Stikeez Part 2.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Stikeez Part 2: With a little help from your friends"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> This [Jupyter](https://jupyter.org/) notebook is a supplement to the corresponding $++ [blog post](http://dollarpp.com). You can play around with it in two ways:\n",
    ">\n",
    ">  - Install Python, Jupyter and the rest of the required Python packages on your computer and run the notebook there. I would recommend a computing-oriented Python distribution such as [Anaconda](http://continuum.io/anaconda/) to get you up and running with the minimum fuss.\n",
    ">\n",
    ">  - Register for a cloud computing service such as [SageMathCloud](https://cloud.sagemath.com), upload this notebook to their site and run it \"in the cloud\". [Wakari](http://wakari.io) also seems like a good option, but their software is currently too old to open this notebook.\n",
    ">\n",
    "> Both options allow you to tweak the values and methods in this notebook and rerun the experiment to your heart's delight."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We start off with some common Python package imports for data analysis but also include [seaborn](http://stanford.edu/~mwaskom/software/seaborn/index.html), which makes some dashing plots on top of matplotlib."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import matplotlib as mpl\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import seaborn as sns"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We pick a minimal seaborn plotting style and pick the same font and text colour as in the blog theme ([Penscratch](http://penscratchdemo.wordpress.com)). This assumes that you installed the Roboto Slab Light font on your system, but it should fall back to the default font if not."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# That Penscratch grey\n",
    "textcolor = str(102 / 255.)\n",
    "sns.set(style='ticks', font='Roboto Slab',\n",
    "        rc={# Increase DPI to ensure sufficient pixels in output PNG\n",
    "            # Remember that notebook figures are actually saved images...\n",
    "            'savefig.dpi': 200,\n",
    "            # The Penscratch theme uses Roboto Slab Light as its base font\n",
    "            'font.weight': 'light', 'axes.labelweight': 'light',\n",
    "            # Change text and axes markings from black to base font colour\n",
    "            'text.color': textcolor, 'axes.edgecolor': textcolor,\n",
    "            'axes.labelcolor': textcolor, 'xtick.color': textcolor})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The heart of the calculation is a routine that simulates the quest to collect multiple sets of 24 Stikeez. It is assumed that the items / Stikeez are uniformly distributed and therefore equally likely to appear in a draw.\n",
    "\n",
    "To speed things up, we always draw a fixed number of items *(max_draws)* that should be more than enough to get the requisite number of complete sets. The final number of draws is expressed per set; in other words, it is what each person in the group of *num_sets* people have to contribute to ensure complete sets for them all."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def collect_items(num_items, num_sets, max_draws):\n",
    "    \"\"\"Collect items until several full sets have been completed.\n",
    "\n",
    "    Parameters\n",
    "    ----------\n",
    "    num_items : integer\n",
    "        Number of unique items that form a set\n",
    "    num_sets : integer\n",
    "        Number of complete sets to collect\n",
    "    max_draws : integer\n",
    "        Maximum number of draws before giving up\n",
    "\n",
    "    Returns\n",
    "    -------\n",
    "    draws_till_success : integer\n",
    "        Number of draws *per set* that resulted in complete sets\n",
    "\n",
    "    \"\"\"\n",
    "    # Generate all random draws at once, up to the maximum\n",
    "    items = np.random.randint(num_items, size=max_draws)\n",
    "    # Each row of indicator corresponds to a draw, and has a single 1\n",
    "    # in the column corresponding to the drawn item (and 0 elsewhere)\n",
    "    indicator = np.zeros((max_draws, num_items), dtype=np.int)\n",
    "    indicator[range(max_draws), items] = 1\n",
    "    # The n'th row of counts contains the tally of items after n draws\n",
    "    counts = indicator.cumsum(axis=0)\n",
    "    # Find the number of rows with no columns less than num_sets,\n",
    "    # i.e. the number of draws that had >= num_sets complete sets\n",
    "    draws_complete = (counts >= num_sets).all(axis=-1).sum()\n",
    "    # The total number of draws is the number of incomplete draws + 1\n",
    "    draws_till_success = max_draws - draws_complete + 1\n",
    "    # Get the number of draws per set (rounded up to nearest integer)\n",
    "    return int(np.ceil(draws_till_success / float(num_sets)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we set the main parameters of the experiment and collect the data. The number of trials *(num_trials)* should be as large as possible to ensure a good estimate of the true underlying statistics. We run a series of experiments, one per number of sets."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "num_items = 24\n",
    "num_sets = [1, 2, 10, 50, 1000]\n",
    "max_draws = [450, 600, 1000, 5000, 30000]\n",
    "num_trials = [100000, 100000, 40000, 20000, 10000]\n",
    "\n",
    "data = [[collect_items(num_items, num_sets[m], max_draws[m])\n",
    "         for n in range(num_trials[m])]\n",
    "        for m in range(len(num_sets))]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We get the median number of items required (50th percentile) for each data set."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[85.0, 65.0, 41.0, 31.0, 26.0]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "medians = [np.median(x) for x in data]\n",
    "medians"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We plot the histograms of each data set overlaid on separate axes on the same figure, in order to make them more visible."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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AZs6cGaeffnr8xV/8RUycODG+9a1vRUdHR/H5SZMmRUTEs88+m1SJVUuABQAAALCT5557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\nWgsAYGwSYAGJamtsKLmN8bR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VZWlsaPH39Nv2726dNHa9eu1dq1a9WrV696fXkcMmSIhgwZYrE+q1atUlpa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\nRK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10d3e9f50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, axes = plt.subplots(len(data), 1, figsize=(7.075, 4.867),\n",
    "                         sharex='all', subplot_kw={'yticks': []})\n",
    "sns.despine(left=True)\n",
    "xlim = (20, 120)\n",
    "# Default colour cycle (list of 6 colours)\n",
    "colors = sns.color_palette()\n",
    "# This transform has data coordinates for x and figure coordinates for y\n",
    "trans = mpl.transforms.blended_transform_factory(axes[0].transData,\n",
    "                                                 fig.transFigure)\n",
    "axes[-1].set_xticks([xlim[1]] + medians + [xlim[0]])\n",
    "axes[-1].set_xlabel('Number of Stikeez required (per set) to get '\n",
    "                    'multiple full sets')\n",
    "axes[len(axes) // 2].set_ylabel('Frequency')\n",
    "\n",
    "# Iterate over data sets\n",
    "for m, x in enumerate(data):\n",
    "    # Set up properties for current plot\n",
    "    ax, sets, color = axes[m], num_sets[m], colors[m % len(data)]\n",
    "    label = '%d %s' % (sets, 'set' if sets == 1 else 'sets')\n",
    "    bins = np.arange(xlim[0] - 0.5, xlim[1] + 1, 1)\n",
    "    # Plot solid white histogram to avoid histogram behind leaking through\n",
    "    sns.distplot(x, ax=ax, bins=bins, color=(1.0, 1.0, 1.0),\n",
    "                 kde=False, norm_hist=True,\n",
    "                 hist_kws={'clip_box': fig.bbox, 'alpha': 1.0})\n",
    "    # This is the actual histogram with transparent / subdued colour\n",
    "    sns.distplot(x, ax=ax, bins=bins, color=color,\n",
    "                 kde=False, norm_hist=True, label=label,\n",
    "                 hist_kws={'clip_box': fig.bbox})\n",
    "    # All axes are set to same limits (based on axis 0)\n",
    "    ax.set_xlim(*xlim)\n",
    "    ax.set_ylim(0, 1.5 * axes[0].get_ylim()[1])\n",
    "    # Bottom of current axes in figure coordinates\n",
    "    ybottom = ax.get_position().extents[1]\n",
    "    # Draw vertical line from bottom of axis to top of figure\n",
    "    ax.plot([medians[m], medians[m]], [ybottom, 1],\n",
    "            color=color, linestyle='--',\n",
    "            transform=trans, clip_on=False)\n",
    "    # Add legend to label histogram appropriately\n",
    "    ax.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally we save the image to a file to use it on the blog."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# fig.savefig('hist_multiset.png', bbox_inches='tight')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### License (BSD 3-Clause)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<small>\n",
    "\n",
    "Copyright (c) 2015, Ludwig Schwardt\n",
    "\n",
    "All rights reserved.\n",
    "\n",
    "Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met:\n",
    "\n",
    "1. Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer.\n",
    "\n",
    "2. Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution.\n",
    "\n",
    "3. Neither the name of the copyright holder nor the names of its contributors may be used to endorse or promote products derived from this software without specific prior written permission.\n",
    "\n",
    "THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS \"AS IS\" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.\n",
    "\n",
    "</small>"
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  }
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