src/chapter6.ipynb

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     "cell_type": "markdown",
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     "source": [
      "## Chapter 6: Feature Selection ##"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from __future__ import division\n",
      "import pandas as pd\n",
      "import numpy as np\n",
      "import scipy as sp\n",
      "import matplotlib.pyplot as plt\n",
      "from sklearn.linear_model import LinearRegression\n",
      "from sklearn.linear_model import Ridge\n",
      "from sklearn.linear_model import Lasso\n",
      "from sklearn.feature_selection import SelectKBest\n",
      "from sklearn.feature_selection import f_regression\n",
      "from sklearn.metrics import mean_squared_error\n",
      "from sklearn.cross_validation import KFold\n",
      "%matplotlib inline"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "hitters_df = pd.read_csv(\"../data/Hitters.csv\")\n",
      "hitters_df.dropna(inplace=True)\n",
      "hitters_df.head()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "html": [
        "<div style=\"max-height:1000px;max-width:1500px;overflow:auto;\">\n",
        "<table border=\"1\" class=\"dataframe\">\n",
        "  <thead>\n",
        "    <tr style=\"text-align: right;\">\n",
        "      <th></th>\n",
        "      <th>AtBat</th>\n",
        "      <th>Hits</th>\n",
        "      <th>HmRun</th>\n",
        "      <th>Runs</th>\n",
        "      <th>RBI</th>\n",
        "      <th>Walks</th>\n",
        "      <th>Years</th>\n",
        "      <th>CAtBat</th>\n",
        "      <th>CHits</th>\n",
        "      <th>CHmRun</th>\n",
        "      <th>CRuns</th>\n",
        "      <th>CRBI</th>\n",
        "      <th>CWalks</th>\n",
        "      <th>League</th>\n",
        "      <th>Division</th>\n",
        "      <th>PutOuts</th>\n",
        "      <th>Assists</th>\n",
        "      <th>Errors</th>\n",
        "      <th>Salary</th>\n",
        "      <th>NewLeague</th>\n",
        "    </tr>\n",
        "  </thead>\n",
        "  <tbody>\n",
        "    <tr>\n",
        "      <th>1</th>\n",
        "      <td> 315</td>\n",
        "      <td>  81</td>\n",
        "      <td>  7</td>\n",
        "      <td> 24</td>\n",
        "      <td> 38</td>\n",
        "      <td> 39</td>\n",
        "      <td> 14</td>\n",
        "      <td> 3449</td>\n",
        "      <td>  835</td>\n",
        "      <td>  69</td>\n",
        "      <td> 321</td>\n",
        "      <td> 414</td>\n",
        "      <td> 375</td>\n",
        "      <td> N</td>\n",
        "      <td> W</td>\n",
        "      <td> 632</td>\n",
        "      <td>  43</td>\n",
        "      <td> 10</td>\n",
        "      <td> 475.0</td>\n",
        "      <td> N</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>2</th>\n",
        "      <td> 479</td>\n",
        "      <td> 130</td>\n",
        "      <td> 18</td>\n",
        "      <td> 66</td>\n",
        "      <td> 72</td>\n",
        "      <td> 76</td>\n",
        "      <td>  3</td>\n",
        "      <td> 1624</td>\n",
        "      <td>  457</td>\n",
        "      <td>  63</td>\n",
        "      <td> 224</td>\n",
        "      <td> 266</td>\n",
        "      <td> 263</td>\n",
        "      <td> A</td>\n",
        "      <td> W</td>\n",
        "      <td> 880</td>\n",
        "      <td>  82</td>\n",
        "      <td> 14</td>\n",
        "      <td> 480.0</td>\n",
        "      <td> A</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>3</th>\n",
        "      <td> 496</td>\n",
        "      <td> 141</td>\n",
        "      <td> 20</td>\n",
        "      <td> 65</td>\n",
        "      <td> 78</td>\n",
        "      <td> 37</td>\n",
        "      <td> 11</td>\n",
        "      <td> 5628</td>\n",
        "      <td> 1575</td>\n",
        "      <td> 225</td>\n",
        "      <td> 828</td>\n",
        "      <td> 838</td>\n",
        "      <td> 354</td>\n",
        "      <td> N</td>\n",
        "      <td> E</td>\n",
        "      <td> 200</td>\n",
        "      <td>  11</td>\n",
        "      <td>  3</td>\n",
        "      <td> 500.0</td>\n",
        "      <td> N</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>4</th>\n",
        "      <td> 321</td>\n",
        "      <td>  87</td>\n",
        "      <td> 10</td>\n",
        "      <td> 39</td>\n",
        "      <td> 42</td>\n",
        "      <td> 30</td>\n",
        "      <td>  2</td>\n",
        "      <td>  396</td>\n",
        "      <td>  101</td>\n",
        "      <td>  12</td>\n",
        "      <td>  48</td>\n",
        "      <td>  46</td>\n",
        "      <td>  33</td>\n",
        "      <td> N</td>\n",
        "      <td> E</td>\n",
        "      <td> 805</td>\n",
        "      <td>  40</td>\n",
        "      <td>  4</td>\n",
        "      <td>  91.5</td>\n",
        "      <td> N</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>5</th>\n",
        "      <td> 594</td>\n",
        "      <td> 169</td>\n",
        "      <td>  4</td>\n",
        "      <td> 74</td>\n",
        "      <td> 51</td>\n",
        "      <td> 35</td>\n",
        "      <td> 11</td>\n",
        "      <td> 4408</td>\n",
        "      <td> 1133</td>\n",
        "      <td>  19</td>\n",
        "      <td> 501</td>\n",
        "      <td> 336</td>\n",
        "      <td> 194</td>\n",
        "      <td> A</td>\n",
        "      <td> W</td>\n",
        "      <td> 282</td>\n",
        "      <td> 421</td>\n",
        "      <td> 25</td>\n",
        "      <td> 750.0</td>\n",
        "      <td> A</td>\n",
        "    </tr>\n",
        "  </tbody>\n",
        "</table>\n",
        "<p>5 rows \u00d7 20 columns</p>\n",
        "</div>"
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       "prompt_number": 2,
       "text": [
        "   AtBat  Hits  HmRun  Runs  RBI  Walks  Years  CAtBat  CHits  CHmRun  CRuns  \\\n",
        "1    315    81      7    24   38     39     14    3449    835      69    321   \n",
        "2    479   130     18    66   72     76      3    1624    457      63    224   \n",
        "3    496   141     20    65   78     37     11    5628   1575     225    828   \n",
        "4    321    87     10    39   42     30      2     396    101      12     48   \n",
        "5    594   169      4    74   51     35     11    4408   1133      19    501   \n",
        "\n",
        "   CRBI  CWalks League Division  PutOuts  Assists  Errors  Salary NewLeague  \n",
        "1   414     375      N        W      632       43      10   475.0         N  \n",
        "2   266     263      A        W      880       82      14   480.0         A  \n",
        "3   838     354      N        E      200       11       3   500.0         N  \n",
        "4    46      33      N        E      805       40       4    91.5         N  \n",
        "5   336     194      A        W      282      421      25   750.0         A  \n",
        "\n",
        "[5 rows x 20 columns]"
       ]
      }
     ],
     "prompt_number": 2
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    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Objective is to predict the hitter's salary given other variables. Because this is a regression problem, we first need to convert the non-numeric input variables to factors."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Converting non-numeric input variables to factors.\n",
      "hitters_df[\"League\"] = pd.factorize(hitters_df[\"League\"])[0]\n",
      "hitters_df[\"Division\"] = pd.factorize(hitters_df[\"Division\"])[0]\n",
      "hitters_df[\"NewLeague\"] = pd.factorize(hitters_df[\"NewLeague\"])[0]\n",
      "hitters_df.head()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "html": [
        "<div style=\"max-height:1000px;max-width:1500px;overflow:auto;\">\n",
        "<table border=\"1\" class=\"dataframe\">\n",
        "  <thead>\n",
        "    <tr style=\"text-align: right;\">\n",
        "      <th></th>\n",
        "      <th>AtBat</th>\n",
        "      <th>Hits</th>\n",
        "      <th>HmRun</th>\n",
        "      <th>Runs</th>\n",
        "      <th>RBI</th>\n",
        "      <th>Walks</th>\n",
        "      <th>Years</th>\n",
        "      <th>CAtBat</th>\n",
        "      <th>CHits</th>\n",
        "      <th>CHmRun</th>\n",
        "      <th>CRuns</th>\n",
        "      <th>CRBI</th>\n",
        "      <th>CWalks</th>\n",
        "      <th>League</th>\n",
        "      <th>Division</th>\n",
        "      <th>PutOuts</th>\n",
        "      <th>Assists</th>\n",
        "      <th>Errors</th>\n",
        "      <th>Salary</th>\n",
        "      <th>NewLeague</th>\n",
        "    </tr>\n",
        "  </thead>\n",
        "  <tbody>\n",
        "    <tr>\n",
        "      <th>1</th>\n",
        "      <td> 315</td>\n",
        "      <td>  81</td>\n",
        "      <td>  7</td>\n",
        "      <td> 24</td>\n",
        "      <td> 38</td>\n",
        "      <td> 39</td>\n",
        "      <td> 14</td>\n",
        "      <td> 3449</td>\n",
        "      <td>  835</td>\n",
        "      <td>  69</td>\n",
        "      <td> 321</td>\n",
        "      <td> 414</td>\n",
        "      <td> 375</td>\n",
        "      <td> 0</td>\n",
        "      <td> 0</td>\n",
        "      <td> 632</td>\n",
        "      <td>  43</td>\n",
        "      <td> 10</td>\n",
        "      <td> 475.0</td>\n",
        "      <td> 0</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>2</th>\n",
        "      <td> 479</td>\n",
        "      <td> 130</td>\n",
        "      <td> 18</td>\n",
        "      <td> 66</td>\n",
        "      <td> 72</td>\n",
        "      <td> 76</td>\n",
        "      <td>  3</td>\n",
        "      <td> 1624</td>\n",
        "      <td>  457</td>\n",
        "      <td>  63</td>\n",
        "      <td> 224</td>\n",
        "      <td> 266</td>\n",
        "      <td> 263</td>\n",
        "      <td> 1</td>\n",
        "      <td> 0</td>\n",
        "      <td> 880</td>\n",
        "      <td>  82</td>\n",
        "      <td> 14</td>\n",
        "      <td> 480.0</td>\n",
        "      <td> 1</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>3</th>\n",
        "      <td> 496</td>\n",
        "      <td> 141</td>\n",
        "      <td> 20</td>\n",
        "      <td> 65</td>\n",
        "      <td> 78</td>\n",
        "      <td> 37</td>\n",
        "      <td> 11</td>\n",
        "      <td> 5628</td>\n",
        "      <td> 1575</td>\n",
        "      <td> 225</td>\n",
        "      <td> 828</td>\n",
        "      <td> 838</td>\n",
        "      <td> 354</td>\n",
        "      <td> 0</td>\n",
        "      <td> 1</td>\n",
        "      <td> 200</td>\n",
        "      <td>  11</td>\n",
        "      <td>  3</td>\n",
        "      <td> 500.0</td>\n",
        "      <td> 0</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>4</th>\n",
        "      <td> 321</td>\n",
        "      <td>  87</td>\n",
        "      <td> 10</td>\n",
        "      <td> 39</td>\n",
        "      <td> 42</td>\n",
        "      <td> 30</td>\n",
        "      <td>  2</td>\n",
        "      <td>  396</td>\n",
        "      <td>  101</td>\n",
        "      <td>  12</td>\n",
        "      <td>  48</td>\n",
        "      <td>  46</td>\n",
        "      <td>  33</td>\n",
        "      <td> 0</td>\n",
        "      <td> 1</td>\n",
        "      <td> 805</td>\n",
        "      <td>  40</td>\n",
        "      <td>  4</td>\n",
        "      <td>  91.5</td>\n",
        "      <td> 0</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>5</th>\n",
        "      <td> 594</td>\n",
        "      <td> 169</td>\n",
        "      <td>  4</td>\n",
        "      <td> 74</td>\n",
        "      <td> 51</td>\n",
        "      <td> 35</td>\n",
        "      <td> 11</td>\n",
        "      <td> 4408</td>\n",
        "      <td> 1133</td>\n",
        "      <td>  19</td>\n",
        "      <td> 501</td>\n",
        "      <td> 336</td>\n",
        "      <td> 194</td>\n",
        "      <td> 1</td>\n",
        "      <td> 0</td>\n",
        "      <td> 282</td>\n",
        "      <td> 421</td>\n",
        "      <td> 25</td>\n",
        "      <td> 750.0</td>\n",
        "      <td> 1</td>\n",
        "    </tr>\n",
        "  </tbody>\n",
        "</table>\n",
        "<p>5 rows \u00d7 20 columns</p>\n",
        "</div>"
       ],
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       "output_type": "pyout",
       "prompt_number": 3,
       "text": [
        "   AtBat  Hits  HmRun  Runs  RBI  Walks  Years  CAtBat  CHits  CHmRun  CRuns  \\\n",
        "1    315    81      7    24   38     39     14    3449    835      69    321   \n",
        "2    479   130     18    66   72     76      3    1624    457      63    224   \n",
        "3    496   141     20    65   78     37     11    5628   1575     225    828   \n",
        "4    321    87     10    39   42     30      2     396    101      12     48   \n",
        "5    594   169      4    74   51     35     11    4408   1133      19    501   \n",
        "\n",
        "   CRBI  CWalks  League  Division  PutOuts  Assists  Errors  Salary  NewLeague  \n",
        "1   414     375       0         0      632       43      10   475.0          0  \n",
        "2   266     263       1         0      880       82      14   480.0          1  \n",
        "3   838     354       0         1      200       11       3   500.0          0  \n",
        "4    46      33       0         1      805       40       4    91.5          0  \n",
        "5   336     194       1         0      282      421      25   750.0          1  \n",
        "\n",
        "[5 rows x 20 columns]"
       ]
      }
     ],
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     "input": [
      "# construct a baseline regressor with all features\n",
      "collist = [col for col in hitters_df.columns if col != \"Salary\"]\n",
      "X = hitters_df[collist]\n",
      "y = hitters_df[\"Salary\"]\n",
      "reg = LinearRegression()\n",
      "reg.fit(X, y)\n",
      "ypred = reg.predict(X)\n",
      "np.sqrt(mean_squared_error(ypred, y))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 4,
       "text": [
        "303.34447253531613"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Best Subset Regression ###\n",
      "\n",
      "R's leap package allows for multiple ways to find the best K features for a model - by exhaustive search and forward (greedy) search. Scikit-Learn offers the SelectKBest method to find the best features, presumably using a greedy search."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "mses = []\n",
      "nfeatures = range(1, len(collist))\n",
      "for nfeature in nfeatures:\n",
      "    # compute MSE for different values of k (top features)\n",
      "    selector = SelectKBest(f_regression, k=nfeature)\n",
      "    selector.fit(X, y)\n",
      "    selected = selector.get_support()\n",
      "    feats = [col for (col,sel) in zip(collist, selected) if sel]\n",
      "    reg = LinearRegression()\n",
      "    X_r = hitters_df[feats]\n",
      "    reg.fit(X_r, y)\n",
      "    ypred = reg.predict(X_r)\n",
      "    mses.append(np.sqrt(mean_squared_error(ypred, y)))\n",
      "plt.plot(nfeatures, mses)\n",
      "plt.xlabel(\"k\")\n",
      "plt.ylabel(\"RMSE\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 5,
       "text": [
        "<matplotlib.text.Text at 0x4a0cad0>"
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jj8O+fXDTTVoDYRUVBxEJOuHhsGIFvP22uWW49Dw12EQkKJ14IqxaBZMnw+mn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\nrZkxSAEAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x49f1ad0>"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Model Selection by Cross-Validation ###\n",
      "\n",
      "The RMSE falls as the number of features increase - this is expected because we are computing the RMSE off the training set (overfitting). We will now use 10-fold cross validation on each model to calculate a cross-validation MSE which will give us a better idea of the best feature size to use for the problem."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "cv_errors = []\n",
      "kfold = KFold(len(hitters_df), n_folds=10)\n",
      "nfeatures = range(1, len(collist))\n",
      "for nfeature in nfeatures:\n",
      "    # build model with varying number of features\n",
      "    selector = SelectKBest(f_regression, k=nfeature)\n",
      "    selector.fit(X, y)\n",
      "    selected = selector.get_support()\n",
      "    feats = [col for (col,sel) in zip(collist, selected) if sel]\n",
      "    X_r = hitters_df[feats].values\n",
      "    y = hitters_df[\"Salary\"].values\n",
      "    rmses = []\n",
      "    for train, test in kfold:\n",
      "        # each model is cross validated 10 times\n",
      "        Xtrain, ytrain, Xtest, ytest = X_r[train], y[train], X_r[test], y[test]   \n",
      "        reg = LinearRegression()\n",
      "        reg.fit(Xtrain, ytrain)\n",
      "        ypred = reg.predict(Xtest)\n",
      "        rmses.append(np.sqrt(mean_squared_error(ypred, ytest)))\n",
      "    cv_errors.append(np.mean(rmses))\n",
      "plt.plot(nfeatures, cv_errors)\n",
      "plt.xlabel(\"k\")\n",
      "plt.ylabel(\"RMSE\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 6,
       "text": [
        "<matplotlib.text.Text at 0x4c84710>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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m5VBZWUlcXBw2mw2r1UpaWhoA8+fPJyYmBpvNRmJiIkVFRQAUFhbSvn177HY7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\n40y57ijuAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x4c75090>"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Ridge Regression and the Lasso ###\n",
      "\n",
      "These two methods improve the model by using all features but shrinking the coefficients. Ridge regression uses the L2-norm and Lasso regression uses the L1-norm. Here we use cross validation to compute the RMSE for a baseline model, a model regularized by Ridge regression and one regularized using Lasso."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def cross_validate(X, y, nfolds, reg_name):\n",
      "    rmses = []\n",
      "    kfold = KFold(X.shape[0], n_folds=nfolds)\n",
      "    for train, test in kfold:\n",
      "        Xtrain, ytrain, Xtest, ytest = X[train], y[train], X[test], y[test]\n",
      "        reg = None\n",
      "        if reg_name == \"ridge\":\n",
      "            reg = Ridge()\n",
      "        elif reg_name == \"lasso\":\n",
      "            reg = Lasso()\n",
      "        else:\n",
      "            reg = LinearRegression()\n",
      "        reg.fit(Xtrain, ytrain)\n",
      "        ypred = reg.predict(Xtest)\n",
      "        rmses.append(np.sqrt(mean_squared_error(ytest, ypred)))\n",
      "    return np.mean(rmses)\n",
      "\n",
      "collist = [col for col in hitters_df.columns if col != \"Salary\"]\n",
      "X = hitters_df[collist].values\n",
      "y = hitters_df[\"Salary\"].values\n",
      "rmse_baseline = cross_validate(X, y, 10, \"baseline\")\n",
      "rmse_ridge = cross_validate(X, y, 10, \"ridge\")\n",
      "rmse_lasso = cross_validate(X, y, 10, \"lasso\")\n",
      "(rmse_baseline, rmse_ridge, rmse_lasso)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 8,
       "text": [
        "(331.00425667935832, 330.72572506106593, 330.00460161612779)"
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Finally, we attempt to find an optimum value of alpha for the Lasso regressor using cross-validation."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "cv_errors = []\n",
      "alphas = [0.1 * alpha for alpha in range(1, 200, 20)]\n",
      "kfold = KFold(X.shape[0], n_folds=10)\n",
      "for alpha in alphas:\n",
      "    rmses = []\n",
      "    for train, test in kfold:\n",
      "        Xtrain, ytrain, Xtest, ytest = X[train], y[train], X[test], y[test]\n",
      "        reg = Lasso(alpha=alpha)\n",
      "        reg.fit(Xtrain, ytrain)\n",
      "        ypred = reg.predict(Xtest)\n",
      "        rmses.append(np.sqrt(mean_squared_error(ytest, ypred)))\n",
      "    cv_errors.append(np.mean(rmses))\n",
      "plt.plot(alphas, cv_errors)\n",
      "plt.xlabel(\"alpha\")\n",
      "plt.ylabel(\"RMSE\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 9,
       "text": [
        "<matplotlib.text.Text at 0x4ef3590>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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ylESuUxKRn7LbIS0NZs2CDh1gyRJjIF5E/k2zs0Qq4eNjnPv++efGSYu9exu7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\ntvD555/z9ttvc/LkSU+HVau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       "text": [
        "<matplotlib.figure.Figure at 0x49a6cd0>"
       ]
      }
     ],
     "prompt_number": 9
    }
   ],
   "metadata": {}
  }
 ]
}