{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Stat 593 : Robust Statistics\n",
    "## Introduction to robustness\n",
    "### *Joseph Salmon*\n",
    "\n",
    "This notebook reproduces the pictures for the course \"IntroRobustness\"\n",
    "\n",
    "- REM:\n",
    "  - you need TeX install on your machine (otherwise errors could appends)\n",
    "  - you need the file share_code somewhere available (for my architecture it was few level up)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# import packages\n",
    "import numpy as np\n",
    "from scipy import stats\n",
    "from statsmodels.nonparametric.kde import KDEUnivariate\n",
    "# dirty local imports:\n",
    "import sys\n",
    "sys.path.append(\"./../../../\")\n",
    "from share_code.utils import plt, sns, my_saving_display\n",
    "from joblib import Memory\n",
    "import scipy as sp\n",
    "\n",
    "# cachedir = './joblib_cache/'\n",
    "# memory = Memory(cachedir=cachedir, verbose=1)\n",
    "\n",
    "sns.set_palette(\"colorblind\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# to get images in the notebook\n",
    "%matplotlib inline "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Plot initialization"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "dirname = \"../prebuiltimages/\"\n",
    "imageformat = \".pdf\"  # should be .pdf or .png    \n",
    "# some colors I'll use\n",
    "brown = (0.64, 0.16, 0.16)\n",
    "purple = (148. / 255, 0, 211. / 255)\n",
    "plt.close(\"all\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Saving display function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "saving = False  # True"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Popular statistics:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "mu, sigma, nb_samples = 1.5, 4, 15\n",
    "\n",
    "# Random setting\n",
    "np.random.seed(seed=2)\n",
    "rgamma = np.random.gamma\n",
    "X = rgamma(mu, sigma, nb_samples)  # Generate sample from a Gamma distribution\n",
    "y = np.ones(nb_samples,)\n",
    "\n",
    "X[np.argmin(X)] = -3\n",
    "X[np.argmax(X)] = 30\n",
    "\n",
    "# Various statistics:\n",
    "meanX = np.mean(X)  # mean\n",
    "minX = np.min(X)  # min\n",
    "maxX = np.max(X)  # max\n",
    "medX = np.median(X)  # median\n",
    "MADX = np.median(np.abs(X - medX))  # mean absolute deviation\n",
    "s = np.std(X, ddof=1)  # unbiased standard deviation\n",
    "\n",
    "alpha_trim = 0.25\n",
    "tmeanX = stats.trim_mean(X, alpha_trim)  # trimmed mean (level: alpha_trim)\n",
    "\n",
    "xmin = minX - 20\n",
    "xmax = 35"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Empirical mean display:\n",
    "\n",
    "REM: meaning of shortcuts such as lw (linewidth), ls (linestyle) etc., are available at: https://matplotlib.org/api/lines_api.html"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "fig1, ax = plt.subplots(figsize=(10, 3))\n",
    "ax.set_ylim(0, 1.5)\n",
    "ax.set_xlim(xmin, xmax)\n",
    "ax.get_xaxis().tick_bottom()\n",
    "ax.axes.get_yaxis().set_visible(False)\n",
    "\n",
    "ax.spines['right'].set_color('none')\n",
    "ax.spines['top'].set_color('none')\n",
    "ax.spines['bottom'].set_position(('data', 0.5))\n",
    "ax.spines['left'].set_color('none')\n",
    "\n",
    "ax.scatter(X, y, c='black', s=300, marker='o', edgecolors=brown, lw='1')\n",
    "ax.plot([meanX, meanX], [0, 1.5], color=brown, lw=1.5, ls=\"--\")\n",
    "\n",
    "plt.xlabel(r'$x$', fontsize=18)\n",
    "plt.annotate(r'$\\overline{x}_n : \\mbox{empirical mean}$',\n",
    "             xy=(meanX, 0.4), xycoords='data', xytext=(+10, +30),\n",
    "             textcoords='offset points', fontsize=18, color=brown)\n",
    "\n",
    "plt.tight_layout()\n",
    "plt.show()\n",
    "\n",
    "if saving:\n",
    "    my_saving_display(fig1, dirname, \"GammaSampleMean\", imageformat)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Empirical median display:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "fig1, ax = plt.subplots(figsize=(10, 3))\n",
    "ax.set_ylim(0, 1.5)\n",
    "ax.set_xlim(xmin, xmax)\n",
    "ax.get_xaxis().tick_bottom()\n",
    "ax.axes.get_yaxis().set_visible(False)\n",
    "\n",
    "ax.spines['right'].set_color('none')\n",
    "ax.spines['top'].set_color('none')\n",
    "ax.spines['bottom'].set_position(('data', 0.5))\n",
    "ax.spines['left'].set_color('none')\n",
    "\n",
    "ax.scatter(X, y, c='black', s=300, marker='o', edgecolors=brown, lw='1')\n",
    "\n",
    "ax.plot([medX, medX], [0, 1.5], color=purple, lw=1.5, ls=\"--\")\n",
    "plt.xlabel(r'$x$', fontsize=18)\n",
    "plt.annotate(r'$\\rm{Med}_n(\\mathbb{x}): \\mbox{empirical median}$',\n",
    "             xy=(medX, 0.4), xycoords='data', xytext=(-210, +30),\n",
    "             textcoords='offset points', fontsize=18, color=purple)\n",
    "\n",
    "plt.tight_layout()\n",
    "plt.show()\n",
    "\n",
    "if saving:\n",
    "    my_saving_display(fig1, dirname, \"GammaSampleMediane\", imageformat)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#  Trimmed mean (level: alpha_trim) display:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "fig1, ax = plt.subplots(figsize=(10, 3))\n",
    "ax.set_ylim(0, 1.5)\n",
    "ax.set_xlim(xmin, xmax)\n",
    "ax.get_xaxis().tick_bottom()\n",
    "ax.axes.get_yaxis().set_visible(False)\n",
    "\n",
    "ax.spines['right'].set_color('none')\n",
    "ax.spines['top'].set_color('none')\n",
    "ax.spines['bottom'].set_position(('data', 0.5))\n",
    "ax.spines['left'].set_color('none')\n",
    "\n",
    "ax.scatter(X, y, c='black', s=300, marker='o', edgecolors=brown, lw='1')\n",
    "\n",
    "ax.plot([tmeanX, tmeanX], [0, 1.5], c='blue', lw=1.5, ls=\":\")\n",
    "plt.xlabel(r'$x$', fontsize=18)\n",
    "tt = \"$\\overline{x}_{n,%s} : \\mbox{trimmed mean}$\" % str(alpha_trim)\n",
    "plt.annotate(tt, xy=(tmeanX - 18, 0.4), xycoords='data', xytext=(+5, +30),\n",
    "             textcoords='offset points', fontsize=18, color='blue')\n",
    "\n",
    "plt.tight_layout()\n",
    "plt.show()\n",
    "\n",
    "if saving:\n",
    "    my_saving_display(fig1, dirname, \"GammaSampleTrimmed\", imageformat)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#  Empirical mean / median / trimmed mean :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "fig1, ax = plt.subplots(figsize=(10, 3))\n",
    "ax.set_ylim(0, 1.5)\n",
    "ax.set_xlim(xmin, xmax)\n",
    "ax.get_xaxis().tick_bottom()\n",
    "ax.axes.get_yaxis().set_visible(False)\n",
    "\n",
    "ax.spines['right'].set_color('none')\n",
    "ax.spines['top'].set_color('none')\n",
    "ax.spines['bottom'].set_position(('data', 0.5))\n",
    "ax.spines['left'].set_color('none')\n",
    "\n",
    "ax.scatter(X, y, c='black', s=300, marker='o', edgecolors=brown, lw='1')\n",
    "\n",
    "ax.plot([meanX, meanX], [0, 1.5], color=brown, lw=1.5, ls=\"--\")\n",
    "ax.plot([medX, medX], [0, 1.5], color=purple, lw=1.5, ls=\"--\")\n",
    "ax.plot([tmeanX, tmeanX], [0, 1.5], color='blue',\n",
    "        lw=1.5, ls=\":\")\n",
    "\n",
    "plt.xlabel(r'$x$', fontsize=18)\n",
    "plt.annotate(r'$\\rm{Med}_n(\\mathbb{x}): \\mbox{empirical median}$',\n",
    "             xy=(medX, 1), xycoords='data', xytext=(-210, +30),\n",
    "             textcoords='offset points', fontsize=18, color=purple)\n",
    "plt.annotate(r'$\\bar{x}_n : \\mbox{empirical mean}$', xy=(meanX, 0.4),\n",
    "             xycoords='data', xytext=(+10, +30), textcoords='offset points',\n",
    "             fontsize=18, color=brown)\n",
    "plt.annotate(tt, xy=(tmeanX - 18, 0.4), xycoords='data', xytext=(+5, +30),\n",
    "             textcoords='offset points', fontsize=18, color='blue')\n",
    "\n",
    "\n",
    "plt.tight_layout()\n",
    "plt.show()\n",
    "\n",
    "if saving:\n",
    "    my_saving_display(fig1, dirname, \"GammaSampleMedianeMean\", imageformat)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "X[np.argmax(X)] = 10\n",
    "\n",
    "fig1, ax = plt.subplots(figsize=(10, 3))\n",
    "ax.set_ylim(0, 1.5)\n",
    "ax.set_xlim(xmin, xmax)\n",
    "ax.get_xaxis().tick_bottom()\n",
    "ax.axes.get_yaxis().set_visible(False)\n",
    "\n",
    "ax.spines['right'].set_color('none')\n",
    "ax.spines['top'].set_color('none')\n",
    "ax.spines['bottom'].set_position(('data', 0.5))\n",
    "ax.spines['left'].set_color('none')\n",
    "\n",
    "ax.scatter(X, y, c='black', s=300, marker='o', edgecolors=brown, lw='1')\n",
    "\n",
    "ax.plot([np.mean(X), np.mean(X)], [0, 1.5], color=brown, lw=1.5, ls=\"--\")\n",
    "ax.plot([medX, medX], [0, 1.5], color=purple, lw=1.5, ls=\"--\")\n",
    "ax.plot([tmeanX, tmeanX], [0, 1.5], color='blue',\n",
    "        lw=1.5, ls=\":\")\n",
    "\n",
    "plt.xlabel(r'$x$', fontsize=18)\n",
    "plt.annotate(r'$\\rm{Med}_n(\\mathbb{x}): \\mbox{empirical median}$',\n",
    "             xy=(medX, 1), xycoords='data', xytext=(-210, +30),\n",
    "             textcoords='offset points', fontsize=18, color=purple)\n",
    "plt.annotate(r'$\\bar{x}_n : \\mbox{empirical mean}$', xy=(meanX, 0.4),\n",
    "             xycoords='data', xytext=(+10, +30), textcoords='offset points',\n",
    "             fontsize=18, color=brown)\n",
    "plt.annotate(tt, xy=(tmeanX - 18, 0.4), xycoords='data', xytext=(+5, +30),\n",
    "             textcoords='offset points', fontsize=18, color='blue')\n",
    "\n",
    "\n",
    "plt.tight_layout()\n",
    "plt.show()\n",
    "\n",
    "if saving:\n",
    "    my_saving_display(fig1, dirname, \"GammaSampleMedianeMeanBig10\", imageformat)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "X[np.argmax(X)] = 20\n",
    "\n",
    "fig1, ax = plt.subplots(figsize=(10, 3))\n",
    "ax.set_ylim(0, 1.5)\n",
    "ax.set_xlim(xmin, xmax)\n",
    "ax.get_xaxis().tick_bottom()\n",
    "ax.axes.get_yaxis().set_visible(False)\n",
    "\n",
    "ax.spines['right'].set_color('none')\n",
    "ax.spines['top'].set_color('none')\n",
    "ax.spines['bottom'].set_position(('data', 0.5))\n",
    "ax.spines['left'].set_color('none')\n",
    "\n",
    "ax.scatter(X, y, c='black', s=300, marker='o', edgecolors=brown, lw='1')\n",
    "\n",
    "ax.plot([np.mean(X), np.mean(X)], [0, 1.5], color=brown, lw=1.5, ls=\"--\")\n",
    "ax.plot([medX, medX], [0, 1.5], color=purple, lw=1.5, ls=\"--\")\n",
    "ax.plot([tmeanX, tmeanX], [0, 1.5], color='blue',\n",
    "        lw=1.5, ls=\":\")\n",
    "\n",
    "plt.xlabel(r'$x$', fontsize=18)\n",
    "plt.annotate(r'$\\rm{Med}_n(\\mathbb{x}): \\mbox{empirical median}$',\n",
    "             xy=(medX, 1), xycoords='data', xytext=(-210, +30),\n",
    "             textcoords='offset points', fontsize=18, color=purple)\n",
    "plt.annotate(r'$\\bar{x}_n : \\mbox{empirical mean}$', xy=(meanX, 0.4),\n",
    "             xycoords='data', xytext=(+10, +30), textcoords='offset points',\n",
    "             fontsize=18, color=brown)\n",
    "plt.annotate(tt, xy=(tmeanX - 18, 0.4), xycoords='data', xytext=(+5, +30),\n",
    "             textcoords='offset points', fontsize=18, color='blue')\n",
    "\n",
    "\n",
    "plt.tight_layout()\n",
    "plt.show()\n",
    "\n",
    "if saving:\n",
    "    my_saving_display(fig1, dirname, \"GammaSampleMedianeMeanBig20\", imageformat)\n",
    "\n",
    "X[np.argmax(X)] = 30"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Standard deviation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "fig1, ax = plt.subplots(figsize=(10, 3))\n",
    "ax.set_ylim(0, 1.5)\n",
    "ax.set_xlim(xmin, xmax)\n",
    "ax.get_xaxis().tick_bottom()\n",
    "ax.axes.get_yaxis().set_visible(False)\n",
    "\n",
    "ax.spines['right'].set_color('none')\n",
    "ax.spines['top'].set_color('none')\n",
    "ax.spines['bottom'].set_position(('data', 0.5))\n",
    "ax.spines['left'].set_color('none')\n",
    "\n",
    "ax.scatter(X, y, c='black', s=300, marker='o', edgecolors=brown, lw='1')\n",
    "ax.plot([meanX, meanX], [0, 1.5], color=brown, lw=1.5, ls=\"--\")\n",
    "\n",
    "plt.arrow(meanX, 1.2, -s, 0, fc=brown, ec=brown,\n",
    "          head_width=0.05, head_length=0.1, length_includes_head=True)\n",
    "plt.arrow(meanX - s, 1.2, s, 0, fc=brown, ec=brown,\n",
    "          head_width=0.05, head_length=0.1, length_includes_head=True)\n",
    "plt.arrow(meanX, 1.2, s, 0, fc=brown, ec=brown,\n",
    "          head_width=0.05, head_length=0.1, length_includes_head=True)\n",
    "plt.arrow(meanX + s, 1.2, -s, 0, fc=brown, ec=brown,\n",
    "          head_width=0.05, head_length=0.1, length_includes_head=True)\n",
    "\n",
    "plt.xlabel(r'$x$', fontsize=18)\n",
    "\n",
    "plt.annotate(r'$\\bar{x}_n : \\mbox{empirical mean}$',\n",
    "             xy=(meanX, 0.4), xycoords='data', xytext=(+10, +30),\n",
    "             textcoords='offset points', fontsize=18, color=brown)\n",
    "plt.annotate(r'$s_n$', xy=(meanX + s * (0.4), 1), xycoords='data',\n",
    "             xytext=(+10, +30), textcoords='offset points', fontsize=18,\n",
    "             color=brown)\n",
    "plt.annotate(r'$s_n$', xy=(meanX - s * (0.6), 1), xycoords='data',\n",
    "             xytext=(+10, +30), textcoords='offset points', fontsize=18,\n",
    "             color=brown)\n",
    "\n",
    "plt.tight_layout()\n",
    "plt.show()\n",
    "\n",
    "if saving:\n",
    "    my_saving_display(fig1, dirname, \"GammaSD\", imageformat)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Mean Absolute Deviation (MAD):\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "fig1, ax = plt.subplots(figsize=(10, 3))\n",
    "ax.set_ylim(0, 1.5)\n",
    "ax.set_xlim(xmin, xmax)\n",
    "ax.get_xaxis().tick_bottom()\n",
    "ax.axes.get_yaxis().set_visible(False)\n",
    "\n",
    "ax.spines['right'].set_color('none')\n",
    "ax.spines['top'].set_color('none')\n",
    "ax.spines['bottom'].set_position(('data', 0.5))\n",
    "ax.spines['left'].set_color('none')\n",
    "\n",
    "ax.scatter(X, y, c='black', s=300, marker='o', edgecolors=brown, lw='1')\n",
    "ax.plot([medX, medX], [0, 1.5], color=purple, lw=1.5, ls=\"--\")\n",
    "\n",
    "plt.arrow(medX, 1.2, -MADX, 0, fc=purple, ec=purple, head_width=0.05,\n",
    "          head_length=0.1, length_includes_head=True)\n",
    "plt.arrow(medX - MADX, 1.2, MADX, 0, fc=purple, ec=purple, head_width=0.05,\n",
    "          head_length=0.1, length_includes_head=True)\n",
    "plt.arrow(medX, 1.2, MADX, 0, fc=purple, ec=purple, head_width=0.05,\n",
    "          head_length=0.1, length_includes_head=True)\n",
    "plt.arrow(medX + MADX, 1.2, -MADX, 0, fc=purple, ec=purple, head_width=0.05,\n",
    "          head_length=0.1, length_includes_head=True)\n",
    "\n",
    "plt.xlabel(r'$x$', fontsize=18)\n",
    "\n",
    "plt.annotate(r'$\\rm{Med}_n(\\mathbb{x}): \\mbox{empirical median}$',\n",
    "             xy=(medX, 0.4), xycoords='data', xytext=(+10, +30),\n",
    "             textcoords='offset points', fontsize=18, color=purple)\n",
    "plt.annotate(r'$\\rm{MAD}_n(\\mathbb{x})$', xy=(medX + MADX * (0.1), 1),\n",
    "             xycoords='data', xytext=(+10, +30), textcoords='offset points',\n",
    "             fontsize=14, color=purple)\n",
    "plt.annotate(r'$\\rm{MAD}_n(\\mathbb{x})$', xy=(medX - MADX * (1.2), 1),\n",
    "             xycoords='data', xytext=(-55, +30), textcoords='offset points',\n",
    "             fontsize=14, color=purple)\n",
    "\n",
    "plt.tight_layout()\n",
    "plt.show()\n",
    "\n",
    "if saving:\n",
    "    my_saving_display(fig1, dirname, \"GammaMAD\", imageformat)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Newcomb's measurements for evaluating the speed of light"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is extracted from Maronna et al. 2006 (see also Stigler 1977)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "X = np.array([28, 26, 33, 24, 34, 29, 22, 24, 21, 25, -44, 27, 16, 40, -2, 30, 23, 29, 31, 19]) * 0.001 + 24.8\n",
    "nb_samples = X.shape[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Various statistics:\n",
    "y = np.ones(nb_samples,)\n",
    "meanX = np.mean(X)  # mean\n",
    "minX = np.min(X)  # min\n",
    "maxX = np.max(X)  # max\n",
    "medX = np.median(X)  # median\n",
    "MADX = np.median(np.abs(X - medX))  # mean absolute deviation\n",
    "MADNX = MADX / sp.stats.norm.ppf(0.75) # 0.6745 correponds to sp.stats.norm.ppf(0.75)\n",
    "s = np.std(X, ddof=1)  # unbiased standard deviation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "fig1, ax = plt.subplots(figsize=(10, 3))\n",
    "ax.set_ylim(0, 1.5)\n",
    "ax.set_xlim(minX-0.01, maxX+0.01)\n",
    "ax.get_xaxis().tick_bottom()\n",
    "ax.axes.get_yaxis().set_visible(False)\n",
    "\n",
    "ax.spines['right'].set_color('none')\n",
    "ax.spines['top'].set_color('none')\n",
    "ax.spines['bottom'].set_position(('data', 0.5))\n",
    "ax.spines['left'].set_color('none')\n",
    "plt.xlabel(r'Raw observations', fontsize=18)\n",
    "\n",
    "ax.scatter(X, y, c='black', s=300, marker='o', edgecolors=brown, lw='1')\n",
    "plt.tight_layout()\n",
    "plt.show()\n",
    "\n",
    "if saving:\n",
    "    my_saving_display(fig1, dirname, \"Newcombe_raw\", imageformat)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Investigation of outliers with t-statistics "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "t = (X - meanX) / s"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from matplotlib.ticker import MaxNLocator\n",
    "fig1, ax = plt.subplots(figsize=(10, 3))\n",
    "ax.set_ylim(-15, 15)\n",
    "ax.set_xlim(-1, t.shape[0])\n",
    "ax.xaxis.set_major_locator(MaxNLocator(integer=True))\n",
    "# ax.yaxis.set_major_locator(MaxNLocator(integer=True))\n",
    "\n",
    "plt.ylabel(r't-statistics', fontsize=18)\n",
    "plt.xlabel(r'index', fontsize=18)\n",
    "ax.axhline(y=3, color='k', lw=1.5, ls=\"--\")\n",
    "ax.axhline(y=-3, color='k', lw=1.5, ls=\"--\")\n",
    "\n",
    "ax.scatter(np.arange(t.shape[0]),t, c='black', s=100, marker='o', edgecolors=brown, lw='1')\n",
    "plt.tight_layout()\n",
    "plt.show()\n",
    "\n",
    "if saving:\n",
    "    my_saving_display(fig1, dirname, \"Newcombe_t\", imageformat)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Conclusion**: with the classical rule that \"outliers\" correspond to values with $|t_i|>3$, only the values t[10] = -3.72, qualifies as an outlier"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Investigation of outliers with \"medianized\" t-statistics"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "t_2 = (X - medX) / MADNX"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from matplotlib.ticker import MaxNLocator\n",
    "fig1, ax = plt.subplots(figsize=(10, 3))\n",
    "ax.set_ylim(-15, 15)\n",
    "ax.set_xlim(-1, t.shape[0])\n",
    "ax.xaxis.set_major_locator(MaxNLocator(integer=True))\n",
    "\n",
    "plt.ylabel(r't-statistics', fontsize=18)\n",
    "plt.xlabel(r'index', fontsize=18)\n",
    "ax.axhline(y=3, color='k', lw=1.5, ls=\"--\")\n",
    "ax.axhline(y=-3, color='k', lw=1.5, ls=\"--\")\n",
    "\n",
    "ax.scatter(np.arange(t_2.shape[0]),t_2, c='black', s=100, marker='o', edgecolors=brown, lw='1')\n",
    "plt.tight_layout()\n",
    "plt.show()\n",
    "\n",
    "if saving:\n",
    "    my_saving_display(fig1, dirname, \"Newcombe_t_2\", imageformat)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Conclusion**: with the robust variant rule the \"outliers\" correspond to values with $|t'_i|>3$, and now two values t[10] = -11.72 and t_2[14] = -4.64 qualify as outliers"
   ]
  }
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