mc.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-06-05T16:18:07.336505Z",
     "start_time": "2018-06-05T16:18:07.003013Z"
    },
    "run_control": {
     "frozen": false,
     "read_only": false
    }
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import scipy.optimize as opt\n",
    "import uncertainties.unumpy as unp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-06-05T16:18:07.342702Z",
     "start_time": "2018-06-05T16:18:07.338800Z"
    },
    "run_control": {
     "frozen": false,
     "read_only": false
    }
   },
   "outputs": [],
   "source": [
    "count_data = 43000 # number of data points used in fit\n",
    "bins = 400 # number of bins after efficiency corrections \n",
    "tau = 2.1969811e-6 # value taken from wiki\n",
    "count_MC_samples = 10000 # higher values increase confidence in end result"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-06-05T16:18:07.350324Z",
     "start_time": "2018-06-05T16:18:07.346203Z"
    },
    "run_control": {
     "frozen": false,
     "read_only": false
    }
   },
   "outputs": [],
   "source": [
    "def random_lifetimes(size,seed=0):\n",
    "    p = np.random.rand(size)\n",
    "    return -np.log(p*tau)*tau"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-06-05T16:18:07.363922Z",
     "start_time": "2018-06-05T16:18:07.353484Z"
    },
    "run_control": {
     "frozen": false,
     "read_only": false
    }
   },
   "outputs": [],
   "source": [
    "def sample_fit():\n",
    "    data = random_lifetimes(count_data)\n",
    "    hist, bin_edges = np.histogram(data, bins = bins)\n",
    "\n",
    "    t = bin_edges[:-1]\n",
    "    hits = hist\n",
    "\n",
    "    N_0 = sum(hits)\n",
    "    e = [(sum(hits[:i])) for i in range(len(hits))]\n",
    "\n",
    "    def fit_function(x,N_0,pr,decay_constant):\n",
    "        a = -N_0*np.exp(-decay_constant*x) + N_0 -pr\n",
    "        return a\n",
    "    \n",
    "    params, _ = opt.curve_fit(fit_function,t,e,p0=[sum(hits)*0.7,1,np.log(2)/2e-6])\n",
    "    \n",
    "    tau_mc = 1/params[-1]\n",
    "    return tau_mc"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-06-05T16:21:47.184067Z",
     "start_time": "2018-06-05T16:18:07.367764Z"
    }
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/usr/lib/python3.6/site-packages/ipykernel_launcher.py:12: RuntimeWarning: overflow encountered in exp\n",
      "  if sys.path[0] == '':\n",
      "/usr/lib/python3.6/site-packages/ipykernel_launcher.py:12: RuntimeWarning: overflow encountered in multiply\n",
      "  if sys.path[0] == '':\n"
     ]
    }
   ],
   "source": [
    "results = np.zeros(count_MC_samples)\n",
    "for i in range(count_MC_samples):\n",
    "    results[i] = sample_fit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-06-05T16:21:47.198052Z",
     "start_time": "2018-06-05T16:21:47.186230Z"
    }
   },
   "outputs": [],
   "source": [
    "average = np.average(results)\n",
    "sor = np.array(sorted(results))\n",
    "median_index = int(len(sor)/2)\n",
    "interval_lenght = 0.95 # two sigma\n",
    "left_border = sor[median_index - int(len(sor)*interval_lenght/2)]\n",
    "right_border = sor[median_index + int(len(sor)*interval_lenght/2)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-06-05T16:22:01.709782Z",
     "start_time": "2018-06-05T16:22:01.123900Z"
    }
   },
   "outputs": [
    {
     "data": {
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      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Relative Abweichung am linken Rand: -1.0420473677118418 %\n",
      "Relative Abweichung am rechten Rand: 1.0458345167097765 %\n"
     ]
    }
   ],
   "source": [
    "plt.hist(results*1e6,label=\"MC\",bins=20)\n",
    "plt.vlines(tau*1e6,0,200,label=\"Literaturwert $\\tau$\")\n",
    "plt.vlines(average*1e6,0,100,label=r\"Mittelwert $\\tau_{MC}$\",color=\"r\")\n",
    "plt.axvspan(left_border*1e6,right_border*1e6,0,250,color=\"y\",alpha=0.5,label=str(interval_lenght*100)+\"% der MC-Daten\")\n",
    "\n",
    "plt.legend()\n",
    "plt.savefig(\"mc.pdf\")\n",
    "plt.show()\n",
    "print(\"Relative Abweichung am linken Rand:\",(left_border-tau)/tau*100,\"%\")\n",
    "print(\"Relative Abweichung am rechten Rand:\",(right_border-tau)/tau*100,\"%\")"
   ]
  }
 ],
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