Mixture_of_Gaussians.ipynb

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   "source": [
    "# style the notebook\n",
    "from IPython.core.display import HTML\n",
    "import urllib.request\n",
    "response = urllib.request.urlopen('http://bit.ly/1LC7EI7')\n",
    "HTML(response.read().decode(\"utf-8\"))"
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   "source": [
    "# Mixture of Gaussians\n",
    "\n",
    "This is a simple concept - we create a mixture of Gaussians as a weighted sum of a set of Gaussians:\n",
    "\n",
    "$$p(x) = \\sum\\limits_{k=1}^N w_k \\mathcal{N}(\\mu_k, \\sigma_k^2)$$\n",
    "\n",
    "with the constraint that the weights are all positive and sum to one: $\\sum\\limits_{k=1}^N w_k=1$.\n",
    "\n",
    "Here is some code to visualize this for 1D Gaussians"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
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sZsiKpOudd6BHD1iwIMbfVzN141SI1avh+OOjb1BLIYhUhu22i9Vla3WrwvYo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\nDMonAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x87c9c88>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "from scipy.stats import norm\n",
    "\n",
    "def plot_mog(gaussians, weights):\n",
    "    ys = []\n",
    "    \n",
    "    # find range for x\n",
    "    min_x, max_x = 1e308, -1e308\n",
    "    for g in gaussians:\n",
    "        min_x = min(min_x, norm.ppf(.001, g[0], g[1]))\n",
    "        max_x = max(max_x, norm.ppf(.999, g[0], g[1]))\n",
    "        \n",
    "    xs = np.linspace(min_x, max_x, 500)\n",
    "    ys = np.zeros(len(xs))\n",
    "    for g, w in zip(gaussians, weights):\n",
    "        ys += w * norm.pdf(xs, loc=g[0], scale=g[1])\n",
    "        \n",
    "    plt.plot(xs, ys)\n",
    "\n",
    "gaussians = [[1, 1.5], \n",
    "             [5, 1],\n",
    "             [8, .8]]\n",
    "\n",
    "weights = [.3, .3, .4]\n",
    "\n",
    "plot_mog(gaussians, weights)"
   ]
  }
 ],
 "metadata": {
  "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": "3.4.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}