{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "**Chapter 8 – Dimensionality Reduction**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "_This notebook contains all the sample code and solutions to the exercices in chapter 8._"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "# Setup"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "First, let's make sure this notebook works well in both python 2 and 3, import a few common modules, ensure MatplotLib plots figures inline and prepare a function to save the figures:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "# To support both python 2 and python 3\n",
    "from __future__ import division, print_function, unicode_literals\n",
    "\n",
    "# Common imports\n",
    "import numpy as np\n",
    "import numpy.random as rnd\n",
    "import os\n",
    "\n",
    "# to make this notebook's output stable across runs\n",
    "rnd.seed(42)\n",
    "\n",
    "# To plot pretty figures\n",
    "%matplotlib inline\n",
    "import matplotlib\n",
    "import matplotlib.pyplot as plt\n",
    "plt.rcParams['axes.labelsize'] = 14\n",
    "plt.rcParams['xtick.labelsize'] = 12\n",
    "plt.rcParams['ytick.labelsize'] = 12\n",
    "\n",
    "# Where to save the figures\n",
    "PROJECT_ROOT_DIR = \".\"\n",
    "CHAPTER_ID = \"dim_reduction\"\n",
    "\n",
    "def save_fig(fig_id, tight_layout=True):\n",
    "    path = os.path.join(PROJECT_ROOT_DIR, \"images\", CHAPTER_ID, fig_id + \".png\")\n",
    "    print(\"Saving figure\", fig_id)\n",
    "    if tight_layout:\n",
    "        plt.tight_layout()\n",
    "    plt.savefig(path, format='png', dpi=300)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "# Projection methods\n",
    "Build 3D dataset:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "rnd.seed(4)\n",
    "m = 60\n",
    "w1, w2 = 0.1, 0.3\n",
    "noise = 0.1\n",
    "\n",
    "angles = rnd.rand(m) * 3 * np.pi / 2 - 0.5\n",
    "X = np.empty((m, 3))\n",
    "X[:, 0] = np.cos(angles) + np.sin(angles)/2 + noise * rnd.randn(m) / 2\n",
    "X[:, 1] = np.sin(angles) * 0.7 + noise * rnd.randn(m) / 2\n",
    "X[:, 2] = X[:, 0] * w1 + X[:, 1] * w2 + noise * rnd.randn(m)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Mean normalize the data:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "X = X - X.mean(axis=0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Apply PCA to reduce to 2D."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "from sklearn.decomposition import PCA\n",
    "\n",
    "pca = PCA(n_components = 2)\n",
    "X2D = pca.fit_transform(X)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Recover the 3D points projected on the plane (PCA 2D subspace)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "X2D_inv = pca.inverse_transform(X2D)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Utility class to draw 3D arrows (copied from http://stackoverflow.com/questions/11140163)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "from matplotlib.patches import FancyArrowPatch\n",
    "from mpl_toolkits.mplot3d import proj3d\n",
    "\n",
    "class Arrow3D(FancyArrowPatch):\n",
    "    def __init__(self, xs, ys, zs, *args, **kwargs):\n",
    "        FancyArrowPatch.__init__(self, (0,0), (0,0), *args, **kwargs)\n",
    "        self._verts3d = xs, ys, zs\n",
    "\n",
    "    def draw(self, renderer):\n",
    "        xs3d, ys3d, zs3d = self._verts3d\n",
    "        xs, ys, zs = proj3d.proj_transform(xs3d, ys3d, zs3d, renderer.M)\n",
    "        self.set_positions((xs[0],ys[0]),(xs[1],ys[1]))\n",
    "        FancyArrowPatch.draw(self, renderer)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Express the plane as a function of x and y."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "axes = [-1.8, 1.8, -1.3, 1.3, -1.0, 1.0]\n",
    "\n",
    "x1s = np.linspace(axes[0], axes[1], 10)\n",
    "x2s = np.linspace(axes[2], axes[3], 10)\n",
    "x1, x2 = np.meshgrid(x1s, x2s)\n",
    "\n",
    "C = pca.components_\n",
    "R = C.T.dot(C)\n",
    "z = (R[0, 2] * x1 + R[1, 2] * x2) / (1 - R[2, 2])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Plot the 3D dataset, the plane and the projections on that plane."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "\n",
    "fig = plt.figure(figsize=(6, 3.8))\n",
    "ax = fig.add_subplot(111, projection='3d')\n",
    "\n",
    "X3D_above = X[X[:, 2] > X2D_inv[:, 2]]\n",
    "X3D_below = X[X[:, 2] <= X2D_inv[:, 2]]\n",
    "\n",
    "ax.plot(X3D_below[:, 0], X3D_below[:, 1], X3D_below[:, 2], \"bo\", alpha=0.5)\n",
    "\n",
    "ax.plot_surface(x1, x2, z, alpha=0.2, color=\"k\")\n",
    "np.linalg.norm(C, axis=0)\n",
    "ax.add_artist(Arrow3D([0, C[0, 0]],[0, C[0, 1]],[0, C[0, 2]], mutation_scale=15, lw=1, arrowstyle=\"-|>\", color=\"k\"))\n",
    "ax.add_artist(Arrow3D([0, C[1, 0]],[0, C[1, 1]],[0, C[1, 2]], mutation_scale=15, lw=1, arrowstyle=\"-|>\", color=\"k\"))\n",
    "ax.plot([0], [0], [0], \"k.\")\n",
    "\n",
    "for i in range(m):\n",
    "    if X[i, 2] > X2D_inv[i, 2]:\n",
    "        ax.plot([X[i][0], X2D_inv[i][0]], [X[i][1], X2D_inv[i][1]], [X[i][2], X2D_inv[i][2]], \"k-\")\n",
    "    else:\n",
    "        ax.plot([X[i][0], X2D_inv[i][0]], [X[i][1], X2D_inv[i][1]], [X[i][2], X2D_inv[i][2]], \"k-\", color=\"#505050\")\n",
    "    \n",
    "ax.plot(X2D_inv[:, 0], X2D_inv[:, 1], X2D_inv[:, 2], \"k+\")\n",
    "ax.plot(X2D_inv[:, 0], X2D_inv[:, 1], X2D_inv[:, 2], \"k.\")\n",
    "ax.plot(X3D_above[:, 0], X3D_above[:, 1], X3D_above[:, 2], \"bo\")\n",
    "ax.set_xlabel(\"$x_1$\", fontsize=18)\n",
    "ax.set_ylabel(\"$x_2$\", fontsize=18)\n",
    "ax.set_zlabel(\"$x_3$\", fontsize=18)\n",
    "ax.set_xlim(axes[0:2])\n",
    "ax.set_ylim(axes[2:4])\n",
    "ax.set_zlim(axes[4:6])\n",
    "\n",
    "save_fig(\"dataset_3d_plot\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "fig = plt.figure()\n",
    "ax = fig.add_subplot(111, aspect='equal')\n",
    "\n",
    "ax.plot(X2D[:, 0], X2D[:, 1], \"k+\")\n",
    "ax.plot(X2D[:, 0], X2D[:, 1], \"k.\")\n",
    "ax.plot([0], [0], \"ko\")\n",
    "ax.arrow(0, 0, 0, 1, head_width=0.05, length_includes_head=True, head_length=0.1, fc='k', ec='k')\n",
    "ax.arrow(0, 0, 1, 0, head_width=0.05, length_includes_head=True, head_length=0.1, fc='k', ec='k')\n",
    "ax.set_xlabel(\"$z_1$\", fontsize=18)\n",
    "ax.set_ylabel(\"$z_2$\", fontsize=18, rotation=0)\n",
    "ax.axis([-1.5, 1.3, -1.2, 1.2])\n",
    "ax.grid(True)\n",
    "save_fig(\"dataset_2d_plot\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "PCA using SVD decomposition"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "m, n = X.shape\n",
    "\n",
    "X_centered = X - X.mean(axis=0)\n",
    "U, s, V = np.linalg.svd(X_centered)\n",
    "c1 = V.T[:, 0]\n",
    "c2 = V.T[:, 1]\n",
    "\n",
    "S = np.zeros(X.shape)\n",
    "S[:n, :n] = np.diag(s)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "np.allclose(X, U.dot(S).dot(V))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "T = X.dot(V.T[:, :2])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "np.allclose(T, U.dot(S)[:, :2])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "from sklearn.decomposition import PCA\n",
    "pca = PCA(n_components = 2)\n",
    "X2D_p = pca.fit_transform(X)\n",
    "np.allclose(X2D_p, T)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "X3D_recover = T.dot(V[:2, :])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "np.allclose(X3D_recover, pca.inverse_transform(X2D_p))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "V"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "pca.components_"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "R = pca.components_.T.dot(pca.components_)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "S[:3]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "pca.explained_variance_ratio_"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "1 - pca.explained_variance_ratio_.sum()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "S[0,0]**2/(S**2).sum(), S[1,1]**2/(S**2).sum()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "np.sqrt((T[:, 1]**2).sum())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "# Manifold learning\n",
    "Swiss roll:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "from sklearn.datasets import make_swiss_roll\n",
    "X, t = make_swiss_roll(n_samples=1000, noise=0.2, random_state=42)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "axes = [-11.5, 14, -2, 23, -12, 15]\n",
    "\n",
    "fig = plt.figure(figsize=(6, 5))\n",
    "ax = fig.add_subplot(111, projection='3d')\n",
    "\n",
    "ax.scatter(X[:, 0], X[:, 1], X[:, 2], c=t, cmap=plt.cm.hot)\n",
    "ax.view_init(10, -70)\n",
    "ax.set_xlabel(\"$x_1$\", fontsize=18)\n",
    "ax.set_ylabel(\"$x_2$\", fontsize=18)\n",
    "ax.set_zlabel(\"$x_3$\", fontsize=18)\n",
    "ax.set_xlim(axes[0:2])\n",
    "ax.set_ylim(axes[2:4])\n",
    "ax.set_zlim(axes[4:6])\n",
    "\n",
    "save_fig(\"swiss_roll_plot\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "plt.figure(figsize=(11, 4))\n",
    "\n",
    "plt.subplot(121)\n",
    "plt.scatter(X[:, 0], X[:, 1], c=t, cmap=plt.cm.hot)\n",
    "plt.axis(axes[:4])\n",
    "plt.xlabel(\"$x_1$\", fontsize=18)\n",
    "plt.ylabel(\"$x_2$\", fontsize=18, rotation=0)\n",
    "plt.grid(True)\n",
    "\n",
    "plt.subplot(122)\n",
    "plt.scatter(t, X[:, 1], c=t, cmap=plt.cm.hot)\n",
    "plt.axis([4, 15, axes[2], axes[3]])\n",
    "plt.xlabel(\"$z_1$\", fontsize=18)\n",
    "plt.grid(True)\n",
    "\n",
    "save_fig(\"squished_swiss_roll_plot\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "from matplotlib import gridspec\n",
    "\n",
    "axes = [-11.5, 14, -2, 23, -12, 15]\n",
    "\n",
    "x2s = np.linspace(axes[2], axes[3], 10)\n",
    "x3s = np.linspace(axes[4], axes[5], 10)\n",
    "x2, x3 = np.meshgrid(x2s, x3s)\n",
    "\n",
    "fig = plt.figure(figsize=(6, 5))\n",
    "ax = plt.subplot(111, projection='3d')\n",
    "\n",
    "positive_class = X[:, 0] > 5\n",
    "X_pos = X[positive_class]\n",
    "X_neg = X[~positive_class]\n",
    "ax.view_init(10, -70)\n",
    "ax.plot(X_neg[:, 0], X_neg[:, 1], X_neg[:, 2], \"y^\")\n",
    "ax.plot_wireframe(5, x2, x3, alpha=0.5)\n",
    "ax.plot(X_pos[:, 0], X_pos[:, 1], X_pos[:, 2], \"gs\")\n",
    "ax.set_xlabel(\"$x_1$\", fontsize=18)\n",
    "ax.set_ylabel(\"$x_2$\", fontsize=18)\n",
    "ax.set_zlabel(\"$x_3$\", fontsize=18)\n",
    "ax.set_xlim(axes[0:2])\n",
    "ax.set_ylim(axes[2:4])\n",
    "ax.set_zlim(axes[4:6])\n",
    "\n",
    "save_fig(\"manifold_decision_boundary_plot1\")\n",
    "plt.show()\n",
    "\n",
    "fig = plt.figure(figsize=(5, 4))\n",
    "ax = plt.subplot(111)\n",
    "\n",
    "plt.plot(t[positive_class], X[positive_class, 1], \"gs\")\n",
    "plt.plot(t[~positive_class], X[~positive_class, 1], \"y^\")\n",
    "plt.axis([4, 15, axes[2], axes[3]])\n",
    "plt.xlabel(\"$z_1$\", fontsize=18)\n",
    "plt.ylabel(\"$z_2$\", fontsize=18, rotation=0)\n",
    "plt.grid(True)\n",
    "\n",
    "save_fig(\"manifold_decision_boundary_plot2\")\n",
    "plt.show()\n",
    "\n",
    "fig = plt.figure(figsize=(6, 5))\n",
    "ax = plt.subplot(111, projection='3d')\n",
    "\n",
    "positive_class = 2 * (t[:] - 4) > X[:, 1]\n",
    "X_pos = X[positive_class]\n",
    "X_neg = X[~positive_class]\n",
    "ax.view_init(10, -70)\n",
    "ax.plot(X_neg[:, 0], X_neg[:, 1], X_neg[:, 2], \"y^\")\n",
    "ax.plot(X_pos[:, 0], X_pos[:, 1], X_pos[:, 2], \"gs\")\n",
    "ax.set_xlabel(\"$x_1$\", fontsize=18)\n",
    "ax.set_ylabel(\"$x_2$\", fontsize=18)\n",
    "ax.set_zlabel(\"$x_3$\", fontsize=18)\n",
    "ax.set_xlim(axes[0:2])\n",
    "ax.set_ylim(axes[2:4])\n",
    "ax.set_zlim(axes[4:6])\n",
    "\n",
    "save_fig(\"manifold_decision_boundary_plot3\")\n",
    "plt.show()\n",
    "\n",
    "fig = plt.figure(figsize=(5, 4))\n",
    "ax = plt.subplot(111)\n",
    "\n",
    "plt.plot(t[positive_class], X[positive_class, 1], \"gs\")\n",
    "plt.plot(t[~positive_class], X[~positive_class, 1], \"y^\")\n",
    "plt.plot([4, 15], [0, 22], \"b-\", linewidth=2)\n",
    "plt.axis([4, 15, axes[2], axes[3]])\n",
    "plt.xlabel(\"$z_1$\", fontsize=18)\n",
    "plt.ylabel(\"$z_2$\", fontsize=18, rotation=0)\n",
    "plt.grid(True)\n",
    "\n",
    "save_fig(\"manifold_decision_boundary_plot4\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "# PCA"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "angle = np.pi / 5\n",
    "stretch = 5\n",
    "m = 200\n",
    "\n",
    "rnd.seed(3)\n",
    "X = rnd.randn(m, 2) / 10\n",
    "X = X.dot(np.array([[stretch, 0],[0, 1]])) # stretch\n",
    "X = X.dot([[np.cos(angle), np.sin(angle)], [-np.sin(angle), np.cos(angle)]]) # rotate\n",
    "\n",
    "u1 = np.array([np.cos(angle), np.sin(angle)])\n",
    "u2 = np.array([np.cos(angle - 2 * np.pi/6), np.sin(angle - 2 * np.pi/6)])\n",
    "u3 = np.array([np.cos(angle - np.pi/2), np.sin(angle - np.pi/2)])\n",
    "\n",
    "X_proj1 = X.dot(u1.reshape(-1, 1))\n",
    "X_proj2 = X.dot(u2.reshape(-1, 1))\n",
    "X_proj3 = X.dot(u3.reshape(-1, 1))\n",
    "\n",
    "plt.figure(figsize=(8,4))\n",
    "plt.subplot2grid((3,2), (0, 0), rowspan=3)\n",
    "plt.plot([-1.4, 1.4], [-1.4*u1[1]/u1[0], 1.4*u1[1]/u1[0]], \"k-\", linewidth=1)\n",
    "plt.plot([-1.4, 1.4], [-1.4*u2[1]/u2[0], 1.4*u2[1]/u2[0]], \"k--\", linewidth=1)\n",
    "plt.plot([-1.4, 1.4], [-1.4*u3[1]/u3[0], 1.4*u3[1]/u3[0]], \"k:\", linewidth=2)\n",
    "plt.plot(X[:, 0], X[:, 1], \"bo\", alpha=0.5)\n",
    "plt.axis([-1.4, 1.4, -1.4, 1.4])\n",
    "plt.arrow(0, 0, u1[0], u1[1], head_width=0.1, linewidth=5, length_includes_head=True, head_length=0.1, fc='k', ec='k')\n",
    "plt.arrow(0, 0, u3[0], u3[1], head_width=0.1, linewidth=5, length_includes_head=True, head_length=0.1, fc='k', ec='k')\n",
    "plt.text(u1[0] + 0.1, u1[1] - 0.05, r\"$\\mathbf{c_1}$\", fontsize=22)\n",
    "plt.text(u3[0] + 0.1, u3[1], r\"$\\mathbf{c_2}$\", fontsize=22)\n",
    "plt.xlabel(\"$x_1$\", fontsize=18)\n",
    "plt.ylabel(\"$x_2$\", fontsize=18, rotation=0)\n",
    "plt.grid(True)\n",
    "\n",
    "plt.subplot2grid((3,2), (0, 1))\n",
    "plt.plot([-2, 2], [0, 0], \"k-\", linewidth=1)\n",
    "plt.plot(X_proj1[:, 0], np.zeros(m), \"bo\", alpha=0.3)\n",
    "plt.gca().get_yaxis().set_ticks([])\n",
    "plt.gca().get_xaxis().set_ticklabels([])\n",
    "plt.axis([-2, 2, -1, 1])\n",
    "plt.grid(True)\n",
    "\n",
    "plt.subplot2grid((3,2), (1, 1))\n",
    "plt.plot([-2, 2], [0, 0], \"k--\", linewidth=1)\n",
    "plt.plot(X_proj2[:, 0], np.zeros(m), \"bo\", alpha=0.3)\n",
    "plt.gca().get_yaxis().set_ticks([])\n",
    "plt.gca().get_xaxis().set_ticklabels([])\n",
    "plt.axis([-2, 2, -1, 1])\n",
    "plt.grid(True)\n",
    "\n",
    "plt.subplot2grid((3,2), (2, 1))\n",
    "plt.plot([-2, 2], [0, 0], \"k:\", linewidth=2)\n",
    "plt.plot(X_proj3[:, 0], np.zeros(m), \"bo\", alpha=0.3)\n",
    "plt.gca().get_yaxis().set_ticks([])\n",
    "plt.axis([-2, 2, -1, 1])\n",
    "plt.xlabel(\"$z_1$\", fontsize=18)\n",
    "plt.grid(True)\n",
    "\n",
    "save_fig(\"pca_best_projection\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "# MNIST compression"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "from six.moves import urllib\n",
    "from sklearn.datasets import fetch_mldata\n",
    "try:\n",
    "    mnist = fetch_mldata('MNIST original')\n",
    "except urllib.error.HTTPError as ex:\n",
    "    print(\"Could not download MNIST data from mldata.org, trying alternative...\")\n",
    "\n",
    "    # Alternative method to load MNIST, if mldata.org is down\n",
    "    from scipy.io import loadmat\n",
    "    mnist_alternative_url = \"https://github.com/amplab/datascience-sp14/raw/master/lab7/mldata/mnist-original.mat\"\n",
    "    mnist_path = \"./mnist-original.mat\"\n",
    "    response = urllib.request.urlopen(mnist_alternative_url)\n",
    "    with open(mnist_path, \"wb\") as f:\n",
    "        content = response.read()\n",
    "        f.write(content)\n",
    "    mnist_raw = loadmat(mnist_path)\n",
    "    mnist = {\n",
    "        \"data\": mnist_raw[\"data\"].T,\n",
    "        \"target\": mnist_raw[\"label\"][0],\n",
    "        \"COL_NAMES\": [\"label\", \"data\"],\n",
    "        \"DESCR\": \"mldata.org dataset: mnist-original\",\n",
    "    }\n",
    "    print(\"Success!\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "from sklearn.model_selection import train_test_split\n",
    "\n",
    "X = mnist[\"data\"]\n",
    "y = mnist[\"target\"]\n",
    "\n",
    "X_train, X_test, y_train, y_test = train_test_split(X, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "X = X_train\n",
    "\n",
    "pca = PCA()\n",
    "pca.fit(X)\n",
    "d = np.argmax(np.cumsum(pca.explained_variance_ratio_) >= 0.95) + 1\n",
    "d"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "pca = PCA(n_components=0.95)\n",
    "X_reduced = pca.fit_transform(X)\n",
    "pca.n_components_"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "np.sum(pca.explained_variance_ratio_)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "X_mnist = X_train\n",
    "\n",
    "pca = PCA(n_components = 154)\n",
    "X_mnist_reduced = pca.fit_transform(X_mnist)\n",
    "X_mnist_recovered = pca.inverse_transform(X_mnist_reduced)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "def plot_digits(instances, images_per_row=5, **options):\n",
    "    size = 28\n",
    "    images_per_row = min(len(instances), images_per_row)\n",
    "    images = [instance.reshape(size,size) for instance in instances]\n",
    "    n_rows = (len(instances) - 1) // images_per_row + 1\n",
    "    row_images = []\n",
    "    n_empty = n_rows * images_per_row - len(instances)\n",
    "    images.append(np.zeros((size, size * n_empty)))\n",
    "    for row in range(n_rows):\n",
    "        rimages = images[row * images_per_row : (row + 1) * images_per_row]\n",
    "        row_images.append(np.concatenate(rimages, axis=1))\n",
    "    image = np.concatenate(row_images, axis=0)\n",
    "    plt.imshow(image, cmap = matplotlib.cm.binary, **options)\n",
    "    plt.axis(\"off\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "plt.figure(figsize=(7, 4))\n",
    "plt.subplot(121)\n",
    "plot_digits(X_mnist[::2100])\n",
    "plt.title(\"Original\", fontsize=16)\n",
    "plt.subplot(122)\n",
    "plot_digits(X_mnist_recovered[::2100])\n",
    "plt.title(\"Compressed\", fontsize=16)\n",
    "\n",
    "save_fig(\"mnist_compression_plot\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "from sklearn.decomposition import IncrementalPCA\n",
    "\n",
    "n_batches = 100\n",
    "inc_pca = IncrementalPCA(n_components=154)\n",
    "for X_batch in np.array_split(X_mnist, n_batches):\n",
    "    print(\".\", end=\"\")\n",
    "    inc_pca.partial_fit(X_batch)\n",
    "\n",
    "X_mnist_reduced_inc = inc_pca.transform(X_mnist)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "X_mnist_recovered_inc = inc_pca.inverse_transform(X_mnist_reduced_inc)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "plt.figure(figsize=(7, 4))\n",
    "plt.subplot(121)\n",
    "plot_digits(X_mnist[::2100])\n",
    "plt.subplot(122)\n",
    "plot_digits(X_mnist_recovered_inc[::2100])\n",
    "plt.tight_layout()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "np.allclose(pca.mean_, inc_pca.mean_)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "np.allclose(X_mnist_reduced, X_mnist_reduced_inc)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "filename = \"my_mnist.data\"\n",
    "\n",
    "X_mm = np.memmap(filename, dtype='float32', mode='write', shape=X_mnist.shape)\n",
    "X_mm[:] = X_mnist\n",
    "del X_mm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "X_mm = np.memmap(filename, dtype='float32', mode='readonly', shape=X_mnist.shape)\n",
    "\n",
    "batch_size = len(X_mnist) // n_batches\n",
    "inc_pca = IncrementalPCA(n_components=154, batch_size=batch_size)\n",
    "inc_pca.fit(X_mm)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "rnd_pca = PCA(n_components=154, random_state=42, svd_solver=\"randomized\")\n",
    "X_reduced = rnd_pca.fit_transform(X_mnist)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "import time\n",
    "\n",
    "for n_components in (2, 10, 154):\n",
    "    print(\"n_components =\", n_components)\n",
    "    regular_pca = PCA(n_components=n_components)\n",
    "    inc_pca = IncrementalPCA(n_components=154, batch_size=500)\n",
    "    rnd_pca = PCA(n_components=154, random_state=42, svd_solver=\"randomized\")\n",
    "\n",
    "    for pca in (regular_pca, inc_pca, rnd_pca):\n",
    "        t1 = time.time()\n",
    "        pca.fit(X_mnist)\n",
    "        t2 = time.time()\n",
    "        print(pca.__class__.__name__, t2 - t1, \"seconds\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "# Kernel PCA"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "from sklearn.decomposition import KernelPCA\n",
    "\n",
    "X, t = make_swiss_roll(n_samples=1000, noise=0.2, random_state=42)\n",
    "\n",
    "lin_pca = KernelPCA(n_components = 2, kernel=\"linear\", fit_inverse_transform=True)\n",
    "rbf_pca = KernelPCA(n_components = 2, kernel=\"rbf\", gamma=0.0433, fit_inverse_transform=True)\n",
    "sig_pca = KernelPCA(n_components = 2, kernel=\"sigmoid\", gamma=0.001, coef0=1, fit_inverse_transform=True)\n",
    "\n",
    "y = t > 6.9\n",
    "\n",
    "plt.figure(figsize=(11, 4))\n",
    "for subplot, pca, title in ((131, lin_pca, \"Linear kernel\"), (132, rbf_pca, \"RBF kernel, $\\gamma=0.04$\"), (133, sig_pca, \"Sigmoid kernel, $\\gamma=10^{-3}, r=1$\")):\n",
    "    X_reduced = pca.fit_transform(X)\n",
    "    if subplot == 132:\n",
    "        X_reduced_rbf = X_reduced\n",
    "    \n",
    "    plt.subplot(subplot)\n",
    "    #plt.plot(X_reduced[y, 0], X_reduced[y, 1], \"gs\")\n",
    "    #plt.plot(X_reduced[~y, 0], X_reduced[~y, 1], \"y^\")\n",
    "    plt.title(title, fontsize=14)\n",
    "    plt.scatter(X_reduced[:, 0], X_reduced[:, 1], c=t, cmap=plt.cm.hot)\n",
    "    plt.xlabel(\"$z_1$\", fontsize=18)\n",
    "    if subplot == 131:\n",
    "        plt.ylabel(\"$z_2$\", fontsize=18, rotation=0)\n",
    "    plt.grid(True)\n",
    "\n",
    "save_fig(\"kernel_pca_plot\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "plt.figure(figsize=(6, 5))\n",
    "\n",
    "X_inverse = pca.inverse_transform(X_reduced_rbf)\n",
    "\n",
    "ax = plt.subplot(111, projection='3d')\n",
    "ax.view_init(10, -70)\n",
    "ax.scatter(X_inverse[:, 0], X_inverse[:, 1], X_inverse[:, 2], c=t, cmap=plt.cm.hot, marker=\"x\")\n",
    "ax.set_xlabel(\"\")\n",
    "ax.set_ylabel(\"\")\n",
    "ax.set_zlabel(\"\")\n",
    "ax.set_xticklabels([])\n",
    "ax.set_yticklabels([])\n",
    "ax.set_zticklabels([])\n",
    "\n",
    "save_fig(\"preimage_plot\", tight_layout=False)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "X_reduced = rbf_pca.fit_transform(X)\n",
    "\n",
    "plt.figure(figsize=(11, 4))\n",
    "plt.subplot(132)\n",
    "plt.scatter(X_reduced[:, 0], X_reduced[:, 1], c=t, cmap=plt.cm.hot, marker=\"x\")\n",
    "plt.xlabel(\"$z_1$\", fontsize=18)\n",
    "plt.ylabel(\"$z_2$\", fontsize=18, rotation=0)\n",
    "plt.grid(True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "from sklearn.pipeline import Pipeline\n",
    "from sklearn.linear_model import LogisticRegression\n",
    "from sklearn.model_selection import GridSearchCV\n",
    "\n",
    "clf = Pipeline([\n",
    "        (\"kpca\", KernelPCA(n_components=2)),\n",
    "        (\"log_reg\", LogisticRegression())\n",
    "    ])\n",
    "\n",
    "param_grid = [\n",
    "        {\"kpca__gamma\": np.linspace(0.03, 0.05, 10), \"kpca__kernel\": [\"rbf\", \"sigmoid\"]}\n",
    "    ]\n",
    "\n",
    "grid_search = GridSearchCV(clf, param_grid, cv=3)\n",
    "grid_search.fit(X, y)\n",
    "grid_search.best_params_"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "rbf_pca = KernelPCA(n_components = 2, kernel=\"rbf\", gamma=0.0433,\n",
    "                    fit_inverse_transform=True)\n",
    "X_reduced = rbf_pca.fit_transform(X)\n",
    "X_preimage = rbf_pca.inverse_transform(X_reduced)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "from sklearn.metrics import mean_squared_error\n",
    "mean_squared_error(X, X_preimage)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "times_rpca = []\n",
    "times_pca = []\n",
    "sizes = [1000, 10000, 20000, 30000, 40000, 50000, 70000, 100000, 200000, 500000]\n",
    "for n_samples in sizes:\n",
    "    X = rnd.randn(n_samples, 5)\n",
    "    pca = PCA(n_components = 2, random_state=42, svd_solver=\"randomized\")\n",
    "    t1 = time.time()\n",
    "    pca.fit(X)\n",
    "    t2 = time.time()\n",
    "    times_rpca.append(t2 - t1)\n",
    "    pca = PCA(n_components = 2)\n",
    "    t1 = time.time()\n",
    "    pca.fit(X)\n",
    "    t2 = time.time()\n",
    "    times_pca.append(t2 - t1)\n",
    "\n",
    "plt.plot(sizes, times_rpca, \"b-o\", label=\"RPCA\")\n",
    "plt.plot(sizes, times_pca, \"r-s\", label=\"PCA\")\n",
    "plt.xlabel(\"n_samples\")\n",
    "plt.ylabel(\"Training time\")\n",
    "plt.legend(loc=\"upper left\")\n",
    "plt.title(\"PCA and Randomized PCA time complexity \")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "times_rpca = []\n",
    "times_pca = []\n",
    "sizes = [1000, 2000, 3000, 4000, 5000, 6000]\n",
    "for n_features in sizes:\n",
    "    X = rnd.randn(2000, n_features)\n",
    "    pca = PCA(n_components = 2, random_state=42, svd_solver=\"randomized\")\n",
    "    t1 = time.time()\n",
    "    pca.fit(X)\n",
    "    t2 = time.time()\n",
    "    times_rpca.append(t2 - t1)\n",
    "    pca = PCA(n_components = 2)\n",
    "    t1 = time.time()\n",
    "    pca.fit(X)\n",
    "    t2 = time.time()\n",
    "    times_pca.append(t2 - t1)\n",
    "\n",
    "plt.plot(sizes, times_rpca, \"b-o\", label=\"RPCA\")\n",
    "plt.plot(sizes, times_pca, \"r-s\", label=\"PCA\")\n",
    "plt.xlabel(\"n_features\")\n",
    "plt.ylabel(\"Training time\")\n",
    "plt.legend(loc=\"upper left\")\n",
    "plt.title(\"PCA and Randomized PCA time complexity \")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "# LLE"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "from sklearn.manifold import LocallyLinearEmbedding\n",
    "\n",
    "X, t = make_swiss_roll(n_samples=1000, noise=0.2, random_state=41)\n",
    "\n",
    "lle = LocallyLinearEmbedding(n_neighbors=10, n_components=2, random_state=42)\n",
    "X_reduced = lle.fit_transform(X)\n",
    "\n",
    "plt.title(\"Unrolled swiss roll using LLE\", fontsize=14)\n",
    "plt.scatter(X_reduced[:, 0], X_reduced[:, 1], c=t, cmap=plt.cm.hot)\n",
    "plt.xlabel(\"$z_1$\", fontsize=18)\n",
    "plt.ylabel(\"$z_2$\", fontsize=18)\n",
    "plt.axis([-0.065, 0.055, -0.1, 0.12])\n",
    "plt.grid(True)\n",
    "\n",
    "save_fig(\"lle_unrolling_plot\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "# MDS, Isomap and t-SNE"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "from sklearn.manifold import MDS\n",
    "mds = MDS(n_components=2, random_state=42)\n",
    "X_reduced_mds = mds.fit_transform(X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "from sklearn.manifold import Isomap\n",
    "isomap = Isomap(n_components=2)\n",
    "X_reduced_isomap = isomap.fit_transform(X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "from sklearn.manifold import TSNE\n",
    "tsne = TSNE(n_components=2)\n",
    "X_reduced_tsne = tsne.fit_transform(X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "from sklearn.discriminant_analysis import LinearDiscriminantAnalysis\n",
    "lda = LinearDiscriminantAnalysis(n_components=2)\n",
    "X_mnist = mnist[\"data\"]\n",
    "y_mnist = mnist[\"target\"]\n",
    "lda.fit(X_mnist, y_mnist)\n",
    "X_reduced_lda = lda.transform(X_mnist)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "titles = [\"MDS\", \"Isomap\", \"t-SNE\"]\n",
    "\n",
    "plt.figure(figsize=(11,4))\n",
    "\n",
    "for subplot, title, X_reduced in zip((131, 132, 133), titles,\n",
    "                                     (X_reduced_mds, X_reduced_isomap, X_reduced_tsne)):\n",
    "    plt.subplot(subplot)\n",
    "    plt.title(title, fontsize=14)\n",
    "    plt.scatter(X_reduced[:, 0], X_reduced[:, 1], c=t, cmap=plt.cm.hot)\n",
    "    plt.xlabel(\"$z_1$\", fontsize=18)\n",
    "    if subplot == 131:\n",
    "        plt.ylabel(\"$z_2$\", fontsize=18, rotation=0)\n",
    "    plt.grid(True)\n",
    "\n",
    "save_fig(\"other_dim_reduction_plot\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "source": [
    "# Exercise solutions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "**Coming soon**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": []
  }
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