EM_algo.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "#loading library\n",
    "import numpy as np\n",
    "import scipy as sp\n",
    "import pandas as pd\n",
    "from sklearn.cluster import KMeans"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 108,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "#loading data\n",
    "data = pd.read_table('EMGaussian.data', header = None, sep='\\s+', names = ('x1', 'x2'))\n",
    "test = pd.read_table('EMGaussian.test', header = None, sep='\\s+', names = ('x1', 'x2'))\n",
    "X = np.array(data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 109,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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       "       [ -2.1     ,   5.7945  ],\n",
       "       [ -4.4638  ,  -5.9922  ],\n",
       "       [ -3.2302  ,  -2.5581  ],\n",
       "       [ -3.5148  ,  -4.3457  ],\n",
       "       [ -0.31993 ,  -1.9771  ],\n",
       "       [ -1.5641  ,  -1.4856  ],\n",
       "       [  2.9451  ,   1.6211  ],\n",
       "       [ -3.2555  ,  -4.3771  ],\n",
       "       [ -5.8734  ,  -6.8819  ],\n",
       "       [ -4.6493  ,  -5.3944  ],\n",
       "       [ -5.1804  ,  -5.8217  ],\n",
       "       [ -0.80557 ,  -1.6731  ],\n",
       "       [ -1.3568  ,  -2.3314  ],\n",
       "       [ -2.1038  ,  -2.7101  ],\n",
       "       [ -1.2224  ,  -1.6403  ],\n",
       "       [ -2.7811  ,  -3.955   ],\n",
       "       [ -3.0974  ,  -3.7423  ],\n",
       "       [ -0.90066 ,  -1.4437  ],\n",
       "       [ -0.88375 ,  -2.0386  ],\n",
       "       [  3.0176  ,   2.5822  ],\n",
       "       [ -3.8465  ,  -3.2908  ],\n",
       "       [ -0.10906 ,  -0.66399 ],\n",
       "       [  2.2317  ,   1.6163  ],\n",
       "       [ -5.3679  ,  -5.2427  ],\n",
       "       [  3.7873  ,   6.7762  ],\n",
       "       [  4.2726  ,   7.414   ],\n",
       "       [ -1.917   ,   6.7981  ],\n",
       "       [  3.5429  ,   3.3781  ],\n",
       "       [ -1.52    ,   3.4396  ],\n",
       "       [ -6.1823  ,  -6.1833  ],\n",
       "       [ -8.4768  ,  -8.6457  ],\n",
       "       [  3.8484  ,  -3.1225  ],\n",
       "       [ -2.6823  ,   6.1854  ],\n",
       "       [  4.003   ,   6.8276  ],\n",
       "       [ -0.22063 ,   4.1701  ],\n",
       "       [  4.1718  ,   0.36369 ],\n",
       "       [ -4.1914  ,   2.3526  ],\n",
       "       [ -2.7859  ,  -3.2817  ],\n",
       "       [ -4.6301  ,  -4.3811  ],\n",
       "       [ -2.9214  ,  -3.1551  ],\n",
       "       [ -6.168   ,  -5.7083  ],\n",
       "       [ -5.0697  ,  -5.8915  ],\n",
       "       [  5.7916  ,  -4.7474  ],\n",
       "       [  4.004   ,  -3.5499  ],\n",
       "       [  3.1497  ,   1.6276  ],\n",
       "       [ -0.88009 ,   2.2903  ],\n",
       "       [  3.2194  ,   7.5905  ],\n",
       "       [ -0.83187 ,   3.5615  ],\n",
       "       [  4.4622  ,   7.5957  ],\n",
       "       [  3.7712  ,  -5.6106  ],\n",
       "       [  4.7404  ,  -3.7217  ],\n",
       "       [  2.7255  ,  -2.0637  ],\n",
       "       [  5.9585  ,  -3.513   ],\n",
       "       [  4.4933  ,  -3.59    ],\n",
       "       [  4.7518  ,  -2.8339  ],\n",
       "       [  4.3798  ,  -2.7528  ],\n",
       "       [  4.7651  ,  -0.84398 ],\n",
       "       [ -1.0548  ,   7.372   ],\n",
       "       [  3.8066  ,  -2.9404  ],\n",
       "       [ -4.9757  ,   4.2238  ],\n",
       "       [  4.0636  ,   4.5207  ],\n",
       "       [ -3.4687  ,   1.8693  ],\n",
       "       [  4.807   ,  10.118   ],\n",
       "       [ -2.607   ,   2.3596  ],\n",
       "       [  3.8759  ,   4.6343  ],\n",
       "       [  1.8351  ,   4.0183  ],\n",
       "       [  3.9743  ,   1.3937  ],\n",
       "       [ -1.047   ,   4.8668  ],\n",
       "       [  4.0907  ,   4.0664  ],\n",
       "       [ -0.72718 ,   6.5097  ],\n",
       "       [  4.631   ,   6.3615  ],\n",
       "       [ -2.018   ,   3.9771  ],\n",
       "       [  4.7558  ,   8.0755  ],\n",
       "       [ -2.9022  ,  -3.2036  ],\n",
       "       [ -2.6121  ,  -3.2505  ],\n",
       "       [ -1.6416  ,  -1.401   ],\n",
       "       [ -1.554   ,  -1.714   ],\n",
       "       [ -4.5754  ,  -4.9314  ],\n",
       "       [ -3.3874  ,  -3.2293  ],\n",
       "       [  0.80596 ,  -0.28273 ],\n",
       "       [  0.29144 ,   0.31055 ],\n",
       "       [ -1.1098  ,  -1.0884  ],\n",
       "       [ -4.0911  ,  -4.0292  ],\n",
       "       [ -3.0079  ,  -3.308   ],\n",
       "       [  4.1569  ,   3.6405  ],\n",
       "       [ -3.8159  ,  -4.7663  ],\n",
       "       [ -0.7173  ,  -1.7444  ],\n",
       "       [ -4.9813  ,  -5.1365  ],\n",
       "       [ -4.1847  ,  -4.8179  ],\n",
       "       [ -1.778   ,  -3.1563  ],\n",
       "       [ -3.7228  ,  -5.3512  ],\n",
       "       [ -4.5548  ,  -4.5827  ],\n",
       "       [  3.9443  ,  -2.3816  ],\n",
       "       [ -4.0841  ,   4.8377  ],\n",
       "       [  4.5484  ,  -0.48432 ],\n",
       "       [ -4.3493  ,   5.7543  ],\n",
       "       [  2.9174  ,   7.7307  ],\n",
       "       [ -4.4949  ,   2.9297  ],\n",
       "       [  5.2867  ,  -2.1644  ],\n",
       "       [  3.4909  ,  -2.7531  ],\n",
       "       [  2.7198  ,  -4.2817  ],\n",
       "       [  2.8146  ,  -3.3142  ],\n",
       "       [  3.1129  ,  -2.485   ],\n",
       "       [  3.3719  ,  -2.4157  ],\n",
       "       [  2.8676  ,  -4.1425  ],\n",
       "       [  2.7679  ,  -3.312   ],\n",
       "       [  3.8195  ,  -3.7047  ],\n",
       "       [  4.7983  ,  -3.3384  ],\n",
       "       [  3.3473  ,  -6.7136  ],\n",
       "       [  5.1468  ,  -4.9632  ],\n",
       "       [  2.7829  ,  -4.9296  ],\n",
       "       [  2.9354  ,  -2.9406  ],\n",
       "       [  4.7228  ,  -4.625   ],\n",
       "       [  4.3851  ,  -2.3614  ],\n",
       "       [  5.278   ,  -4.9131  ],\n",
       "       [  3.2037  ,  -1.7062  ],\n",
       "       [  4.0592  ,  -2.3924  ],\n",
       "       [ -6.3549  ,  -6.6346  ],\n",
       "       [ -5.917   ,  -6.7503  ],\n",
       "       [ -2.6437  ,  -2.5488  ],\n",
       "       [ -0.56818 ,  -1.2981  ],\n",
       "       [ -4.6186  ,  -4.2745  ],\n",
       "       [ -2.5813  ,  -2.9707  ],\n",
       "       [ -1.1014  ,  -0.84597 ],\n",
       "       [  3.485   ,   7.9973  ],\n",
       "       [ -3.3649  ,   3.43    ],\n",
       "       [  3.7886  ,  -1.2331  ],\n",
       "       [ -4.1737  ,   5.451   ],\n",
       "       [  3.7375  ,   3.4877  ],\n",
       "       [ -1.4053  ,   8.7257  ],\n",
       "       [  4.5037  ,   0.30935 ],\n",
       "       [ -5.7265  ,   4.2021  ],\n",
       "       [  3.7219  ,   2.0385  ],\n",
       "       [ -2.1572  ,   2.9838  ],\n",
       "       [  4.5102  ,   1.8433  ],\n",
       "       [ -8.9079  ,  -9.806   ],\n",
       "       [ -1.9095  ,  -2.7737  ],\n",
       "       [ -2.9942  ,  -2.0706  ],\n",
       "       [ -3.8083  ,  -4.0658  ],\n",
       "       [ -7.5274  ,  -8.1222  ],\n",
       "       [ -3.8115  ,  -3.5646  ],\n",
       "       [ -2.0464  ,  -2.1179  ],\n",
       "       [ -1.7796  ,  -2.7828  ],\n",
       "       [ -0.3628  ,  -0.6534  ],\n",
       "       [ -6.452   ,  -6.6973  ],\n",
       "       [ -7.3222  ,  -7.5441  ],\n",
       "       [  0.13535 ,   0.1454  ],\n",
       "       [ -6.9113  ,  -6.4961  ],\n",
       "       [ -1.7182  ,  -1.9256  ]])"
      ]
     },
     "execution_count": 109,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 110,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "#initialization\n",
    "K = 4\n",
    "[N,d] = data.shape #dimension of x\n",
    "kmean_res = KMeans(init = 'random', n_clusters=4).fit(data)\n",
    "lmu = kmean_res.cluster_centers_\n",
    "lsigma = [np.eye(2,2) for k in range(K)]\n",
    "lpi = [1/K for k in range(K)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-2.24034753,  4.12744772],\n",
       "       [ 3.80280826,  5.10467248],\n",
       "       [ 3.33557966, -2.644529  ],\n",
       "       [-3.81879354, -4.27423441]])"
      ]
     },
     "execution_count": 111,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lmu"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 112,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def compute_loglikelihood(X, Q, lpi, lmu, lsigma):\n",
    "    (N,d)=X.shape\n",
    "    lglklhd=0\n",
    "    for i in range(N):\n",
    "        for k in range(K):\n",
    "            lglklhd += Q[i,k]*np.log(sp.stats.multivariate_normal.pdf(X[i,:],lmu[k,:], lsigma[k])) + Q[i,k]*np.log(lpi[k])\n",
    "    return lglklhd"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 113,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "#lpi is 1d-array; lmu, X are 2d-arrays; lsig is a list\n",
    "\n",
    "Q = np.zeros((N,K))\n",
    "condition = False\n",
    "count = 0\n",
    "tol=0.001\n",
    "\n",
    "lglklhd=compute_loglikelihood(X,Q,lpi,lmu,lsigma)\n",
    "\n",
    "while(condition==False & count < 30):\n",
    "    \n",
    "    count += 1\n",
    "#E-step\n",
    "    for i in range(N):\n",
    "        denom=0\n",
    "        for k in range(K):\n",
    "            denom=denom+lpi[k]*sp.stats.multivariate_normal.pdf(X[i,:],lmu[k,:], lsigma[k])\n",
    "        for k in range(K):\n",
    "            Q[i,k]=lpi[k]*sp.stats.multivariate_normal.pdf(X[i,:],lmu[k,:], lsigma[k])/denom\n",
    "\n",
    "#M-step\n",
    "    for k in range(K):\n",
    "        #compute mu\n",
    "        denom_mu=np.zeros(2)\n",
    "        for i in range(N):\n",
    "            denom_mu += X[i,:]*Q[i,k]\n",
    "        lmu[k,:]= denom_mu/np.sum(Q[:,k])\n",
    "        #compute sigma\n",
    "        denom_sigma=np.matlib.zeros((2,2))\n",
    "        for i in range(N):\n",
    "            A=np.asmatrix(X[i,:]-lmu[k])\n",
    "            denom_sigma += A.T.dot(A)\n",
    "        lsigma[k]=np.asarray(denom_sigma/np.sum(Q[:,k]))\n",
    "        #compute pi\n",
    "        lpi[k]=np.sum(Q[:,k])/np.sum(Q)\n",
    "        \n",
    "        lg_past=lglklhd\n",
    "        lglklhd=compute_loglikelihood(X,Q,lpi,lmu,lsigma)\n",
    "        if (np.abs(lglklhd-lg_past<tol)):\n",
    "            condition=True\n",
    "        \n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 114,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-2.22375874,  4.12517673],\n",
       "       [ 3.78817488,  5.10711606],\n",
       "       [ 3.35897274, -2.66609607],\n",
       "       [-3.7969143 , -4.23540821]])"
      ]
     },
     "execution_count": 114,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lmu"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 115,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 115,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "count"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.91029 , -2.531   ,  4.3602  , -0.70489 , -5.7383  , -4.4819  ,\n",
       "        0.90842 ,  4.4273  , -5.5724  ,  4.256   , -1.8763  ,  3.9002  ,\n",
       "       -1.6512  ,  3.4097  ,  2.7509  ,  4.0017  , -1.3762  ,  3.1672  ,\n",
       "        2.5706  ,  4.1783  , -1.4094  ,  3.3032  , -0.6899  ,  4.4551  ,\n",
       "        0.43072 ,  4.2493  , -2.2102  ,  4.5083  , -1.5381  ,  3.8053  ,\n",
       "       -3.6955  ,  4.0747  , -3.3814  ,  4.3867  , -3.2273  ,  3.5652  ,\n",
       "        0.98228 ,  4.3048  , -5.6913  ,  4.3524  , -3.3028  ,  4.2079  ,\n",
       "       -2.8765  ,  4.4197  , -4.3528  ,  3.7617  , -2.0124  ,  3.7219  ,\n",
       "       -4.1637  ,  3.7035  , -1.5331  ,  4.4484  , -2.8353  ,  4.0611  ,\n",
       "       -2.0325  ,  3.4695  , -0.75983 ,  3.8583  ,  3.4517  , -0.41852 ,\n",
       "        3.7967  , -1.4054  ,  4.723   , -0.60948 ,  4.6102  , -2.4528  ,\n",
       "        4.1376  , -1.4837  ,  3.9858  ,  0.29761 ,  5.0873  , -3.3855  ,\n",
       "        3.8461  , -0.19813 ,  3.8222  ,  1.4774  ,  3.7977  , -1.8651  ,\n",
       "        4.2754  , -0.45499 ,  4.4675  , -2.916   ,  4.6984  ,  0.16431 ,\n",
       "       -2.2341  ,  1.046   , -1.9258  , -4.1876  , -2.4265  ,  2.089   ,\n",
       "       -1.6956  , -4.9665  , -7.6645  , -2.4237  ,  4.5974  ,  0.41252 ,\n",
       "       -5.5943  , -1.921   , -2.3281  ,  4.1444  , -1.7756  ,  3.725   ,\n",
       "       -2.543   ,  4.2798  , -2.105   ,  3.7389  , -2.1966  ,  4.1613  ,\n",
       "        0.49415 ,  4.145   , -3.4003  ,  3.562   , -3.1733  ,  4.2408  ,\n",
       "       -4.5667  ,  3.4199  ,  3.8368  ,  2.2181  ,  3.2454  ,  3.3015  ,\n",
       "        3.3484  ,  3.6584  ,  4.0736  ,  5.5095  ,  4.0373  ,  3.673   ,\n",
       "        2.6925  ,  3.5127  ,  4.1824  ,  3.9281  ,  2.1866  ,  4.5942  ,\n",
       "        5.1001  ,  1.6937  ,  3.9808  ,  3.3     ,  3.3678  ,  5.6717  ,\n",
       "        4.8229  ,  1.9228  ,  4.0438  , -4.3256  ,  4.5552  , -2.3859  ,\n",
       "        3.784   , -0.63884 ,  0.19277 ,  3.81    , -1.959   ,  4.1755  ,\n",
       "       -1.1374  ,  3.494   ,  2.023   ,  4.7368  ,  3.6569  ,  3.3891  ,\n",
       "        1.1641  ,  3.6771  ,  2.0863  , -5.6459  ,  3.7341  , -0.80097 ,\n",
       "        4.0098  , -0.15353 ,  4.0991  ,  3.4908  , -1.2162  ,  4.7011  ,\n",
       "       -1.6408  ,  4.1221  , -4.2697  ,  2.6435  , -0.33739 ,  4.5352  ,\n",
       "       -4.6471  ,  3.2218  , -3.5667  ,  3.6457  , -3.1236  ,  3.4015  ,\n",
       "       -2.2361  ,  3.5774  , -2.8059  ,  4.6763  ,  2.8623  ,  2.4022  ,\n",
       "        4.0854  ,  2.2737  ,  4.2111  , -4.8184  ,  4.211   , -0.61964 ,\n",
       "        4.541   , -4.9658  ,  3.6774  , -0.93472 ,  4.0845  , -3.1372  ,\n",
       "        3.4721  , -2.9124  ,  4.274   ,  0.097471,  3.6908  , -2.9873  ,\n",
       "        4.0399  , -2.8669  ,  4.1302  , -1.6036  ,  3.5711  ,  4.4288  ,\n",
       "        5.3364  ,  4.8365  ,  2.913   , -2.9483  , -4.8586  , -4.4156  ,\n",
       "        0.65682 , -6.0976  , -4.0427  , -2.0815  , -3.8932  ,  3.352   ,\n",
       "       -4.2808  ,  4.1862  ,  1.2056  , -1.2377  ,  3.2969  , -0.70217 ,\n",
       "        4.1146  , -7.9094  , -1.5287  , -3.0705  , -3.6431  , -5.4596  ,\n",
       "       -0.49322 , -1.7353  , -0.030879, -3.5208  , -2.0409  , -4.5003  ,\n",
       "       -2.5341  , -0.79956 , -3.9905  , -6.2081  , -6.2917  , -1.14    ,\n",
       "       -4.5251  , -6.6058  , -3.6462  ,  4.0095  ,  4.9832  , -3.1972  ,\n",
       "        3.7497  , -2.6378  ,  3.2878  , -3.3728  ,  3.2631  , -2.5654  ,\n",
       "        3.8138  , -3.075   , -7.4818  , -7.7904  , -2.6232  , -4.3258  ,\n",
       "       -1.5386  , -1.3164  , -3.4128  ,  3.7795  , -2.2473  ,  4.2704  ,\n",
       "       -2.8967  ,  3.4568  , -1.9547  ,  4.9374  , -3.8674  ,  4.6291  ,\n",
       "       -2.2421  ,  4.1042  ,  1.4501  ,  4.601   , -2.5942  ,  4.2225  ,\n",
       "       -0.44259 ,  3.5856  , -2.8579  ,  3.9053  , -0.78559 ,  3.5789  ,\n",
       "       -1.1239  , -2.3201  , -2.5773  ,  1.9398  , -7.969   , -5.1357  ,\n",
       "       -3.7722  , -1.8879  , -0.87956 , -5.0074  , -0.17428 , -2.8674  ,\n",
       "       -0.49909 , -5.7375  ,  2.4581  , -8.8572  , -2.2742  , -2.0785  ,\n",
       "       -2.1301  , -4.8923  , -4.1633  , -4.9662  , -1.7976  , -4.797   ,\n",
       "       -4.0304  ,  3.9319  ,  2.9496  ,  4.4533  ,  3.6769  ,  3.7776  ,\n",
       "       -2.1358  , -3.3866  , -2.4989  ,  5.4275  ,  3.2714  ,  3.6667  ,\n",
       "        3.1996  ,  3.9487  ,  3.323   ,  3.4077  ,  2.5883  ,  4.9396  ,\n",
       "        3.4996  ,  3.9549  ,  4.4672  ,  3.9771  ,  3.2576  ,  3.6935  ,\n",
       "       -1.2999  ,  4.3998  , -2.1732  ,  3.5987  , -4.2041  ,  3.9004  ,\n",
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       "       -0.72718 ,  4.631   , -2.018   ,  4.7558  , -2.9022  , -2.6121  ,\n",
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       "       -4.0841  ,  4.5484  , -4.3493  ,  2.9174  , -4.4949  ,  5.2867  ,\n",
       "        3.4909  ,  2.7198  ,  2.8146  ,  3.1129  ,  3.3719  ,  2.8676  ,\n",
       "        2.7679  ,  3.8195  ,  4.7983  ,  3.3473  ,  5.1468  ,  2.7829  ,\n",
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       "       -6.9113  , -1.7182  ])"
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      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#now let's plot \n",
    "from plotly.offline import download_plotlyjs, init_notebook_mode, plot, iplot\n",
    "\n",
    "from plotly.graph_objs import Scatter, Figure, Layout\n",
    "\n",
    "trace = Scatter(\n",
    "    x = X[:,0],\n",
    "    y = X[:,1],\n",
    "    mode = 'markers'\n",
    ")\n",
    "\n",
    "centroids = Scatter(\n",
    "    x = lmu[:,0]\n",
    "    y = lmu[:,1]\n",
    ")\n",
    "\n",
    "data = [trace]\n",
    "\n",
    "# Plot and embed in ipython notebook!\n",
    "iplot(data, filename='basic-scatter')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 108,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[  6811.90148996,   2270.04417975],\n",
       "       [  2270.04417975,  10389.73236583]])"
      ]
     },
     "execution_count": 108,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "k=1\n",
    "np.dot(np.transpose((X - lmu[k])),(X - lmu[k]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.70680149,  0.70680149])"
      ]
     },
     "execution_count": 101,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lmu[1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.70746067,  0.        ],\n",
       "       [ 0.        ,  0.70746067]])"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mu0\n",
    "sigma0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div id=\"7184f990-5f88-43e5-ba84-beebe81d017f\" style=\"height: 525px; width: 100%;\" class=\"plotly-graph-div\"></div><script type=\"text/javascript\">require([\"plotly\"], function(Plotly) { window.PLOTLYENV=window.PLOTLYENV || {};window.PLOTLYENV.BASE_URL=\"https://plot.ly\";Plotly.newPlot(\"7184f990-5f88-43e5-ba84-beebe81d017f\", [{\"type\": \"scatter\", \"x\": [1, 2, 3], \"y\": [3, 1, 6]}], {}, {\"linkText\": \"Export to plot.ly\", \"showLink\": true})});</script>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from plotly.offline import download_plotlyjs, init_notebook_mode, plot, iplot\n",
    "\n",
    "from plotly.graph_objs import Scatter, Figure, Layout\n",
    "\n",
    "iplot([Scatter(x=[1, 2, 3], y=[3, 1, 6])])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
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  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "Python [conda root]",
   "language": "python",
   "name": "conda-root-py"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.5.2"
  }
 },
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 "nbformat_minor": 1
}