{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "import seaborn as sns\n",
    "sns.set()\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "w = np.random.randn(500, 2)\n",
    "plt.figure(figsize=(6, 6))\n",
    "plt.scatter(w[:,0], w[:,1])\n",
    "plt.axis('equal')\n",
    "plt.show()\n",
    "w = np.random.randn(500, 3)\n",
    "fig = plt.figure(figsize=(6, 6))\n",
    "ax = fig.add_subplot(111, projection='3d')\n",
    "ax.scatter(w[:,0], w[:,1], w[:,2])\n",
    "ax.axis('equal')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "dims = [1, 3, 5, 10, 30, 100]\n",
    "\n",
    "def norms(x):\n",
    "    return np.sqrt(np.sum(np.square(x), axis=1))\n",
    "\n",
    "# Distribution of ||x||\n",
    "plt.figure()\n",
    "for d in dims:\n",
    "    x = np.random.randn(1000, d)\n",
    "    sns.distplot(norms(x), kde_kws={'label' : 'dim={0:d}'.format(d)})\n",
    "    plt.xlabel('Norm')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def gen_gaussian_clusters(mu0, dim, n_samples):\n",
    "    \"\"\"\n",
    "    Generate three Gaussian clusters centered at:\n",
    "    (mu0[0], 0, 0, ...)\n",
    "    (mu0[1], 0, 0, ...)\n",
    "    (mu0[2], 0, 0, ...)\n",
    "    :param mu0: Determines the center of the clusters.\n",
    "    :param dim: Dimension of the Gaussian samples.\n",
    "    :param n_samples: Number of samples to generate for each cluster.\n",
    "    \"\"\"\n",
    "    mu = np.zeros((3, dim))\n",
    "    mu[:,0] = mu0\n",
    "    x1 = np.random.randn(n_samples, dim) + mu[0,:]\n",
    "    x2 = np.random.randn(n_samples, dim) + mu[1,:]\n",
    "    x3 = np.random.randn(n_samples, dim) + mu[2,:]\n",
    "    return x1, x2, x3\n",
    "\n",
    "dim = 500\n",
    "n_samples = 500\n",
    "mu0 = np.array([0, 5, 10])\n",
    "\n",
    "x1, x2, x3 = gen_gaussian_clusters(mu0, dim, n_samples)\n",
    "\n",
    "# Visualize if possible\n",
    "plt.figure()\n",
    "if dim == 1:\n",
    "    plt.hist(x1)\n",
    "    plt.hist(x2)\n",
    "    plt.hist(x3)\n",
    "elif dim >= 2:\n",
    "    plt.scatter(x1[:,0], x1[:,1])\n",
    "    plt.scatter(x2[:,0], x2[:,1])\n",
    "    plt.scatter(x3[:,0], x3[:,1])\n",
    "    plt.axis('equal')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Plot the distribution of distances for various # of dimensions\n",
    "dims = [3, 10, 50, 100, 200, 500]\n",
    "\n",
    "# Distance distribution\n",
    "plt.figure(1, figsize=(15, 10))\n",
    "for i, d in enumerate(dims, 1):\n",
    "    x1, x2, x3 = gen_gaussian_clusters(mu0, d, n_samples)\n",
    "    plt.subplot(2, 3, i)\n",
    "    sns.distplot(norms(x1))\n",
    "    sns.distplot(norms(x2))\n",
    "    sns.distplot(norms(x3))\n",
    "    plt.legend(['x1', 'x2', 'x3'])\n",
    "    plt.xlabel('Distance to origin')\n",
    "    plt.title('dim = {0:d}'.format(d))\n",
    "    plt.xlim(0, plt.xlim()[1])\n",
    "plt.show()\n",
    "\n",
    "from scipy.spatial.distance import pdist, squareform\n",
    "\n",
    "# Distance matrix\n",
    "plt.figure(figsize=(16, 10))\n",
    "for i, d in enumerate(dims, 1):\n",
    "    x1, x2, x3 = gen_gaussian_clusters(mu0, d, n_samples)\n",
    "    plt.subplot(2, 3, i)\n",
    "    D = squareform(pdist(np.concatenate((x1, x2, x3), axis=0), 'euclidean'))\n",
    "    sns.heatmap(D)\n",
    "    plt.title('dim = {0:d}'.format(d))\n",
    "plt.show()"
   ]
  }
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