grinnbearit/Markov Chain Monte Carlo

## Markov Chain Monte Carlo
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 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# Markov Chain Monte Carlo\n",
      "\n"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import numpy as np\n",
      "import pandas as pd\n",
      "from pandas import Series, DataFrame\n",
      "import matplotlib.pyplot as plt\n",
      "%matplotlib inline"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "P = Series([0.00007, # P(A -> B)\n",
      "            0.0005, # P(B -> C)\n",
      "            0.00013, # P(C -> D),\n",
      "            0.00022, # P(D -> S),\n",
      "            0]) # S doesn't decay\n",
      "\n",
      "atoms = Series([1000, # 1000 A\n",
      "                0, # 0 B\n",
      "                0, # 0 C\n",
      "                0, # 0 D\n",
      "                0]) # 0 S\n",
      "\n",
      "s = 5000\n",
      "np.random.seed(4215)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "np.random.binomial(1, 0.5)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 3,
       "text": [
        "1"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "np.random.binomial(atoms[0], 0.5)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 4,
       "text": [
        "467"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "second_interval = Series([56, 44, 0, 0, 0])\n",
      "a_decay = np.random.binomial(second_interval[0], 0.5)\n",
      "b_decay = np.random.binomial(second_interval[1], 0.5)\n",
      "\n",
      "third_interval = second_interval - Series([a_decay, b_decay-a_decay, -b_decay, 0, 0])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 5
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "third_interval"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 6,
       "text": [
        "0    32\n",
        "1    48\n",
        "2    20\n",
        "3     0\n",
        "4     0\n",
        "dtype: int64"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def sim_decay(atoms, s, p):\n",
      "    num_atoms = len(atoms)\n",
      "    results = DataFrame(np.zeros((s, num_atoms))).astype(int)\n",
      "    results.ix[0] = atoms\n",
      "    \n",
      "    for i in range(1, s):\n",
      "        decayed = np.random.binomial(results.ix[i-1], p, num_atoms)\n",
      "        results.ix[i] = results.ix[i-1] - decayed\n",
      "        results.ix[i, 1:] += decayed[:-1]\n",
      "\n",
      "    return results"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 7
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "original_sim = sim_decay(atoms, s, P)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 8
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "original_sim[-1:]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "html": [
        "<div style=\"max-height:1000px;max-width:1500px;overflow:auto;\">\n",
        "<table border=\"1\" class=\"dataframe\">\n",
        "  <thead>\n",
        "    <tr style=\"text-align: right;\">\n",
        "      <th></th>\n",
        "      <th>0</th>\n",
        "      <th>1</th>\n",
        "      <th>2</th>\n",
        "      <th>3</th>\n",
        "      <th>4</th>\n",
        "    </tr>\n",
        "  </thead>\n",
        "  <tbody>\n",
        "    <tr>\n",
        "      <th>4999</th>\n",
        "      <td> 716</td>\n",
        "      <td> 95</td>\n",
        "      <td> 151</td>\n",
        "      <td> 33</td>\n",
        "      <td> 5</td>\n",
        "    </tr>\n",
        "  </tbody>\n",
        "</table>\n",
        "</div>"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 9,
       "text": [
        "        0   1    2   3  4\n",
        "4999  716  95  151  33  5"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "fig, ax = plt.subplots(1, 1)\n",
      "ax.set_xlabel(\"Seconds Elapsed\")\n",
      "ax.set_ylabel(\"Number of Atoms\")\n",
      "ax.plot(original_sim)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 10,
       "text": [
        "[<matplotlib.lines.Line2D at 0x10a5fb8d0>,\n",
        " <matplotlib.lines.Line2D at 0x10a5fba50>,\n",
        " <matplotlib.lines.Line2D at 0x10a5fbc90>,\n",
        " <matplotlib.lines.Line2D at 0x10a5fbe50>,\n",
        " <matplotlib.lines.Line2D at 0x10a5ff050>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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Dk4H17r7fzHoDRe6+tC0K2FiatCgi0nRtdtled4+ZWS0ww8xS63cDHSpMRESk/XxsmJjZ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\nEWk2hYmIiDSbwkRERJpNYSIiIs32/wHA2wfKUT2giQAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x1070bf590>"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "second_sim = sim_decay(Series([900, 200, 1000, 50, 0]),\n",
      "                       500,                         \n",
      "                       Series([0.01, 0.03, 0.09, 0.02, 0])) "
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 11
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "second_sim[-1:]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "html": [
        "<div style=\"max-height:1000px;max-width:1500px;overflow:auto;\">\n",
        "<table border=\"1\" class=\"dataframe\">\n",
        "  <thead>\n",
        "    <tr style=\"text-align: right;\">\n",
        "      <th></th>\n",
        "      <th>0</th>\n",
        "      <th>1</th>\n",
        "      <th>2</th>\n",
        "      <th>3</th>\n",
        "      <th>4</th>\n",
        "    </tr>\n",
        "  </thead>\n",
        "  <tbody>\n",
        "    <tr>\n",
        "      <th>499</th>\n",
        "      <td> 6</td>\n",
        "      <td> 1</td>\n",
        "      <td> 0</td>\n",
        "      <td> 5</td>\n",
        "      <td> 2138</td>\n",
        "    </tr>\n",
        "  </tbody>\n",
        "</table>\n",
        "</div>"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 12,
       "text": [
        "     0  1  2  3     4\n",
        "499  6  1  0  5  2138"
       ]
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "fig, ax = plt.subplots(1, 1)\n",
      "ax.set_xlabel(\"Seconds Elapsed\")\n",
      "ax.set_ylabel(\"Number of Atoms\")\n",
      "ax.plot(second_sim)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 13,
       "text": [
        "[<matplotlib.lines.Line2D at 0x10a6397d0>,\n",
        " <matplotlib.lines.Line2D at 0x10a639a90>,\n",
        " <matplotlib.lines.Line2D at 0x10a639cd0>,\n",
        " <matplotlib.lines.Line2D at 0x10a639e90>,\n",
        " <matplotlib.lines.Line2D at 0x10a63c090>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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nyefjko0b2drRwVXp6fwgN5dJYbYFblnVwpartzB7x2wstlNtmEchxOlAAghd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\nRK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10a5e6510>"
       ]
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def trans_mat(p):\n",
      "    num_p = len(p)\n",
      "    T = DataFrame(np.zeros((num_p, num_p)))\n",
      "    for i in range(num_p):\n",
      "        T.ix[i, i] = 1 - p[i]\n",
      "        if i != num_p - 1:\n",
      "            T.ix[i, i+1] = p[i]\n",
      "    return T\n",
      "\n",
      "trans_mat(P)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "html": [
        "<div style=\"max-height:1000px;max-width:1500px;overflow:auto;\">\n",
        "<table border=\"1\" class=\"dataframe\">\n",
        "  <thead>\n",
        "    <tr style=\"text-align: right;\">\n",
        "      <th></th>\n",
        "      <th>0</th>\n",
        "      <th>1</th>\n",
        "      <th>2</th>\n",
        "      <th>3</th>\n",
        "      <th>4</th>\n",
        "    </tr>\n",
        "  </thead>\n",
        "  <tbody>\n",
        "    <tr>\n",
        "      <th>0</th>\n",
        "      <td> 0.99993</td>\n",
        "      <td> 0.00007</td>\n",
        "      <td> 0.00000</td>\n",
        "      <td> 0.00000</td>\n",
        "      <td> 0.00000</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>1</th>\n",
        "      <td> 0.00000</td>\n",
        "      <td> 0.99950</td>\n",
        "      <td> 0.00050</td>\n",
        "      <td> 0.00000</td>\n",
        "      <td> 0.00000</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>2</th>\n",
        "      <td> 0.00000</td>\n",
        "      <td> 0.00000</td>\n",
        "      <td> 0.99987</td>\n",
        "      <td> 0.00013</td>\n",
        "      <td> 0.00000</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>3</th>\n",
        "      <td> 0.00000</td>\n",
        "      <td> 0.00000</td>\n",
        "      <td> 0.00000</td>\n",
        "      <td> 0.99978</td>\n",
        "      <td> 0.00022</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>4</th>\n",
        "      <td> 0.00000</td>\n",
        "      <td> 0.00000</td>\n",
        "      <td> 0.00000</td>\n",
        "      <td> 0.00000</td>\n",
        "      <td> 1.00000</td>\n",
        "    </tr>\n",
        "  </tbody>\n",
        "</table>\n",
        "</div>"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 14,
       "text": [
        "         0        1        2        3        4\n",
        "0  0.99993  0.00007  0.00000  0.00000  0.00000\n",
        "1  0.00000  0.99950  0.00050  0.00000  0.00000\n",
        "2  0.00000  0.00000  0.99987  0.00013  0.00000\n",
        "3  0.00000  0.00000  0.00000  0.99978  0.00022\n",
        "4  0.00000  0.00000  0.00000  0.00000  1.00000"
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def markov_decay(atoms, s, p):\n",
      "    num_atoms = len(atoms)\n",
      "    T = trans_mat(p)\n",
      "    M = DataFrame(np.zeros((s, num_atoms)))\n",
      "    M.ix[0] = atoms\n",
      "    \n",
      "    for i in range(1, s):\n",
      "        M.ix[i] = M.ix[i-1].dot(T)\n",
      "    \n",
      "    return M"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 15
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "original_markov = markov_decay(atoms, s, P)\n",
      "original_markov[-1:]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "html": [
        "<div style=\"max-height:1000px;max-width:1500px;overflow:auto;\">\n",
        "<table border=\"1\" class=\"dataframe\">\n",
        "  <thead>\n",
        "    <tr style=\"text-align: right;\">\n",
        "      <th></th>\n",
        "      <th>0</th>\n",
        "      <th>1</th>\n",
        "      <th>2</th>\n",
        "      <th>3</th>\n",
        "      <th>4</th>\n",
        "    </tr>\n",
        "  </thead>\n",
        "  <tbody>\n",
        "    <tr>\n",
        "      <th>4999</th>\n",
        "      <td> 704.728788</td>\n",
        "      <td> 101.362288</td>\n",
        "      <td> 150.965462</td>\n",
        "      <td> 31.944745</td>\n",
        "      <td> 10.998717</td>\n",
        "    </tr>\n",
        "  </tbody>\n",
        "</table>\n",
        "</div>"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 16,
       "text": [
        "               0           1           2          3          4\n",
        "4999  704.728788  101.362288  150.965462  31.944745  10.998717"
       ]
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "fig, ax = plt.subplots(1, 1)\n",
      "ax.set_xlabel(\"Seconds Elapsed\")\n",
      "ax.set_ylabel(\"Number of Atoms\")\n",
      "ax.plot(original_markov)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 17,
       "text": [
        "[<matplotlib.lines.Line2D at 0x10a6ec810>,\n",
        " <matplotlib.lines.Line2D at 0x10a6ecad0>,\n",
        " <matplotlib.lines.Line2D at 0x10a6ecd10>,\n",
        " <matplotlib.lines.Line2D at 0x10a6eced0>,\n",
        " <matplotlib.lines.Line2D at 0x10a6f10d0>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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CJLeevh52H9/NzmM7Q1hEAZEJjt3Hd1OTqmHB9AUsnL6QBdMWsGBaVJ4eyk3T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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10a608e90>"
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "second_markov = sim_decay(Series([900, 200, 1000, 50, 0]),\n",
      "                       500,                         \n",
      "                       Series([0.01, 0.03, 0.09, 0.02, 0])) \n",
      "second_markov[-1:]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "html": [
        "<div style=\"max-height:1000px;max-width:1500px;overflow:auto;\">\n",
        "<table border=\"1\" class=\"dataframe\">\n",
        "  <thead>\n",
        "    <tr style=\"text-align: right;\">\n",
        "      <th></th>\n",
        "      <th>0</th>\n",
        "      <th>1</th>\n",
        "      <th>2</th>\n",
        "      <th>3</th>\n",
        "      <th>4</th>\n",
        "    </tr>\n",
        "  </thead>\n",
        "  <tbody>\n",
        "    <tr>\n",
        "      <th>499</th>\n",
        "      <td> 10</td>\n",
        "      <td> 4</td>\n",
        "      <td> 1</td>\n",
        "      <td> 5</td>\n",
        "      <td> 2130</td>\n",
        "    </tr>\n",
        "  </tbody>\n",
        "</table>\n",
        "</div>"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 18,
       "text": [
        "      0  1  2  3     4\n",
        "499  10  4  1  5  2130"
       ]
      }
     ],
     "prompt_number": 18
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "fig, ax = plt.subplots(1, 1)\n",
      "ax.set_xlabel(\"Seconds Elapsed\")\n",
      "ax.set_ylabel(\"Number of Atoms\")\n",
      "ax.plot(second_markov)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 19,
       "text": [
        "[<matplotlib.lines.Line2D at 0x10a795650>,\n",
        " <matplotlib.lines.Line2D at 0x10a795910>,\n",
        " <matplotlib.lines.Line2D at 0x10a795b50>,\n",
        " <matplotlib.lines.Line2D at 0x10a795d10>,\n",
        " <matplotlib.lines.Line2D at 0x10a795ed0>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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MOKgJay9fIMDLaTt5duxYikssfP3r8PHHMGnSoS9lUYoHR43i9pIS5n35Jd9O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\nx8aitabU7WZabCyzY2MZfazRTQxIQyK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       "text": [
        "<matplotlib.figure.Figure at 0x10a627190>"
       ]
      }
     ],
     "prompt_number": 19
    }
   ],
   "metadata": {}
  }
 ]
}