10_Monte-Carlo-Calculating-Pi.ipynb

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     "source": [
      "#Monte Carlo: Calculating \u03c0\n",
      "\n",
      "In this tutorial we will introduce Monte-Carlo simulations, which are an incredibly useful for simulating statistical systems. We will work through 2 simple examples to demonstrate the principles.\n",
      "\n",
      "1. calculation of pi\n",
      "\n",
      "2. Maxwell-Boltzmann velocity and speed distributions"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Calculation of pi\n",
      "\n",
      "Take a dart board. Draw a square, and a circle inside the square whose diameter equals the length of the side of the square. Throw some darts at it. Assuming you're as good at darts as me, they will be randomly populated inside the square (ignore any that land outside the square...). How many darts land inside the circle as well? It's a question of relative area:\n",
      "\n",
      "The area of the square is $d^2$, and the area of circle is $\\pi (d/2)^2$. The circle therefore covers $\\pi/4$ of the area of the square. So assuming uniform distributions, the ratio of darts that land inside the circle to the total inside the square will also be $\\pi/4$, which we can then rearrange to find pi.\n",
      "\n",
      "So how many darts do you need to throw to get an accurate answer? Let's set up the program to find out."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%matplotlib inline\n",
      "import numpy as np\n",
      "import matplotlib.pyplot as plt\n",
      "from matplotlib.patches import Rectangle, Circle\n",
      "from IPython.display import display\n",
      "\n",
      "#create the plot so we can visualise it\n",
      "fig = plt.figure(figsize=(6,6))\n",
      "ax = fig.add_subplot(111)\n",
      "#add a square\n",
      "Rpatch = Rectangle((0,0),1,1,ec='k',fc='grey',alpha=0.35)\n",
      "ax.add_patch(Rpatch)\n",
      "#add circle\n",
      "Cpatch = Circle((0.5,0.5),0.5,ec='k',fc='b',alpha=0.25)\n",
      "ax.add_patch(Cpatch)\n",
      "plt.draw()\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
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       "output_type": "display_data",
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09HTXfb+Oh/vWH0rhLiKdZ2sKZJVQqD6Lhe3keLiHQiEikYoevSciHSeXW2do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\nN4VCjnR6hmBwlddfT3DkiEK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       "text": [
        "<matplotlib.figure.Figure at 0x376b0f0>"
       ]
      }
     ],
     "prompt_number": 1
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now we make use of random number generators to simulate our dart thrower. Let's add a few to the plot as an example:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "xs = np.random.random(20)\n",
      "ys = np.random.random(20)\n",
      "\n",
      "ax.plot(xs,ys,'ro')\n",
      "display(fig)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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pncVu33vLvcrhLi13IYw2cOolCfM6oFQOq3XvjeEq97lvPpgjhBDicTlsts3H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\nPwgcBfRyHmSlAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x376b0f0>"
       ]
      }
     ],
     "prompt_number": 2
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "To decide if the points are inside the circle, we just need to look at their radius (relative to the center coordinate of the circle):"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "rad = np.sqrt((xs-0.5)**2+(ys-0.5)**2)\n",
      "inside = rad[rad<0.5] # use logical statements to index lists, this selects the elements of rad whose value is less than 0.5\n",
      "\n",
      "print len(inside), len(rad)\n",
      "\n",
      "ratio = float(len(inside))/len(rad)\n",
      "pi_guess = 4 * ratio\n",
      "\n",
      "print 'Estimate of pi:',pi_guess\n",
      "print 'Relative error (%):', 100*abs(pi_guess-np.pi)/np.pi"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "17 20\n",
        "Estimate of pi: 3.4\n",
        "Relative error (%): 8.22536130249\n"
       ]
      }
     ],
     "prompt_number": 49
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Not too bad for 20 points! Let's try increasing the numbers of darts though...\n",
      "\n",
      "We'll rewrite the above code into a routine we can call with an argument that represents the number of points:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def estimate_pi(n):\n",
      "    \"\"\" Estimate pi using the drunken dart player model. \n",
      "        Argument n (integer) is the number of 'darts' thrown\n",
      "    \"\"\"\n",
      "    xs = np.random.random(n)\n",
      "    ys = np.random.random(n)\n",
      "    rad = np.sqrt((xs-0.5)**2+(ys-0.5)**2)\n",
      "    inside = rad[rad<0.5]\n",
      "    ratio = float(len(inside))/len(rad)\n",
      "    pi_guess = 4 * ratio\n",
      "    \n",
      "    return pi_guess\n",
      "    "
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 50
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Ok, let's try with a few different values:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "for n in [20,200,2e3,2e4,2e6]:\n",
      "    pi_guess = estimate_pi(n)\n",
      "    print 'Number of points:', n\n",
      "    print 'Estimate of pi:',pi_guess\n",
      "    print 'Relative error (%):', 100*abs(pi_guess-np.pi)/np.pi\n",
      "    print ''"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Number of points: 20\n",
        "Estimate of pi: 3.4\n",
        "Relative error (%): 8.22536130249\n",
        "\n",
        "Number of points: 200\n",
        "Estimate of pi: 3.12\n",
        "Relative error (%): 0.687315510657\n",
        "\n",
        "Number of points: 2000.0\n",
        "Estimate of pi: 3.134\n",
        "Relative error (%): 0.24168167\n",
        "\n",
        "Number of points: 20000.0\n",
        "Estimate of pi: 3.1438\n",
        "Relative error (%): 0.0702620184601\n",
        "\n",
        "Number of points:"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        " 2000000.0\n",
        "Estimate of pi: 3.141448\n",
        "Relative error (%): 0.00460446677031\n",
        "\n"
       ]
      }
     ],
     "prompt_number": 51
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "So as more points are added, the estimate gets closer to the real value. Sometimes low numbers appear to get lucky, so let's have a look at the variability:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "pi_guesses = []\n",
      "for i in range(20):\n",
      "    pi_guesses.append(estimate_pi(200))\n",
      "\n",
      "print 'Number of points: 200'\n",
      "print pi_guesses\n",
      "print 'Mean value:', np.array(pi_guesses).mean()\n",
      "print 'Standard deviation:', np.array(pi_guesses).std()\n",
      "\n",
      "pi_guesses = []\n",
      "for i in range(20):\n",
      "    pi_guesses.append(estimate_pi(2e6))\n",
      "\n",
      "print 'Number of points: 2M'\n",
      "print pi_guesses\n",
      "print 'Mean value:', np.array(pi_guesses).mean()\n",
      "print 'Standard deviation:', np.array(pi_guesses).std()\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Number of points: 200\n",
        "[3.06, 3.16, 3.0, 3.02, 2.92, 3.02, 3.16, 3.18, 3.08, 3.12, 3.26, 3.06, 2.98, 3.14, 3.24, 3.16, 3.2, 3.26, 3.1, 3.12]\n",
        "Mean value: 3.112\n",
        "Standard deviation: 0.0923904756996\n",
        "Number of points: 2M"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n",
        "[3.141804, 3.142372, 3.141678, 3.141238, 3.141286, 3.142784, 3.140546, 3.14064, 3.142424, 3.139482, 3.141902, 3.141988, 3.141348, 3.142602, 3.142088, 3.142558, 3.14093, 3.14298, 3.1401, 3.14212]\n",
        "Mean value: 3.1416435\n",
        "Standard deviation: 0.000915795146307\n"
       ]
      }
     ],
     "prompt_number": 52
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Ok, so not only does the average get closer, but the standard deviation also decreases considerably when more points are used. Let's look at that graphically, using histograms to analyse the spread of values:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "pi_guesses = []\n",
      "for i in range(500):\n",
      "    pi_guesses.append(estimate_pi(200))\n",
      "\n",
      "print '200 darts (blue), 10000 darts (red)'\n",
      "fig2 = plt.figure(figsize=(6,5))\n",
      "ax = fig2.add_subplot(111)\n",
      "ax.axvline(np.pi,lw=3,linestyle='dashed',color='k')\n",
      "bad_hist = ax.hist(pi_guesses,bins=30,color='b',alpha=0.65,lw=0.2)\n",
      "ax.set_xlim(2.7,3.5)\n",
      "\n",
      "\n",
      "pi_guesses = []\n",
      "for i in range(500):\n",
      "    pi_guesses.append(estimate_pi(1e4))\n",
      "    \n",
      "good_hist = ax.hist(pi_guesses,bins=30,color='r',alpha=0.65,lw=0.2)\n",
      "ax.set_xlabel('Estimate of $\\pi$')\n",
      "ax.set_ylabel('Number of occurrences')\n",
      "plt.draw()\n",
      "\n",
      "#display(fig2)\n",
      "#display(fig3)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "200 darts (blue), 10000 darts (red)\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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f56KRDdX93Io92tZoNN6wEzG1fZKFYOHChRg/fjwSExMhhMBHH32EZ599VnLHVVVVGDVq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\nISAi0rj/B04L7Pz2FQl8AAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x9e65898>"
       ]
      }
     ],
     "prompt_number": 53
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "How many darts do you have to throw until you get a value accurate to a certain amount of significant figures? In other words, how does the relative error scale with the number of darts $n$? We leave that as an excercise to the reader (though a quick Googling will tell you the answer, I find it's always nicer if I have a code to test the theory!).\n",
      "\n",
      "More information on this problem can be found here:\n",
      "http://en.wikipedia.org/wiki/Monte_Carlo_integration"
     ]
    }
   ],
   "metadata": {}
  }
 ]
}