2015-10-28-Velocity-Amsterdam/Episode 4 -- Quantiles and Outliers.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Episode 4 -- Quantiles and Outliers"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* Classical summary statistics good for describing the body of the distribution\n",
    "* Need information about the tail of the distributions, e.g. for writing good SLAs\n",
    "* Determine outliers in a dataset"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import math\n",
    "import random\n",
    "import numpy as np\n",
    "import matplotlib as mpl\n",
    "from matplotlib import pyplot as plt\n",
    "import seaborn as sns\n",
    "mpl.rcParams['figure.figsize'] = (15,5)\n",
    "%matplotlib inline\n",
    "X = np.loadtxt(\"DataSets/RequestRates.csv\", delimiter=\",\")[:,1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Comulative Distribution Functions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The (empirical) cumulative distribution function of a dataset $X$, is defined as:\n",
    "\n",
    "$$ CDF(y) = \\# \\{ i \\, | \\, x_i \\leq y \\} / \\# X $$\n",
    "\n",
    "So CDF(y) = the ratio of samples that are lower than $y$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def CDF(y,X):\n",
    "    lower_count = 0\n",
    "    for x in X:\n",
    "        if x <= y: lower_count += 1\n",
    "    return float(lower_count) / len(X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.axes._subplots.AxesSubplot at 0x7fa5759cd0d0>"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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8HfhomaeX1FySJEnShBkSSZIkrbzjgAS4Ajih3lIkSZJWjGsSSd1hn6sJ7HM1wbh9nmTF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\nAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fa586381f10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "grid = np.linspace(0,3000,100)\n",
    "plt.figure(figsize=(20,5))\n",
    "plt.plot(grid, [CDF(y,X) for y in grid])\n",
    "sns.rugplot(X)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Properties:\n",
    "\n",
    "* $0 \\leq CDF(y) \\leq 1$\n",
    "* CDF is monotonically increasing"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example: Using CDF to monitor SLAs\n",
    "\n",
    "CDFs can be used to determine service levels.\n",
    "\n",
    "1. Measure response time latencies each minute over 1h\n",
    "2. Calculate `CDF(<max tolerable latency>)` for each 1h window\n",
    "3. Good service when `CDF = 1`\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<figure>\n",
    "<img src=\"img/example_cdf_sla.png\">\n",
    "<figcaption>CDF(0.05) over 1h windows for Twitter Ping Latencies</figcaption>\n",
    "</figure>\n",
    "\n",
    "* share: https://share.circonus.com/shared/graphs/65d896dd-2be3-4be9-a76d-6fec209358b1/ZrBa3u\n",
    "* graph: https://parlette.circonus.com/trending/graphs/view/65d896dd-2be3-4be9-a76d-6fec209358b1#no-set"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example: CDF for a histogram metric"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* Histograms are a much better representation of actual API usage\n",
    "* Calculate CDF over all requestst that arrived in 1h\n",
    "* Catch delays that affect only a small fraction of the requests\n",
    "\n",
    "<figure>\n",
    "<img src=\"img/example_histogram_IVP.png\">\n",
    "<figcaption>Histogram metric with CFD(3) over 1h windows</figcaption>\n",
    "</figure>\n",
    "\n",
    "* share:https://share.circonus.com/shared/graphs/0455bb94-1fbd-4128-9364-41d756d98933/iZntwd\n",
    "\n",
    "* graph: https://circonus-ops.circonus.com/trending/graphs/view/0455bb94-1fbd-4128-9364-41d756d98933#no-set"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Quantiles and Percentiles\n",
    "\n",
    "* Complement or Inverse to CDFs:\n",
    "  - CDF: The ratio of samples was below 100 was `<CDF>`\n",
    "  - Quantile: 90% of all queries where faster than `<quantile>`\n",
    "* Qunatile parameter is indpendent of value range"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Examples:\n",
    "\n",
    "* The minimum is a 0-quantile\n",
    "* A median is a 0.5-quantile\n",
    "* The maximum is a 1-quantile\n",
    "\n",
    "Special names:\n",
    "\n",
    "* Quartiles: $k/4$-quantiles\n",
    "* Percentiles: $k/100$-quantiles"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Median\n",
    "\n",
    "A _median value_ for $x_1, \\dots, x_n$ is number $m$ such that\n",
    "  \n",
    " $$ \\# \\{ i \\,|\\, x_i \\leq m \\} = \\# \\{ i \\,|\\, x_i \\geq m \\}. $$\n",
    "\n",
    "So the number of samples smaller than $m$ is equal to the number of samples larger than $m$.\n",
    "(Both should be roughly $n/2$).\n",
    "  \n",
    "Remark:\n",
    "\n",
    "* A Median always exists\n",
    "* Median is not unique\n",
    "* Can be computed in linear time\n",
    "* Not influenced by outliers (robust)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Computation of the median\n",
    "\n",
    "def median_range(X):\n",
    "    S = sorted(X)\n",
    "    n = len(X) # S[0] < ... < S[n-1]\n",
    "    lower_mid = int(math.floor((n-1)/2.))\n",
    "    upper_mid = int(math.ceil((n-1)/2.))\n",
    "    return (S[lower_mid], S[upper_mid])\n",
    "\n",
    "def plot_median_range(X):\n",
    "    plt.figure(figsize=(20,5))\n",
    "    sns.rugplot(X, color=\"green\", height=1)\n",
    "    sns.rugplot(median_range(X), color=\"red\", height=-1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fa55e7b8650>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_median_range([1,2,3,4])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Median range (112.2, 112.2)\n"
     ]
    },
    {
     "data": {
      "image/png": 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AkOAIAAAAgCHBEQAAAABDgiMAAAAAhgRHAAAAAAwJjgAAAAAYEhwBAAAAMCQ4AgAAAGBI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\nHAEAAAAwJDgCAAAAYEhwBAAAAMDQ/wf0hfqQKzaRUwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fa568d00550>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "print \"Median range\", median_range(X)\n",
    "plot_median_range(X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(112.2, 112.2)\n"
     ]
    }
   ],
   "source": [
    "# Median is little affected by outliers\n",
    "print median_range(list(X) + [20000])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fa5686b4710>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fa5686ef550>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Median is always a in the dataset\n",
    "# -> Mean value was in the middle of the graph\n",
    "plt.figure(figsize=(14,2))\n",
    "plot_median_range([1,2,0.4,1.2,100,110,112])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# General Definition of Quantiles\n",
    "Let $0\\leq q \\leq 1$ be a real number. A $q$-quantile for $X$ is a value $y$ such that, if\n",
    "\n",
    "$$ \\#\\{i \\,|\\, x_i \\leq y \\} \\geq q \\cdot n  $$\n",
    "\n",
    "and\n",
    "\n",
    "$$ \\#\\{i \\,|\\, x_i \\geq y \\} \\geq (1-q) \\cdot n $$\n",
    "\n",
    "Roughly speaking, $y$ divides $X$ in $q \\cdot n$ samples that are lower than $y$ and $(1-q) \\cdot n$ samples that are larger than $y$.\n",
    "\n",
    "Remarks:\n",
    "\n",
    "* Quantiles always exists\n",
    "* Non unique (like median)\n",
    "* Lot's of ways to choose a quantile function, i.e. interpolate between $s_a$ and $s_b$ cf.  \n",
    "  http://en.wikipedia.org/wiki/Quantile#Estimating_the_quantiles_of_a_population"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Computation of Quantiles\n",
    "\n",
    "def quantile_range(q,X):\n",
    "    S = sorted(X)\n",
    "    n = len(X)\n",
    "    upper_d = int(math.ceil(q * (n + 1)))\n",
    "    lower_d = int(math.floor(q * (n + 1)))\n",
    "    # edge cases\n",
    "    if upper_d == 0: upper_d = 1\n",
    "    if upper_d == n+1: upper_d = n\n",
    "    if lower_d == 0: lower_d = 1\n",
    "    if lower_d == n+1: lower_d = n\n",
    "    return (S[lower_d-1], S[upper_d-1])\n",
    "\n",
    "def quantile(q,X):\n",
    "    a,b = quantile_range(q,X)\n",
    "    return (a+b)/2 # mid point quantile\n",
    "\n",
    "def plot_quantile_range(q,X):\n",
    "    plt.figure(figsize=(20,5))\n",
    "    sns.rugplot(X, color=\"green\", height=1)\n",
    "    sns.rugplot(quantile_range(q,X), color=\"red\", height=-1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fa572de7510>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_quantile_range(.25,[1,2,3,4])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Example: Ping Latency Quantiles\n",
    "\n",
    "* Plot of (Twitter) ping latencies\n",
    "* With Quantiles as horizontal lines\n",
    "  - Q(1) = max\n",
    "  - Q(0.99)\n",
    "  - Q(0.95)\n",
    "\n",
    "<img src=\"img/Circonus_Quantiles.png\">"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Example: Quantiles for a Hisogram metric\n",
    "\n",
    "* Q(0.25) \n",
    "* Q(0.5) - Median\n",
    "* Q(0.75)\n",
    "\n",
    "<img src=\"img/Circonus_Histogram_Box.png\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# IQR and Outliers\n",
    "\n",
    "The interquartile range of a sample X is defined as:\n",
    "\n",
    "`IQR(X) = Q(0.75,X) - Q(0.25,X)`\n",
    "\n",
    "It is a robust measure for variance of the data. Good alternative to standard / mean deviation.\n",
    "\n",
    "**Def.** (Tukey, 1969) a k-outlier is a data point X which is either\n",
    "\n",
    "* larger than `Q(0.75) + k * IQR(X)` or\n",
    "* smaller than `Q(0.25) - k * IQR(X)`.\n",
    "\n",
    "An outlier (without k) is an 1.5-outlier."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def IQR(X): return quantile(0.75,X)-quantile(0.25,X)\n",
    "\n",
    "def outliers(X, k=1.5):\n",
    "    iqr = IQR(X)\n",
    "    lower_bound = quantile(0.25,X) - k * iqr\n",
    "    upper_bound = quantile(0.75,X) + k * iqr\n",
    "    out = []\n",
    "    for x in X:\n",
    "        if x < lower_bound: out.append(x)\n",
    "        if x > upper_bound: out.append(x)\n",
    "    return out"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "22\n"
     ]
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fa55afabfd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "X = np.genfromtxt(\"DataSets/WebLatency.csv\", delimiter=\",\")[:,2][:200]\n",
    "O = outliers(X,1.5)\n",
    "print len(O)\n",
    "\n",
    "plt.figure(figsize=(20,3))\n",
    "sns.rugplot(X, height=1, color=\"blue\")\n",
    "sns.rugplot([quantile(0.25,X), quantile(0.5,X),quantile(0.75,X)], color=\"green\", height=-1)\n",
    "if len(O) > 0: sns.rugplot(O, height=-1, color=\"red\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Tukey's Boxplots\n",
    "\n",
    "Show:\n",
    "\n",
    "* Median\n",
    "* Box around 0.25 and 0.75 Quantiles\n",
    "* \"whiskers\" from min to max\n",
    "* points for outliers\n",
    "\n",
    "Allows visual clues:\n",
    "\n",
    "* Where is the data concentracted?\n",
    "* How far is it spread?\n",
    "* How skew is the data?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA2YAAABXCAYAAABvG1ZrAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nAGhMMAMAAGhMMAMAAGhMMAMAAGjsf/fBadFoRcqHAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fa572fc6150>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Tukey Boxplots show outliers as 'points' outside the whiskers\n",
    "plt.figure(figsize=(15,1))\n",
    "sns.boxplot(X, vert=False).set_xlim(0,200)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA2MAAABXCAYAAACJMp0vAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nAKADYQwAAKADYQwAAKCD/wJUr7cXHPGnSwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fa5712e65d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Normal data\n",
    "N = [np.random.normal() for x in xrange(100)]\n",
    "\n",
    "# Add outlier\n",
    "NO = N + [7]\n",
    "\n",
    "plt.figure(figsize=(15,1))\n",
    "sns.boxplot(NO, vert=False).set_xlim(-10,10)\n",
    "plt.show()\n",
    "\n",
    "# Note, that IQR and Median are not affected by the outlier!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "Python [Root]",
   "language": "python",
   "name": "Python [Root]"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.11"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}