat3</title>\n",
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"</g>\n",
"<!-- Nbkg_cat3&#45;&gt;lumi -->\n",
"<g class=\"edge\" id=\"edge33\"><title>Nbkg_cat3-&gt;lumi</title>\n",
"<path d=\"M985.462,-101.925C969.533,-97.6008 951.601,-93.1556 935,-90 799.604,-64.2632 760.845,-86.8645 627,-54 621.707,-52.7003 616.267,-51.1006 610.881,-49.3389\" fill=\"none\" stroke=\"black\"/>\n",
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"</g>\n",
"<!-- bkg_xsec_cat3 -->\n",
"<g class=\"node\" id=\"node24\"><title>bkg_xsec_cat3</title>\n",
"<ellipse cx=\"1033\" cy=\"-27\" fill=\"none\" rx=\"68.3838\" ry=\"26.7407\" stroke=\"blue\"/>\n",
"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"1033\" y=\"-30.8\">RooRealVar</text>\n",
"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"1033\" y=\"-15.8\">bkg_xsec_cat3</text>\n",
"</g>\n",
"<!-- Nbkg_cat3&#45;&gt;bkg_xsec_cat3 -->\n",
"<g class=\"edge\" id=\"edge34\"><title>Nbkg_cat3-&gt;bkg_xsec_cat3</title>\n",
"<path d=\"M1033,-90.0731C1033,-81.9993 1033,-72.9428 1033,-64.2961\" fill=\"none\" stroke=\"black\"/>\n",
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"</g>\n",
"</g>\n",
"</svg>"
],
"text/plain": [
"<IPython.core.display.SVG object>"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"model_sim_uncorr.graphVizTree(\"model_sim_uncorr.dot\")\n",
"!dot -Tsvg model_sim_uncorr.dot > model_sim_uncorr.svg; rm model_sim_uncorr.dot\n",
"s = SVG(\"model_sim_uncorr.svg\")\n",
"s.data = re.sub(r'width=\"[0-9]+pt\"', r'width=\"90%\"', s.data)\n",
"s.data = re.sub(r'height=\"[0-9]+pt\"', r'height=\"\"', s.data)\n",
"s"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Correlate signal, uncorrelate background"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$$\\sigma_{s,1} = \\sigma_s k_{s,1}\\qquad\\sigma_{s,2} = \\sigma_s k_{s,2}\\qquad\\sigma_{s,3} = \\sigma_s k_{s,3}$$\n",
"$$\\sigma_{b,1}, \\sigma_{b,2}, \\sigma_{b,3}$$\n",
"with $k_{s, 1}, k_{s, 2}, k_{s, 3}$ fixed"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"model_sim_corr = ws.factory(\"EDIT:model_sim_corr(model_sim_uncorr, signal_xsec_cat1=prod:signal_xsec_cor1(signal_xsec, ksig1), \\\n",
" signal_xsec_cat2=prod:signal_xsec_cor2(signal_xsec, ksig2), \\\n",
" signal_xsec_cat3=prod:signal_xsec_cor3(signal_xsec, ksig3))\")"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {
"collapsed": false
},
"outputs": [
{
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"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"771\" y=\"-390.8\">RooSimultaneous</text>\n",
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"</g>\n",
"<!-- cat -->\n",
"<g class=\"node\" id=\"node2\"><title>cat</title>\n",
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"</g>\n",
"<!-- model_sim_corr&#45;&gt;cat -->\n",
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"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"637\" y=\"-300.8\">RooAddPdf</text>\n",
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"<!-- model_sim_corr&#45;&gt;model_cat1_model_sim_corr -->\n",
"<g class=\"edge\" id=\"edge24\"><title>model_sim_corr-&gt;model_cat1_model_sim_corr</title>\n",
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"<!-- model_sim_corr&#45;&gt;model_cat2_model_sim_corr -->\n",
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"</g>\n",
"<!-- model_sim_corr&#45;&gt;model_cat3_model_sim_corr -->\n",
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"<g class=\"node\" id=\"node4\"><title>Nbkg_cat1</title>\n",
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"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"68\" y=\"-210.8\">RooProduct</text>\n",
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"</g>\n",
"<!-- model_cat1_model_sim_corr&#45;&gt;Nbkg_cat1 -->\n",
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"<polygon fill=\"black\" points=\"119.217,-226.071 108.622,-226.034 116.881,-232.67 119.217,-226.071\" stroke=\"black\"/>\n",
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"<ellipse cx=\"1429\" cy=\"-207\" fill=\"none\" rx=\"62.3993\" ry=\"26.7407\" stroke=\"red\"/>\n",
"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"1429\" y=\"-210.8\">RooGaussian</text>\n",
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"</g>\n",
"<!-- model_cat1_model_sim_corr&#45;&gt;sig -->\n",
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"<g class=\"node\" id=\"node11\"><title>bkg</title>\n",
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"</g>\n",
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"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"262\" y=\"-210.8\">RooProduct</text>\n",
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"</g>\n",
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"</g>\n",
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"<g class=\"node\" id=\"node5\"><title>bkg_xsec_cat1</title>\n",
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"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"68\" y=\"-120.8\">RooRealVar</text>\n",
"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"68\" y=\"-105.8\">bkg_xsec_cat1</text>\n",
"</g>\n",
"<!-- Nbkg_cat1&#45;&gt;bkg_xsec_cat1 -->\n",
"<g class=\"edge\" id=\"edge14\"><title>Nbkg_cat1-&gt;bkg_xsec_cat1</title>\n",
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"<ellipse cx=\"566\" cy=\"-117\" fill=\"none\" rx=\"59.096\" ry=\"26.7407\" stroke=\"blue\"/>\n",
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"<text font-family=\"Times,serif\" font-size=\"14.00\" text-anchor=\"middle\" x=\"876\" y=\"-105.8\">signal_xsec_cor3</text>\n",
"</g>\n",
"<!-- Nsig_cat3_model_sim_corr&#45;&gt;signal_xsec_cor3 -->\n",
"<g class=\"edge\" id=\"edge26\"><title>Nsig_cat3_model_sim_corr-&gt;signal_xsec_cor3</title>\n",
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"text/plain": [
"<IPython.core.display.SVG object>"
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"model_sim_corr.graphVizTree(\"model_sim_corr.dot\")\n",
"!dot -Tsvg model_sim_corr.dot > model_sim_corr.svg; rm model_sim_corr.dot\n",
"s = SVG(\"model_sim_corr.svg\")\n",
"s.data = re.sub(r'width=\"[0-9]+pt\"', r'width=\"90%\"', s.data)\n",
"s.data = re.sub(r'height=\"[0-9]+pt\"', r'height=\"\"', s.data)\n",
"s"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Set prefit values"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"True"
]
},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"ws.loadSnapshot('prefit')\n",
"# bkg xsection is uncorrelated for both\n",
"ws.var('bkg_xsec_cat1').setVal(ws.var('bkg_xsec').getVal() * kbkg1)\n",
"ws.var('bkg_xsec_cat2').setVal(ws.var('bkg_xsec').getVal() * kbkg2)\n",
"ws.var('bkg_xsec_cat3').setVal(ws.var('bkg_xsec').getVal() * kbkg3)\n",
"ws.saveSnapshot('prefit_sim_corr', ws.allVars())\n",
"\n",
"ws.var('signal_xsec_cat1').setVal(ws.function('signal_xsec_cor1').getVal())\n",
"ws.var('signal_xsec_cat2').setVal(ws.function('signal_xsec_cor2').getVal())\n",
"ws.var('signal_xsec_cat3').setVal(ws.function('signal_xsec_cor3').getVal())\n",
"\n",
"ws.saveSnapshot('prefit_sim_uncorr', ws.allVars())"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Generate data, from the correlated model, use them everywhere"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"12059.9172402\n"
]
}
],
"source": [
"ws.loadSnapshot('prefit_sim_corr')\n",
"data_simul = model_sim_corr.generate(ROOT.RooArgSet(mgg, cat), ROOT.RooFit.ExpectedData())\n",
"print data_simul.sumEntries()\n",
"data_simul.table(cat).Print(\"v\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Expected significance with uncorrelated model\n",
"\n",
"Null hypothesis: $\\sigma_1 = \\sigma_2 = \\sigma_3 = 0$ (3 degrees of freedom)"
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"1.67394616802\n"
]
}
],
"source": [
"ws.loadSnapshot('prefit_sim_uncorr')\n",
"qvalue = compute_qvalue(ws, model_sim_uncorr, {'signal_xsec_cat1': 0, 'signal_xsec_cat2': 0, 'signal_xsec_cat3':0}, data_simul)\n",
"pvalue = stats.chi2(3).sf(qvalue) / 2. # 3 degree of freedom\n",
"zvalue_uncor = stats.norm(0, 1).isf(pvalue) \n",
"print zvalue_uncor"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Is correct? How many degrees of freedom are there? I am using three, since the likelihood ratio is\n",
"\n",
"$$\\frac{L_1(\\sigma_{s,1}=0) L_2(\\sigma_{s,2}=0) L_3(\\sigma_{s,3}=0)}{L_1(\\hat\\sigma_{s,1}) L_2(\\hat\\sigma_{s,2}) L_3(\\hat\\sigma_{s,3})}$$\n",
"\n",
"but what if I define a model where I profile the fractions between categories:\n",
"\n",
"$$\\frac{L_1(\\sigma_S=0) L_2(\\sigma_S=0) L_3(\\sigma_S=0)}{\\max_{\\vec k_S, \\sigma_S}L_1(k_{S,1}\\sigma_S) L_2(k_{S,2}\\sigma_S) L_3(k_{S,3}\\sigma_S)}$$\n",
"\n",
"*how many degrees of freedom do I have?*"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Null hypothesis: $\\sigma_i = 0$"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[2.3567863960450954, 0.94975924760511654, 0.22558159309235346]\n"
]
}
],
"source": [
"zvalues_uncor_cat = []\n",
"for poi in 'signal_xsec_cat1', 'signal_xsec_cat2', 'signal_xsec_cat3':\n",
" ws.loadSnapshot('prefit_sim_uncorr')\n",
" qvalue = compute_qvalue(ws, model_sim_uncorr, {poi: 0}, data_simul)\n",
" pvalue = stats.chi2(1).sf(qvalue) / 2. \n",
" zvalue_uncor_cat = stats.norm(0, 1).isf(pvalue) \n",
" zvalues_uncor_cat += [zvalue_uncor_cat]\n",
"print zvalues_uncor_cat "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Expected significance with correlated model\n",
"Null hypothesis $\\sigma=0$"
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"2.51273741769\n"
]
}
],
"source": [
"ws.loadSnapshot('prefit_sim_corr')\n",
"data_simul_cor = model_sim_corr.generate(ROOT.RooArgSet(mgg, cat), ROOT.RooFit.ExpectedData())\n",
"qvalue = compute_qvalue(ws, model_sim_corr, {'signal_xsec': 0}, data_simul)\n",
"pvalue = stats.chi2(1).sf(qvalue) / 2.\n",
"zvalue_corr = stats.norm(0, 1).isf(pvalue) \n",
"print zvalue_corr"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Summary of significances"
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"inclusive 1.93\n",
"each category separately 2.43 0.71 0.31\n",
"uncorrelated (bkg correlated) 1.67\n",
"uncorrelated (bkg correlated) each category 2.36 0.95 0.23\n",
"correlated 2.51\n"
]
}
],
"source": [
"print \"inclusive {:.2f}\".format(zvalue_inclusive)\n",
"print \"each category separately {}\".format(' '.join('%.2f' % x for x in zvalues_cat))\n",
"print \"uncorrelated (bkg correlated) {:.2f}\".format(zvalue_uncor)\n",
"print \"uncorrelated (bkg correlated) each category {}\".format(' '.join('%.2f' % x for x in zvalues_uncor_cat))\n",
"print \"correlated {:.2f}\".format(zvalue_corr)"
]
}
],
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"file_extension": ".py",
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