04.3-Density-GMM.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<small><i>This notebook was put together by [Jake Vanderplas](http://www.vanderplas.com). Source and license info is on [GitHub](https://github.com/jakevdp/sklearn_tutorial/).</i></small>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Density Estimation: Gaussian Mixture Models"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here we'll explore **Gaussian Mixture Models**, which is an unsupervised clustering & density estimation technique.\n",
    "\n",
    "We'll start with our standard set of initial imports"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy import stats\n",
    "\n",
    "plt.style.use('seaborn')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Introducing Gaussian Mixture Models\n",
    "\n",
    "We previously saw an example of K-Means, which is a clustering algorithm which is most often fit using an expectation-maximization approach.\n",
    "\n",
    "Here we'll consider an extension to this which is suitable for both **clustering** and **density estimation**.\n",
    "\n",
    "For example, imagine we have some one-dimensional data in a particular distribution:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/matplotlib/axes/_axes.py:6462: UserWarning: The 'normed' kwarg is deprecated, and has been replaced by the 'density' kwarg.\n",
      "  warnings.warn(\"The 'normed' kwarg is deprecated, and has been \"\n"
     ]
    },
    {
     "data": {
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      "text/plain": [
       "<Figure size 576x396 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "np.random.seed(2)\n",
    "x = np.concatenate([np.random.normal(0, 2, 2000),\n",
    "                    np.random.normal(5, 5, 2000),\n",
    "                    np.random.normal(3, 0.5, 600)])\n",
    "plt.hist(x, 80, normed=True)\n",
    "plt.xlim(-10, 20);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Gaussian mixture models will allow us to approximate this density:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:58: DeprecationWarning: Class GMM is deprecated; The class GMM is deprecated in 0.18 and will be  removed in 0.20. Use class GaussianMixture instead.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function distribute_covar_matrix_to_match_covariance_type is deprecated; The function distribute_covar_matrix_to_match_covariance_typeis deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/matplotlib/axes/_axes.py:6462: UserWarning: The 'normed' kwarg is deprecated, and has been replaced by the 'density' kwarg.\n",
      "  warnings.warn(\"The 'normed' kwarg is deprecated, and has been \"\n"
     ]
    },
    {
     "data": {
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PnsyoUaMYO3Ysubm5x9wmPz+fSy65BK+3fG+mpKSE2267jTFjxnDTTTeRn59f/cml3rJ+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\nYII=\n",
      "text/plain": [
       "<Figure size 576x396 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from sklearn.mixture import GMM\n",
    "X = x[:, np.newaxis]\n",
    "clf = GMM(4, n_iter=500, random_state=3).fit(X)\n",
    "xpdf = np.linspace(-10, 20, 1000)\n",
    "density = np.exp(clf.score(xpdf[:, np.newaxis]))\n",
    "\n",
    "plt.hist(x, 80, normed=True, alpha=0.5)\n",
    "plt.plot(xpdf, density, '-r')\n",
    "plt.xlim(-10, 20);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that this density is fit using a **mixture of Gaussians**, which we can examine by looking at the ``means_``, ``covars_``, and ``weights_`` attributes:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.25306823],\n",
       "       [ 3.96038601],\n",
       "       [ 1.45350312],\n",
       "       [ 9.56303626]])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "clf.means_"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[  6.4601766 ],\n",
       "       [ 23.19014431],\n",
       "       [  5.80884786],\n",
       "       [ 14.79998213]])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "clf.covars_"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.31823886,  0.21876416,  0.35341345,  0.10958354])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "clf.weights_"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/matplotlib/axes/_axes.py:6462: UserWarning: The 'normed' kwarg is deprecated, and has been replaced by the 'density' kwarg.\n",
      "  warnings.warn(\"The 'normed' kwarg is deprecated, and has been \"\n"
     ]
    },
    {
     "data": {
      "image/png": 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sXwPVYkXteaONLqVb0/v0gXzpZZB2/BvmDzYZXQ4RdUHYZV1utxsOh6P1a5PJ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\nDAGkYEskEolEMgSQgi2RSCQSyRDg35kKrQy4OQu3AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 576x396 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(x, 80, normed=True, alpha=0.3)\n",
    "plt.plot(xpdf, density, '-r')\n",
    "\n",
    "for i in range(clf.n_components):\n",
    "    pdf = clf.weights_[i] * stats.norm(clf.means_[i, 0],\n",
    "                                       np.sqrt(clf.covars_[i, 0])).pdf(xpdf)\n",
    "    plt.fill(xpdf, pdf, facecolor='gray',\n",
    "             edgecolor='none', alpha=0.3)\n",
    "plt.xlim(-10, 20);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "These individual Gaussian distributions are fit using an expectation-maximization method, much as in K means, except that rather than explicit cluster assignment, the **posterior probability** is used to compute the weighted mean and covariance.\n",
    "Somewhat surprisingly, this algorithm **provably** converges to the optimum (though the optimum is not necessarily global)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## How many Gaussians?\n",
    "\n",
    "Given a model, we can use one of several means to evaluate how well it fits the data.\n",
    "For example, there is the Aikaki Information Criterion (AIC) and the Bayesian Information Criterion (BIC)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "25911.1937804\n",
      "25840.421853\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n"
     ]
    }
   ],
   "source": [
    "print(clf.bic(X))\n",
    "print(clf.aic(X))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's take a look at these as a function of the number of gaussians:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:58: DeprecationWarning: Class GMM is deprecated; The class GMM is deprecated in 0.18 and will be  removed in 0.20. Use class GaussianMixture instead.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function distribute_covar_matrix_to_match_covariance_type is deprecated; The function distribute_covar_matrix_to_match_covariance_typeis deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:58: DeprecationWarning: Class GMM is deprecated; The class GMM is deprecated in 0.18 and will be  removed in 0.20. Use class GaussianMixture instead.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function distribute_covar_matrix_to_match_covariance_type is deprecated; The function distribute_covar_matrix_to_match_covariance_typeis deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:58: DeprecationWarning: Class GMM is deprecated; The class GMM is deprecated in 0.18 and will be  removed in 0.20. Use class GaussianMixture instead.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function distribute_covar_matrix_to_match_covariance_type is deprecated; The function distribute_covar_matrix_to_match_covariance_typeis deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:58: DeprecationWarning: Class GMM is deprecated; The class GMM is deprecated in 0.18 and will be  removed in 0.20. Use class GaussianMixture instead.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function distribute_covar_matrix_to_match_covariance_type is deprecated; The function distribute_covar_matrix_to_match_covariance_typeis deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:58: DeprecationWarning: Class GMM is deprecated; The class GMM is deprecated in 0.18 and will be  removed in 0.20. Use class GaussianMixture instead.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function distribute_covar_matrix_to_match_covariance_type is deprecated; The function distribute_covar_matrix_to_match_covariance_typeis deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:58: DeprecationWarning: Class GMM is deprecated; The class GMM is deprecated in 0.18 and will be  removed in 0.20. Use class GaussianMixture instead.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function distribute_covar_matrix_to_match_covariance_type is deprecated; The function distribute_covar_matrix_to_match_covariance_typeis deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:58: DeprecationWarning: Class GMM is deprecated; The class GMM is deprecated in 0.18 and will be  removed in 0.20. Use class GaussianMixture instead.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function distribute_covar_matrix_to_match_covariance_type is deprecated; The function distribute_covar_matrix_to_match_covariance_typeis deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:58: DeprecationWarning: Class GMM is deprecated; The class GMM is deprecated in 0.18 and will be  removed in 0.20. Use class GaussianMixture instead.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function distribute_covar_matrix_to_match_covariance_type is deprecated; The function distribute_covar_matrix_to_match_covariance_typeis deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:58: DeprecationWarning: Class GMM is deprecated; The class GMM is deprecated in 0.18 and will be  removed in 0.20. Use class GaussianMixture instead.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function distribute_covar_matrix_to_match_covariance_type is deprecated; The function distribute_covar_matrix_to_match_covariance_typeis deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n"
     ]
    },
    {
     "data": {
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3FhGREVJ4j7PB28WM5C5A171FRGTkFN7jLM+Rg9VsxWdEm9a0xrmIiIyU9XQv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\nSUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 576x396 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "n_estimators = np.arange(1, 10)\n",
    "clfs = [GMM(n, n_iter=1000).fit(X) for n in n_estimators]\n",
    "bics = [clf.bic(X) for clf in clfs]\n",
    "aics = [clf.aic(X) for clf in clfs]\n",
    "\n",
    "plt.plot(n_estimators, bics, label='BIC')\n",
    "plt.plot(n_estimators, aics, label='AIC')\n",
    "plt.legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It appears that for both the AIC and BIC, 4 components is preferred."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example: GMM For Outlier Detection\n",
    "\n",
    "GMM is what's known as a **Generative Model**: it's a probabilistic model from which a dataset can be generated.\n",
    "One thing that generative models can be useful for is **outlier detection**: we can simply evaluate the likelihood of each point under the generative model; the points with a suitably low likelihood (where \"suitable\" is up to your own bias/variance preference) can be labeld outliers.\n",
    "\n",
    "Let's take a look at this by defining a new dataset with some outliers:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.random.seed(0)\n",
    "\n",
    "# Add 20 outliers\n",
    "true_outliers = np.sort(np.random.randint(0, len(x), 20))\n",
    "y = x.copy()\n",
    "y[true_outliers] += 50 * np.random.randn(20)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:58: DeprecationWarning: Class GMM is deprecated; The class GMM is deprecated in 0.18 and will be  removed in 0.20. Use class GaussianMixture instead.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function distribute_covar_matrix_to_match_covariance_type is deprecated; The function distribute_covar_matrix_to_match_covariance_typeis deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n",
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/matplotlib/axes/_axes.py:6462: UserWarning: The 'normed' kwarg is deprecated, and has been replaced by the 'density' kwarg.\n",
      "  warnings.warn(\"The 'normed' kwarg is deprecated, and has been \"\n"
     ]
    },
    {
     "data": {
      "image/png": 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RFba0PJcL2xef4xuSrWtgh4PNhvf8C7Ht3YOtaJPZaUQkQuxmB5D4Zy/ahMUw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\n9Sr+jJ7Unj3S7DgSRyo7d+c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9998PwO7du+nYsWOT34uIFfYLL7zAxRdffNQ/hYWFXHTR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\nAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 576x396 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "clf = GMM(4, n_iter=500, random_state=0).fit(y[:, np.newaxis])\n",
    "xpdf = np.linspace(-10, 20, 1000)\n",
    "density_noise = np.exp(clf.score(xpdf[:, np.newaxis]))\n",
    "\n",
    "plt.hist(y, 80, normed=True, alpha=0.5)\n",
    "plt.plot(xpdf, density_noise, '-r')\n",
    "plt.xlim(-15, 30);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's evaluate the log-likelihood of each point under the model, and plot these as a function of ``y``:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/jakevdp/anaconda/lib/python3.6/site-packages/sklearn/utils/deprecation.py:77: DeprecationWarning: Function log_multivariate_normal_density is deprecated; The function log_multivariate_normal_density is deprecated in 0.18 and will be removed in 0.20.\n",
      "  warnings.warn(msg, category=DeprecationWarning)\n"
     ]
    },
    {
     "data": {
      "image/png": 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IAAAYgEAGAMAABDIAAK2I9kMtCGQAAC5ixzOaCWQAAC5ixzOaCWQAAC5ix/Pj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\n",
      "text/plain": [
       "<Figure size 576x396 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "log_likelihood = clf.score_samples(y[:, np.newaxis])[0]\n",
    "plt.plot(y, log_likelihood, '.k');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "true outliers:\n",
      "[  99  537  705 1033 1653 1701 1871 2046 2135 2163 2222 2496 2599 2607 2732\n",
      " 2893 2897 3264 3468 4373]\n",
      "\n",
      "detected outliers:\n",
      "[  99  537  705 1653 2046 2135 2163 2222 2496 2732 2893 2897 3067 3173 3253\n",
      " 3468 3483 4373]\n"
     ]
    }
   ],
   "source": [
    "detected_outliers = np.where(log_likelihood < -9)[0]\n",
    "\n",
    "print(\"true outliers:\")\n",
    "print(true_outliers)\n",
    "print(\"\\ndetected outliers:\")\n",
    "print(detected_outliers)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The algorithm misses a few of these points, which is to be expected (some of the \"outliers\" actually land in the middle of the distribution!)\n",
    "\n",
    "Here are the outliers that were missed:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{1033, 1701, 1871, 2599, 2607, 3264}"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "set(true_outliers) - set(detected_outliers)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And here are the non-outliers which were spuriously labeled outliers:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{3067, 3173, 3253, 3483}"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "set(detected_outliers) - set(true_outliers)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, we should note that although all of the above is done in one dimension, GMM does generalize to multiple dimensions, as we'll see in the breakout session."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Other Density Estimators\n",
    "\n",
    "The other main density estimator that you might find useful is *Kernel Density Estimation*, which is available via ``sklearn.neighbors.KernelDensity``. In some ways, this can be thought of as a generalization of GMM where there is a gaussian placed at the location of *every* training point!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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SwhZCSGAL0SK872BHXUxgVzVHSUKINkYCW4gW4J3lrKnPsE1mFJsNnM7mKEsI0YZIYAvR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\nTkSuQmCC\n",
      "text/plain": [
       "<Figure size 576x396 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from sklearn.neighbors import KernelDensity\n",
    "kde = KernelDensity(0.15).fit(x[:, None])\n",
    "density_kde = np.exp(kde.score_samples(xpdf[:, None]))\n",
    "\n",
    "plt.hist(x, 80, density=True, alpha=0.5)\n",
    "plt.plot(xpdf, density, '-b', label='GMM')\n",
    "plt.plot(xpdf, density_kde, '-r', label='KDE')\n",
    "plt.xlim(-10, 20)\n",
    "plt.legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "All of these density estimators can be viewed as **Generative models** of the data: that is, that is, the model tells us how more data can be created which fits the model."
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