{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Investigating Gaussian Processes for di-jet invariant mass fit function\n", "\n", "Kyle Cranmer, Meghan Frate, Daniel Whiteson 2015" ] }, { "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": [ "import george\n", "from george.kernels import ExpSquaredKernel, MyLocalGaussianKernel, ExpKernel, MyLocalExpSquared\n", "from scipy.optimize import curve_fit\n", "import triangle\n", "from lmfit import Parameters, minimize, fit_report\n", "import emcee\n", "pl = plt" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Load data" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": true }, "outputs": [], "source": [ "# https://github.com/mfrate28/GuassianProcesses/blob/master/DijetsResiduals.ipynb\n", "# originally from HEPdata\n", "#np.savetxt('dijet_data.csv', (xval, xerrminus, xerrplus, yval, yerr, xerr))" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [], "source": [ "#cut out first 24 bins b/c fits seem crazy there\n", "xval, xerrminus, xerrplus, yval, yerr, xerr = np.loadtxt('dijet_data.csv', usecols=range(24,60))\n", "#scale data by bin width since curve-fit can't have the model take in an argument\n", "#newy = yval/(xerrplus+xerrminus) " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Define standard CDF dijet fitting function" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Notice that curve_fit is a least squares fit and minimize will minimize what is given, but residual_CDF2009 isn't a -logL" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def model2_CDF2009(t, params): \n", " global xerrplus, xerrminus\n", " p0, p1, p2, p3 = params\n", " sqrts = 8000.\n", " return (p0 * (1.-t/sqrts)**p1 * (t/sqrts)**(p2+p3*np.log(t/sqrts)))*(xerrplus+xerrminus)" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def nll(params, t,data=None): #gaussian distribution\n", " #prediction = p0 * (1.-t/sqrts)**p1 * (t/sqrts)**(p2+p3*np.log(t/sqrts))\n", " prediction = model2_CDF2009(t,params.valuesdict().values() )\n", " return (prediction-data)/np.sqrt(prediction)" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "Parameters([('p0', ), ('p1', ), ('p2', ), ('p3', )])" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "#initialize parameters\n", "fit_params = Parameters()\n", "fit_params.add('p0', value=0.0490546530927)\n", "fit_params.add('p1', value=7.3087377889)\n", "fit_params.add('p2', value=-6.00446454843)\n", "fit_params.add('p3', value=0.1347386042)\n", "#try a different set of parameter values\n", "fit_params['p0'].value = 0.55765009832017132\n", "fit_params['p1'].value = 8.6753672190427995\n", "fit_params['p2'].value = -5.4038968232544793\n", "fit_params['p3'].value = -0.042639640230075333\n", "fit_params" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "('do fit', '[[Variables]]\\n p0: 0.55765009 (init= 0.04905465)\\n p1: 8.67536721 (init= 7.308738)\\n p2: -5.40389682 (init=-6.004465)\\n p3: -0.04263964 (init= 0.1347386)\\n[[Correlations]] (unreported correlations are < 0.100)')\n" ] }, { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAYIAAAEDCAYAAAA4FgP0AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", 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(prediction-y)*(prediction-y) / prediction) \n", " # return p1*math.log(1-t)+(p2+p3*math.log(t)\n", "\n", "def lnprior_ind(p):\n", " p0, p1, p2, p3 = p\n", " if (0 < p0 < 3 and 0 < p1 < 10 and -10 < p2 < 0 and -1 < p3 < 1):\n", " return 0.0\n", " return -np.inf\n", "\n", "def lnprob_ind(p, x, y, yerr): \n", " lp = lnprior_ind(p) \n", " return lp + lnlike_ind(p, x, y, yerr) if np.isfinite(lp) else -np.inf" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def fit_ind(initial, data, nwalkers=32):\n", " ndim = len(initial)\n", " p0 = [np.array(initial) + 1e-8 * np.random.randn(ndim)\n", " for i in xrange(nwalkers)]\n", " sampler = emcee.EnsembleSampler(nwalkers, ndim, lnprob_ind, args=data)\n", " print(\"Running burn-in\")\n", " p0, lnp, _ = sampler.run_mcmc(p0, 500)\n", " sampler.reset()\n", " \n", " best_fit = p0[np.argmax(lnp)]\n", " print best_fit\n", "\n", " #print(\"Running second burn-in\")\n", " #p = p0[np.argmax(lnp)]\n", " #p0 = [p + 1e-8 * np.random.randn(ndim) for i in xrange(nwalkers)]\n", " #p0, _, _ = sampler.run_mcmc(p0, 500)\n", " #sampler.reset()\n", "\n", " print(\"Running production\")\n", " p0, _, _ = sampler.run_mcmc(p0, 1000)\n", " return sampler, best_fit" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Fitting ind\n", "Running burn-in\n", "[ 0.70227265 8.73040196 -5.16589499 0.01853028]\n", "Running production\n" ] } ], "source": [ "np.random.seed(1234)\n", "init_params = fit_params.valuesdict().values()\n", "data = (xval, yval, np.sqrt(yval))\n", "\n", "# Fit assuming GP.\n", "print(\"Fitting ind\")\n", "sampler, best_fit = fit_ind(init_params, data)\n", "samples = sampler.flatchain" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Look at residuals for some sample curves from the posterior\n", "kind of useless b/c big jumps come from fluctuation of data that no model can accomodate" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "(-1000, 1000)" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAZwAAAERCAYAAABPbxE/AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzs3XecW1eZ8PHfvdKVpjdNsT1uiXsSp8cJIcWkiBQRwsKG\n", "ABvYJewLBAIfWJYA4RWKlt2FXcoLCyTwArsJnXcJ2UQpKCEVSJxGSHHvnrE9vRe1e94/zpVHHmvG\n", "0hRryvPlI6S5unN1RuPomXPOc54DQgghhBBCCCGEEEIIIYQQQgghhBBCCCGEEEIIISbhx0AL8FrG\n", "sRrgUWA7EAWqMp77PLAD2Ar4M46f41xjB/CtaWyvEEKIWepi4CyODjj/BnzWeXwb8BXn8SnAK4AF\n", "LAd2Aobz3PPABufxQ8BV09ZiIYQQs9Zyjg44W4EG5/EC52vQvZvbMs57BLgAWAhsyTh+I3DXdDRU\n", "CCHExJmFbkAWDehhNpz7dPBZBDRlnNcENGY53uwcF0IIMYO4C92A41DObSrsBFZM0bWEEGK+2AWs\n", "nIoLzcQeTgt6KA30cFmr87gZWJJx3mJ0z6bZeZx5vDnLdVeg53xm6+2OGdAGaf/svc3m9s/mts+F\n", 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"output_type": "display_data" } ], "source": [ "import corner\n", "#p = [ 0.6797052, 8.64475999, -5.17048789, 0.02135271]\n", "fig = corner.corner(samples, labels=[\"$p0$\", \"p1\", \"p2\", \"p3\"],\n", " truths=best_fit)" ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "(-78.227968039392508, -19.453318271555524)" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "lnlike_ind(init_params,xval,yval, yval), lnlike_ind(best_fit,xval,yval, yval)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Based on MCMC samples, build covariance and correlation matrix" ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "collapsed": true }, "outputs": [], "source": [ "modelMatrix=np.empty([samples.shape[0], xval.size])\n", "for i,s in enumerate(samples):\n", " modelMatrix[i,:]= model2_CDF2009(xval,s)\n", "covMat = np.cov(modelMatrix.T)\n", "corMat = np.corrcoef(modelMatrix.T)" ] }, { "cell_type": "code", "execution_count": 18, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "[]" ] }, "execution_count": 18, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAZEAAAESCAYAAAA8BeghAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzt3XmYXFWZOOC3srHILsoabRYRRVQcFdFB2lEkiIALyqBB\n", "FgVRAZEtRMAcZVgFYZARBxAGtwCioo4LP1yiqAPIIKJsBgTZE8RlXHHh/P44t9PdSXenq7u6qm73\n", "9z5PPal7q+rWx004p87ynUMIIYQQQgghhBBCCCGEEEIIIYQQQgghhBBCCCGEEEIIIYQQprTpnQ5g\n", "BA2cjNfhKfhJZ8MJIYSwvGmdDmAEr8Mm+Cse6HAsIYQQusDFWIKfLnd+Du7AYsyrzs3DQdXzz7Ul\n", "uhBCCE1pd0vkEqXCGGg6zqvOPxv74FlK6+O31XueaFeAIYQQRq/dlci1+M1y516Mu3Av/obLsCe+\n", "gF1wLha1LcIQQgijNqPTASjjHvcPOH4A2+PPeMdKPnsXtpiguEIIYbK6G1u24kLdMLCex/HZLZRZ\n", "XHV9fLALYoj46/uI+CP2sT5a9uO7GyqRBzF7wPFszc3GSuhtYTwhhDBZ9SplZst0QyVyI56BHszC\n", 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"plt.plot(xval,np.diagonal(covMat), c='b')\n", "decay = 200\n", "plt.plot(xval, 1e5*np.exp(-(xval-1000)/decay), c='g') # basic \"fit\" to the variance vs. m_jj\n", "plt.ylabel('covariance')\n", "plt.xlabel('m_jj')\n", "plt.semilogy()" ] }, { "cell_type": "code", "execution_count": 19, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAToAAADtCAYAAAAvDrl/AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAGUBJREFUeJzt3X+MFOd9x/H3gg0mDQHOdmx+HD+snGXRRo1DgqNEhbEo\n", "P9w/AEuOoVVt2qIqySlxUzeVIU3wHhdVoZXt4lbGf5jEh1VjaGgxlh1+xHjqVBEGu9eEhlDAGIlb\n", "/KMFgxMlJHfc9o/vs7fDsrM7uzt73D33eUmjnXt2Z2bv2P3wzPPM7hdERERERERERERERERERERE\n", "ROozH/Jo0aJl0Jf58+bladCk2o55rmTzVuAV4KfAfwMPuPYWYB9wDNgLTIxssxY4DhwFFkXa5wCH\n", "3X0bI+1jgW2u/QAwI3LfKneMY8D9tfze5WSq3J8v/LWzbgHga18r/+glS+L3dNtt5ds/+tGyzb/u\n", "v3ZgvbMzyze/aUfv7Y0/RD7mpZGJ+S1Hjy7fPmpUcT167Gh7pW2SSPL4bDZLNput+rgMDb8n6j52\n", 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "#plot covariance matrix\n", "crop=0\n", "maxVal = covMat[crop,crop]\n", "plt.imshow(covMat[crop:,crop:], cmap=\"bwr\",vmin=-maxVal, vmax=maxVal, interpolation=\"nearest\")\n", "plt.colorbar()\n", "plt.gca().set_xticklabels([])\n", "plt.gca().set_yticklabels([]);" ] }, { "cell_type": "code", "execution_count": 20, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAASQAAAD3CAYAAAC0JCnZAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAGh1JREFUeJztnX2QZFV5xn8z+wXD7GfBfrC7sIkh1lpEM6yyi2SzF0Mq\n", "ShKNqYqBaJkCs5V/MCRUEmVTZXq0QqmlBCytBAcxGpNQCWqCiiaa0FghSsmC4Mey+BFhcRcCLPsF\n", "y37nj9MzND3nPX3Pvbenb595flW3tvvce8853dM8nHuf+74vCCGEEEIIIYQQQgghhBBCCCGEEEII\n", "IRLiVuBJ4DuBYz4C/AB4EBibiUkJIWYnm3EiYwnSZcCdrdcbgW/OxKSEELOXddiC9LfA77a9fxhY\n", "UXbA4bIdCCFmJauBXW3vHwfWlO1UgiSEKMpQx/tTZTsMCtIWN4A2bdpmeNsyNnaKkiyNG/NAZPc/\n", 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] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "#plot values for rows of correlation vs. m_jj\n", "for i in range(xval.size):\n", " plt.plot(xval,corMat[i,:])\n", "plt.xlabel(\"m_jj\");" ] }, { "cell_type": "code", "execution_count": 22, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAYMAAAEQCAYAAABSlhj/AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzsnXd4FNXawH+z2fQCoTcBkd6lKSDIoRexIyCICIiCvTfE\n", "VWyIWLCg4lVEEBQU6SDl0Euo0kFa6KEF0pMt5/vjnU1Cu5/fJ1fAO7/nyZOd3dmZs7Oz73veesDB\n", "wcHBwcHBwcHBwcHBwcHBwcHBwcHBwcHBwcHBwcHBwcHBwcHBwcHBweGS8Q2QBGy6yOs9gN+BjcAy\n", "oPbfNC4HBwcHh7+RZsD1XFwZNAYK2I/bAyv/jkE5ODg4OPz9lOfiyiA/8cDB/+xQHBwcHBz+r7j+\n", "5vP1BWb+zed0cHBwcPibKM//bhkoYCtiHTg4ODg4XEG4/6bz1AZGITGD5Ivsswu47m8aj4ODg8M/\n", "hd1Axcs9iCDlubhlUBYR9Df+L8cwl3JA/0E8l3sAfxLP5R7An8BzuQfwJ/Fc7gH8STyXewB/Es/l\n", "HsCfxHO5B/AnuSSy81JYBuOBm4EiwAHgNSDUfu1LYDDiGhppP+cFGl2C8zo4ODg4XCIuhTLo/r+8\n", 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"cell_type": "markdown", "metadata": {}, "source": [ "The kernel is defined using the new george yml approach\n", "http://dan.iel.fm/george/dev/tutorials/new_kernel/\n", "\n", "**First try MyDijetKernel**\n", "\n", "The key part is this:\n", "```\n", "value: |\n", " double xbar = 0.5*(std::abs(x1) + std::abs(x2));\n", " double r = std::abs(x1-x2); \n", " return exp(-xbar/a) * exp(-r*r/b/b);\n", "```" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#Try with Gaussian Process\n", "\n", "two different notions of what the GP is describing.\n", " * the GP kernel has a diagonal piece yerr=sqrt(yval) that describes the independent bin-by-bin Poisson fluctuations. In this case, we would fit the actual data and should expect the GP to bounce around a lot b/c it's describing both statistical fluctuations and how the mean moves around\n", " \n", " * we use the GP to describe the mean, so it should be very smooth. Then the yerr part shoudl basically be zero. In this case we don't fit the data, but the diferent mean itself. To start with this we could try to fit the different samples of the CDF fit function or some Monte Carlo samples. \n", "\n", "The first approach can still work when we add a signal component b/c any deviation bigger than the bin-by-bin Poisson fluctuations and the flexibility of the GP kernel will be ascribed to the new signal.\n", " " ] }, { "cell_type": "code", "execution_count": 25, "metadata": { "collapsed": false }, "outputs": [], "source": [ "example_residual_model = best_fit_model-model2_CDF2009(xval,samples[1])" ] }, { "cell_type": "code", "execution_count": 26, "metadata": { "collapsed": true }, "outputs": [], "source": [ "kernel = 500/np.exp(-1200/decay)*MyDijetKernel(a=decay,b=100)\n", "#kernel = 500*ExpSquaredKernel(50)\n", "gp = george.GP(kernel)\n", "# Pre-compute the factorization of the matrix.\n", "gp.compute(xval,yerr=10)" ] }, { "cell_type": "code", "execution_count": 27, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "" ] }, 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"data_residual = yval-model2_CDF2009(xval,samples[1])\n", "gp.compute(xval,yerr=np.sqrt(yval))\n", "meanGP, covGP = gp.predict(data_residual, xval)\n", "stdGP = np.sqrt(np.diag(covGP))\n", "#pl.plot(xval, data_residual, c='r')\n", "plt.errorbar(xval,data_residual,marker='o',ls='None', yerr=np.sqrt(yval),c='black')\n", "pl.plot(xval, meanGP, color=\"#4682b4\", alpha=0.5)\n", "plt.fill_between(xval,meanGP-stdGP,meanGP+stdGP,facecolor='lightgray')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Do MCMC with CDF model as mean of GP" ] }, { "cell_type": "code", "execution_count": 31, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def lnprob_gp(p, x, y, yerr): \n", " decay = 200\n", " kernel = 15000/np.exp(-1200/decay)*MyDijetKernel(a=decay,b=100)\n", " gp = george.GP(kernel)\n", " gp.compute(x, yerr)\n", " gpl = gp.lnlikelihood(y - model2_CDF2009(x,p))\n", " return gpl " ] }, { "cell_type": "code", "execution_count": 32, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def fit_gp(initial, data, nwalkers=32):\n", " ndim = len(initial)\n", " p0 = [np.array(initial) + 1e-8 * np.random.randn(ndim)\n", " for i in xrange(nwalkers)]\n", " sampler = emcee.EnsembleSampler(nwalkers, ndim, lnprob_gp, args=data)\n", " print(\"Running burn-in\")\n", " p0, lnp, _ = sampler.run_mcmc(p0, 500)\n", " sampler.reset()\n", " \n", " best_fit = p0[np.argmax(lnp)]\n", " print best_fit\n", "\n", " #print(\"Running second burn-in\")\n", " #p = p0[np.argmax(lnp)]\n", " #p0 = [p + 1e-8 * np.random.randn(ndim) for i in xrange(nwalkers)]\n", " #p0, _, _ = sampler.run_mcmc(p0, 500)\n", " #sampler.reset()\n", "\n", " print(\"Running production\")\n", " p0, _, _ = sampler.run_mcmc(p0, 1000)\n", " return sampler, best_fit" ] }, { "cell_type": "code", "execution_count": 33, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Fitting gp\n", "Running burn-in\n", "[ 0.86600782 8.90666434 -5.00591501 0.05262614]\n", "Running production\n" ] } ], "source": [ "np.random.seed(1234)\n", "init_params = best_fit\n", "data = (xval, yval, np.sqrt(yval))\n", "\n", "# Fit assuming GP.\n", "print(\"Fitting gp\")\n", "sampler, best_fit = fit_gp(init_params, data)\n", "samples = sampler.flatchain" ] }, { "cell_type": "code", "execution_count": 34, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAApkAAAKdCAYAAABoPnJEAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzs3XlYlFX/BvB7hm2GGZZhU3BFRMVSEJTCFSy3XMsUy1fT\n", "yiXLtFcz00qytFwyX800LfcN/VWuuWAK5o5bULigiCjIvi8zMMzz+2OAQGYQbGBE7s91zeXwnGf5\n", "MqHdnOc55wBEREREREREREREREREREREREREREREREREREREREREREREREQ1JjJ2AUYQCqCXsYug\n", "eisMgL+xiyAiIqInj1DN/YJqs4gaCKqrC/Wdf+BRn01QXdRRDUFGvHZ1f36IiIgaNLGxCyAiIiKi\n", "pw9DJhEREREZHEOmfqHGLqBEqLELKCfU2AWUCDV2AURERFQ1hkz9Qo1dQIlQYxdQTqixCygRauwC\n", "iIiIqGoMmURERERkcAyZRERERGRwDJlEREREZHAMmURERERkcAyZRERERGRwpsYuwEiCyr0PBUcr\n", 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"2J6X51+r/Xl5/iMom/r9gAeBi8nP86/V/mypnI78/LNcQDmzJy/PPyXb9rz87R+Q24HVqKbSCuAj\n", "KF3st1RPF+toA+Ratv+jqLN6Fvn57qTso4SO1f7zkCvgGcrpbZeRn+dfrf1vJz/P/2TgKdT+Z5G/\n", "GPLz/Gu1Py/PP8sFlDN78vL8U2ZRbvv3yd+zN8YYY4wxxhhjjDHGGGOMMcYYY4wxxhhjjDHGGGOM\n", "McYYYw7M/weDb9MOtEkOJgAAAABJRU5ErkJggg==\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "for s in samples[:20]:\n", " kernel = 15000/np.exp(-1200/decay)*MyDijetKernel(a=decay,b=100)\n", " gp = george.GP(kernel)\n", " gp.compute(xval, yerr=np.sqrt(yval))\n", " meanGP = gp.sample_conditional(data_residual, xval) + model2_CDF2009(xval,s)\n", " pl.plot(xval, model2_CDF2009(xval,best_fit)-meanGP, color=\"#4682b4\", alpha=0.5)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Inject a signal see how it goes" ] }, { "cell_type": "code", "execution_count": 38, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def sig_model(x, N=500., mass=3000., width=100.):\n", " return N*np.exp(-(x-mass)**2/2/width/width)/np.sqrt(2*np.pi)/width*(xerrplus+xerrminus)" ] }, { "cell_type": "code", "execution_count": 39, "metadata": { "collapsed": false }, "outputs": [], "source": [ "fake_data = yval+sig_model(xval)" ] }, { "cell_type": "code", "execution_count": 40, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "[]" ] }, "execution_count": 40, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAYAAAAEDCAYAAAA849PJAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzt3Xl8VOW9x/FPSAIqIrsUMDUKpSylSLWACvUkAZKwQwiL\n", "su+yhkAIYELGra29eq3V7i5t1ast1uutVbEupLV1qdaitAIKFmUTrGUTEEIy948zmElI4kzmzDxz\n", "znzfr9e8mDlJnvk9J+H3m+d5zgIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIhLXkqPQZhJwKzAG\n", "aA+8FYX3EBGRCDWJQptjgM7AKWB3FNoXEZEYuh/YD2yutT0H2Aq8BxQHthUDcwPP18ckOhERCVuo\n", "I4AHsJN9sGTgnsD2nsBkoAf2p/5Dge+pciBGERGJglALwEvAwVrb+gHbgZ1ABfAoMBp4HMgGfgCU\n", 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"RERERERERERERERkPP8fednk6a5FUCcAAAAASUVORK5CYII=\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# do a GP fit to the fake data residual\n", "# note, the GP cannot fill in the signal near 3000 GeV\n", "kernel = 15000/np.exp(-1200/decay)*MyDijetKernel(a=decay,b=100)\n", "gp = george.GP(kernel)\n", "gp.compute(xval,yerr=np.sqrt(fake_data))\n", "meanGP, covGP = gp.predict(fake_data_residual, xval)\n", "stdGP = np.sqrt(np.diag(covGP))\n", "#pl.plot(xval, data_residual, c='r')\n", "plt.errorbar(xval,fake_data_residual,marker='o',ls='None', yerr=np.sqrt(fake_data),c='black')\n", "pl.plot(xval, meanGP, color=\"#4682b4\", alpha=0.5)\n", "plt.fill_between(xval,meanGP-stdGP,meanGP+stdGP,facecolor='lightgray')" ] }, { "cell_type": "code", "execution_count": 44, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 44, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": [ 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5\n", "# note, the GP now CAN fill in the signal near 3000 GeV\n", "# this shows that the nominal model for the GP above is powerful\n", "# but we can't let the hyperparameters of the kernel be out of control\n", "# or the GP will fit the signal\n", "\n", "kernel = 15000/np.exp(-1200/decay)*MyDijetKernel(a=5*decay,b=100)\n", "gp = george.GP(kernel)\n", "\n", "fake_data_residual = fake_data-model2_CDF2009(xval,samples[1])\n", "gp.compute(xval,yerr=np.sqrt(fake_data))\n", "meanGP, covGP = gp.predict(fake_data_residual, xval)\n", "stdGP = np.sqrt(np.diag(covGP))\n", "#pl.plot(xval, data_residual, c='r')\n", "plt.errorbar(xval,fake_data_residual,marker='o',ls='None', yerr=np.sqrt(fake_data),c='black')\n", "pl.plot(xval, meanGP, color=\"#4682b4\", alpha=0.5)\n", "plt.fill_between(xval,meanGP-stdGP,meanGP+stdGP,facecolor='lightgray')" ] }, { "cell_type": "code", "execution_count": 45, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def sig_cdf(t, params): \n", " global xerrplus, xerrminus\n", " p0, p1, p2, p3, N, mass, width = params\n", " sqrts = 8000.\n", " return (p0 * (1.-t/sqrts)**p1 * (t/sqrts)**(p2+p3*np.log(t/sqrts)) \\\n", " + N*np.exp(-(t-mass)**2/2/width/width)/np.sqrt(2*np.pi)/width)\\\n", " *(xerrplus+xerrminus)" ] }, { "cell_type": "code", "execution_count": 46, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def lnprior_signal(p):\n", " N, mass, width = p\n", " if (0 < N < 3000 and 1000 < mass < 4500 and 20 < width < 1000 ):\n", " return 0.0\n", " return -np.inf" ] }, { "cell_type": "code", "execution_count": 47, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def lnprob_gp_with_sig(p, x, y, yerr): \n", " decay = 200\n", " kernel = 15000/np.exp(-1200/decay)*MyDijetKernel(a=decay,b=100)\n", " gp = george.GP(kernel)\n", " gp.compute(x, yerr)\n", " gpl = gp.lnlikelihood(y - sig_cdf(x,p))\n", " return gpl " ] }, { "cell_type": "code", "execution_count": 48, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def fit_gp_with_sig(initial, data, nwalkers=32):\n", " ndim = len(initial)\n", " p0 = [np.array(initial) + 1e-8 * np.random.randn(ndim)\n", " for i in xrange(nwalkers)]\n", " sampler = emcee.EnsembleSampler(nwalkers, ndim, lnprob_gp_with_sig, args=data)\n", " print(\"Running burn-in\")\n", " p0, lnp, _ = sampler.run_mcmc(p0, 500)\n", " sampler.reset()\n", " \n", " best_fit = p0[np.argmax(lnp)]\n", " print best_fit\n", "\n", " print(\"Running production\")\n", " p0, _, _ = sampler.run_mcmc(p0, 1000)\n", " return sampler, best_fit" ] }, { "cell_type": "code", "execution_count": 49, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Fitting gp\n", "Running burn-in\n", "[ 8.20697060e-01 8.66994700e+00 -4.96057225e+00 8.05462654e-02\n", " 1.99047360e+02 3.00341642e+03 9.34115046e+01]\n", "Running production\n" ] } ], "source": [ "np.random.seed(1234)\n", "init_params = np.hstack((best_fit,[200, 3000, 100]))\n", "data = 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yerr=np.sqrt(fake_data))\n", " meanGP = gp.sample_conditional(fake_data_residual, xval) + sig_cdf(xval,s)\n", " pl.plot(xval, meanGP, color=\"#4682b4\", alpha=0.5)\n", "plt.plot(xval, sig_cdf(xval,best_fit_with_sig), c='r')\n", "bkg_only_params = np.hstack((best_fit,[0, 3000, 100]))\n", "plt.plot(xval, sig_cdf(xval,bkg_only_params), c='g')\n", "plt.semilogy()\n", "plt.xlim(2500,3500)\n", "plt.ylim(10,1500)" ] }, { "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 }