09-sklearn-lecture-solutions.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Linear Modeling with `scikit-learn`\n",
    "---\n",
    "\n",
    "The two most popular packages in Python for linear modeling (and more) are `scikit-learn` and `statsmodels`.  For people comfortable with the output of linear models in R, you may prefer the `statsmodels` package.  However, `scikit-learn` has much more functionality including the most popular machine learning methods, evaluation techniques, and data preprocessing.  It also makes it easy to spread your model building and evaluation across multiple CPU cores.\n",
    "\n",
    "This being the case, we'll cover them both, but with more emphasis on `scikit-learn`.  You may end up using parts of both packages in your analyses.  Both are compatible with NumPy arrays or Pandas data frames, so your knowledge of those packages will help you here.\n",
    "\n",
    "Unlike NumPy or Pandas, functionality is split up into many different `modules` that each must be imported separately.  Because of this, there isn't a simple `import numpy as np`-type import we can use.  Instead, we import each module we want to use separately, and to reduce the amount of typing, we'll use the `from` syntax to import functions and variables."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "import seaborn as sns\n",
    "import sqlalchemy\n",
    "\n",
    "from sklearn.datasets import *\n",
    "from sklearn.dummy import *\n",
    "from sklearn.linear_model import *\n",
    "from sklearn.metrics import *\n",
    "from sklearn.preprocessing import *\n",
    "\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This will import everything (indicated by the asterisk) from these modules into our namespace.  In general this is considered sub-ideal and you would instead list by hand each function you want to import, but we'll use this shortcut for convenience here.\n",
    "\n",
    "Since everyone has experience with linear modeling, I'll focus here on their application in Python rather than the underlying math."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Boston Housing Dataset\n",
    "---\n",
    "\n",
    "`scikit-learn` comes with several commonly used datasets in the `sklearn.datasets` module.  Each dataset has load function, for example `load_boston()`, which returns a dictionary:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "dict_keys(['data', 'target', 'feature_names', 'DESCR'])"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "boston = load_boston()\n",
    "boston.keys()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The first 3 are Numpy arrays and the last is a string describing the dataset.\n",
    "\n",
    "We have a 2D feature matrix with 516 samples having 13 features each:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(506, 13)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "boston['data'].shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And a single quantitative value we can use for modeling, one value for each of the 506 samples:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(506,)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "boston['target'].shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "These are NumPy arrays, so unlike a data frame, we won't have column or row names, but column names are provided for reference:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array(['CRIM', 'ZN', 'INDUS', 'CHAS', 'NOX', 'RM', 'AGE', 'DIS', 'RAD',\n",
       "       'TAX', 'PTRATIO', 'B', 'LSTAT'], \n",
       "      dtype='<U7')"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "boston['feature_names']"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, a description of all the independent and dependent variables is provided:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Boston House Prices dataset\n",
      "===========================\n",
      "\n",
      "Notes\n",
      "------\n",
      "Data Set Characteristics:  \n",
      "\n",
      "    :Number of Instances: 506 \n",
      "\n",
      "    :Number of Attributes: 13 numeric/categorical predictive\n",
      "    \n",
      "    :Median Value (attribute 14) is usually the target\n",
      "\n",
      "    :Attribute Information (in order):\n",
      "        - CRIM     per capita crime rate by town\n",
      "        - ZN       proportion of residential land zoned for lots over 25,000 sq.ft.\n",
      "        - INDUS    proportion of non-retail business acres per town\n",
      "        - CHAS     Charles River dummy variable (= 1 if tract bounds river; 0 otherwise)\n",
      "        - NOX      nitric oxides concentration (parts per 10 million)\n",
      "        - RM       average number of rooms per dwelling\n",
      "        - AGE      proportion of owner-occupied units built prior to 1940\n",
      "        - DIS      weighted distances to five Boston employment centres\n",
      "        - RAD      index of accessibility to radial highways\n",
      "        - TAX      full-value property-tax rate per $10,000\n",
      "        - PTRATIO  pupil-teacher ratio by town\n",
      "        - B        1000(Bk - 0.63)^2 where Bk is the proportion of blacks by town\n",
      "        - LSTAT    % lower status of the population\n",
      "        - MEDV     Median value of owner-occupied homes in $1000's\n",
      "\n",
      "    :Missing Attribute Values: None\n",
      "\n",
      "    :Creator: Harrison, D. and Rubinfeld, D.L.\n",
      "\n",
      "This is a copy of UCI ML housing dataset.\n",
      "http://archive.ics.uci.edu/ml/datasets/Housing\n",
      "\n",
      "\n",
      "This dataset was taken from the StatLib library which is maintained at Carnegie Mellon University.\n",
      "\n",
      "The Boston house-price data of Harrison, D. and Rubinfeld, D.L. 'Hedonic\n",
      "prices and the demand for clean air', J. Environ. Economics & Management,\n",
      "vol.5, 81-102, 1978.   Used in Belsley, Kuh & Welsch, 'Regression diagnostics\n",
      "...', Wiley, 1980.   N.B. Various transformations are used in the table on\n",
      "pages 244-261 of the latter.\n",
      "\n",
      "The Boston house-price data has been used in many machine learning papers that address regression\n",
      "problems.   \n",
      "     \n",
      "**References**\n",
      "\n",
      "   - Belsley, Kuh & Welsch, 'Regression diagnostics: Identifying Influential Data and Sources of Collinearity', Wiley, 1980. 244-261.\n",
      "   - Quinlan,R. (1993). Combining Instance-Based and Model-Based Learning. In Proceedings on the Tenth International Conference of Machine Learning, 236-243, University of Massachusetts, Amherst. Morgan Kaufmann.\n",
      "   - many more! (see http://archive.ics.uci.edu/ml/datasets/Housing)\n",
      "\n"
     ]
    }
   ],
   "source": [
    "print(boston['DESCR'])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It's common to see `X` as the name of the 2D NumPy array of predictors / features / independent variables, and `y` as the name of our targets / labels / dependent variable / response variable.  This is the naming convention used in the `scikit-learn` tutorials.  There are unfortunately many different names for what `X` and `y` represent, depending on the community (statistics vs. machine learning), and even within the same community.  The terms above may be used interchangeably here.\n",
    "\n",
    "Let's create a Pandas data frame instead of a NumPy array so we can access columns by name:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
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       "      <td>242.0</td>\n",
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       "      <td>4.03</td>\n",
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       "      <td>0.458</td>\n",
       "      <td>6.998</td>\n",
       "      <td>45.8</td>\n",
       "      <td>6.0622</td>\n",
       "      <td>3.0</td>\n",
       "      <td>222.0</td>\n",
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       "      <td>2.94</td>\n",
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       "      <th>4</th>\n",
       "      <td>0.06905</td>\n",
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       "      <td>0.0</td>\n",
       "      <td>0.458</td>\n",
       "      <td>7.147</td>\n",
       "      <td>54.2</td>\n",
       "      <td>6.0622</td>\n",
       "      <td>3.0</td>\n",
       "      <td>222.0</td>\n",
       "      <td>18.7</td>\n",
       "      <td>396.90</td>\n",
       "      <td>5.33</td>\n",
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       "      CRIM    ZN  INDUS  CHAS    NOX     RM   AGE     DIS  RAD    TAX  \\\n",
       "0  0.00632  18.0   2.31   0.0  0.538  6.575  65.2  4.0900  1.0  296.0   \n",
       "1  0.02731   0.0   7.07   0.0  0.469  6.421  78.9  4.9671  2.0  242.0   \n",
       "2  0.02729   0.0   7.07   0.0  0.469  7.185  61.1  4.9671  2.0  242.0   \n",
       "3  0.03237   0.0   2.18   0.0  0.458  6.998  45.8  6.0622  3.0  222.0   \n",
       "4  0.06905   0.0   2.18   0.0  0.458  7.147  54.2  6.0622  3.0  222.0   \n",
       "\n",
       "   PTRATIO       B  LSTAT  \n",
       "0     15.3  396.90   4.98  \n",
       "1     17.8  396.90   9.14  \n",
       "2     17.8  392.83   4.03  \n",
       "3     18.7  394.63   2.94  \n",
       "4     18.7  396.90   5.33  "
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "features_df = pd.DataFrame(boston['data'], columns = boston['feature_names'])\n",
    "features_df.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The response variable can go into a 1D series:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0    24.0\n",
       "1    21.6\n",
       "2    34.7\n",
       "3    33.4\n",
       "4    36.2\n",
       "Name: MEDV, dtype: float64"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "labels = pd.Series(boston['target'], name = 'MEDV')\n",
    "labels.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, it can be useful to combine the features and labels together into a single dataframe using `concat()` with `axis = 1` to indicate concatenation across rows:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
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       "      <td>2.94</td>\n",
       "      <td>33.4</td>\n",
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       "      <th>4</th>\n",
       "      <td>0.06905</td>\n",
       "      <td>0.0</td>\n",
       "      <td>2.18</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.458</td>\n",
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       "      <td>54.2</td>\n",
       "      <td>6.0622</td>\n",
       "      <td>3.0</td>\n",
       "      <td>222.0</td>\n",
       "      <td>18.7</td>\n",
       "      <td>396.90</td>\n",
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       "      CRIM    ZN  INDUS  CHAS    NOX     RM   AGE     DIS  RAD    TAX  \\\n",
       "0  0.00632  18.0   2.31   0.0  0.538  6.575  65.2  4.0900  1.0  296.0   \n",
       "1  0.02731   0.0   7.07   0.0  0.469  6.421  78.9  4.9671  2.0  242.0   \n",
       "2  0.02729   0.0   7.07   0.0  0.469  7.185  61.1  4.9671  2.0  242.0   \n",
       "3  0.03237   0.0   2.18   0.0  0.458  6.998  45.8  6.0622  3.0  222.0   \n",
       "4  0.06905   0.0   2.18   0.0  0.458  7.147  54.2  6.0622  3.0  222.0   \n",
       "\n",
       "   PTRATIO       B  LSTAT  MEDV  \n",
       "0     15.3  396.90   4.98  24.0  \n",
       "1     17.8  396.90   9.14  21.6  \n",
       "2     17.8  392.83   4.03  34.7  \n",
       "3     18.7  394.63   2.94  33.4  \n",
       "4     18.7  396.90   5.33  36.2  "
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "training_df = pd.concat([features_df, labels], axis = 1)\n",
    "training_df.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>CRIM</th>\n",
       "      <th>ZN</th>\n",
       "      <th>INDUS</th>\n",
       "      <th>CHAS</th>\n",
       "      <th>NOX</th>\n",
       "      <th>RM</th>\n",
       "      <th>AGE</th>\n",
       "      <th>DIS</th>\n",
       "      <th>RAD</th>\n",
       "      <th>TAX</th>\n",
       "      <th>PTRATIO</th>\n",
       "      <th>B</th>\n",
       "      <th>LSTAT</th>\n",
       "      <th>MEDV</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>count</th>\n",
       "      <td>506.000000</td>\n",
       "      <td>506.000000</td>\n",
       "      <td>506.000000</td>\n",
       "      <td>506.000000</td>\n",
       "      <td>506.000000</td>\n",
       "      <td>506.000000</td>\n",
       "      <td>506.000000</td>\n",
       "      <td>506.000000</td>\n",
       "      <td>506.000000</td>\n",
       "      <td>506.000000</td>\n",
       "      <td>506.000000</td>\n",
       "      <td>506.000000</td>\n",
       "      <td>506.000000</td>\n",
       "      <td>506.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>mean</th>\n",
       "      <td>3.593761</td>\n",
       "      <td>11.363636</td>\n",
       "      <td>11.136779</td>\n",
       "      <td>0.069170</td>\n",
       "      <td>0.554695</td>\n",
       "      <td>6.284634</td>\n",
       "      <td>68.574901</td>\n",
       "      <td>3.795043</td>\n",
       "      <td>9.549407</td>\n",
       "      <td>408.237154</td>\n",
       "      <td>18.455534</td>\n",
       "      <td>356.674032</td>\n",
       "      <td>12.653063</td>\n",
       "      <td>22.532806</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>std</th>\n",
       "      <td>8.596783</td>\n",
       "      <td>23.322453</td>\n",
       "      <td>6.860353</td>\n",
       "      <td>0.253994</td>\n",
       "      <td>0.115878</td>\n",
       "      <td>0.702617</td>\n",
       "      <td>28.148861</td>\n",
       "      <td>2.105710</td>\n",
       "      <td>8.707259</td>\n",
       "      <td>168.537116</td>\n",
       "      <td>2.164946</td>\n",
       "      <td>91.294864</td>\n",
       "      <td>7.141062</td>\n",
       "      <td>9.197104</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>min</th>\n",
       "      <td>0.006320</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.460000</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.385000</td>\n",
       "      <td>3.561000</td>\n",
       "      <td>2.900000</td>\n",
       "      <td>1.129600</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>187.000000</td>\n",
       "      <td>12.600000</td>\n",
       "      <td>0.320000</td>\n",
       "      <td>1.730000</td>\n",
       "      <td>5.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25%</th>\n",
       "      <td>0.082045</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>5.190000</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.449000</td>\n",
       "      <td>5.885500</td>\n",
       "      <td>45.025000</td>\n",
       "      <td>2.100175</td>\n",
       "      <td>4.000000</td>\n",
       "      <td>279.000000</td>\n",
       "      <td>17.400000</td>\n",
       "      <td>375.377500</td>\n",
       "      <td>6.950000</td>\n",
       "      <td>17.025000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>50%</th>\n",
       "      <td>0.256510</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>9.690000</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.538000</td>\n",
       "      <td>6.208500</td>\n",
       "      <td>77.500000</td>\n",
       "      <td>3.207450</td>\n",
       "      <td>5.000000</td>\n",
       "      <td>330.000000</td>\n",
       "      <td>19.050000</td>\n",
       "      <td>391.440000</td>\n",
       "      <td>11.360000</td>\n",
       "      <td>21.200000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>75%</th>\n",
       "      <td>3.647423</td>\n",
       "      <td>12.500000</td>\n",
       "      <td>18.100000</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.624000</td>\n",
       "      <td>6.623500</td>\n",
       "      <td>94.075000</td>\n",
       "      <td>5.188425</td>\n",
       "      <td>24.000000</td>\n",
       "      <td>666.000000</td>\n",
       "      <td>20.200000</td>\n",
       "      <td>396.225000</td>\n",
       "      <td>16.955000</td>\n",
       "      <td>25.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>max</th>\n",
       "      <td>88.976200</td>\n",
       "      <td>100.000000</td>\n",
       "      <td>27.740000</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.871000</td>\n",
       "      <td>8.780000</td>\n",
       "      <td>100.000000</td>\n",
       "      <td>12.126500</td>\n",
       "      <td>24.000000</td>\n",
       "      <td>711.000000</td>\n",
       "      <td>22.000000</td>\n",
       "      <td>396.900000</td>\n",
       "      <td>37.970000</td>\n",
       "      <td>50.000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "             CRIM          ZN       INDUS        CHAS         NOX          RM  \\\n",
       "count  506.000000  506.000000  506.000000  506.000000  506.000000  506.000000   \n",
       "mean     3.593761   11.363636   11.136779    0.069170    0.554695    6.284634   \n",
       "std      8.596783   23.322453    6.860353    0.253994    0.115878    0.702617   \n",
       "min      0.006320    0.000000    0.460000    0.000000    0.385000    3.561000   \n",
       "25%      0.082045    0.000000    5.190000    0.000000    0.449000    5.885500   \n",
       "50%      0.256510    0.000000    9.690000    0.000000    0.538000    6.208500   \n",
       "75%      3.647423   12.500000   18.100000    0.000000    0.624000    6.623500   \n",
       "max     88.976200  100.000000   27.740000    1.000000    0.871000    8.780000   \n",
       "\n",
       "              AGE         DIS         RAD         TAX     PTRATIO           B  \\\n",
       "count  506.000000  506.000000  506.000000  506.000000  506.000000  506.000000   \n",
       "mean    68.574901    3.795043    9.549407  408.237154   18.455534  356.674032   \n",
       "std     28.148861    2.105710    8.707259  168.537116    2.164946   91.294864   \n",
       "min      2.900000    1.129600    1.000000  187.000000   12.600000    0.320000   \n",
       "25%     45.025000    2.100175    4.000000  279.000000   17.400000  375.377500   \n",
       "50%     77.500000    3.207450    5.000000  330.000000   19.050000  391.440000   \n",
       "75%     94.075000    5.188425   24.000000  666.000000   20.200000  396.225000   \n",
       "max    100.000000   12.126500   24.000000  711.000000   22.000000  396.900000   \n",
       "\n",
       "            LSTAT        MEDV  \n",
       "count  506.000000  506.000000  \n",
       "mean    12.653063   22.532806  \n",
       "std      7.141062    9.197104  \n",
       "min      1.730000    5.000000  \n",
       "25%      6.950000   17.025000  \n",
       "50%     11.360000   21.200000  \n",
       "75%     16.955000   25.000000  \n",
       "max     37.970000   50.000000  "
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "training_df.describe()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's look at the distribution of our response variable, which is in units of $1000 US dollars:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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7r+vtBsY1X1XzMvPxzOyOiE7gKopfw215LHpl5uqI+BLwWeCrtPHxiIh3A8sy88ZKc7se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\nJAwDSRKGgSQJw0CSBPw/XClvKfcJj9sAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x114614358>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "labels.hist(grid = False)\n",
    "plt.xlabel('Price ($1000 USD)')\n",
    "plt.ylabel('Count');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The correlation between all pairs of numeric variables can be calculated using the `corr()` function of a data frame:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style>\n",
       "    .dataframe thead tr:only-child th {\n",
       "        text-align: right;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: left;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>CRIM</th>\n",
       "      <th>ZN</th>\n",
       "      <th>INDUS</th>\n",
       "      <th>CHAS</th>\n",
       "      <th>NOX</th>\n",
       "      <th>RM</th>\n",
       "      <th>AGE</th>\n",
       "      <th>DIS</th>\n",
       "      <th>RAD</th>\n",
       "      <th>TAX</th>\n",
       "      <th>PTRATIO</th>\n",
       "      <th>B</th>\n",
       "      <th>LSTAT</th>\n",
       "      <th>MEDV</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>CRIM</th>\n",
       "      <td>1.000000</td>\n",
       "      <td>-0.199458</td>\n",
       "      <td>0.404471</td>\n",
       "      <td>-0.055295</td>\n",
       "      <td>0.417521</td>\n",
       "      <td>-0.219940</td>\n",
       "      <td>0.350784</td>\n",
       "      <td>-0.377904</td>\n",
       "      <td>0.622029</td>\n",
       "      <td>0.579564</td>\n",
       "      <td>0.288250</td>\n",
       "      <td>-0.377365</td>\n",
       "      <td>0.452220</td>\n",
       "      <td>-0.385832</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>ZN</th>\n",
       "      <td>-0.199458</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>-0.533828</td>\n",
       "      <td>-0.042697</td>\n",
       "      <td>-0.516604</td>\n",
       "      <td>0.311991</td>\n",
       "      <td>-0.569537</td>\n",
       "      <td>0.664408</td>\n",
       "      <td>-0.311948</td>\n",
       "      <td>-0.314563</td>\n",
       "      <td>-0.391679</td>\n",
       "      <td>0.175520</td>\n",
       "      <td>-0.412995</td>\n",
       "      <td>0.360445</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>INDUS</th>\n",
       "      <td>0.404471</td>\n",
       "      <td>-0.533828</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.062938</td>\n",
       "      <td>0.763651</td>\n",
       "      <td>-0.391676</td>\n",
       "      <td>0.644779</td>\n",
       "      <td>-0.708027</td>\n",
       "      <td>0.595129</td>\n",
       "      <td>0.720760</td>\n",
       "      <td>0.383248</td>\n",
       "      <td>-0.356977</td>\n",
       "      <td>0.603800</td>\n",
       "      <td>-0.483725</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>CHAS</th>\n",
       "      <td>-0.055295</td>\n",
       "      <td>-0.042697</td>\n",
       "      <td>0.062938</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.091203</td>\n",
       "      <td>0.091251</td>\n",
       "      <td>0.086518</td>\n",
       "      <td>-0.099176</td>\n",
       "      <td>-0.007368</td>\n",
       "      <td>-0.035587</td>\n",
       "      <td>-0.121515</td>\n",
       "      <td>0.048788</td>\n",
       "      <td>-0.053929</td>\n",
       "      <td>0.175260</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>NOX</th>\n",
       "      <td>0.417521</td>\n",
       "      <td>-0.516604</td>\n",
       "      <td>0.763651</td>\n",
       "      <td>0.091203</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>-0.302188</td>\n",
       "      <td>0.731470</td>\n",
       "      <td>-0.769230</td>\n",
       "      <td>0.611441</td>\n",
       "      <td>0.668023</td>\n",
       "      <td>0.188933</td>\n",
       "      <td>-0.380051</td>\n",
       "      <td>0.590879</td>\n",
       "      <td>-0.427321</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>RM</th>\n",
       "      <td>-0.219940</td>\n",
       "      <td>0.311991</td>\n",
       "      <td>-0.391676</td>\n",
       "      <td>0.091251</td>\n",
       "      <td>-0.302188</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>-0.240265</td>\n",
       "      <td>0.205246</td>\n",
       "      <td>-0.209847</td>\n",
       "      <td>-0.292048</td>\n",
       "      <td>-0.355501</td>\n",
       "      <td>0.128069</td>\n",
       "      <td>-0.613808</td>\n",
       "      <td>0.695360</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>AGE</th>\n",
       "      <td>0.350784</td>\n",
       "      <td>-0.569537</td>\n",
       "      <td>0.644779</td>\n",
       "      <td>0.086518</td>\n",
       "      <td>0.731470</td>\n",
       "      <td>-0.240265</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>-0.747881</td>\n",
       "      <td>0.456022</td>\n",
       "      <td>0.506456</td>\n",
       "      <td>0.261515</td>\n",
       "      <td>-0.273534</td>\n",
       "      <td>0.602339</td>\n",
       "      <td>-0.376955</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>DIS</th>\n",
       "      <td>-0.377904</td>\n",
       "      <td>0.664408</td>\n",
       "      <td>-0.708027</td>\n",
       "      <td>-0.099176</td>\n",
       "      <td>-0.769230</td>\n",
       "      <td>0.205246</td>\n",
       "      <td>-0.747881</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>-0.494588</td>\n",
       "      <td>-0.534432</td>\n",
       "      <td>-0.232471</td>\n",
       "      <td>0.291512</td>\n",
       "      <td>-0.496996</td>\n",
       "      <td>0.249929</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>RAD</th>\n",
       "      <td>0.622029</td>\n",
       "      <td>-0.311948</td>\n",
       "      <td>0.595129</td>\n",
       "      <td>-0.007368</td>\n",
       "      <td>0.611441</td>\n",
       "      <td>-0.209847</td>\n",
       "      <td>0.456022</td>\n",
       "      <td>-0.494588</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.910228</td>\n",
       "      <td>0.464741</td>\n",
       "      <td>-0.444413</td>\n",
       "      <td>0.488676</td>\n",
       "      <td>-0.381626</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>TAX</th>\n",
       "      <td>0.579564</td>\n",
       "      <td>-0.314563</td>\n",
       "      <td>0.720760</td>\n",
       "      <td>-0.035587</td>\n",
       "      <td>0.668023</td>\n",
       "      <td>-0.292048</td>\n",
       "      <td>0.506456</td>\n",
       "      <td>-0.534432</td>\n",
       "      <td>0.910228</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.460853</td>\n",
       "      <td>-0.441808</td>\n",
       "      <td>0.543993</td>\n",
       "      <td>-0.468536</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>PTRATIO</th>\n",
       "      <td>0.288250</td>\n",
       "      <td>-0.391679</td>\n",
       "      <td>0.383248</td>\n",
       "      <td>-0.121515</td>\n",
       "      <td>0.188933</td>\n",
       "      <td>-0.355501</td>\n",
       "      <td>0.261515</td>\n",
       "      <td>-0.232471</td>\n",
       "      <td>0.464741</td>\n",
       "      <td>0.460853</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>-0.177383</td>\n",
       "      <td>0.374044</td>\n",
       "      <td>-0.507787</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>B</th>\n",
       "      <td>-0.377365</td>\n",
       "      <td>0.175520</td>\n",
       "      <td>-0.356977</td>\n",
       "      <td>0.048788</td>\n",
       "      <td>-0.380051</td>\n",
       "      <td>0.128069</td>\n",
       "      <td>-0.273534</td>\n",
       "      <td>0.291512</td>\n",
       "      <td>-0.444413</td>\n",
       "      <td>-0.441808</td>\n",
       "      <td>-0.177383</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>-0.366087</td>\n",
       "      <td>0.333461</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>LSTAT</th>\n",
       "      <td>0.452220</td>\n",
       "      <td>-0.412995</td>\n",
       "      <td>0.603800</td>\n",
       "      <td>-0.053929</td>\n",
       "      <td>0.590879</td>\n",
       "      <td>-0.613808</td>\n",
       "      <td>0.602339</td>\n",
       "      <td>-0.496996</td>\n",
       "      <td>0.488676</td>\n",
       "      <td>0.543993</td>\n",
       "      <td>0.374044</td>\n",
       "      <td>-0.366087</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>-0.737663</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>MEDV</th>\n",
       "      <td>-0.385832</td>\n",
       "      <td>0.360445</td>\n",
       "      <td>-0.483725</td>\n",
       "      <td>0.175260</td>\n",
       "      <td>-0.427321</td>\n",
       "      <td>0.695360</td>\n",
       "      <td>-0.376955</td>\n",
       "      <td>0.249929</td>\n",
       "      <td>-0.381626</td>\n",
       "      <td>-0.468536</td>\n",
       "      <td>-0.507787</td>\n",
       "      <td>0.333461</td>\n",
       "      <td>-0.737663</td>\n",
       "      <td>1.000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "             CRIM        ZN     INDUS      CHAS       NOX        RM       AGE  \\\n",
       "CRIM     1.000000 -0.199458  0.404471 -0.055295  0.417521 -0.219940  0.350784   \n",
       "ZN      -0.199458  1.000000 -0.533828 -0.042697 -0.516604  0.311991 -0.569537   \n",
       "INDUS    0.404471 -0.533828  1.000000  0.062938  0.763651 -0.391676  0.644779   \n",
       "CHAS    -0.055295 -0.042697  0.062938  1.000000  0.091203  0.091251  0.086518   \n",
       "NOX      0.417521 -0.516604  0.763651  0.091203  1.000000 -0.302188  0.731470   \n",
       "RM      -0.219940  0.311991 -0.391676  0.091251 -0.302188  1.000000 -0.240265   \n",
       "AGE      0.350784 -0.569537  0.644779  0.086518  0.731470 -0.240265  1.000000   \n",
       "DIS     -0.377904  0.664408 -0.708027 -0.099176 -0.769230  0.205246 -0.747881   \n",
       "RAD      0.622029 -0.311948  0.595129 -0.007368  0.611441 -0.209847  0.456022   \n",
       "TAX      0.579564 -0.314563  0.720760 -0.035587  0.668023 -0.292048  0.506456   \n",
       "PTRATIO  0.288250 -0.391679  0.383248 -0.121515  0.188933 -0.355501  0.261515   \n",
       "B       -0.377365  0.175520 -0.356977  0.048788 -0.380051  0.128069 -0.273534   \n",
       "LSTAT    0.452220 -0.412995  0.603800 -0.053929  0.590879 -0.613808  0.602339   \n",
       "MEDV    -0.385832  0.360445 -0.483725  0.175260 -0.427321  0.695360 -0.376955   \n",
       "\n",
       "              DIS       RAD       TAX   PTRATIO         B     LSTAT      MEDV  \n",
       "CRIM    -0.377904  0.622029  0.579564  0.288250 -0.377365  0.452220 -0.385832  \n",
       "ZN       0.664408 -0.311948 -0.314563 -0.391679  0.175520 -0.412995  0.360445  \n",
       "INDUS   -0.708027  0.595129  0.720760  0.383248 -0.356977  0.603800 -0.483725  \n",
       "CHAS    -0.099176 -0.007368 -0.035587 -0.121515  0.048788 -0.053929  0.175260  \n",
       "NOX     -0.769230  0.611441  0.668023  0.188933 -0.380051  0.590879 -0.427321  \n",
       "RM       0.205246 -0.209847 -0.292048 -0.355501  0.128069 -0.613808  0.695360  \n",
       "AGE     -0.747881  0.456022  0.506456  0.261515 -0.273534  0.602339 -0.376955  \n",
       "DIS      1.000000 -0.494588 -0.534432 -0.232471  0.291512 -0.496996  0.249929  \n",
       "RAD     -0.494588  1.000000  0.910228  0.464741 -0.444413  0.488676 -0.381626  \n",
       "TAX     -0.534432  0.910228  1.000000  0.460853 -0.441808  0.543993 -0.468536  \n",
       "PTRATIO -0.232471  0.464741  0.460853  1.000000 -0.177383  0.374044 -0.507787  \n",
       "B        0.291512 -0.444413 -0.441808 -0.177383  1.000000 -0.366087  0.333461  \n",
       "LSTAT   -0.496996  0.488676  0.543993  0.374044 -0.366087  1.000000 -0.737663  \n",
       "MEDV     0.249929 -0.381626 -0.468536 -0.507787  0.333461 -0.737663  1.000000  "
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "training_df.corr()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Pandas supports Pearson correlation (the default), Spearman rank correlation, and Kendall's rank correlation via the `method` parameter.\n",
    "\n",
    "It can be difficult to find highly correlated variables in a large array, so let's visualize it using `sns.heatmap()`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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7sCpwObAGMDci7szMUavtRU3YU8pKegB4CvhQZj66gPcfHBE7U3SP/w3YJzNn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\noqiu35+Zt1cVp7Q/5bhiZl4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      "text/plain": [
       "<matplotlib.figure.Figure at 0x114644588>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize = (8, 8))\n",
    "g = sns.heatmap(\n",
    "    training_df.corr(),\n",
    "    cmap = 'PuOr_r',\n",
    "    square = True\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We are telling the Seaborn plotting library to map each element of the array to a color, with low values mapping to dark purple, zero values mapping to white, and high values mapping to orange.  The `square = True` parameter ensures that cells are rendered as squares instead of rectangles.\n",
    "\n",
    "Even this can be somewhat hard to see.  We can use a variant called a *cluster heatmap* that reorders the rows and/or columns hierarchically, so rows are placed next to similar rows and columns next to similar columns.  This makes it much easier to see related variables."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/williamchiu/anaconda/lib/python3.6/site-packages/matplotlib/cbook.py:136: MatplotlibDeprecationWarning: The axisbg attribute was deprecated in version 2.0. Use facecolor instead.\n",
      "  warnings.warn(message, mplDeprecation, stacklevel=1)\n"
     ]
    },
    {
     "data": {
      "image/png": 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VEXsAJwA7Z2YnIh4DnAe8PDP/0uTX4IibJEnqJ0uAr0TErHHbTwGuzswXZeaL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\nYAHa8DfFAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x107a6acf8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "g = sns.clustermap(\n",
    "    training_df.corr(),\n",
    "    cmap = 'PuOr_r'\n",
    ")\n",
    "plt.setp(g.ax_heatmap.get_yticklabels(), rotation = 0);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There's an extra line of code here to rotate y-axis labels so they are easier to read.\n",
    "\n",
    "It's clear from the cluster heatmap that the number of rooms is highly correlated with the median home price.  Let's try doing a simple linear regression using `scikit-learn` with the top 2 correlated predictors, and measure our performance using the correlation between the predicted and actual labels (R$^2$):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.50633812954246937"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "model = LinearRegression()\n",
    "\n",
    "model.fit(features_df[['ZN', 'RM']], labels)\n",
    "predictions = model.predict(features_df[['ZN', 'RM']])\n",
    "\n",
    "r2_score(labels, predictions)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here we create our model and train it by passing `fit()` our features and labels.  This causes it to learn a set of weights to minimize the squared sum of errors, and these weights are stored in the model object.\n",
    "\n",
    "Next we pass `predict()` our set of features again so that it can generate predicted labels.\n",
    "\n",
    "Finally, we measure the correlation between true and predicted labels using R$^2$.\n",
    "\n",
    "We can look at the increase or decrease in the response variable per increase in our predictors by examining the learned coefficients stored in the `coef_` attribute of the model.  These are stored in order of the features the model was fit on.  Thus, we can use feature names as the index to a series with values taken from `coef_`.  Finally, we can call `sort_values` to obtain a sorted list of coefficients."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "ZN    0.062691\n",
       "RM    8.452878\n",
       "dtype: float64"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pd.Series(model.coef_, index = ['ZN', 'RM']).sort_values()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's repeat this process using the full set of features, rather than the 2 most correlated."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.7406077428649428"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "model = LinearRegression()\n",
    "\n",
    "model.fit(features_df, labels)\n",
    "predictions = model.predict(features_df)\n",
    "\n",
    "r2_score(labels, predictions)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Quite a big increase in $R^2$ -- those other features are useful together, even if not highly correlated with the response individually.\n",
    "\n",
    "Let's look again at the sorted coefficients, this time taking our names from the columns of the features data frame:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "NOX       -17.795759\n",
       "DIS        -1.475759\n",
       "PTRATIO    -0.953464\n",
       "LSTAT      -0.525467\n",
       "CRIM       -0.107171\n",
       "TAX        -0.012329\n",
       "AGE         0.000751\n",
       "B           0.009393\n",
       "INDUS       0.020860\n",
       "ZN          0.046395\n",
       "RAD         0.305655\n",
       "CHAS        2.688561\n",
       "RM          3.804752\n",
       "dtype: float64"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pd.Series(model.coef_, index = features_df.columns).sort_values()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The number of rooms has a high coefficient again, but whether or not the house is by the Charles river also gets a high coefficient.  Was it highly correlated with the response?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.091251225043455952"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "features_df.corr().loc['RM', 'CHAS']"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "No.  Was our previously included feature `ZN`?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.31199058737409197"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "features_df.corr().loc['RM', 'ZN']"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "No, but moreso than `CHAS`.  Thus it might not be wise to exclude features simply based on individual correlation with the response.\n",
    "\n",
    "Often we'd like to pre-process our data somehow, such as applying principal component analysis or *standardizing* our variables (subtracting the mean and dividing by the standard deviation).  There is a convenient method for this in `scikit-learn` using a *pipeline*.\n",
    "\n",
    "The easiest way of making a pipeline is with the `make_pipeline` function, giving it a sequence of objects that perform transformations of the data, and ending with a classification or regression object.  For example, the `StandardScaler()` class performs standardization of its input and can be placed in a pipeline before our regressor:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.7406077428649428"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sklearn.pipeline import make_pipeline\n",
    "\n",
    "estimator = make_pipeline(StandardScaler(), LinearRegression())\n",
    "\n",
    "estimator.fit(features_df, labels)\n",
    "predictions = estimator.predict(features_df)\n",
    "\n",
    "r2_score(labels, predictions)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This didn't affect our R$^2$ but often it does.  It likely affected our coefficients, however.\n",
    "\n",
    "Since our regressor is now wrapped in a pipeline, we don't have immediate access to it.  We can still access it via the pipeline, which we have called an `estimator` since it is now more than simply a model.  The `steps` attribute returns a list of objects in the pipeline:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[('standardscaler', StandardScaler(copy=True, with_mean=True, with_std=True)),\n",
       " ('linearregression',\n",
       "  LinearRegression(copy_X=True, fit_intercept=True, n_jobs=1, normalize=False))]"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "estimator.steps"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Each element of the list is thus a tuple, with the first element a name for the step and the second element the object.  If we want to access the `coef_` attribute of the model, we can thus use last element in the pipeline, and access the second element of the resulting tuple to get at our regressor object:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "LSTAT     -3.748680\n",
       "DIS       -3.104448\n",
       "TAX       -2.075898\n",
       "PTRATIO   -2.062156\n",
       "NOX       -2.060092\n",
       "CRIM      -0.920411\n",
       "AGE        0.021121\n",
       "INDUS      0.142967\n",
       "CHAS       0.682203\n",
       "B          0.856640\n",
       "ZN         1.080981\n",
       "RAD        2.658787\n",
       "RM         2.670641\n",
       "dtype: float64"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pd.Series(estimator.steps[-1][1].coef_, index = features_df.columns).sort_values()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note how `RM` still has the highest coefficient, but `CHAS` has now fallen down the list and `RAD` (accessibility to radial highways) has taken 2nd place.\n",
    "\n",
    "It's often useful to build a *baseline* (\"dummy\") model to see how well a model that doesn't use any features performs.  In the case of regression, this can be done by always predicting either the mean, the median, or some other constant.  If our linear regression cannot clearly out-perform a baseline model, it means our model hasn't learned much if anything from the data."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "model = DummyRegressor(strategy = 'mean')\n",
    "\n",
    "model.fit(features_df, labels)\n",
    "predictions = model.predict(features_df)\n",
    "\n",
    "r2_score(labels, predictions)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here our baseline model has no correlation between its predictions (always the mean) and the true labels.  However, baseline models can often have non-zero performance, and can be a sobering test of whether your model is performing well on a problem where you don't know what value of $R^2$ (or some other metric) represents \"good\" performance."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Pandas Database Integration\n",
    "---\n",
    "\n",
    "There are many ways to connect with databases in Python, but Pandas provides an abstraction using the [SQLAlchemy](https://www.sqlalchemy.org) package that makes interfacing with different SQL databases a breeze.\n",
    "\n",
    "SQLAlchemy supports the following databases:\n",
    "- Firebird\n",
    "- Microsoft SQL Server\n",
    "- MySQL\n",
    "- Oracle\n",
    "- PostgreSQL\n",
    "- SQLite\n",
    "- Sybase\n",
    "\n",
    "These are called [*dialects*](http://docs.sqlalchemy.org/en/latest/dialects/index.html) and can be implemented by one or more *drivers*.  To interface with Pandas, we simply create an SQLAlchemy *engine* from a string that specifies where and how to connect to the desired database.\n",
    "\n",
    "This takes the form of:\n",
    "```\n",
    "dialect+driver://username:password@hostname:port/dbname\n",
    "```\n",
    "\n",
    "For example, the following will create a SQLite database from our Pandas data frame:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "engine = sqlalchemy.create_engine('sqlite:///boston.sqlite')\n",
    "training_df.to_sql('training', engine, if_exists = 'replace')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that we didn't have to write **any** SQL to achieve this, as SQLAlchemy takes care of translating from Pandas to the SQLite database specified by `create_engine()`.  Let's verify this worked by reading back from the SQLite database into a data frame:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style>\n",
       "    .dataframe thead tr:only-child th {\n",
       "        text-align: right;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: left;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>CRIM</th>\n",
       "      <th>ZN</th>\n",
       "      <th>INDUS</th>\n",
       "      <th>CHAS</th>\n",
       "      <th>NOX</th>\n",
       "      <th>RM</th>\n",
       "      <th>AGE</th>\n",
       "      <th>DIS</th>\n",
       "      <th>RAD</th>\n",
       "      <th>TAX</th>\n",
       "      <th>PTRATIO</th>\n",
       "      <th>B</th>\n",
       "      <th>LSTAT</th>\n",
       "      <th>MEDV</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>index</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>0.00632</td>\n",
       "      <td>18.0</td>\n",
       "      <td>2.31</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.538</td>\n",
       "      <td>6.575</td>\n",
       "      <td>65.2</td>\n",
       "      <td>4.0900</td>\n",
       "      <td>1.0</td>\n",
       "      <td>296.0</td>\n",
       "      <td>15.3</td>\n",
       "      <td>396.90</td>\n",
       "      <td>4.98</td>\n",
       "      <td>24.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>0.02731</td>\n",
       "      <td>0.0</td>\n",
       "      <td>7.07</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.469</td>\n",
       "      <td>6.421</td>\n",
       "      <td>78.9</td>\n",
       "      <td>4.9671</td>\n",
       "      <td>2.0</td>\n",
       "      <td>242.0</td>\n",
       "      <td>17.8</td>\n",
       "      <td>396.90</td>\n",
       "      <td>9.14</td>\n",
       "      <td>21.6</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0.02729</td>\n",
       "      <td>0.0</td>\n",
       "      <td>7.07</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.469</td>\n",
       "      <td>7.185</td>\n",
       "      <td>61.1</td>\n",
       "      <td>4.9671</td>\n",
       "      <td>2.0</td>\n",
       "      <td>242.0</td>\n",
       "      <td>17.8</td>\n",
       "      <td>392.83</td>\n",
       "      <td>4.03</td>\n",
       "      <td>34.7</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>0.03237</td>\n",
       "      <td>0.0</td>\n",
       "      <td>2.18</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.458</td>\n",
       "      <td>6.998</td>\n",
       "      <td>45.8</td>\n",
       "      <td>6.0622</td>\n",
       "      <td>3.0</td>\n",
       "      <td>222.0</td>\n",
       "      <td>18.7</td>\n",
       "      <td>394.63</td>\n",
       "      <td>2.94</td>\n",
       "      <td>33.4</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>0.06905</td>\n",
       "      <td>0.0</td>\n",
       "      <td>2.18</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.458</td>\n",
       "      <td>7.147</td>\n",
       "      <td>54.2</td>\n",
       "      <td>6.0622</td>\n",
       "      <td>3.0</td>\n",
       "      <td>222.0</td>\n",
       "      <td>18.7</td>\n",
       "      <td>396.90</td>\n",
       "      <td>5.33</td>\n",
       "      <td>36.2</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "          CRIM    ZN  INDUS  CHAS    NOX     RM   AGE     DIS  RAD    TAX  \\\n",
       "index                                                                       \n",
       "0      0.00632  18.0   2.31   0.0  0.538  6.575  65.2  4.0900  1.0  296.0   \n",
       "1      0.02731   0.0   7.07   0.0  0.469  6.421  78.9  4.9671  2.0  242.0   \n",
       "2      0.02729   0.0   7.07   0.0  0.469  7.185  61.1  4.9671  2.0  242.0   \n",
       "3      0.03237   0.0   2.18   0.0  0.458  6.998  45.8  6.0622  3.0  222.0   \n",
       "4      0.06905   0.0   2.18   0.0  0.458  7.147  54.2  6.0622  3.0  222.0   \n",
       "\n",
       "       PTRATIO       B  LSTAT  MEDV  \n",
       "index                                \n",
       "0         15.3  396.90   4.98  24.0  \n",
       "1         17.8  396.90   9.14  21.6  \n",
       "2         17.8  392.83   4.03  34.7  \n",
       "3         18.7  394.63   2.94  33.4  \n",
       "4         18.7  396.90   5.33  36.2  "
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "training_df = pd.read_sql_query('SELECT * from training', engine, index_col = 'index')\n",
    "training_df.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Looks good!  We did have to provide some SQL here, but Pandas and SQLAlchemy were able to handle most of the dirty details.  Without them, reading and writing from databases is a much more code-intensive and error-prone process.\n",
    "\n",
    "SQLite is a widely-used embedded database that operates outside of a client-server model.  All we have to do in order to use a different database is change the string passed to `create_engine()`.  Example strings for PostgreSQL, MySQL, and MS SQL Server:\n",
    "\n",
    "```\n",
    "engine = sqlalchemy.create_engine('postgresql://scott:tiger@hostname:port/dbname')\n",
    "engine = sqlalchemy.create_engine('mysql://scott:tiger@hostname:port/dbname')\n",
    "engine = sqlalchemy.create_engine('mssql+pyodbc://scott:tiger@hostname:port/dbname')\n",
    "engine = sqlalchemy.create_engine('mssql+pymssql://scott:tiger@hostname:port/dbname')\n",
    "```\n",
    "\n",
    "`username:password` can be excluded if there is no authentication, and `:port` can be excluded if a default port is used.  `+driver` is only needed to select one from multiple sets of drivers that may have different functionality or performance.\n",
    "\n",
    "You could install the `pymssql` driver by running:\n",
    "```\n",
    "pip install pymssql\n",
    "```\n",
    "from your terminal.  This requires the FreeTDS package; follow [setup instructions for pymssql](http://pymssql.org/en/stable/intro.html) to install it on your OS.  Looks like there is currently a bug that makes it harder to install on macOS.\n",
    "\n",
    "We don't have these servers running in the cloud.  But, if you wanted to repeat the analysis here in Microsoft SQL Server (for example), you would simply change the string passed to `create_engine()`."
   ]
  }
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