{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "## Testing look-elsewhere effect by creating 2d chi-square random fields with a Gaussian Process\n", "\n", "by Kyle Cranmer, Dec 7, 2015\n", "\n", "The correction for 2d look-elsewhere effect presented in \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", "is based on the fact that the test statistic\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", "is a chi-square random field (with 1 degree of freedom). That means that, for any point in $\\nu_1, \\nu_2$, the quantity $q(\\nu_1, \\nu_2)$ would have a chi-square distribution if you repeated the experiment many times.\n", "\n", "That is what you expect if you have a background model $p_b(x|\\theta)$ and you look for a signal on top of it with signal strength $\\mu$. Creating that scan is somewhat time consuming, so here we make realizations of a chi-square random field by using a Gaussian Process. \n", "The main trick we will use is that a chi-square distribution for one degree of freedom is the same as the distribution of $x^2$ if $x$ is normally distributed. As you might have guessed, a Gaussian Process (GP) is like a chi-square random field, but it is Gaussian-distributed at each point.\n", "\n", "Note, the distributions are not independent at each point, there is some covaraince. So if the $q(\\nu_1, \\nu_2)$ is high at one point, you can expect it to be high near by. We can control this behavior via the GP's kernel.\n", "\n", "For more on the theory of Gaussian Processes, the best resource is available for free online: [Rasmussen & Williams (2006)](http://www.gaussianprocess.org/gpml/). We will [`george`](http://dan.iel.fm/george/current/) -- a nice python package for Gaussian Processes (GP).\n", "\n" ] }, { "cell_type": "code", "execution_count": 4, "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": "markdown", "metadata": {}, "source": [ "The main trick we will use is that a chi-square distribution for one degree of freedom is the same as the distribution of $x^2$ if $x$ is normally distributed. Here's a quick demonstration of that:" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAXkAAAEDCAYAAADQunSaAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAG/lJREFUeJzt3X983HWd4PFXmkzSlElaGqgNoTAchVJOXDAV1FYZK+5W\n", "9wc5Oc/lVlesuy4o3t5j94S6elhl5VF1XXdXTm9/BOq6XpFb91hcV1HUrDz0FqkHbpGfpQRokV9t\n", "UpqUNkkn98dnhkzTJJ3JZPL9zndez8djHsx8Z77febdl3vOZ9+cXSJIkSZIkSZIkSZIkSZIkSZIk\n", "SVINaKzSdU8Afgw8DTxapfeQJB3Hgipd9xrga1W6tiSpRKUm+ZuAZ4Edk45vAB4itNavzR97C/AA\n", "8PxcBChJqr43ABdwdJJvBHYCGSAF3AesBv4Y+DxwB3Ab0DCfgUqSZifD0Un+dcC3ix5vyt8K3gO8\n", "rfphSZKm01TBuV3AU0WPdwMXFT3+cgXXliTNgUqS/HiF770TOLPCa0hSvXkMWFmNC2c4ulzzWo4u\n", "13yEic7XUlT6JTFfNkcdQIk2Rx1ACTZHHUCJNkcdQIk2Rx1AiTZHHUCJNkcdQInKyp2VDKHcDpxF\n", 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"display_data" } ], "source": [ "from scipy.stats import chi2, norm\n", "chi2_array = chi2.rvs(1, size=10000)\n", "norm_array = norm.rvs(size=10000)\n", "_ = plt.hist(chi2_array, bins=100, alpha=.5, label='chi-square')\n", "_ = plt.hist(norm_array**2, bins=100, alpha=.5, color='r', label='x^2')\n", "plt.yscale('log', nonposy='clip')\n", "plt.legend(('chi-square', 'x^2'))\n", "#plt.semilogy()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Ok, now to the Gaussian processes." ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": true }, "outputs": [], "source": [ "import george\n", "from george.kernels import ExpSquaredKernel" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [], "source": [ "length_scale_of_correaltion=0.1\n", "kernel = ExpSquaredKernel(length_scale_of_correaltion, ndim=2)" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": true }, "outputs": [], "source": [ "# Create the Gaussian process\n", "# gp = george.GP(kernel)\n", "gp = george.GP(kernel, solver=george.HODLRSolver) #faster" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false }, "outputs": [], "source": [ "n_scan_points=50\n", "aspect_ratio = 10. # make excesses look like stripes\n", "x_scan = np.arange(0,aspect_ratio,aspect_ratio/n_scan_points)\n", "y_scan = np.arange(0,1,1./n_scan_points)\n", "xx, yy = np.meshgrid(x_scan, y_scan)" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": true }, "outputs": [], "source": [ "# reformat the independent coordinates where we evaluate the GP\n", "indep = np.vstack((np.hstack(xx),np.hstack(yy))).T" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "array([[1, 3],\n", " [2, 4]])" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# illustration of what is being done here\n", "np.vstack([[1,2],[3,4]]).T" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false }, "outputs": [], "source": [ "# slow part: pre-compute internal stuff for the GP\n", "gp.compute(indep)" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": false }, "outputs": [], "source": [ "# evaluate one realization of the GP\n", "z = gp.sample(indep)" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": false }, "outputs": [], "source": [ "# reformat output for plotting\n", "zz = z.reshape((n_scan_points,n_scan_points))" ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAASgAAAD+CAYAAACJF5ifAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", 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"display_data" } ], "source": [ "# plot the chi-square random field\n", "plt.imshow(zz**2, cmap='gray')\n", "plt.colorbar()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now lets histogram the values of the random field.\n", "Don't get confused here... if you pick a single point and histogram the value of over many instances, you expect a Gaussian. However, for a single instance, you don't expect the histogram for the value of the field to be Gaussian (because of the correlations). Thought experiments: if you make `length_scale_of_correaltion` very small, then each point is essentially independent and you do expect to see a Gaussian; however, if `length_scale_of_correaltion` is very large then you expect the field to be nearly constant and the histogram below would be a delta function." ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAXIAAAESCAYAAADg0F5TAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAHCJJREFUeJzt3X/QZXV92PH3shdEdFeymjy7gW3XATaoUxVxwKYarxEz\n", "aAnYdKI1yRR/pElb4482VZZkWtbJdLKSWqfTjJOqAVeroBJi1JiGhfIIrVMMumtAXAGRBrS7a4Ty\n", "oCarT3n6x/fc557n7rn3nt+/7vs1c+c55zz3fM/3Oec8n/u93/P9AZIkSZIkSZIkSZIkSZIkSZIk\n", "SZI6Y3PTGZBa5lzgd4DLga3AwWazI5XjSuCrwF3Ax4AnAduAA8C9wE3A6Y3lTqrGScAnms6EVIZd\n", "wAOE4A3wcUJJ5WrgndG2K4B9tedMSu8a4CihMBJ3MXAYuI9wH4/8PPBnwC/UkjupYtuArwM/BgyA\n", 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binary_closing\n", "\n", "def fill_holes(array):\n", " zero_array = array==0.\n", " temp = grey_closing(array, size=2)*zero_array\n", " print np.sum(temp)\n", " return temp+array\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Generate 25 realizations of the GP, calculate the Euler characteristic for two thresholds, and use the mean of those Euler characteristics to estimate $N_1$ and $N_2$" ] }, { "cell_type": "code", "execution_count": 39, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "0.84008251292\n", "1.07421037292\n", "9.45671105564\n", "3.67675031418\n", "2.39271880675\n", "4.41673419794\n", "2.74658486914\n", "3.98855998155\n", "2.42989489918\n", "5.19320087685\n", "0.465411161491\n", "1.26797652899\n", "4.81617769522\n", "1.23888787035\n", "5.04054081538\n", "9.03239355699\n", "2.56575467823\n", "2.2710098149\n", "1.12810529755\n", "1.59491819727\n", "2.24852081996\n", "3.10801460564\n", 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enumerate(z_array):\n", "\n", " scan = z.reshape((n_scan_points,n_scan_points))**2\n", "\n", " q_max[scan_no] = np.max(scan)\n", " \n", " # set a point to 0 to simulate misc failure\n", " scan[n_scan_points/2,n_scan_points/2] = 0. \n", " scan[n_scan_points/4,n_scan_points/2] = 0. \n", " scan[n_scan_points/2,n_scan_points/4] = 0. \n", "\n", " newscan = fill_holes(scan)\n", " diffscan = scan-newscan\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", " #print 'diff = ', np.sum(exc1), np.sum(exc2)\n", " \n", " if scan_no < n_plots:\n", " aspect = 1.\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, interpolation='none')\n", " plt.imshow(newscan.T, cmap='gray', aspect=aspect, interpolation='none')\n", " plt.subplot(n_plots,3,3*scan_no+3)\n", " plt.imshow(diffscan.T, cmap='gray', aspect=aspect, interpolation='none')\n", "\n", "\n", " phi1 = calculate_euler_characteristic(exc1)\n", " phi2 = calculate_euler_characteristic(exc2)\n", " #print 'phi1, phi2 = ', phi1, phi2\n", " #print 'q_max = ', np.max(scan)\n", "\n", " phis[scan_no] = [phi1, phi2]\n", "\n", "plt.savefig('chi-square-random-fields.png')" ] }, { "cell_type": "code", "execution_count": 40, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "(14.32, 13.4)" ] }, "execution_count": 40, "metadata": {}, "output_type": "execute_result" } ], "source": [ "exp_phi_1, exp_phi_2 = np.mean(phis[:,0]), np.mean(phis[:,1])\n", "exp_phi_1, exp_phi_2" ] }, { "cell_type": "code", "execution_count": 41, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "8.60199674222 12.9677117014\n" ] } ], "source": [ "n1, n2 = get_coefficients(u1=u1, u2=u2, exp_phi_1=exp_phi_1, exp_phi_2=exp_phi_2)\n", "print n1, n2" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "With estimates of $N_1$ and $N_2$ predict the global p-value vs. u" ] }, { "cell_type": "code", "execution_count": 42, "metadata": { "collapsed": false }, "outputs": [], "source": [ "u = np.linspace(5,25,100)\n", "global_p = global_pvalue(u,n1,n2)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Generate 5000 instances of the Gaussian Process, find maximum local significance for each, and check the prediction for the LEE-corrected global p-value" ] }, { "cell_type": "code", "execution_count": 43, "metadata": { "collapsed": true }, "outputs": [], "source": [ "n_samples = 5000\n", "z_array = gp.sample(indep,n_samples)\n", "\n", "q_max = np.zeros(n_samples)\n", "\n", "for scan_no, z in enumerate(z_array):\n", " scan = z.reshape((n_scan_points,n_scan_points))**2\n", " q_max[scan_no] = np.max(scan)" ] }, { "cell_type": 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label='prediction')\n", "plt.xlabel('u')\n", "plt.legend(('toys', 'prediction'))\n", "#plt.ylabel('P(q>u)')\n", "plt.ylim(1E-3,10)\n", "plt.xlim(0,25)\n", "plt.semilogy()" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] } ], "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 }