{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# HLMA 408: Quantiles, densités et fonctions de répartition \n",
    "\n",
    "***\n",
    "> __Auteur__: Joseph Salmon <joseph.salmon@umontpellier.fr>\n",
    "\n",
    "On va illustrer la notion de quantile sur des lois classiques en affichant la densité, la fonction de répartion et la fonction quantile associée.\n",
    "On observera ainsi l'impact du centrage (en: *centering*) et de la mise à l'échelle (en: *mise à l'échelle*).\n",
    "\n",
    "La liste exhaustive des lois disponibles sous `scipy` se trouve ici: https://docs.scipy.org/doc/scipy/reference/stats.html\n",
    "\n",
    "Ce notebook est inspiré des posts suivants:\n",
    "- https://medium.com/kapernikov/ipywidgets-with-matplotlib-93646718eb84\n",
    "- https://medium.com/@jdchipox/how-to-interact-with-jupyter-33a98686f24e"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy import stats \n",
    "import ipywidgets  # ipywidgets>=7.5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib widget"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "## Visualisation, cas continu: densités, fonctions de répartition et quantiles "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def make_box_layout():\n",
    "    return ipywidgets.Layout(\n",
    "        border='solid 1px black',\n",
    "        margin='0px 10px 10px 0px',\n",
    "        padding='5px 5px 5px 5px',\n",
    "    )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "def keep_no_param_distribution():\n",
    "    distributions = stats._continuous_distns._distn_names\n",
    "    distributions_0 = []\n",
    "    for i, name in enumerate(distributions):\n",
    "        dist = getattr(stats, name)\n",
    "        if not dist.shapes or len(dist.shapes)==0:\n",
    "            distributions_0.append(name)\n",
    "    distributions_0_val = [getattr(stats.distributions, string) for string in distributions_0 ]\n",
    "    distributions_0_dict = dict(zip(distributions_0, distributions_0_val))\n",
    "    return distributions_0_dict\n",
    "\n",
    "# inspired by: https://stackoverflow.com/questions/30453097/getting-the-parameter-names-of-scipy-stats-distributions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "distributions_0_dict = keep_no_param_distribution()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "solution": "hidden",
    "solution_first": true
   },
   "outputs": [],
   "source": [
    "class RandomWidgetQuantiles(ipywidgets.HBox):\n",
    "\n",
    "    def __init__(self):\n",
    "        super().__init__()\n",
    "        output = ipywidgets.Output()\n",
    "        output0 = ipywidgets.Output()\n",
    "        self.xranges = (-6, 6)  # Bornes d'observation\n",
    "        self.yranges = (0, 0.5)  # Bornes d'observation\n",
    "\n",
    "        self.x = np.linspace(self.xranges[0], self.xranges[1], num=400)\n",
    "        self.quantiles_value = 0.5\n",
    "\n",
    "        self.mu, self.sigma = 0, 1\n",
    "        self.distribution = distributions_0_dict['norm']\n",
    "        self.size = 5\n",
    "        self.initial_color = '#1a60e1'\n",
    "#         self.jitter = 0.10\n",
    "        self.params = dict(\n",
    "            color=self.initial_color, alpha=0.50, linewidth=0.2, edgecolor=\"black\"\n",
    "        )\n",
    "        self.fontsize = 4\n",
    "\n",
    "        with output:\n",
    "            self.fig, self.ax = plt.subplots(2, 2, sharey='row', sharex='col',\n",
    "                                             num='Quantiles et fonctions de répartition',\n",
    "                                             constrained_layout=True, figsize=(4, 4))\n",
    "        with output0:\n",
    "            self.fig0, self.ax0 = plt.subplots(1, 1,\n",
    "                                               num='Densité',\n",
    "                                               constrained_layout=True, figsize=(3, 1.5))\n",
    "\n",
    "        self.fig0.canvas.toolbar_visible = False\n",
    "        self.fig.canvas.toolbar_visible = False\n",
    "\n",
    "        self.pdf, = self.ax0.plot(self.x,\n",
    "                                  self.distribution.pdf(\n",
    "                                      self.x,\n",
    "                                      loc=self.mu, scale=self.sigma),\n",
    "                                  self.initial_color)\n",
    "        self.ax0.set_xlim(self.xranges)\n",
    "\n",
    "        self.area = self.ax0.fill_between(self.x, 0, self.distribution.pdf(\n",
    "            self.x, 0, 1), where=self.x <= self.quantiles_value, alpha=0.25, color=self.initial_color)\n",
    "\n",
    "        self.quantile_bar = self.ax0.axvline(\n",
    "            x=self.quantiles_value, c='k', ls=\"--\", lw=2)\n",
    "\n",
    "        self.ticks = np.arange(self.xranges[0], self.xranges[1] + 0.1, step=2.)\n",
    "\n",
    "        self.mytext = self.ax[0, 1].text(np.min(self.x), np.mean(self.x), \"Quantile d'ordre {:.2f} : \\n {:.2f}\".format(self.quantiles_value, self.distribution.ppf(\n",
    "            self.quantiles_value, loc=self.mu, scale=self.sigma)))\n",
    "        self.ax[0, 1].axis('Off')\n",
    "\n",
    "        self.diago, = self.ax[1, 0].plot(\n",
    "            [-0.1, 1.1], [-0.1, 1.1], self.initial_color)\n",
    "\n",
    "        self.ax[1, 0].set_xlim((-0.1, 1.1))\n",
    "        self.ax[1, 0].set_ylim((-0.1, 1.1))\n",
    "        self.ax[1, 0].set_xticks([0, 0.25, 0.5, 0.75, 1])\n",
    "        self.ax[1, 0].set_yticks([0, 0.25, 0.5, 0.75, 1])\n",
    "        self.points_q10, = self.ax[1, 0].plot(np.array([self.quantiles_value, self.quantiles_value, 1.1]), np.array([\n",
    "                                              1.1, self.quantiles_value, self.quantiles_value]), '--', color='k')\n",
    "\n",
    "        self.quantiles_range = self.distribution.cdf(\n",
    "            self.x,\n",
    "            loc=self.mu, scale=self.sigma)\n",
    "\n",
    "        self.cdf, = self.ax[1, 1].plot(self.x,\n",
    "                                       self.quantiles_range,\n",
    "                                       self.initial_color)\n",
    "\n",
    "        self.ax[1, 1].set_xlabel(r'Fonction de répartition')\n",
    "        self.ax[1, 1].set_xlim(self.xranges)\n",
    "        self.ax[1, 1].set_ylim((-0.1, 1.1))\n",
    "        self.ax[1, 1].set_xticks(self.ticks)\n",
    "\n",
    "        self.points_q11, = self.ax[1, 1].plot(np.array([self.xranges[0], self.distribution.ppf(\n",
    "            self.quantiles_value, loc=self.mu, scale=self.sigma)]), np.array([self.quantiles_value, self.quantiles_value]), '--', color='k')\n",
    "        self.points_cdf, = self.ax[1, 1].plot(self.distribution.ppf(\n",
    "            self.quantiles_value, loc=self.mu, scale=self.sigma), self.quantiles_value, 'o', color=self.initial_color)\n",
    "        \n",
    "        self.ax[0, 0].set_xlabel(r'Quantiles')\n",
    "        self.ax[0, 0].set_ylim(self.xranges)\n",
    "        self.ax[0, 0].set_xticks([0, 0.25, 0.5, 0.75, 1])\n",
    "\n",
    "        self.ppf, = self.ax[0, 0].plot(self.quantiles_range, self.x)\n",
    "        self.ax[0, 0].set_ylim(self.xranges)\n",
    "        self.ax[0, 0].set_yticks(self.ticks)\n",
    "        self.points_q00, = self.ax[0, 0].plot(np.array([self.quantiles_value, self.quantiles_value]), np.array([self.distribution.ppf(\n",
    "            self.quantiles_value, loc=self.mu, scale=self.sigma), self.xranges[0]]), '--', color='k')\n",
    "        self.points, = self.ax[0, 0].plot(self.quantiles_value, self.distribution.ppf(\n",
    "            self.quantiles_value, loc=self.mu, scale=self.sigma), 'o', color=self.initial_color)\n",
    "        self.fig.canvas.toolbar_position = 'bottom'\n",
    "\n",
    "        # define widgets\n",
    "        style = {'description_width': '100px'}\n",
    "        layout = {'width': '300px'}\n",
    "\n",
    "        mu_slider = ipywidgets.FloatSlider(\n",
    "            value=0, min=-2, max=2, step=0.1, description='$\\mu$',\n",
    "            style=style, layout=layout)\n",
    "        sigma_slider = ipywidgets.FloatSlider(\n",
    "            value=1, min=0.5, max=3, step=0.1, description='$\\sigma$',\n",
    "            style=style, layout=layout)\n",
    "        color_picker = ipywidgets.ColorPicker(\n",
    "            value=self.initial_color,\n",
    "            description='Couleur', style=style, layout=layout)\n",
    "        quantile_slider = ipywidgets.FloatSlider(\n",
    "            value=self.quantiles_value, min=0.001, max=0.999, step=0.001,\n",
    "            description=\"Quantile d'ordre \", style=style, layout=layout)\n",
    "        text_distribution = ipywidgets.Dropdown(\n",
    "            options=list(distributions_0_dict),\n",
    "            value='norm',\n",
    "            description='Distribution', style=style, layout=layout)\n",
    "        int_xrange_slider = ipywidgets.FloatRangeSlider(\n",
    "            value=self.xranges,\n",
    "            min=-5, max=5, step=0.1,\n",
    "            description=\"Zoom en x\", style=style, layout=layout)\n",
    "        int_yrange_slider_pdf = ipywidgets.FloatRangeSlider(\n",
    "            value=(self.yranges),\n",
    "            min=-1, max=3, step=0.1,\n",
    "            description=\"Zoom en y\", style=style, layout=layout)\n",
    "\n",
    "        controls = ipywidgets.VBox([\n",
    "            mu_slider,\n",
    "            sigma_slider,\n",
    "            text_distribution,\n",
    "            color_picker,\n",
    "            int_xrange_slider,\n",
    "            int_yrange_slider_pdf,\n",
    "            quantile_slider,\n",
    "            output0\n",
    "        ])\n",
    "\n",
    "        controls.layout = make_box_layout()\n",
    "        out_box = ipywidgets.Box([output])\n",
    "        output.layout = make_box_layout()\n",
    "\n",
    "        # A Afficher\n",
    "        mu_slider.observe(self.update_mu, 'value')\n",
    "        sigma_slider.observe(self.update_sigma, 'value')\n",
    "        color_picker.observe(self.line_color, 'value')\n",
    "        text_distribution.observe(self.update_text_distribution, 'value')\n",
    "        int_xrange_slider.observe(self.update_xrange_slider, 'value')\n",
    "        int_yrange_slider_pdf.observe(self.update_yrange_slider_pdf, 'value')\n",
    "        quantile_slider.observe(self.update_quantile_slider, 'value')\n",
    "\n",
    "        self.children = [controls, output]\n",
    "\n",
    "    def update_mu(self, change):\n",
    "        \"\"\"Evolution with the mu parameter.\"\"\"\n",
    "        self.mu = change.new\n",
    "        self.quantiles_range = self.distribution.cdf(\n",
    "            self.x,\n",
    "            loc=self.mu, scale=self.sigma)\n",
    "\n",
    "        self.pdf.set_ydata(self.distribution.pdf(\n",
    "            self.x, loc=self.mu, scale=self.sigma))\n",
    "        self.cdf.set_ydata(self.distribution.cdf(\n",
    "            self.x, loc=self.mu, scale=self.sigma))\n",
    "        self.ppf.set_xdata(self.quantiles_range)\n",
    "\n",
    "        self.points.set_ydata(self.distribution.ppf(\n",
    "            self.quantiles_value, loc=self.mu, scale=self.sigma))\n",
    "        self.points_cdf.set_xdata(self.distribution.ppf(\n",
    "            self.quantiles_value, loc=self.mu, scale=self.sigma))\n",
    "\n",
    "        self.points_q11.set_xdata(np.array([self.xranges[0], self.distribution.ppf(\n",
    "            self.quantiles_value, loc=self.mu, scale=self.sigma)]))\n",
    "        self.points_q11.set_ydata(\n",
    "            np.array([self.quantiles_value, self.quantiles_value]))\n",
    "        self.points_q00.set_xdata(\n",
    "            np.array([self.quantiles_value, self.quantiles_value]))\n",
    "        self.points_q00.set_ydata(np.array([self.xranges[0], self.distribution.ppf(\n",
    "            self.quantiles_value, loc=self.mu, scale=self.sigma)]))\n",
    "\n",
    "        self.area.remove()\n",
    "        self.quantile_bar.remove()\n",
    "\n",
    "        self.mytext.set_text(\"Quantile d'ordre {:.2f} : \\n {:.2f}\".format(self.quantiles_value, self.distribution.ppf(\n",
    "            self.quantiles_value, loc=self.mu, scale=self.sigma)))\n",
    "\n",
    "        self.area = self.ax0.fill_between(self.x, 0, self.distribution.pdf(\n",
    "            self.x, loc=self.mu, scale=self.sigma), where=self.x <= self.distribution.ppf(\n",
    "            self.quantiles_value, loc=self.mu, scale=self.sigma), alpha=0.25, color=self.initial_color)\n",
    "        self.quantile_bar = self.ax0.axvline(\n",
    "            x=self.distribution.ppf(\n",
    "                self.quantiles_value, loc=self.mu, scale=self.sigma), c='k', ls=\"--\", lw=2)\n",
    "        self.fig.canvas.draw()\n",
    "\n",
    "    def update_sigma(self, change):\n",
    "        \"\"\"Evolution with the sigma parameter.\"\"\"\n",
    "        self.sigma = change.new\n",
    "        self.quantiles_range = self.distribution.cdf(\n",
    "            self.x,\n",
    "            loc=self.mu, scale=self.sigma)\n",
    "\n",
    "        self.pdf.set_ydata(self.distribution.pdf(\n",
    "            self.x, loc=self.mu, scale=self.sigma))\n",
    "        self.cdf.set_ydata(self.distribution.cdf(\n",
    "            self.x, loc=self.mu, scale=self.sigma))\n",
    "        self.ppf.set_xdata(self.quantiles_range)\n",
    "\n",
    "        self.points.set_ydata(self.distribution.ppf(\n",
    "            self.quantiles_value, loc=self.mu, scale=self.sigma))\n",
    "        self.points_cdf.set_xdata(self.distribution.ppf(\n",
    "            self.quantiles_value, loc=self.mu, scale=self.sigma))\n",
    "        self.mytext.set_text(\"Quantile d'ordre {:.2f} : \\n {:.2f}\".format(self.quantiles_value, self.distribution.ppf(\n",
    "            self.quantiles_value, loc=self.mu, scale=self.sigma)))\n",
    "\n",
    "        self.points_q11.set_xdata(np.array([self.xranges[0], self.distribution.ppf(\n",
    "            self.quantiles_value, loc=self.mu, scale=self.sigma)]))\n",
    "        self.points_q11.set_ydata(\n",
    "            np.array([self.quantiles_value, self.quantiles_value]))\n",
    "        self.points_q00.set_xdata(\n",
    "            np.array([self.quantiles_value, self.quantiles_value]))\n",
    "        self.points_q00.set_ydata(np.array([self.xranges[0], self.distribution.ppf(\n",
    "            self.quantiles_value, loc=self.mu, scale=self.sigma)]))\n",
    "\n",
    "        self.area.remove()\n",
    "        self.quantile_bar.remove()\n",
    "\n",
    "        self.area = self.ax0.fill_between(self.x, 0, self.distribution.pdf(\n",
    "            self.x, loc=self.mu, scale=self.sigma), where=self.x <= self.distribution.ppf(\n",
    "            self.quantiles_value, loc=self.mu, scale=self.sigma), alpha=0.25, color=self.initial_color)\n",
    "        self.quantile_bar = self.ax0.axvline(\n",
    "            x=self.distribution.ppf(\n",
    "                self.quantiles_value, loc=self.mu, scale=self.sigma), c='k', ls=\"--\", lw=2)\n",
    "        self.fig.canvas.draw()\n",
    "\n",
    "    def update_quantile_slider(self, change):\n",
    "        \"\"\"Evolution with the quantile parameter.\"\"\"\n",
    "        self.quantiles_value = change.new\n",
    "\n",
    "        self.ppf.set_xdata(self.quantiles_range)\n",
    "\n",
    "        self.points.set_ydata(self.distribution.ppf(\n",
    "            self.quantiles_value, loc=self.mu, scale=self.sigma))\n",
    "        self.points.set_xdata(self.quantiles_value)\n",
    "\n",
    "        self.points_cdf.set_xdata(self.distribution.ppf(\n",
    "            self.quantiles_value, loc=self.mu, scale=self.sigma))\n",
    "        self.points_cdf.set_ydata(self.quantiles_value)\n",
    "\n",
    "        self.mytext.set_text(\"Quantile d'ordre {:.2f} : \\n {:.2f}\".format(self.quantiles_value, self.distribution.ppf(\n",
    "            self.quantiles_value, loc=self.mu, scale=self.sigma)))\n",
    "\n",
    "        self.points_q11.set_xdata(np.array([self.xranges[0], self.distribution.ppf(\n",
    "            self.quantiles_value, loc=self.mu, scale=self.sigma)]))\n",
    "        self.points_q11.set_ydata(\n",
    "            np.array([self.quantiles_value, self.quantiles_value]))\n",
    "        self.points_q00.set_xdata(\n",
    "            np.array([self.quantiles_value, self.quantiles_value]))\n",
    "        self.points_q00.set_ydata(np.array([self.xranges[0], self.distribution.ppf(\n",
    "            self.quantiles_value, loc=self.mu, scale=self.sigma)]))\n",
    "\n",
    "        self.points_q10.set_xdata(\n",
    "            np.array([self.quantiles_value, self.quantiles_value, 1.1]))\n",
    "        self.points_q10.set_ydata(\n",
    "            np.array([1.1, self.quantiles_value, self.quantiles_value]))\n",
    "\n",
    "        self.area.remove()\n",
    "        self.quantile_bar.remove()\n",
    "\n",
    "        self.area = self.ax0.fill_between(self.x, 0, self.distribution.pdf(\n",
    "            self.x, loc=self.mu, scale=self.sigma), where=self.x <= self.distribution.ppf(\n",
    "            self.quantiles_value, loc=self.mu, scale=self.sigma), alpha=0.25, color=self.initial_color)\n",
    "        self.quantile_bar = self.ax0.axvline(\n",
    "            x=self.distribution.ppf(\n",
    "                self.quantiles_value, loc=self.mu, scale=self.sigma), c='k', ls=\"--\", lw=2)\n",
    "        self.fig.canvas.draw()\n",
    "\n",
    "    def line_color(self, change):\n",
    "        self.initial_color = change.new\n",
    "        self.pdf.set_color(self.initial_color)\n",
    "        self.cdf.set_color(self.initial_color)\n",
    "        self.ppf.set_color(self.initial_color)\n",
    "        self.area.set_color(self.initial_color)\n",
    "        self.params['color'] = self.initial_color\n",
    "        self.points_cdf.set_color(self.initial_color)\n",
    "        self.points.set_color(self.initial_color)\n",
    "        self.diago.set_color(self.initial_color)\n",
    "        self.fig.canvas.draw()\n",
    "\n",
    "    def update_xrange_slider(self, change):\n",
    "        self.xranges = change.new\n",
    "        self.ax[1, 1].set_xlim(self.xranges)\n",
    "        self.ax[0, 0].set_ylim(self.xranges)\n",
    "        self.ax0.set_xlim(self.xranges)\n",
    "\n",
    "        self.mytext.set_position((self.xranges[0], np.mean(self.xranges)))\n",
    "        self.fig.canvas.draw()\n",
    "\n",
    "    def update_yrange_slider_pdf(self, change):\n",
    "        self.yranges_pdf = change.new\n",
    "        self.ax0.set_ylim(self.yranges_pdf)\n",
    "        self.fig.canvas.draw()\n",
    "\n",
    "    def update_text_distribution(self, change):\n",
    "        self.distribution = distributions_0_dict[change.new]\n",
    "\n",
    "        self.quantiles_range = self.distribution.cdf(\n",
    "            self.x,\n",
    "            loc=self.mu, scale=self.sigma)\n",
    "\n",
    "        self.pdf.set_ydata(self.distribution.pdf(\n",
    "            self.x, loc=self.mu, scale=self.sigma))\n",
    "        self.cdf.set_ydata(self.distribution.cdf(\n",
    "            self.x, loc=self.mu, scale=self.sigma))\n",
    "        self.ppf.set_xdata(self.quantiles_range)\n",
    "\n",
    "        self.points.set_ydata(self.distribution.ppf(\n",
    "            self.quantiles_value, loc=self.mu, scale=self.sigma))\n",
    "        self.points_cdf.set_xdata(self.distribution.ppf(\n",
    "            self.quantiles_value, loc=self.mu, scale=self.sigma))\n",
    "\n",
    "        self.mytext.set_text(\"Quantile d'ordre {:.2f} : \\n {:.2f}\".format(self.quantiles_value, self.distribution.ppf(\n",
    "            self.quantiles_value, loc=self.mu, scale=self.sigma)))\n",
    "\n",
    "        self.area.remove()\n",
    "        self.quantile_bar.remove()\n",
    "\n",
    "        self.area = self.ax0.fill_between(self.x, 0, self.distribution.pdf(\n",
    "            self.x, loc=self.mu, scale=self.sigma), where=self.x <= self.distribution.ppf(\n",
    "            self.quantiles_value, loc=self.mu, scale=self.sigma), alpha=0.25, color=self.initial_color)\n",
    "        self.quantile_bar = self.ax0.axvline(\n",
    "            x=self.distribution.ppf(\n",
    "                self.quantiles_value, loc=self.mu, scale=self.sigma), c='k', ls=\"--\", lw=2)\n",
    "        self.fig.canvas.draw()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "442f8158cf824febbdd940545f3db4ee",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "RandomWidgetQuantiles(children=(VBox(children=(FloatSlider(value=0.0, description='$\\\\mu$', layout=Layout(widt…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "RandomWidgetQuantiles()"
   ]
  }
 ],
 "metadata": {
  "hide_input": false,
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.5"
  },
  "latex_envs": {
   "LaTeX_envs_menu_present": true,
   "autoclose": false,
   "autocomplete": true,
   "bibliofile": "biblio.bib",
   "cite_by": "apalike",
   "current_citInitial": 1,
   "eqLabelWithNumbers": true,
   "eqNumInitial": 1,
   "hotkeys": {
    "equation": "Ctrl-E",
    "itemize": "Ctrl-I"
   },
   "labels_anchors": false,
   "latex_user_defs": false,
   "report_style_numbering": false,
   "user_envs_cfg": false
  },
  "toc": {
   "base_numbering": 1,
   "nav_menu": {},
   "number_sections": false,
   "sideBar": true,
   "skip_h1_title": false,
   "title_cell": "Table of Contents",
   "title_sidebar": "Contents",
   "toc_cell": false,
   "toc_position": {},
   "toc_section_display": true,
   "toc_window_display": true
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}