pmf_intro.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Introduction to PMFs\n",
    "====================\n",
    "\n",
    "Copyright 2015 Allen Downey\n",
    "\n",
    "License: [Creative Commons Attribution 4.0 International](http://creativecommons.org/licenses/by/4.0/)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# this line makes the code compatible with Python 2 and 3\n",
    "from __future__ import print_function, division\n",
    "\n",
    "# this line makes Jupyter show figures in the notebook\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Hist objects\n",
    "\n",
    "A histogram is a map from each possible value to the number of times it appears.  A map can be a mathematical function or, as in the examples below, a Python data structure that provides the ability to look up a value and get its probability.\n",
    "\n",
    "`Counter` is a data structure provided by Python; I am defining a new data structure, called a `Hist`, that has all the features of a Counter, plus a few more that I define."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import random\n",
    "import matplotlib.pyplot as plt\n",
    "from collections import Counter\n",
    "from itertools import izip\n",
    "\n",
    "class Hist(Counter):\n",
    "    \n",
    "    def __add__(self, other):\n",
    "        \"\"\"Returns the Pmf of the sum of elements from self and other.\"\"\"\n",
    "        return Hist(x + y for x, y in product(self.elements(), other.elements()))\n",
    "    \n",
    "    def choice(self):\n",
    "        \"\"\"Chooses a random element.\"\"\"\n",
    "        return random.choice(list(self.elements()))\n",
    "    \n",
    "    def plot(self, **options):\n",
    "        \"\"\"Plots the Pmf.\"\"\"\n",
    "        plt.bar(*zip(*self.items()), **options)\n",
    "        plt.xlabel('Values')\n",
    "        plt.ylabel('Counts')\n",
    "    \n",
    "    def ranks(self):\n",
    "        \"\"\"Returns ranks and counts as lists.\"\"\"\n",
    "        return izip(*enumerate(sorted(self.values(), reverse=True)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As an example, I'll make a Hist of the letters in my name:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Hist({'a': 1, 'e': 1, 'l': 2, 'n': 1})"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "hist = Hist('allen')\n",
    "hist"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can look up a letter and get the corresponding count:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "hist['l']"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Or loop through all the letters and print their counts:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "a 1\n",
      "e 1\n",
      "l 2\n",
      "n 1\n"
     ]
    }
   ],
   "source": [
    "for letter in hist:\n",
    "    print(letter, hist[letter])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`Counter` provides `most_common`, which makes a list of (element, count) pairs:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[('l', 2), ('a', 1), ('e', 1), ('n', 1)]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "hist.most_common()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here they are in a more readable form:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "l 2\n",
      "a 1\n",
      "e 1\n",
      "n 1\n"
     ]
    }
   ],
   "source": [
    "for letter, count in hist.most_common():\n",
    "    print(letter, count)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "I defined `choice`, which returns a random element from the Hist.  On average, 'l' should appear twice as often as the other letters."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "a\n",
      "a\n",
      "e\n",
      "a\n",
      "e\n",
      "n\n",
      "l\n",
      "l\n",
      "l\n",
      "l\n"
     ]
    }
   ],
   "source": [
    "for i in range(10):\n",
    "    print(hist.choice())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "One (perhaps surprising) thing you can use Hists for: checking whether two words are anagrams of each  other.  If two words are anagrams, they have the same Hist: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def is_anagram(word1, word2):\n",
    "    return Hist(word1) == Hist(word2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here's a simple test:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "is_anagram('allen', 'nella')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And my favorite anagram pair:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "is_anagram('tachymetric', 'mccarthyite')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And here's a false one, just to make sure:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "is_anagram('abcd', 'abccd')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So far the elements in the Hists have been letters (actually strings), but in statistics it is more common to work with numerical elements.  Here's a Hist that represents the possible outcomes of a six-sided die:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Hist({1: 1, 2: 1, 3: 1, 4: 1, 5: 1, 6: 1})"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d6 = Hist([1,2,3,4,5,6])\n",
    "d6"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`Hist` provides a plot function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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AhiPmAAAYjpgDAGA4Yg4AgOGIOQAAhiPmAAAYjpgDAGA4Yg4AgOGIOQAAhiPmAAAYjpgD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\nhiPmAAAYjpgDAGC4/wNfN1FpXeEFYgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6c18b719d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import seaborn as sns\n",
    "COLORS = sns.color_palette()\n",
    "\n",
    "d6.plot(color=COLORS[3])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`elements` returns an iterator"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<itertools.chain at 0x7f6c15eaa710>"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d6.elements()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Which is easier to see if you convert to a list:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1, 2, 3, 4, 5, 6]"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "list(d6.elements())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The product of two iterators is an iterator that enumerates all pairs:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<itertools.product at 0x7f6c160a1050>"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from itertools import product\n",
    "\n",
    "product(d6.elements(), d6.elements())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here are the elements of the product:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(1, 1),\n",
       " (1, 2),\n",
       " (1, 3),\n",
       " (1, 4),\n",
       " (1, 5),\n",
       " (1, 6),\n",
       " (2, 1),\n",
       " (2, 2),\n",
       " (2, 3),\n",
       " (2, 4),\n",
       " (2, 5),\n",
       " (2, 6),\n",
       " (3, 1),\n",
       " (3, 2),\n",
       " (3, 3),\n",
       " (3, 4),\n",
       " (3, 5),\n",
       " (3, 6),\n",
       " (4, 1),\n",
       " (4, 2),\n",
       " (4, 3),\n",
       " (4, 4),\n",
       " (4, 5),\n",
       " (4, 6),\n",
       " (5, 1),\n",
       " (5, 2),\n",
       " (5, 3),\n",
       " (5, 4),\n",
       " (5, 5),\n",
       " (5, 6),\n",
       " (6, 1),\n",
       " (6, 2),\n",
       " (6, 3),\n",
       " (6, 4),\n",
       " (6, 5),\n",
       " (6, 6)]"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "list(product(d6.elements(), d6.elements()))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can compute the sum of all pairs:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[2,\n",
       " 3,\n",
       " 4,\n",
       " 5,\n",
       " 6,\n",
       " 7,\n",
       " 3,\n",
       " 4,\n",
       " 5,\n",
       " 6,\n",
       " 7,\n",
       " 8,\n",
       " 4,\n",
       " 5,\n",
       " 6,\n",
       " 7,\n",
       " 8,\n",
       " 9,\n",
       " 5,\n",
       " 6,\n",
       " 7,\n",
       " 8,\n",
       " 9,\n",
       " 10,\n",
       " 6,\n",
       " 7,\n",
       " 8,\n",
       " 9,\n",
       " 10,\n",
       " 11,\n",
       " 7,\n",
       " 8,\n",
       " 9,\n",
       " 10,\n",
       " 11,\n",
       " 12]"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "list(x + y for x, y in product(d6.elements(), d6.elements()))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And finally make a Hist of the sums:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Hist({2: 1, 3: 2, 4: 3, 5: 4, 6: 5, 7: 6, 8: 5, 9: 4, 10: 3, 11: 2, 12: 1})"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Hist(x + y for x, y in product(d6.elements(), d6.elements()))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "But all of that is provided by `__add__`, which we can call using the `+` operator:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Hist({2: 1, 3: 2, 4: 3, 5: 4, 6: 5, 7: 6, 8: 5, 9: 4, 10: 3, 11: 2, 12: 1})"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "twice = d6 + d6\n",
    "twice"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can plot the histogram of outcomes from rolling two dice:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAe4AAAFmCAYAAACr9HnjAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nAAAMIdwAABhCuAEAMIRwAwBgyP8Dd7bJ8T4AxeYAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6c15eaa690>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "twice.plot(color=COLORS[2])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Or three dice:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Hist({3: 1,\n",
       "      4: 3,\n",
       "      5: 6,\n",
       "      6: 10,\n",
       "      7: 15,\n",
       "      8: 21,\n",
       "      9: 25,\n",
       "      10: 27,\n",
       "      11: 27,\n",
       "      12: 25,\n",
       "      13: 21,\n",
       "      14: 15,\n",
       "      15: 10,\n",
       "      16: 6,\n",
       "      17: 3,\n",
       "      18: 1})"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "thrice = twice + d6\n",
    "thrice"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Notice that this is looking more and more like a bell curve:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6c15ef7750>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "thrice.plot(color=COLORS[1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As the number of dice increases, the result converges to a normal distribution, also known as a Gaussian distribution."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Are first babies more likely to be late?\n",
    "----------------------------------------\n",
    "\n",
    "This is one of the first topics I wrote about in my blog, and still the most popular, with more than 100,000 page views:\n",
    "\n",
    "http://allendowney.blogspot.com/2011/02/are-first-babies-more-likely-to-be-late.html\n",
    "\n",
    "I used data from the National Survey of Family Growth (NSFG):\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>caseid</th>\n",
       "      <th>pregordr</th>\n",
       "      <th>howpreg_n</th>\n",
       "      <th>howpreg_p</th>\n",
       "      <th>moscurrp</th>\n",
       "      <th>nowprgdk</th>\n",
       "      <th>pregend1</th>\n",
       "      <th>pregend2</th>\n",
       "      <th>nbrnaliv</th>\n",
       "      <th>multbrth</th>\n",
       "      <th>...</th>\n",
       "      <th>poverty_i</th>\n",
       "      <th>laborfor_i</th>\n",
       "      <th>religion_i</th>\n",
       "      <th>metro_i</th>\n",
       "      <th>basewgt</th>\n",
       "      <th>adj_mod_basewgt</th>\n",
       "      <th>finalwgt</th>\n",
       "      <th>secu_p</th>\n",
       "      <th>sest</th>\n",
       "      <th>cmintvw</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>6.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>1.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>...</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>3410.389399</td>\n",
       "      <td>3869.349602</td>\n",
       "      <td>6448.271112</td>\n",
       "      <td>2</td>\n",
       "      <td>9</td>\n",
       "      <td>1231</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>1</td>\n",
       "      <td>2</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>6.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>1.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>...</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>3410.389399</td>\n",
       "      <td>3869.349602</td>\n",
       "      <td>6448.271112</td>\n",
       "      <td>2</td>\n",
       "      <td>9</td>\n",
       "      <td>1231</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>2</td>\n",
       "      <td>1</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>5.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>3.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>...</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>7226.301740</td>\n",
       "      <td>8567.549110</td>\n",
       "      <td>12999.542264</td>\n",
       "      <td>2</td>\n",
       "      <td>12</td>\n",
       "      <td>1231</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>2</td>\n",
       "      <td>2</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>6.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>1.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>...</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>7226.301740</td>\n",
       "      <td>8567.549110</td>\n",
       "      <td>12999.542264</td>\n",
       "      <td>2</td>\n",
       "      <td>12</td>\n",
       "      <td>1231</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>2</td>\n",
       "      <td>3</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>6.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>1.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>...</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>7226.301740</td>\n",
       "      <td>8567.549110</td>\n",
       "      <td>12999.542264</td>\n",
       "      <td>2</td>\n",
       "      <td>12</td>\n",
       "      <td>1231</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>5 rows × 243 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "   caseid  pregordr  howpreg_n  howpreg_p  moscurrp  nowprgdk  pregend1  \\\n",
       "0       1         1        NaN        NaN       NaN       NaN       6.0   \n",
       "1       1         2        NaN        NaN       NaN       NaN       6.0   \n",
       "2       2         1        NaN        NaN       NaN       NaN       5.0   \n",
       "3       2         2        NaN        NaN       NaN       NaN       6.0   \n",
       "4       2         3        NaN        NaN       NaN       NaN       6.0   \n",
       "\n",
       "   pregend2  nbrnaliv  multbrth   ...     poverty_i  laborfor_i  religion_i  \\\n",
       "0       NaN       1.0       NaN   ...             0           0           0   \n",
       "1       NaN       1.0       NaN   ...             0           0           0   \n",
       "2       NaN       3.0       5.0   ...             0           0           0   \n",
       "3       NaN       1.0       NaN   ...             0           0           0   \n",
       "4       NaN       1.0       NaN   ...             0           0           0   \n",
       "\n",
       "   metro_i      basewgt  adj_mod_basewgt      finalwgt  secu_p  sest  cmintvw  \n",
       "0        0  3410.389399      3869.349602   6448.271112       2     9     1231  \n",
       "1        0  3410.389399      3869.349602   6448.271112       2     9     1231  \n",
       "2        0  7226.301740      8567.549110  12999.542264       2    12     1231  \n",
       "3        0  7226.301740      8567.549110  12999.542264       2    12     1231  \n",
       "4        0  7226.301740      8567.549110  12999.542264       2    12     1231  \n",
       "\n",
       "[5 rows x 243 columns]"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import thinkstats2\n",
    "\n",
    "dct_file = '2002FemPreg.dct'\n",
    "dat_file = '2002FemPreg.dat.gz'\n",
    "\n",
    "dct = thinkstats2.ReadStataDct(dct_file)\n",
    "preg = dct.ReadFixedWidth(dat_file, compression='gzip')\n",
    "\n",
    "preg.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The variable `outcome` encodes the outcome of the pregnancy.  Outcome 1 is a live birth."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1    9148\n",
       "2    1862\n",
       "3     120\n",
       "4    1921\n",
       "5     190\n",
       "6     352\n",
       "Name: outcome, dtype: int64"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "preg.outcome.value_counts().sort_index()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`pregorder` is 1 for first pregnancies, 2 for others."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1     5033\n",
       "2     3766\n",
       "3     2334\n",
       "4     1224\n",
       "5      613\n",
       "6      308\n",
       "7      158\n",
       "8       78\n",
       "9       38\n",
       "10      17\n",
       "11       8\n",
       "12       5\n",
       "13       3\n",
       "14       3\n",
       "15       1\n",
       "16       1\n",
       "17       1\n",
       "18       1\n",
       "19       1\n",
       "Name: pregordr, dtype: int64"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "preg.pregordr.value_counts().sort_index()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "I selected live births, then split into first babies and others."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(4413, 4735)"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "live = preg[preg.outcome == 1]\n",
    "firsts = live[live.birthord == 1]\n",
    "others = live[live.birthord != 1]\n",
    "len(firsts), len(others)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The mean pregnancy lengths are slightly different:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(38.60095173351461, 38.52291446673706)"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "firsts.prglngth.mean(), others.prglngth.mean()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The difference is 0.078 weeks:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.07803726677754952"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff = firsts.prglngth.mean() - others.prglngth.mean()\n",
    "diff"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Which is 13 hours.  Note: the best units to report are often not the units you computed."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "13.11026081862832"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff * 7 * 24"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's see if we can visualize the difference in the histograms:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "first_hist = Hist(firsts.prglngth)\n",
    "other_hist = Hist(others.prglngth)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "I used some plotting options to put two bar charts side-by-side:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def plot_distributions(dist1, dist2):\n",
    "    dist1.plot(width=-0.45, align='edge', color=COLORS[3], label='firsts')\n",
    "    dist2.plot(width=0.45, align='edge', color=COLORS[4], label='others')\n",
    "    plt.xlim(33.5, 43.5)\n",
    "    plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here are the two histograms:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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cytUqhw9XnfX5W2LN4N+Lb2voy06Th/zw4cM1c+ZMvfvuu/J6vfrVr36lXr16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\nYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6c15d1bf50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_distributions(first_hist, other_hist)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Remember that the vertical axis is counts.  In this case, we are comparing counts with different totals, which might be misleading.\n",
    "\n",
    "An alternative is to compute a probability mass function (PMF), which divides the counts by the totals, yielding a map from each element to its probability.\n",
    "\n",
    "The probabilities are \"normalized\" to add up to 1.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "\n",
    "class Pmf(Hist):\n",
    "    \n",
    "    def normalize(self):\n",
    "        total = sum(self.values())\n",
    "        for element in self:\n",
    "            self[element] /= total\n",
    "        return self\n",
    "    \n",
    "    def plot_cumulative(self, **options):\n",
    "        xs, ps = zip(*sorted(self.iteritems()))\n",
    "        cs = np.cumsum(ps, dtype=np.float)\n",
    "        cs /= cs[-1]\n",
    "        plt.plot(xs, cs, **options)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can compare PMFs fairly."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "first_pmf = Pmf(firsts.prglngth).normalize()\n",
    "other_pmf = Pmf(others.prglngth).normalize()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Using PMFs, we see that some of the difference at 39 weeks was an artifact of the different samples sizes:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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s8HWZRqvtszx37lylpqbqxRdfVGxsbI17D+DULVmyRG3atJEkPf3000pMTNTK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\nAwBgIAIcAAADEeAAABiIAAcAwED/D8UBSTL827U6AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6c157d0a10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_distributions(first_pmf, other_pmf)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Even so, it is not easy to compare PMFs.  One more alternative is the cumulative mass function (CMF), which shows, for each $t$, the total probability up to and including $t$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f6c1558c750>"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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iWsLcPYliYAgTEcWZruvRz4TZEqYYGMJERHGmqX7oWihyrCgiZLMdZoucxFpR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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6c1558cfd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "first_pmf.plot_cumulative(linewidth=4, color=COLORS[3], label='firsts')\n",
    "other_pmf.plot_cumulative(linewidth=4, color=COLORS[4], label='others')\n",
    "plt.xlim(23.5, 44.5)\n",
    "plt.legend(loc='upper left')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The CDFs are similar up to week 38.  After that, first babies are more likely to be born late."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note: don't be afraid of thick lines.  Differences that are only visible with thin lines are unlikely to be real.\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Word Frequencies\n",
    "----------------\n",
    "\n",
    "Next topic: let's look at histograms of words, bigrams and trigrams.\n",
    "\n",
    "The following function reads lines from a file or URL and splits them into words:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def iterate_words(filename):\n",
    "    \"\"\"Read lines from a file and split them into words.\"\"\"\n",
    "    for line in open(filename):\n",
    "        for word in line.split():\n",
    "            yield word.strip()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here's an example using a book from Project Gutenberg.  `wc` is a histogram of word counts:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# FAIRY TALES\n",
    "# By The Brothers Grimm\n",
    "# http://www.gutenberg.org/cache/epub/2591/pg2591.txt'\n",
    "wc = Hist(iterate_words('pg2591.txt'))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here are the 20 most common words:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[('the', 6507),\n",
       " ('and', 5250),\n",
       " ('to', 2707),\n",
       " ('a', 1932),\n",
       " ('he', 1817),\n",
       " ('of', 1450),\n",
       " ('was', 1337),\n",
       " ('in', 1080),\n",
       " ('she', 1049),\n",
       " ('that', 1021),\n",
       " ('his', 1014),\n",
       " ('you', 941),\n",
       " ('it', 881),\n",
       " ('her', 880),\n",
       " ('had', 827),\n",
       " ('I', 755),\n",
       " ('they', 751),\n",
       " ('for', 721),\n",
       " ('with', 720),\n",
       " ('as', 718)]"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "wc.most_common(20)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Word frequencies in natural languages follow a predictable pattern called Zipf's law (which is an instance of Stigler's law, which is also an instance of Stigler's law).\n",
    "\n",
    "We can see the pattern by lining up the words in descending order of frequency and plotting their ranks (1st, 2nd, 3rd, ...) versus counts (6507, 5250, 2707):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x7f6c1548c750>"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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dO3bod7/7ne6+++7Y8ePHjyscDjteHAAAGH+fGw5uv/12zZo1S8uXL1dlZWXsuMvl0rRp\n0xwvDgAAjL8vvHxyTk6OfvrTn2pwcFADAwOx44ODg5oyZYqjxZ0NzCoAABAfW/dWePrpp1VfX69A\nIBC7pbJhGHrjjTccLQ4AAIw/W+Fgz549euutt+Tz+ZyuBwAAJJit3QqXX345wQAAgBRha+Tgoosu\n0t13362ZM2fK7XbHjj/88MOOFXb2sOoAAIB42AoHU6ZM0ezZs52uBQAAnANshYMlS5Y4XQcAADhH\n2AoHX/3qV2UYRuyxYRjKyMjQnj17HCvsbGFSAQCA+NgKB/v27Yt9PTo6qra2Nv3mN79xrCgAAJA4\ncd94KS0tTTfeeKNaWlqcqAcAACSYrZGDXbt2WR4fPHhQwWDQkYIAAEBi2QoHnZ2dlsfp6en6/ve/\n70hBAAAgsWyFg7Vr10qSBgYGZBiGJk+e7GhRAAAgcWyFg66uLq1cuVJHjhyRaZqaMmWK1q9fr+nT\npztdHwAAGGe2wsHGjRtVV1env/iLv5Ak/c///I++973v6cUXX3S0uLPBZC8jAABxsbVbweVyxYKB\ndOK6BydfRhkAACQP2+GgqalJQ0NDGhoa0u7duwkHAAAkKVvTCt/97nf1T//0T1qzZo1cLpeuuuoq\nPf30007XBgAAEsDWyEFLS4vS0tLU0dGhPXv2KBqN6s0333S6NgAAkAC2wkFDQ4Oee+652OOf/OQn\neu211xwrCgAAJI6tcDA2NmZZY+ByxX3VZQAAMEHYWnMwd+5c3XXXXZo5c6ai0ajeeustzZ8/3+na\nAABAAtgKB0uWLNH111+v//7v/5ZhGHriiSf0l3/5l07XdlZwmQMAAOJjKxxI0nXXXafrrrvOyVoA\nAMA5gMUDAADAgnAAAAAsbE8rnImjR4+qqqpK/f39Gh0d1YMPPqirrrpKK1askGmaysrK0rp16+T1\netXQ0KCtW7fK7XZr0aJFKi0tVSQSUVVVlXp7e+V2u7V27Vrl5eU5WTIAACnP0XDQ3Nys6dOn6957\n71Vvb6/+/u//XjNmzNDixYt1yy23qLa2VvX19brjjjtUV1en+vp6eTwelZaWav78+WpubtbkyZO1\nYcMGtbS0aOPGjaqtrXWyZAAAUp6j0woLFizQvffeK0nq7e3VxRdfrI6ODs2dO1eSVFJSotbWVvX0\n9KigoEB+v18+n08zZsxQZ2en2traNG/ePElSUVGRurq6nCwXAADI4ZGDj911113q6+vTv/7rv6qi\nokJer1eSlJmZqb6+PvX39ysQCMSeHwgEFAqFFA6HY8cNw5DL5VIkEpHHY79sk82MAADEZVzCwY4d\nO7Rv3z4tX75cpvnJL+uTvz7Z6Y5Ho9G439vtcikrKyPu16UC+mIfvbKHPtlDn+yjV4nhaDjYu3ev\nMjMzddFFF+mqq65SNBqV3+/X6Oio0tLSFAwGlZOTo+zsbIVCodjrgsGgCgsLlZ2drXA4rPz8fEUi\nkRMFxzFqIElj0ahCocGzel7JICsrg77YRK/soU/20Cf76JU9TgQoR9ccdHR06Cc/+YkkKRwOa3h4\nWLNnz1ZjY6MkqampSXPmzFFBQYHeffddDQ0N6ciRI+ru7tbMmTNVXFwce25zc7NmzZoVfxHMKgAA\nEBdHRw7+9m//VqtXr9bdd9+tY8eO6cknn9TVV1+tlStXaufOncrNzdXChQvldru1bNkyVVRUyOVy\nqbKyUunp6VqwYIFaWlpUVlYmn8+nmpoaJ8sFAACSDPN0E/wT1H27rTsaLkzzaMW1VyaomnMXw3X2\n0St76JM99Mk+emXPhJtWAAAAE0/Sh4OkGhYBAGAcJH04AAAA8SEcAAAAC8IBAACwIBwAAAALwgEA\nALAgHAAAAIukDwdsZQQAID5JHw4AAEB8CAcAAMAi+cMB8woAAMQl+cMBAACIC+EAAABYEA4AAIBF\n0ocDlhwAABCfpA8HAAAgPoQDAABgQTgAAAAWKRAOWHUAAEA8UiAcAACAeBAOAACARdKHAyYVAACI\nT9KHAwAAEB/CAQAAsCAcAAAAC8IBAACwIBwAAAALwgEAALBI+nBgspcRAIC4JH04AAAA8SEcAAAA\nC8IBAACwSPpwwJIDAADi43H6DdatW6euri6NjY3p/vvv1/Tp07VixQqZpqmsrCytW7dOXq9XDQ0N\n2rp1q9xutxYtWqTS0lJFIhFVVVWpt7dXbrdba9euVV5entMlAwCQ0hwNB3v27NF7772nHTt2aGBg\nQAsXLtQNN9ygxYsX65ZbblFtba3q6+t1xx13qK6uTvX19fJ4PCotLdX8+fPV3NysyZMna8OGDWpp\nadHGjRtVW1vrZMkAAKQ8R6cVrr/+em3atEmSdMEFF2h4eFgdHR2aO3euJKmkpEStra3q6elRQUGB\n/H6/fD6fZsyYoc7OTrW1tWnevHmSpKKiInV1dTlZLgAAkMPhwDAMTZo0SZK0a9cu3XTTTRoZGZHX\n65UkZWZmqq+vT/39/QoEArHXBQIBhUIhhcPh2HHDMORyuRSJRJwsGQCAlDcuCxJ/8YtfqL6+Xo89\n9pjMk65KZJ7mCkWnOx6NRh2pDwAAfMLxBYm/+tWv9KMf/UgvvPCC0tPT5ff7NTo6qrS0NAWDQeXk\n5Cg7O1uhUCj2mmAwqMLCQmVnZyscDis/Pz82YuDxxFeyy2UoKyvjrJ5TsqAv9tEre+iTPfTJPnqV\nGI6Gg6GhIa1fv17//u//royME/+DZ8+eraamJt12221qamrSnDlzVFBQoDVr1mhoaEiGYai7u1vV\n1dUaHBxUY2OjiouL1dzcrFmzZsVdw1jUVCg0eLZPbcLLysqgLzbRK3vokz30yT56ZY8TAcrRcLB7\n924NDAzokUcekWmaMgxDzzzzjKqrq/XSSy8pNzdXCxculNvt1rJly1RRUSGXy6XKykqlp6drwYIF\namlpUVlZmXw+n2pqapwsFwAASDLM003wT1D37bbuaDjf49aawq8kqJpzF4ncPnplD32yhz7ZR6/s\ncWLkIOmvkMg1EgEAiE8KhAMAABAPwgEAALAgHAAAAIukDwfJtdwSAADnJX04AAAA8SEcAAAAC8IB\nAACwIBwAAAALwgEAALAgHAAAAIukDwfsZAQAID5JHw4AAEB8CAcAAMCCcAAAACwIBwAAwIJwAAAA\nLAgHAADAIunDAVsZAQCIT9KHAwAAEB/CAQAAsCAcAAAAi+QPByw6AAAgLskfDgAAQFwIBwAAwCLp\nw4HJvAIAAHFJ+nAAAADiQzgAAAAWhAMAAGBBOAAAABaEAwAAYEE4AAAAFkkfDtjICABAfJI+HAAA\ngPgQDgAAgIXj4eC3v/2tbr75Zr344ouSpIMHD6q8vFyLFy/Wd77zHR0/flyS1NDQoNLSUt15553a\ntWuXJCkSiWj58uUqKytTeXm5PvjgA6fLBQAg5TkaDkZGRvT0009r9uzZsWObNm1SeXm5tm3bpssu\nu0z19fUaGRlRXV2dtmzZoq1bt2rLli06fPiwXn/9dU2ePFnbt2/XAw88oI0bN8Zdg8miAwAA4uJo\nOPD5fNq8ebOys7Njx9rb21VSUiJJKikpUWtrq3p6elRQUCC/3y+fz6cZM2aos7NTbW1tmjdvniSp\nqKhIXV1dcdcQMU2ZJAQAAGxzNBy4XC6lpaVZjo2MjMjr9UqSMjMz1dfXp/7+fgUCgdhzAoGAQqGQ\nwuFw7LhhGHK5XIpEInHXMUY4AADAtoQuSDzdv+hPdzwajZ7R+0SihAMAAOzyjPcb+v1+jY6OKi0t\nTcFgUDk5OcrOzlYoFIo9JxgMqrCwUNnZ2QqHw8rPz4+NGHg88Zc8OeDXBT7vWTuHZJGVlZHoEiYM\nemUPfbKHPtlHrxJj3MPB7Nmz1dTUpNtuu01NTU2aM2eOCgoKtGbNGg0NDckwDHV3d6u6ulqDg4Nq\nbGxUcXGxmpubNWvWrDN6z4OhQR0jHFhkZWUoFBpMdBkTAr2yhz7ZQ5/so1f2OBGgHA0He/fuVU1N\njXp7e+XxeNTU1KQNGzaoqqpKL730knJzc7Vw4UK53W4tW7ZMFRUVcrlcqqysVHp6uhYsWKCWlhaV\nlZXJ5/OppqbmjOpgWgEAAPsMM8mW8t+3+7M7GiqvvkwXn+9LQDXnLhK5ffTKHvpkD32yj17Z48TI\nQUpcIZGRAwAA7EuJcHD8DHc5AACQilIiHESSa+YEAABHpUY4YFoBAADbCAcAAMAiNcIB0woAANiW\nEuHgOCMHAADYlhLhIMJuBQAAbEuNcMC0AgAAtqVEOGBaAQAA+1IiHLBbAQAA+1IjHDCtAACAbakR\nDliQCACAbSkSDhg5AADArpQIB8eZVgAAwLaUCAeMHAAAYF9KhIOjY6w5AADArtQIB5GxRJcAAMCE\nkRLhYJiRAwAAbEuJcDDCyAEAALalRDgYjZosSgQAwKaUCAeSNDLG6AEAAHakTjiIsO4AAAA7Uigc\nMHIAAIAdqRMOmFYAAMCWlAkHw0wrAABgS8qEgyNMKwAAYEvKhIPW4ECiSwAAYEJImXDgMhJdAQAA\nE0PKhIOPjkU0eDyS6DIAADjnpUw4kKSG9/sSXQIAAOe8lAoHez86op7+wUSXAQDAOS2lwoEkvbT/\noJ7f+38KjhxLdCkAAJyTPIkuIBEODB/Tpnf/T16XoUv9k3Shz6uAz6vJaR6d53Ypw+uRYUg+t0t+\nj1uGpDS3Sy6DVY0AgOSXkuHgY8ejpvYPjkiDI1/4XLdhyOMy5JKU5nLpfI9LUUku48Qxl2Eo3evW\nmGnKkGQYhgyd2CVhyJBhSIYkj8vQeW73ibtE/unYicjxyXM+jiAf/4wTX39y3G0YchvGSTswjE/+\na5x85LNff/xz04+MaGjo2Cm+d6rXWev4vJ/rNiTXn87FaWf+FvG9cvJYRIcOHT2jcxrPOHnG73WW\nzqs3GtXhw8Nn/c3GtYfj8GZBmTp0aHicPxtn+G4J/sz3u6SBz/lMjVcPDcPQ1EleZXhT51fmOX+m\na9euVU9PjwzD0OrVqzV9+vSE1DFmmhobO3Hb55GxqA4dT0gZAIAEuTDNo1svz9L/m5Ke6FIcd06v\nOejo6ND777+vHTt26Omnn9b3vve9RJcEAEhRH41G9NPffahfH/wo0aU47pwOB21tbZo3b54k6c/+\n7M90+PBhHTly5HNfk+Y+p08JADDBNX3Qr76R0USX4ahz+jdpOBxWIBCIPb7wwgsVDoc/9zVPfO3/\nOV0WACCFjZmm3vljcm+LP6fDwaeZpvmFz8n2+/Stq/LGoRoAQKr6cDi5t8Of0wsSs7OzLSMFfX19\nysrK+sLXXfeVbF33lWwnSwMAIGmd0yMHxcXFampqkiTt3btXOTk5Ov/88xNcFQAAye2cHjkoLCzU\n1Vdfrbvuuktut1uPP/54oksCACDpGaadiXwAAJAyzulpBQAAMP4IBwAAwIJwAAAALM7pBYnxOFfu\nwZBo69atU1dXl8bGxnT//fdr+vTpWrFihUzTVFZWltatWyev16uGhgZt3bpVbrdbixYtUmlpqSKR\niKqqqtTb2yu32621a9cqLy+5rxlx7Ngx3XrrrVq6dKluuOEGenUKDQ0NeuGFF+TxePTQQw8pPz+f\nPn3K8PCwVq1apUOHDun48eNaunSppk2bRp9O8tvf/lZLly7V3/3d3+nuu+/WwYMHv3R/9u3bpyef\nfFIul0v5+fl64oknEn2aZ8Wne/Xhhx9q9erVikQi8nq9Wr9+vTIzM53tlZkE2tvbzW9961umaZrm\n//7v/5p33nlngitKjLfeesu8//77TdM0zY8++si86aabzKqqKrOxsdE0TdN89tlnzf/4j/8wh4eH\nzVtuucUcGhoyjx49at56663moUOHzFdeecV86qmnTNM0zV//+tfmI488krBzGS/PPvusWVpaar7y\nyitmVVWV2dTUFDtOr058jubPn28ODw+boVDIfOyxx+jTKWzbts189tlnTdM0zWAwaP7VX/0Vf/ZO\nMjw8bJaXl5uPPfaYuW3bNtM0zbPyOSovLzffffdd0zRN8x//8R/N//qv/0rA2Z1dp+rVqlWrYp+l\nbdu2mevXr3e8V0kxrXAm92BIRtdff702bdokSbrgggs0PDysjo4OzZ07V5JUUlKi1tZW9fT0qKCg\nQH6/Xz5POjxZAAAGGklEQVSfTzNmzFBnZ6elj0VFRerq6krYuYyH/fv3a//+/brxxhtlmqY6OjpU\nUlIiiV59rLW1VcXFxTrvvPM0depUPfXUU2pvb6dPn3LhhRfqo49O3Izn0KFDCgQC/Nk7ic/n0+bN\nm5Wd/cnF6b7M56i7u1vHjx/XBx98oKuvvlqSNHfuXLW2to7/yZ1lp+rVk08+qfnz50uSAoGABgYG\nHO9VUoSDM7kHQzIyDEOTJk2SJO3atUs33XSTRkZG5PV6JUmZmZnq6+tTf3+/pV+BQEChUMjSR8Mw\n5HK5FIlExv9Exskzzzyjqqqq2GN69VkHDhzQyMiIHnzwQS1evFhtbW06evQoffqUBQsWqLe3V/Pn\nz1d5eblWrlzJ5+kkLpdLaWlplmNfpj+GYSgcDmvKlCmfee5Ed6peTZo0SYZhKBqNavv27br11ls/\n83vvbPcqadYcnMxM8Us3/OIXv1B9fb1eeOGFWNqUTt+X0x2PRqOO1HcuePXVV1VYWKhLLrnklN+n\nVyeYpqmBgQE9//zzOnDggO655x5LD+jTCQ0NDcrNzdXmzZv1m9/8Ro8++qjl+/Tp88XbH9M0ZRhG\nSv1dH41GtWLFCs2ePVs33HCDXn/9dcv3z3avkmLk4EzvwZCMfvWrX+lHP/qRNm/erPT0dPn9fo2O\nnri1aDAYVE5OjrKzsy2p8eTjH/fx43+1eDxJmR/15ptv6o033tCdd96pXbt2qa6uTueffz69+pSp\nU6eqsLBQLpdLl156qfx+P5+pU+jq6tKcOXMkSfn5+QqFQjrvvPPo0+f4Mp8j80+LGAcGBizPPXko\nPtk8+uijuvLKK7VkyRJJcrxXSREOuAfDCUNDQ1q/fr1+8IMfKCMjQ5I0e/bsWG+ampo0Z84cFRQU\n6N1339XQ0JCOHDmi7u5uzZw5U8XFxWpsbJQkNTc3a9asWQk7F6fV1tbq5Zdf1ksvvaTS0lItXbpU\ns2fPjp0/vTqhuLhYe/bskWma+uijjzQ8PEyfTuHyyy/X22+/LenEVIzf71dRURF9+hxf9u8mt9ut\nr3zlK7H1GT//+c9jAS3ZNDQ0KC0tTd/+9rdjx6699lpHe5U0l09+9tln1d7eHrsHQ35+fqJLGnc7\nd+7Uc889pyuuuCI2lPTMM8+ourpao6Ojys3N1dq1a+V2u/Xzn/9cmzdvlsvlUnl5uf76r/9a0WhU\n1dXVev/99+Xz+VRTU6OcnJxEn5bjnnvuOeXl5elrX/uaVq5cSa8+ZefOnXr55ZdlGIaWLFmia665\nhj59yvDwsFavXq3+/n6NjY3pkUce0ZVXXqlVq1bRJ534R1tNTY16e3vl8XiUk5OjDRs2qKqq6kv1\n57333tPjjz8u0zR17bXXatWqVYk+1S/tVL364x//qLS0NPn9fhmGoWnTpunxxx93tFdJEw4AAMDZ\nkRTTCgAA4OwhHAAAAAvCAQAAsCAcAAAAC8IBAACwIBwAAAALwgEAx7S3t6usrCzRZQCIE+EAgKMM\nw0h0CQDilNwX7wbwpbW3t6uurk6TJk3SzJkz9dZbb2lsbEyDg4MqLy/XN77xDb3yyitqbW1VNBrV\n73//e+Xl5emf//mfLT9n3759WrFihTZv3pxUV/8DkhHhAMAX2rt3r5qbm3XgwAFNmzZNJSUlCoVC\nuu222/SNb3xDkvT222/rZz/7mdLS0nTzzTdr3759sdcHg0FVVVXpX/7lXwgGwARAOADwha688kpl\nZGQoKytLP/7xj/XjH/9Ybrdbhw4dij2noKAgdh/6iy66SAMDA3K5XBoaGtJ9992nRx55RFdccUWC\nzgBAPFhzAOALeb1eSdL3v/99XXHFFdq+fbt++MMfWp7jdrstjz++bcuBAwdUXFysf/u3fxufYgF8\naYQDALaFw2FNmzZNkvTaa6/J5XJpdHT0c1+Tn5+vVatW6aKLLlJdXd14lAngSyIcALBt8eLF2rRp\nk+69915lZGTohhtu0PLlyz+zI+FUOxSeeOIJNTQ06O233x6vcgGcIW7ZDAAALBg5AAAAFoQDAABg\nQTgAAAAWhAMAAGBBOAAAABaEAwAAYEE4AAAAFoQDAABg8f8B3rgqpCgMim0AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6c15614dd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ranks, counts = wc.ranks()\n",
    "plt.plot(ranks, counts, linewidth=10, color=COLORS[5])\n",
    "plt.xlabel('rank')\n",
    "plt.ylabel('count')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Huh.  Maybe that's not so clear after all.  The problem is that the counts drop off very quickly.  If we use the highest count to scale the figure, most of the other counts are indistinguishable from zero.\n",
    "\n",
    "Also, there are more than 10,000 words, but most of them appear only a few times, so we are wasting most of the space in the figure in a regime where nothing is happening.\n",
    "\n",
    "This kind of thing happens a lot.  A common way to deal with it is to compute the log of the quantities or to plot them on a log scale:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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ujDJrX5KeO9qiw9yuB6QVAh/IYPMKcnTPwqqofU8datZgIBjnigDECoEPZLjS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\nGMsAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6c1545c2d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ranks, counts = wc.ranks()\n",
    "plt.plot(ranks, counts, linewidth=4, color=COLORS[5])\n",
    "plt.xlabel('rank')\n",
    "plt.ylabel('count')\n",
    "plt.xscale('log')\n",
    "plt.yscale('log')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This (approximately) straight line is characteristic of Zipf's law.\n",
    "\n",
    "n-grams\n",
    "-------\n",
    "\n",
    "On to the next topic: bigrams and trigrams."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from itertools import tee\n",
    "\n",
    "def pairwise(iterator):\n",
    "    \"\"\"Iterates through a sequence in overlapping pairs.\n",
    "    \n",
    "    If the sequence is 1, 2, 3, the result is (1, 2), (2, 3), (3, 4), etc.\n",
    "    \"\"\"\n",
    "    a, b = tee(iterator)\n",
    "    next(b, None)\n",
    "    return izip(a, b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`bigrams` is the histogram of word pairs:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "bigrams = Hist(pairwise(iterate_words('pg2591.txt')))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And here are the 20 most common:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(('to', 'the'), 444),\n",
       " (('in', 'the'), 399),\n",
       " (('of', 'the'), 369),\n",
       " (('and', 'the'), 349),\n",
       " (('into', 'the'), 294),\n",
       " (('said', 'the'), 251),\n",
       " (('on', 'the'), 199),\n",
       " (('and', 'when'), 168),\n",
       " (('he', 'was'), 164),\n",
       " (('he', 'had'), 164),\n",
       " (('to', 'be'), 163),\n",
       " (('it', 'was'), 152),\n",
       " (('Then', 'the'), 151),\n",
       " (('I', 'will'), 149),\n",
       " (('that', 'he'), 143),\n",
       " (('at', 'the'), 142),\n",
       " (('came', 'to'), 138),\n",
       " (('and', 'he'), 135),\n",
       " (('she', 'was'), 129),\n",
       " (('all', 'the'), 125)]"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bigrams.most_common(20)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Similarly, we can iterate the trigrams:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def triplewise(iterator):\n",
    "    a, b, c = tee(iterator, 3)\n",
    "    next(b)\n",
    "    next(c)\n",
    "    next(c)\n",
    "    return izip(a, b, c)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And make a histogram:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "trigrams = Hist(triplewise(iterate_words('pg2591.txt')))\n",
    "\n",
    "# Uncomment this line to run the analysis with Elvis Presley lyrics\n",
    "#trigrams = Hist(triplewise(iterate_words('lyrics-elvis-presley.txt')))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here are the 20 most common:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(('came', 'to', 'the'), 65),\n",
       " (('and', 'when', 'he'), 50),\n",
       " (('out', 'of', 'the'), 50),\n",
       " (('said', 'to', 'the'), 34),\n",
       " (('he', 'came', 'to'), 33),\n",
       " (('and', 'when', 'she'), 33),\n",
       " (('went', 'into', 'the'), 32),\n",
       " (('went', 'to', 'the'), 31),\n",
       " (('and', 'said', 'to'), 31),\n",
       " (('came', 'to', 'a'), 30),\n",
       " (('one', 'of', 'the'), 30),\n",
       " (('and', 'as', 'he'), 29),\n",
       " (('they', 'came', 'to'), 29),\n",
       " (('he', 'did', 'not'), 28),\n",
       " (('there', 'was', 'a'), 28),\n",
       " (('that', 'he', 'had'), 28),\n",
       " (('and', 'I', 'will'), 27),\n",
       " (('that', 'it', 'was'), 25),\n",
       " (('and', 'at', 'last'), 24),\n",
       " (('and', 'when', 'the'), 24)]"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "trigrams.most_common(20)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And now for a little fun.  I'll make a dictionary that maps from each word pair to a Hist of the words that can follow."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from collections import defaultdict\n",
    "\n",
    "d = defaultdict(Hist)\n",
    "for a, b, c in trigrams:\n",
    "    d[a, b][c] += trigrams[a, b, c]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can look up a pair and see what might come next:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Hist({'came,': 1,\n",
       "      'fell': 1,\n",
       "      'might': 1,\n",
       "      'of': 2,\n",
       "      'on': 1,\n",
       "      'ran': 2,\n",
       "      'ran.': 1,\n",
       "      'streamed': 1,\n",
       "      'that': 1})"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d['the', 'blood']"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here are the most common words that follow \"into the\":"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[('forest', 15),\n",
       " ('forest,', 13),\n",
       " ('garden', 9),\n",
       " ('kitchen,', 8),\n",
       " ('cellar', 8),\n",
       " ('room,', 7),\n",
       " ('wide', 7),\n",
       " ('water,', 7),\n",
       " ('wood', 6),\n",
       " ('kitchen', 6)]"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d['into', 'the'].most_common(10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here are the words that follow \"said the\":"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[('old', 13),\n",
       " ('man,', 12),\n",
       " ('little', 10),\n",
       " ('fisherman,', 8),\n",
       " ('father,', 7),\n",
       " ('ass,', 6),\n",
       " ('tailor,', 5),\n",
       " ('wife,', 5),\n",
       " ('fish;', 5),\n",
       " ('other;', 5)]"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d['said', 'the'].most_common(10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`Hist` provides `choice`, which chooses a random word with probability proportional to count:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'wife,'"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d['said', 'the'].choice()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Given a prefix, we can choose a random suffix:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'sparrow;'"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "prefix = 'said', 'the'\n",
    "suffix = d[prefix].choice()\n",
    "suffix"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Then we can shift the words and compute the next prefix:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "('the', 'sparrow;')"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "prefix = prefix[1], suffix\n",
    "prefix"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Repeating this process, we can generate random new text that has the same correlation structure between words as the original:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "but killed his horse with his finger, and three times with the sack of pearls and precious stones for his wife, being awakened by the hand up to her, 'Take us out, or alas! we shall see what sort of game lies there.' And the king all that had been with the assistance they need, is critical to reaching Project Gutenberg-tm's goals and ensuring that the wife was not dead, but had been made to fetch this ring up from the gate,' as if it cost what it is said that it was the wolf, who lived in great comfort, "
     ]
    }
   ],
   "source": [
    "for i in range(100):\n",
    "    suffix = d[prefix].choice()\n",
    "    print(suffix, end=' ')\n",
    "    prefix = prefix[1], suffix"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "With a prefix of two words, we typically get text that flirts with sensibility."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": []
  }
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