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 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# - compatibility with Python 3\n",
      "from __future__ import print_function  # print('me') instead of print 'me'\n",
      "from __future__ import division  # 1/2 == 0.5, not 0"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# - show figures inside the notebook\n",
      "%matplotlib inline"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# - import common modules\n",
      "import numpy as np  # the Python array package\n",
      "import matplotlib.pyplot as plt  # the Python plotting package"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Range"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "`range` is a built-in function in Python.\n",
      "\n",
      "\"built-in\" means you can use `range` without having to `import` it.\n",
      "\n",
      "In Python 2, `range` returns a list:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "range(10)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 4,
       "text": [
        "[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "type(range(10))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 5,
       "text": [
        "list"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "In Python 3, range returns a special object that behaves rather like a list, but doesn't create the values until you need them.   This is more efficient in memory.\n",
      "\n",
      "Here is `range` in Python 3:\n",
      "\n",
      "```\n",
      "In [1]: range(10)\n",
      "Out[1]: range(0, 10)\n",
      "\n",
      "In [2]: type(range(10))\n",
      "Out[2]: range\n",
      "\n",
      "In [3]: list(range(10))\n",
      "Out[3]: [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]\n",
      "```"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "There are three ways of calling range - with 1, 2 or 3 arguments.\n",
      "\n",
      "With just one argument, the argument value is the length of the returned list, or, equivalently, the largest value in the list, plus 1 (as when you do slicing), or, equivalently, the value `stop` such that no list element is `>= stop`:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "range(5)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 6,
       "text": [
        "[0, 1, 2, 3, 4]"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "range(2)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 7,
       "text": [
        "[0, 1]"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "If you pass 2 arguments, the first value is the first value in the returned list, and the second value is the stop value:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "range(4, 10)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 8,
       "text": [
        "[4, 5, 6, 7, 8, 9]"
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "If you pass 3 arguments, the first value is the first returned value, the second is the stop value, and the third is the step size."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "range(4, 10, 2)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 9,
       "text": [
        "[4, 6, 8]"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "range(4, 11, 2)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 10,
       "text": [
        "[4, 6, 8, 10]"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## np.arange"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "`arange` in numpy is very like range with two important differences:\n",
      "\n",
      "* `arange` returns an array rather than a list (in Python 2 or Python 3);\n",
      "* `arange` arguments can be floating point values."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "np.arange(4, 11, 2)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 11,
       "text": [
        "array([ 4,  6,  8, 10])"
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "np.arange(4, 11, 0.5)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 12,
       "text": [
        "array([  4. ,   4.5,   5. ,   5.5,   6. ,   6.5,   7. ,   7.5,   8. ,\n",
        "         8.5,   9. ,   9.5,  10. ,  10.5])"
       ]
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Because `arange` returns arrays, you can also multiply by the step size and maybe add the start value to create equally spaced vectors:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "np.arange(10) * 0.5 + 4"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 13,
       "text": [
        "array([ 4. ,  4.5,  5. ,  5.5,  6. ,  6.5,  7. ,  7.5,  8. ,  8.5])"
       ]
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Exponential floating point format"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We have so far seen floating point numbers in this kind of format in text files : ``314.15926``.\n",
      "\n",
      "It is common for numpy and MATLAB to write floating point numbers in exponential format, like this ``3.1415926e02``.  You can read this as meaning `3.1415926 * 10**2` == `3.1415926 * 100` == `314.15926`.  Both `np.loadtxt` and the `float` function will happily convert these to floating point values in memory:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "float('2.2e02')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 14,
       "text": [
        "220.0"
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Legends on plots"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We have already seen that we can plot two things onto the same plot:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x = np.arange(0, 2*np.pi, 0.1)\n",
      "y1 = np.sin(x)\n",
      "y2 = np.cos(x)\n",
      "plt.plot(x, y1)\n",
      "plt.plot(x, y2)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 15,
       "text": [
        "[<matplotlib.lines.Line2D at 0x105fc1ed0>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
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UejZ3LK/cCRRQSuVRSiUHmgOL//GYxUDr2ECVgev/LPLCcbLU0nFBQdbBNRMm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\nFM2KNuN+9H0WH1lsOortlSl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       "text": [
        "<matplotlib.figure.Figure at 0x105fc1e50>"
       ]
      }
     ],
     "prompt_number": 15
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "It can be hard to remember which plot is which (is green the sin or the cos)?  `label` and `legend` are useful for that:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.plot(x, y1, label='sin')\n",
      "plt.plot(x, y2, label='cos')\n",
      "plt.legend()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 16,
       "text": [
        "<matplotlib.legend.Legend at 0x1062bdb10>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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VXgAlyqw9s6j3eD0p8h4myYVeKVVfKXVIKXVEKfXuIx4zPubre5VSFR71XBER\nUE/6JsUqsFog47eNl6WWDujRw9pVffeu6STuJVpHM2HbBAKrypJKT5OkQq+U8gUmAvWBkkArpVSJ\nfz2mAVBYa10EeAuY8qjnkyWVj1a/cH1u3L3B5jObTUexvZIlreWWCxaYTuJeVh5dSbqU6aiZt6bp\nKMLJklpWqwJHtdYntdb3gfnAvzvHNwZmAWittwIBSqlYN6q3a5fENB7MR/lYXS2lV71DHiy1lBdA\njnuwQUqWVHqepBb63ED4Qx+fiflcfI+JdQJQllTG7fXyr7Pq2CrO3DhjOortNWgAV67IUktHhV0J\nY9f5XbKk0kMl9WBSR8dL/x4ixPp9gwcP/uv92rVrU7t27USF8lQPL7Uc8swQ03Fs7eGlltWrm05j\nf7Kk0n0EBwcTHBycoO9J0vJKpVR1YLDWun7MxwOAaK318IceMxUI1lrPj/n4EPC01vriv57rkYeD\ni7/JUkvHXbsGBQvCgQOQK5fpNPYlSyrdW3Isr9wBFFFKFVBKpQRaAD/96zE/Ae1iAlUHrv27yAvH\nyVJLxwUEWAfXTJtmOom9fbXnK1lS6eGSVOi11pFAd2AlcBD4VmsdqpTqrJTqHPOY5cBxpdRRYBrQ\nNYmZvV5gNelq6aju3WWpZVxkSaV3SPJiRq31z1rrYlrrwlrroTGfm6a1nvbQY7rHfL2c1npXUq/p\n7eoXrs/NezfZFL7JdBTbK1kSypaVpZaPsuLoCjKkyiBLKj2crFp3Qz7Khx5Ve0hXSwfJUstHG791\nPIFVZUmlp5NC76bal2/PL8d+kaWWDmjQwOqBs0Xa+v/DocuH2H1hNy1KtzAdRbiYFHo3lSFVBtqW\nbcvk7ZNNR7E9Hx85QDw2E7dN5K2Kb8nqLS9gq+6VdsniLo5cOcITXz7BqV6nSJ0itek4tnb9urXU\nMiQEcv97S58Xuh5xnYJBBQnpEkLuDPIDiUtUlHUzP00a00li53bdK0XCFMlShCq5qzBv/zzTUWwv\nY0Z47TXpavnAzD0zeb7w81LkHbB0Kbz0kukUSSOF3s0FVg1k/FbpaumIHj1g+nSrS6o3i9bRTNw2\nUZZUOmj8ePc/2lQKvZur93g9IiIjWH96vekotlesGFSsCPPnm05i1s9HfiZT6kxUzyO9IeITEgKh\nofDyy6aTJI0Uejfno3z+2kAl4hcYaI3QvPkFUNDWIHpU7SFLKh0wfjx06QIpU5pOkjRS6D1Au3Lt\nCD4ZzKmjhzr0AAAcV0lEQVRrp0xHsb3nn4dbt2DDBtNJzDj4+0FCLoXQopQsqYzP5cuwcCF07mw6\nSdJJofcA6VKmo3259kzaPsl0FNvz8bHm6sd76V6z8VvH83alt0nll8p0FNubMQOaNoXHHjOdJOlk\neaWHOH71OFU/r8qpXqdImzKt6Ti29uefkD8/7NkD+fKZTpN8/rjzB4+Pf5xD3Q6RPV2sZ/+IGPfv\nW4fM//QTVHjk4af2IMsrvUihTIV4Mt+TfL3va9NRbC99eus0s8lettfs852f07hYYynyDli82Np3\nYfci7ygp9B6kV/VeBG0NIlpHm45ie927wxdfwO3bppMkj8joSCZtn0TPaj1NR3ELQUHQ04N+VFLo\nPcjT+Z8mpW9Kfjn2i+kotle4MNSoAd98YzpJ8lgcupgCAQWomLOi6Si2t2MHnDkDTf59+rUbk0Lv\nQZRS9KrWi3Fbx5mO4hZ69YJx47xjqWXQ1iAZzTto/HirN5JfUg9atREp9B6mVZlW7Dq/i0OXD5mO\nYnt16kCKFPCLh78A2nFuB+E3wmlS3IOGqC5y4QIsWQJvvmk6iXNJofcw/n7+dK7UmfFbvXT9YAIo\nZY3qx441ncS1grYG0b1Kd/x8PGiI6iKTJ0PLlpA5s+kkziXLKz3Q+T/PU3JySY4HHidT6kym49ha\nRIS11DI4GEqUMJ3G+c7/eZ5Sk0txLPCY/C7E484dKFAA1q2z2mW4C1le6aVyps9Jo6KNmLFrhuko\ntufvD2+/7bkbqKbumErL0i2lyDtgzhyoXNm9iryjZETvoXac20Hz75pzLPCYvGSPx4UL1mj+2DHP\nesl+5/4dCgQVYN3r6yiW1QOrlxNpDWXKWNN49eqZTpMwMqL3YpVzVSZvhrwsDl1sOort5cgBjRvD\n55+bTuJcc0LmUCVXFSnyDvj1V+ueTd26ppO4hhR6D9a7em/GbvHwO41O0qsXTJxobX33BFprxmwe\nQ58afUxHcQtjx1q/A57a0FMKvQdrWrwpF25eYHP4ZtNRbK9CBXj8cVi0yHQS51h5bCUpfVNSp0Ad\n01FsLzQUdu6E1q1NJ3EdKfQezNfHl17VezFmyxjTUdzCg6WWnnCraOyWsfSu3lt6zjsgKMi6Ie/v\nwWekS6H3cB3Kd2DNiTWcuHrCdBTbe/FFuHIFNm0ynSRp9l/aT8jFEFqWbmk6iu1duQLffgtdu5pO\n4lpS6D1c+lTp6Viho2ygcoCvL/TuDaNGmU6SNGM3j6VblW7Sc94B06dbPeeze3hDT1le6QXCr4dT\nbmo5TvQ8QUb/jKbj2NqtW9ammU2boEgR02kS7uLNixSfVJwjPY6QNU1W03Fs7d49qxXx8uVQrpzp\nNIknyysFAHkz5qV+4fqygcoBadPCW29Zzc7c0ZQdU2hRqoUUeQfMmwelSrl3kXeUjOi9xI5zO2j2\nbTOOBR4jhW8K03Fs7fx5qwAcOQJZsphO47iIyAjyj8vPb6//RvGsxU3HsTWtoWxZGD0annvOdJqk\nkRG9+EvlXJUpmKkgi0I9ZP2gC+XMac3bTp1qOknCfLPvGyrnqixF3gGrVllr5t1tF2xiSaH3In2q\n92H05tHIK6f49eljbaCKiDCdxDHROppRm0bRr2Y/01HcwsiR8M47nrtB6t+k0HuRRkUbcT3iOutO\nrTMdxfZKl7bmbufONZ3EMUsOLyF9qvQ8nf9p01Fsb/duOHTIakfsLaTQexFfH1/61ujLyE0jTUdx\nC337wpgx7rGBauSmkfSr2U82SDlg9GgIDISUKU0nST5S6L1M+/Lt2Xl+J/sv7Tcdxfbq1rXW1q9c\naTpJ3DaFb+L8zfM0K9HMdBTbCw+3llO+9ZbpJMlLCr2X8ffzp0fVHoza5Oa7gpKBUtY87kibvwAa\nuWkkfar3kXbUDggKgg4dICDAdJLkJcsrvdDVO1d5fPzj7OuyjzwZ8piOY2v37//d7KxKFdNp/uvw\n5cPUmlmLk71OkiZFGtNxbO36dShUyJqjz5fPdBrnkeWVIlaZUmfi9fKvM26Lm+4KSkYpUlhz9cOH\nm04Su9GbR9O1Slcp8g6YPh3q1/esIu8oGdF7qfDr4ZSfVp5jgccI8Pey17EJdOuWtVV+wwYoWtR0\nmr9duHmBkpNKcrj7YbKlzWY6jq3dvWuN5pcutVpSexIZ0YtHypsxL42KNmLK9immo9he2rRWd0O7\nzdVP2DqBVqVbSZF3wKxZ1k5YTyvyjpIRvRcLuRjCc988x4meJ/D38+Bm3E5w+bI1mt+/H3LlMp0G\nbt67ScGggmzpuIXHMz9uOo6tRUZC8eLw5Zfw1FOm0zifjOhFnMpkL0PFnBX5eu/XpqPYXtas0K6d\nfZqdTd0xlWcKPiNF3gELF1ptiGvVMp3EHBnRe7nfTv5GpyWdCO0Wiq+Pr+k4tnb6tPXS/9gxs8vz\nIiIjKBRUiJ9b/0y5HF7QejEJtLb+PxsyBBo1Mp3GNWREL+L1VP6nyJY2GwsPLjQdxfby5bOKxRTD\ntzVm7p5JpVyVpMg7YMUKiI6Ghg1NJzFLRvSCn4/8zLu/vsuet/fgo+Rvf1wOHLB2zB4/DqlTJ//1\n70fdp8iEIsxrPo8aeWskfwA389RT1nmwr71mOonryIheOKR+4fqk8E3B0rClpqPYXqlSULUqzJxp\n5vrz9s+jUKZCUuQdsHEjnD0Lr75qOol5UugFSik+qPUBQ9YNkRbGDvjgA2sD1b17yXvdaB3N0A1D\n+aDWB8l7YTc1dCj06wd+0hlCCr2wNC3elFv3b/HL8V9MR7G9qlWhRAmYPTt5r7s4dDEZUmXgmYLP\nJO+F3dC+fbBrF7z+uukk9pDoQq+UyqyU+kUpFaaUWqWUinUdglLqpFJqn1Jqt1JqW+KjClfyUT68\n/+T7DFk3xHQUt/C//8Fnn1m9cJKD1ppP13/KB7U+kFbEDhgyxDo8xl+2hwBJG9G/B/yitS4KrI75\nODYaqK21rqC1rpqE6wkXa1G6Bef+PCcHkzjgiSestghz5iTP9VYcXUFkdCSNinroGkEn2r8f1q2D\nLl1MJ7GPpBT6xsCsmPdnAU3jeKwMQdyAn48fA54cwKfrPzUdxS3873/w6afWzktXejCaf7/W+7Iq\nygGffGI1okub1nQS+0jKb012rfXFmPcvAtkf8TgN/KqU2qGU6pSE64lk0LZcW0J/D2X72e2mo9je\n009bB4l/+61rr7P6xGou377MKyVfce2FPMCBAxAcbPUmEn+L8360UuoXIEcsX/rHbX+ttVZKPWq5\nxhNa6/NKqWzAL0qpQ1rr9bE9cPDgwX+9X7t2bWrXrh1XPOECKX1T0v+J/ny87mOWtFpiOo6tKQUD\nB0KPHtb5o74u2FistWZQ8CAGPT1Idi47wBtG88HBwQQHByfoexK9YUopdQhr7v2CUionsFZrXTye\n7xkE3NRaj47la7JhyiYiIiMoMqEIi15dRNXcclslLlpDzZrQqxe0aOH85195dCV9VvVh39v7pNDH\n4+BBqFPHalGRLp3pNMnH1RumfgLax7zfHvghlgBplFLpY95PCzwHhCThmiIZ+Pv580GtDxi4dqDp\nKLb3YFQ/ZIi11d6ZtNYMDB4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oEEKIxEn061al1EtKqXCgOrBMKfVzzOdzKaWWAWitI4Hu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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10602af90>"
       ]
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from scipy.stats import gamma"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 17
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Scipy"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "*Scipy* is a large library of scientific routines that builds on top of numpy. \n",
      "\n",
      "You can think of numpy as being a subset of MATLAB, and numpy + scipy as being as being roughly equivalent to MATLAB plus the MATLAB toolboxes."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Scipy has many sub-packages, for doing things like reading MATLAB `.mat` files (`scipy.io`) or working with sparse matrices (`scipy.sparse`). We are going to be using the functions and objects for working with statistical distributions in `scipy.stats`:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import scipy.stats"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 18
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "`scipy.stats` contains objects for working with many different distributions.  We are going to be working with `scipy.stats.gamma`, which implements the [gamma distribution](https://en.wikipedia.org/wiki/Gamma_distribution).\n",
      "\n",
      "In particular we are interested in the [probability density function](https://en.wikipedia.org/wiki/Probability_density_function) (PDF) of the gamma distribution.\n",
      "\n",
      "Because this is a function, we need to pass it an array of values at which it will evaluate.\n",
      "\n",
      "We can also pass various parameters which change the shape, location and width of the gamma PDF.  The most important is the first parameter (after the input array) known as the [shape parameter](https://en.wikipedia.org/wiki/Shape_parameter) ($k$ in the [wikipedia page on gamma distributions](https://en.wikipedia.org/wiki/Gamma_distribution))."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "First we chose some x values at which to sample from the gamma PDF:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x = np.arange(0, 25, 0.1)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 19
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Next we plot the gamma PDF for shape values of 2, 4, 8, 12."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.plot(x, gamma.pdf(x, 2), label='k=2')\n",
      "plt.plot(x, gamma.pdf(x, 4), label='k=4')\n",
      "plt.plot(x, gamma.pdf(x, 8), label='k=8')\n",
      "plt.plot(x, gamma.pdf(x, 12), label='k=12')\n",
      "plt.legend()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 20,
       "text": [
        "<matplotlib.legend.Legend at 0x107630d90>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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+1OG8Dh6EgQMhLc25zdk3fnMjA1oMYNbFs2qWweLFMH06PPss3H23OYU9dgwe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\nXkQkABV7EZEAVOxFRAL4BmYfGXPfxFmBAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10759d110>"
       ]
      }
     ],
     "prompt_number": 20
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We will see more from the statistics distributions next week."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Correlation coefficient"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "By \"correlation coefficient\", we mean the [Pearson product moment correlation coefficient](https://en.wikipedia.org/wiki/Pearson_product-moment_correlation_coefficient).\n",
      "\n",
      "Let's say I have two arrays, `x` and `y`:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x = np.array([ 20.57,  18.33,  20.8 ,  18.77,  18.46,  21.09,  19.96,  20.61,\n",
      "               18.73,  19.6 ,  18.15,  19.7 ,  20.36,  20.39,  19.67,  20.73])\n",
      "y = np.array([ 28.98,  30.13,  29.64,  29.87,  28.78,  33.48,  31.36,  31.04,\n",
      "               30.43,  31.69,  27.08,  29.25,  29.5 ,  30.04,  29.74,  30.06])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 21
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.plot(x, y, 'o')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 22,
       "text": [
        "[<matplotlib.lines.Line2D at 0x10788d410>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAXYAAAEACAYAAACnJV25AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10745f610>"
       ]
      }
     ],
     "prompt_number": 22
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The numpy routine `np.corrcoef` gives me the correlation *matrix* between `x` and `y`:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "corr_matrix = np.corrcoef(x, y)\n",
      "corr_matrix"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 23,
       "text": [
        "array([[ 1.        ,  0.47920906],\n",
        "       [ 0.47920906,  1.        ]])"
       ]
      }
     ],
     "prompt_number": 23
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "`corr_matrix[0, 0]` is the correlation of `x` with itself, `corr_matrix[1, 0]` (and `corr_matrix[0, 1]`) is the correlation of `x` with `y`."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "See the [correlation exercises](more_correlation.ipynb) for more on correlation."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Be careful - `np.correlate` is something else entirely."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## dot products"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "If I have two vectors $\\mathbf{a}$ with elements $a_0, a_1, ... a_{n-1}$, and $\\mathbf{b}$ with elements $b_0, b_1, ... b_{n-1}$ then the [dot product](https://en.wikipedia.org/wiki/Dot_product) is defined as:\n",
      "\n",
      "$$\n",
      "\\mathbf{a}\\cdot \\mathbf{b} = \\sum_{i=0}^{n-1} a_ib_i = a_0b_0 + a_1b_1 + \\cdots + a_{n-1}b_{n-1}\n",
      "$$\n",
      "\n",
      "In code:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "a = np.arange(5)\n",
      "b = np.arange(10, 15)\n",
      "np.dot(a, b)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 24,
       "text": [
        "130"
       ]
      }
     ],
     "prompt_number": 24
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# The same thing as\n",
      "np.sum(a * b)  # Elementwise multiplication"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 25,
       "text": [
        "130"
       ]
      }
     ],
     "prompt_number": 25
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Matrix multiplication operates by taking dot products of the rows of the first array (matrix) with the columns of the second.\n",
      "\n",
      "Let's say I have a matrix $\\mathbf{X}$, and $X_{i,:}$ is row $i$ in $\\mathbf{X}$.  I have a matrix $\\mathbf{Y}$, and $Y_{:,j}$ is column $j$ in $\\mathbf{Y}$.  The output matrix $\\mathbf{Z} = \\mathbf{X} \\mathbf{Y}$ has entry $Z_{i,j} = X_{i,:} \\cdot Y_{:, j}$.   We will see this often over the next few weeks."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "X = np.array([[0, 1, 2], [3, 4, 5]])\n",
      "X"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 26,
       "text": [
        "array([[0, 1, 2],\n",
        "       [3, 4, 5]])"
       ]
      }
     ],
     "prompt_number": 26
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Y = np.array([[7, 8], [9, 10], [11, 12]])\n",
      "Y"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 27,
       "text": [
        "array([[ 7,  8],\n",
        "       [ 9, 10],\n",
        "       [11, 12]])"
       ]
      }
     ],
     "prompt_number": 27
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "X.dot(Y)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 28,
       "text": [
        "array([[ 31,  34],\n",
        "       [112, 124]])"
       ]
      }
     ],
     "prompt_number": 28
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "X[0, :].dot(Y[:, 0])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 29,
       "text": [
        "31"
       ]
      }
     ],
     "prompt_number": 29
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "X[1, :].dot(Y[:, 0])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 30,
       "text": [
        "112"
       ]
      }
     ],
     "prompt_number": 30
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Convolution and the tail values"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's say I have a neural vector with a couple of spikes:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "times = np.arange(0, 60, 0.5)  # samples every 0.5 seconds\n",
      "neural_vector = np.zeros(times.shape)\n",
      "neural_vector[10] = 1  # At 5 seconds\n",
      "neural_vector[20] = 1  # At 10 seconds\n",
      "plt.plot(times, neural_vector)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 31,
       "text": [
        "[<matplotlib.lines.Line2D at 0x1079a55d0>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x1076b9ed0>"
       ]
      }
     ],
     "prompt_number": 31
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Then I have my HRF function, sampled every half second, to match:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def hrf(t):\n",
      "    \"A hemodynamic response function\"\n",
      "    values = t ** 8.6 * np.exp(-t / 0.547)\n",
      "    # Scale max to 1\n",
      "    return values / np.max(values)\n",
      "\n",
      "hrf_times = np.arange(0, 20, 0.5)\n",
      "hrf_samples = hrf(hrf_times)\n",
      "plt.plot(hrf_times, hrf_samples)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 32,
       "text": [
        "[<matplotlib.lines.Line2D at 0x107bbae50>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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P9Eecc7/OPl3U57PUof7ZyUtm1hF/8tLTJd5nYplZFzM7NHv/YGAS8E7zPyUF\neBqYmr0/Ffh1M9tKC7LBk3MR+owWxMwMeABY5py7Pe9bRX0+Sz5P3czO48B67A84524t6Q4TzMwG\n40fn4E8ce1TvZ3HM7HFgPNAL35/8LvAb4BfAIGAN8L+cc1tD1RgnjbyfNwMZfOvFAauBaaVatTVJ\nzOxM4FXgbQ60WGYCb1LE51MnH4mIJEjKViEXEUk2hbqISIIo1EVEEkShLiKSIAp1EZEEUaiLiCSI\nQl1EJEEU6iIiCfL/AQ1iaM09Y09yAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x1076a4310>"
       ]
      }
     ],
     "prompt_number": 32
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The input neural vector is length 120, and the HRF vector is length 40:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print(len(neural_vector))\n",
      "print(len(hrf_samples))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "120\n",
        "40\n"
       ]
      }
     ],
     "prompt_number": 33
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "For reasons that may be familiar to you now, when we convolve the neural vector with the hrf signal, we get an output that is length 120 + 40 - 1 = 159:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "hemodynamic_prediction = np.convolve(neural_vector, hrf_samples)\n",
      "len(hemodynamic_prediction)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 34,
       "text": [
        "159"
       ]
      }
     ],
     "prompt_number": 34
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This is because of the HRF falling off the end of the input vector.  The value at index 120 in the new vector refers to time 60, and value 121 refers to time 60.5 seconds.  To retain only the values in the new hemodynamic vector that refer to times up to (not including) 60s, we can just drop the last `len(hrf_signal) - 1` values:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "hemodynamic_for_60s = hemodynamic_prediction[:len(neural_vector)]\n",
      "plt.plot(times, neural_vector, label='neural vector')\n",
      "plt.plot(times, hemodynamic_for_60s, label='hemodynamic prediction')\n",
      "plt.legend()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 35,
       "text": [
        "<matplotlib.legend.Legend at 0x1079e3150>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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X6t7utDRQqFu9UhcRcpJzqGmusew1lVLDh21D3Yrul+ZjzSTFJhETFdPvPt66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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x1060340d0>"
       ]
      }
     ],
     "prompt_number": 35
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "##  Getting the time series from a voxel "
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's say I have a 4D image:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import nibabel as nib\n",
      "img = nib.load('ds114_sub009_t2r1.nii', mmap=False)\n",
      "data = img.get_data()\n",
      "data.shape"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 36,
       "text": [
        "(64, 64, 30, 173)"
       ]
      }
     ],
     "prompt_number": 36
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Each element slicing over the last dimension, is a 3D volume:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "vol0 = data[..., 0]\n",
      "vol1 = data[..., 1]\n",
      "vol1.shape"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 37,
       "text": [
        "(64, 64, 30)"
       ]
      }
     ],
     "prompt_number": 37
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The term *voxel* refers to a 3 dimensional pixel - a position in three dimensional space.\n",
      "\n",
      "These 3D volumes contain voxel values, for all of the 64, 64, 30 voxels, for a single time point.\n",
      "\n",
      "For example, let's say I was interested at the voxel at (i, j, k) coordinates (10, 23, 9):"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "vox_value = vol0[10, 23, 9]\n",
      "vox_value"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 38,
       "text": [
        "30"
       ]
      }
     ],
     "prompt_number": 38
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This value is the value for this voxel (voxel at 10, 23, 9) for this time point (first time point).  This would be the value for the same voxel at the second time point (volume index 1):"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "vol1[10, 23, 9]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 39,
       "text": [
        "17"
       ]
      }
     ],
     "prompt_number": 39
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Of course I could get the value for this voxel, at every time point, by doing a loop:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "vox_values = []\n",
      "for i in range(data.shape[-1]):\n",
      "    vox_values.append(data[10, 23, 9, i])\n",
      "len(vox_values)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 40,
       "text": [
        "173"
       ]
      }
     ],
     "prompt_number": 40
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "vox_values[:15]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 41,
       "text": [
        "[30, 17, 22, 30, 40, 38, 26, 45, 47, 27, 32, 38, 33, 35, 21]"
       ]
      }
     ],
     "prompt_number": 41
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "I can get the same effect by slicing over the last dimension:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "sliced_vox_values = data[10, 23, 9, :]\n",
      "sliced_vox_values[:15]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 42,
       "text": [
        "array([30, 17, 22, 30, 40, 38, 26, 45, 47, 27, 32, 38, 33, 35, 21], dtype=int16)"
       ]
      }
     ],
     "prompt_number": 42
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "np.all(vox_values == sliced_vox_values)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 43,
       "text": [
        "True"
       ]
      }
     ],
     "prompt_number": 43
    }
   ],
   "metadata": {}
  }
 ]
}