{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Look Elsewhere Effect in 2-d\n",
    "\n",
    "Kyle Cranmer, Nov 19, 2015\n",
    "\n",
    "Based on\n",
    "*Estimating the significance of a signal in a multi-dimensional search* by  Ofer Vitells and Eilam Gross http://arxiv.org/pdf/1105.4355v1.pdf\n",
    "\n",
    "This is for the special case of a likelihood function of the form \n",
    "$L(\\mu, \\nu_1, \\nu_2)$ where $\\mu$ is a single parameter of interest and\n",
    "$\\nu_1,\\nu_2$ are two nuisance parameters that are not identified under the null.\n",
    "For example, $\\mu$ is the signal strength of a new particle and $\\nu_1,\\nu_2$ are the\n",
    "unknown mass and width of the new particle. Under the null hypothesis, those parameters \n",
    "don't mean anything... aka they \"are not identified under the null\" in the statistics jargon.\n",
    "This introduces a 2-d look elsewhere effect.\n",
    "\n",
    "The LEE correction in this case is based on \n",
    "\n",
    "\\begin{equation}\n",
    "E[ \\phi(A_u) ] = P(\\chi^2_1 > u) + e^{-u/2} (N_1 + \\sqrt{u} N_2) \\,\n",
    "\\end{equation}\n",
    "where \n",
    "   * $A_u$ is the 'excursion set above level $u$ (eg. the set of parameter points in $(\\nu_1,\\nu_2)$ that have a -2 log-likelihood ratio greater than $u$ )\n",
    "   * $\\phi(A_u)$ is the Euler characteristic of the excursion set\n",
    "   * $E[ \\phi(A_u) ]$ is the expectation of the Euler characteristic of those excursion sets under the null\n",
    "   * $P(\\chi^2_1 > u)$ is the standard chi-square probability \n",
    "   * and $N_1$ and $N_2$ are two coefficients that characterize the chi-square random field.\n",
    "   \n",
    "### structure of the notebook\n",
    "\n",
    "The notebook is broken into two parts.\n",
    "  * create a ROOT file with some toy histograms for this example\n",
    "  * read in those toy histograms used to estimate $N_1$ and $N_2$\n",
    "  * calculate the global p-value (eg. correct for the look-elsewhere effect)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Populating the interactive namespace from numpy and matplotlib\n"
     ]
    }
   ],
   "source": [
    "%pylab inline --no-import-all"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from lee2d import *"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## Create some test histograms\n",
    "\n",
    "These are toy histograms, but they are supposed to represent scans of $ q(\\nu_1, \\nu_2)$ where\n",
    "\n",
    "\\begin{equation}\n",
    "q(\\nu_1, \\nu_2) = -2 \\log \\frac{ \\max_\\theta L(\\mu=0, \\nu_1, \\nu_2, \\theta)}{ \\max_{\\mu, \\theta} L(\\mu, \\nu_1, \\nu_2, \\theta)}\n",
    "\\end{equation}\n",
    "\n",
    "and $\\theta$ are nuisance parameters for the background model, $\\mu$ is parameter of interest, and $\\nu_1, \\nu_2$ are nuisance parameters for the signal that are not meaningful for the null (eg. the mass and width of a hypothetical particle).\n",
    "\n",
    "In this setting, the maximum local significance is given by \n",
    "\n",
    "\\begin{equation}\n",
    "Z_{local} = \\max_{\\nu_1, \\nu_2} \\sqrt{q(\\nu_1, \\nu_2)}\n",
    "\\end{equation}\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "application/javascript": [
       "require(['codemirror/mode/clike/clike'], function(Clike) { console.log('ROOTaaS - C++ CodeMirror module loaded'); });"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/javascript": [
       "IPython.CodeCell.config_defaults.highlight_modes['magic_text/x-c++src'] = {'reg':[/^%%cpp/]};"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Welcome to ROOTaaS 6.05/03\n"
     ]
    }
   ],
   "source": [
    "from create_test_histograms import *\n",
    "create_test_histograms()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "HiggsLWA.root    HiggsLWANew.root test_hists.root\r\n"
     ]
    }
   ],
   "source": [
    "#check to make sure test_hists.root exists\n",
    "!ls *root"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Calculate global p-value based on maximum local significance and several toy likelihood scans"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Specify the necessary info\n",
    "max_local_significance = 4. \n",
    "# choose u1, u2 thresholds for doing the scan.\n",
    "# these are arbitrary \n",
    "# if there are enough toys the choice shouldn't matter, but \n",
    "# we may want to do some tests with other choices\n",
    "this_u1, this_u2 = 0.1, 0.9"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Specify the root file with the histograms and their names\n",
    "root_file_name = 'test_hists.root'\n",
    "num_toy_scans = 25\n",
    "names_of_toy_likleihood_scans = [('scan_toy_%d' %(i)) for i in range(11,num_toy_scans)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def convert_hist_to_numpy(hist):\n",
    "    \"\"\"a little helper script\"\"\"\n",
    "    temp = np.zeros((hist.GetNbinsX(), hist.GetNbinsY()))\n",
    "    for i in range(temp.shape[0]):\n",
    "        for j in range(temp.shape[1]):\n",
    "            temp[i,j] = hist.GetBinContent(i+1, j+1)\n",
    "    return temp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Read in histograms, convert them to numpy arrays\n",
    "inFile = ROOT.TFile(root_file_name, 'READ')\n",
    "likelihoodScans = []\n",
    "for histName in names_of_toy_likleihood_scans:\n",
    "    inHist = inFile.Get(histName)\n",
    "    temp = convert_hist_to_numpy(inHist)\n",
    "    likelihoodScans.append(temp)   "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from scipy.ndimage import grey_closing, binary_closing\n",
    "def fill_holes(array):\n",
    "    zero_array = array==0.\n",
    "    temp = grey_closing(array, size=2)*zero_array\n",
    "    return temp+array"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def get_euler_characteristics(listOfScans, u1=0.1, u2=0.9):\n",
    "    \"\"\"\n",
    "    loop through the likleihood scans and calculate expectation\n",
    "    of Euler characteristic for excursion sets above levels u1, u2\n",
    "    \"\"\"\n",
    "    \n",
    "    n_plots = 3\n",
    "    plt.figure(figsize=(9,n_plots*3))\n",
    "    phis = np.zeros((len(listOfScans),2))\n",
    "    for scan_no, scan in enumerate(listOfScans):\n",
    "\n",
    "        # fill holes from failures in original likelihood\n",
    "        scan = fill_holes(scan)\n",
    "        \n",
    "        #get excursion sets above those two levels\n",
    "        exc1 = (scan>u1) + 0. #add 0. to convert from bool to double\n",
    "        exc2 = (scan>u2) + 0.\n",
    "        #print '\\nu1,u2 = ', u1, u2\n",
    "\n",
    "        if scan_no < n_plots:\n",
    "            plt.subplot(n_plots,3,3*scan_no+1)\n",
    "            aspect = 1.*scan.shape[0]/scan.shape[1]\n",
    "            plt.imshow(scan.T, cmap='gray', aspect=aspect)\n",
    "            plt.subplot(n_plots,3,3*scan_no+2)\n",
    "            plt.imshow(exc1.T, cmap='gray', aspect=aspect)\n",
    "            plt.subplot(n_plots,3,3*scan_no+3)\n",
    "            plt.imshow(exc2.T, cmap='gray', aspect=aspect)\n",
    "        \n",
    "        phi1 = calculate_euler_characteristic(exc1)\n",
    "        phi2 = calculate_euler_characteristic(exc2)\n",
    "        #print 'phi1, phi2 = ', phi1, phi2\n",
    "        phis[scan_no] = [phi1, phi2]\n",
    "\n",
    "    plt.savefig('islands.png')\n",
    "    print 'Exp phi_0=%f, phi_2=%f' %(mean(phis[:,0]), mean(phis[:,1]))\n",
    "    return mean(phis[:,0]), mean(phis[:,1])\n",
    "                    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The columns of the below are for \n",
    "\n",
    "$\\left| \\hspace{2em} q(\\nu_1, \\nu_2) \\hspace{2em}\\right| \\hspace{2em}q(\\nu_1, \\nu_2)>u_1 \\hspace{2em} \\left| \\hspace{2em} q(\\nu_1, \\nu_2) > u_2 \\hspace{2em} \\right|$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Exp phi_0=4.857143, phi_2=4.571429\n"
     ]
    },
    {
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       "ERERERERERERERERERERERERERERERGRDeT/+TsQU8mC2FsAAAAASUVORK5CYII=\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1144be3d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "expphi1, expphi2 = get_euler_characteristics(likelihoodScans, u1=this_u1, u2=this_u2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " n1, n2 = 3.15777649996 3.66198277791\n",
      " local p_value = 0.000032,  local significance = 4.000000\n",
      "global p_value = 0.006036, global significance = 2.510004\n"
     ]
    }
   ],
   "source": [
    "global_p_value = do_LEE_correction(max_local_significance, this_u1, this_u2, expphi1, expphi2)"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.10"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}