{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# SD-TSIA204 : Linear models\n",
    "## Least squares definition: 2 variables and 3D visualization\n",
    "### *Joseph Salmon*\n",
    "\n",
    "This notebook reproduces the pictures for the course \"LeastSquare_Def\"\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import statsmodels.api as sm\n",
    "import matplotlib.pyplot as plt\n",
    "from matplotlib import rc\n",
    "import seaborn as sns\n",
    "from os import mkdir, path\n",
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "# interaction mode better for 3D\n",
    "%matplotlib notebook"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Plot initialization"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "dirname = \"../srcimages/\"\n",
    "if not path.exists(dirname):\n",
    "    mkdir(dirname)\n",
    "\n",
    "imageformat = '.pdf'\n",
    "rc('font', **{'family': 'sans-serif', 'sans-serif': ['Computer Modern Roman']})\n",
    "params = {'axes.labelsize': 12,\n",
    "          'font.size': 12,\n",
    "          'legend.fontsize': 12,\n",
    "          'xtick.labelsize': 10,\n",
    "          'ytick.labelsize': 10,\n",
    "          'text.usetex': True,\n",
    "          'figure.figsize': (8, 6)}\n",
    "plt.rcParams.update(params)\n",
    "plt.close(\"all\")\n",
    "\n",
    "# sns.set_context(\"poster\")\n",
    "sns.set_palette(\"colorblind\")\n",
    "sns.axes_style()\n",
    "sns.set_style({'legend.frameon': True})\n",
    "color_blind_list = sns.color_palette(\"colorblind\", 8)\n",
    "my_orange = color_blind_list[2]\n",
    "my_green = color_blind_list[1]\n",
    "\n",
    "###############################################################################\n",
    "# display function:\n",
    "\n",
    "saving = False\n",
    "\n",
    "\n",
    "def my_saving_display(fig, dirname, filename, imageformat):\n",
    "    \"\"\"\"Saving with personal function.\"\"\"\n",
    "    filename = filename.replace('.', 'pt')  # remove \".\" to avoid floats issues\n",
    "    if saving is True:\n",
    "        dirname + filename + imageformat\n",
    "        image_name = dirname + filename + imageformat\n",
    "        fig.savefig(image_name)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 3D case drawing"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "plt.close(\"all\")\n",
    "\n",
    "# Load data\n",
    "url = 'http://vincentarelbundock.github.io/Rdatasets/csv/datasets/trees.csv'\n",
    "dat3 = pd.read_csv(url)\n",
    "\n",
    "# Fit regression model\n",
    "X = dat3[['Girth', 'Height']]\n",
    "X = sm.add_constant(X)\n",
    "y = dat3['Volume']\n",
    "results = sm.OLS(y, X).fit().params\n",
    "\n",
    "\n",
    "XX = np.arange(8, 22, 0.5)\n",
    "YY = np.arange(64, 90, 0.5)\n",
    "xx, yy = np.meshgrid(XX, YY)\n",
    "zz = results[0] + results[1] * xx + results[2] * yy\n",
    "\n",
    "\n",
    "fig = plt.figure()\n",
    "ax = Axes3D(fig)\n",
    "\n",
    "ax.set_xlabel('Girth')\n",
    "ax.set_ylabel('Height')\n",
    "ax.set_zlabel('Volume')\n",
    "ax.set_zlim(5, 80)\n",
    "ax.plot(X['Girth'], X['Height'], y, 'o')\n",
    "ax.plot_wireframe(xx, yy, zz, rstride=10, cstride=10, alpha=0.3)\n",
    "ax.plot_surface(xx, yy, zz, alpha=0.3)\n",
    "my_saving_display(fig, dirname, \"tree_data_plot_regression\", imageformat)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Non trivial minima : 3D visualisation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "sns.set_style(\"white\")\n",
    "\n",
    "XX = np.arange(-1, 1, 0.05)\n",
    "YY = XX\n",
    "xx, yy = np.meshgrid(XX, YY)\n",
    "zz = (xx - yy) ** 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "fig = plt.figure()\n",
    "ax = Axes3D(fig)\n",
    "\n",
    "ax.view_init(elev=20., azim=50)\n",
    "ax.set_xlabel('$x$')\n",
    "ax.set_ylabel('$y$')\n",
    "ax.set_zlabel('$z$')\n",
    "\n",
    "plt.axis('off')\n",
    "\n",
    "ax.set_xticks([])\n",
    "ax.set_yticks([])\n",
    "ax.set_zticks([])\n",
    "\n",
    "surf = ax.plot_surface(xx, yy, zz, rstride=2, cstride=2,\n",
    "                       antialiased=False, alpha=0.5)\n",
    "my_saving_display(fig, dirname, \"CN0_2d_non_trivial1\", imageformat)\n",
    "\n",
    "ax.view_init(elev=20., azim=90)\n",
    "surf = ax.plot_surface(xx, yy, zz, rstride=2, cstride=2,\n",
    "                       antialiased=False, alpha=0.5)\n",
    "my_saving_display(fig, dirname, \"CN0_2d_non_trivial2\", imageformat)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "fig = plt.figure()\n",
    "ax = Axes3D(fig)\n",
    "\n",
    "ax.view_init(elev=20., azim=50)\n",
    "ax.set_xlabel('$x$')\n",
    "ax.set_ylabel('$y$')\n",
    "ax.set_zlabel('$z$')\n",
    "\n",
    "plt.axis('off')\n",
    "\n",
    "ax.set_xticks([])\n",
    "ax.set_yticks([])\n",
    "ax.set_zticks([])\n",
    "surf = ax.plot_surface(xx, yy, zz, rstride=2, cstride=2,\n",
    "                       antialiased=False, alpha=0.5)\n",
    "ax.view_init(elev=20., azim=130)\n",
    "surf = ax.plot_surface(xx, yy, zz, rstride=2, cstride=2,\n",
    "                       antialiased=False, alpha=0.5)\n",
    "my_saving_display(fig, dirname, \"CN0_2d_non_trivial3\", imageformat)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "fig = plt.figure()\n",
    "ax = Axes3D(fig)\n",
    "\n",
    "ax.view_init(elev=20., azim=50)\n",
    "ax.set_xlabel('$x$')\n",
    "ax.set_ylabel('$y$')\n",
    "ax.set_zlabel('$z$')\n",
    "\n",
    "plt.axis('off')\n",
    "\n",
    "ax.set_xticks([])\n",
    "ax.set_yticks([])\n",
    "ax.set_zticks([])\n",
    "surf = ax.plot_surface(xx, yy, zz, rstride=2, cstride=2,\n",
    "                       antialiased=False, alpha=0.5)\n",
    "ax.view_init(elev=20., azim=170)\n",
    "surf = ax.plot_surface(xx, yy, zz, rstride=2, cstride=2,\n",
    "                       antialiased=False, alpha=0.5)\n",
    "my_saving_display(fig, dirname, \"CN0_2d_non_trivial4\", imageformat)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "fig = plt.figure()\n",
    "ax = Axes3D(fig)\n",
    "\n",
    "ax.view_init(elev=20., azim=50)\n",
    "ax.set_xlabel('$x$')\n",
    "ax.set_ylabel('$y$')\n",
    "ax.set_zlabel('$z$')\n",
    "\n",
    "plt.axis('off')\n",
    "\n",
    "ax.set_xticks([])\n",
    "ax.set_yticks([])\n",
    "ax.set_zticks([])\n",
    "surf = ax.plot_surface(xx, yy, zz, rstride=2, cstride=2,\n",
    "                       antialiased=False, alpha=0.5)\n",
    "ax.view_init(elev=20., azim=210)\n",
    "surf = ax.plot_surface(xx, yy, zz, rstride=2, cstride=2,\n",
    "                       antialiased=False, alpha=0.5)\n",
    "my_saving_display(fig, dirname, \"CN0_2d_non_trivial5\", imageformat)"
   ]
  }
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