Tutorial11 Regression Splines.ipynb

{
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<center><img src=\"http://sydney.edu.au/images/content/about/logo-mono.jpg\"></center>\n",
    "\n",
    "<center><h1>Statistical Learning and Data Mining (QBUS6810)</h1></center>\n",
    "<center><h2>Tutorial 11: Regression Splines</h2></center>\n",
    "<br>\n",
    "\n",
    "In this tutorial we study how to implement regression splines and generalised additive models in Python.\n",
    "\n",
    "<a class=\"buttom\" href=\">#Data:-customer-analytics\">Data: Customer Analytics</a> <br>\n",
    "<a class=\"buttom\" href=\"#Exploratory-data-analysis\">Exploratory data analysis</a> <br>\n",
    "<a class=\"buttom\" href=\"#Regression-splines\">Regression splines</a> <br>\n",
    "<a class=\"buttom\" href=\"#Natural-splines\">Natural splines</a> <br>\n",
    "<a class=\"buttom\" href=\"#Generalised-additive-model\">Generalised additive model</a> <br>\n",
    "<a class=\"buttom\" href=\"#Model-evaluation\">Model Evaluation</a> <br>\n",
    "<a class=\"buttom\" href=\"#Exercise\">Exercise</a> <br>\n",
    "\n",
    "This notebook relies on the following imports and settings."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Packages\n",
    "import warnings\n",
    "warnings.filterwarnings(\"ignore\")\n",
    "import numpy as np\n",
    "from scipy import stats\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "import seaborn as sns\n",
    "import statsmodels.api as sm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Plot settings\n",
    "sns.set_context('notebook') \n",
    "sns.set_style('ticks')\n",
    "blue='#1F77B4'\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##Data: Customer Analytics\n",
    "\n",
    "We again focus on customer analytics dataset from Tutorials 6 and 10."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Acquisition</th>\n",
       "      <th>First_Purchase</th>\n",
       "      <th>CLV</th>\n",
       "      <th>Duration</th>\n",
       "      <th>Censor</th>\n",
       "      <th>Acq_Expense</th>\n",
       "      <th>Acq_Expense_SQ</th>\n",
       "      <th>Industry</th>\n",
       "      <th>Revenue</th>\n",
       "      <th>Employees</th>\n",
       "      <th>Ret_Expense</th>\n",
       "      <th>Ret_Expense_SQ</th>\n",
       "      <th>Crossbuy</th>\n",
       "      <th>Frequency</th>\n",
       "      <th>Frequency_SQ</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Customer</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
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       "      <th></th>\n",
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       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>1</td>\n",
       "      <td>433.64</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>384</td>\n",
       "      <td>0</td>\n",
       "      <td>760.36</td>\n",
       "      <td>578147.33</td>\n",
       "      <td>1</td>\n",
       "      <td>30.16</td>\n",
       "      <td>1240</td>\n",
       "      <td>2309.79</td>\n",
       "      <td>5335129.84</td>\n",
       "      <td>5</td>\n",
       "      <td>2</td>\n",
       "      <td>4</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0</td>\n",
       "      <td>0.00</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>147.70</td>\n",
       "      <td>21815.29</td>\n",
       "      <td>1</td>\n",
       "      <td>39.80</td>\n",
       "      <td>166</td>\n",
       "      <td>0.00</td>\n",
       "      <td>0.00</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
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       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>0</td>\n",
       "      <td>0.00</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>252.56</td>\n",
       "      <td>63786.55</td>\n",
       "      <td>1</td>\n",
       "      <td>54.93</td>\n",
       "      <td>1016</td>\n",
       "      <td>0.00</td>\n",
       "      <td>0.00</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>1</td>\n",
       "      <td>225.84</td>\n",
       "      <td>5.7316</td>\n",
       "      <td>730</td>\n",
       "      <td>1</td>\n",
       "      <td>609.73</td>\n",
       "      <td>371770.67</td>\n",
       "      <td>1</td>\n",
       "      <td>45.83</td>\n",
       "      <td>122</td>\n",
       "      <td>2192.59</td>\n",
       "      <td>4807450.91</td>\n",
       "      <td>2</td>\n",
       "      <td>12</td>\n",
       "      <td>144</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>1</td>\n",
       "      <td>363.04</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>579</td>\n",
       "      <td>0</td>\n",
       "      <td>672.36</td>\n",
       "      <td>452067.97</td>\n",
       "      <td>1</td>\n",
       "      <td>69.03</td>\n",
       "      <td>313</td>\n",
       "      <td>801.14</td>\n",
       "      <td>641825.30</td>\n",
       "      <td>4</td>\n",
       "      <td>7</td>\n",
       "      <td>49</td>\n",
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       "    <tr>\n",
       "      <th>6</th>\n",
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       "      <td>0.0000</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>435.57</td>\n",
       "      <td>189721.22</td>\n",
       "      <td>0</td>\n",
       "      <td>22.54</td>\n",
       "      <td>359</td>\n",
       "      <td>0.00</td>\n",
       "      <td>0.00</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>0</td>\n",
       "      <td>0.00</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>362.90</td>\n",
       "      <td>131696.41</td>\n",
       "      <td>0</td>\n",
       "      <td>32.97</td>\n",
       "      <td>902</td>\n",
       "      <td>0.00</td>\n",
       "      <td>0.00</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
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       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>0</td>\n",
       "      <td>0.00</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>883.54</td>\n",
       "      <td>780642.93</td>\n",
       "      <td>0</td>\n",
       "      <td>22.48</td>\n",
       "      <td>264</td>\n",
       "      <td>0.00</td>\n",
       "      <td>0.00</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
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       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>1</td>\n",
       "      <td>599.30</td>\n",
       "      <td>6.9161</td>\n",
       "      <td>730</td>\n",
       "      <td>1</td>\n",
       "      <td>452.35</td>\n",
       "      <td>204620.52</td>\n",
       "      <td>1</td>\n",
       "      <td>17.98</td>\n",
       "      <td>1782</td>\n",
       "      <td>1340.75</td>\n",
       "      <td>1797610.56</td>\n",
       "      <td>1</td>\n",
       "      <td>11</td>\n",
       "      <td>121</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>1</td>\n",
       "      <td>271.18</td>\n",
       "      <td>6.0839</td>\n",
       "      <td>730</td>\n",
       "      <td>1</td>\n",
       "      <td>786.72</td>\n",
       "      <td>618928.36</td>\n",
       "      <td>1</td>\n",
       "      <td>38.91</td>\n",
       "      <td>539</td>\n",
       "      <td>2265.93</td>\n",
       "      <td>5134438.76</td>\n",
       "      <td>1</td>\n",
       "      <td>14</td>\n",
       "      <td>196</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "          Acquisition  First_Purchase     CLV  Duration  Censor  Acq_Expense  \\\n",
       "Customer                                                                       \n",
       "1                   1          433.64  0.0000       384       0       760.36   \n",
       "2                   0            0.00  0.0000         0       0       147.70   \n",
       "3                   0            0.00  0.0000         0       0       252.56   \n",
       "4                   1          225.84  5.7316       730       1       609.73   \n",
       "5                   1          363.04  0.0000       579       0       672.36   \n",
       "6                   0            0.00  0.0000         0       0       435.57   \n",
       "7                   0            0.00  0.0000         0       0       362.90   \n",
       "8                   0            0.00  0.0000         0       0       883.54   \n",
       "9                   1          599.30  6.9161       730       1       452.35   \n",
       "10                  1          271.18  6.0839       730       1       786.72   \n",
       "\n",
       "          Acq_Expense_SQ  Industry  Revenue  Employees  Ret_Expense  \\\n",
       "Customer                                                              \n",
       "1              578147.33         1    30.16       1240      2309.79   \n",
       "2               21815.29         1    39.80        166         0.00   \n",
       "3               63786.55         1    54.93       1016         0.00   \n",
       "4              371770.67         1    45.83        122      2192.59   \n",
       "5              452067.97         1    69.03        313       801.14   \n",
       "6              189721.22         0    22.54        359         0.00   \n",
       "7              131696.41         0    32.97        902         0.00   \n",
       "8              780642.93         0    22.48        264         0.00   \n",
       "9              204620.52         1    17.98       1782      1340.75   \n",
       "10             618928.36         1    38.91        539      2265.93   \n",
       "\n",
       "          Ret_Expense_SQ  Crossbuy  Frequency  Frequency_SQ  \n",
       "Customer                                                     \n",
       "1             5335129.84         5          2             4  \n",
       "2                   0.00         0          0             0  \n",
       "3                   0.00         0          0             0  \n",
       "4             4807450.91         2         12           144  \n",
       "5              641825.30         4          7            49  \n",
       "6                   0.00         0          0             0  \n",
       "7                   0.00         0          0             0  \n",
       "8                   0.00         0          0             0  \n",
       "9             1797610.56         1         11           121  \n",
       "10            5134438.76         1         14           196  "
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data = pd.read_excel('CustomerAcquisition.xls', index_col=[0])\n",
    "data.head(10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Our first goal in this tutorial is to extend of customer lifetime value (CLV) analysis from Tutorial 6 to account for nonlinearities. We therefore process the dataset in a similar way."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "data = data[data['Censor']==1] # our goal is to predict the CLV of retained customers, identified by Censor=1 \n",
    "\n",
    "response = ['CLV']\n",
    "predictors=['Acq_Expense', 'Ret_Expense', 'First_Purchase', 'Revenue', 'Employees', 'Crossbuy', 'Frequency', 'Industry']\n",
    "    \n",
    "data = data[response+predictors] # keeping only the variables that we are going to use\n",
    "\n",
    "train = data.sample(frac=0.7, random_state=1)\n",
    "test = data[data.index.isin(train.index)==False].copy()\n",
    "\n",
    "X_train= train[predictors]\n",
    "X_test= test[predictors]\n",
    "y_train= train['CLV']\n",
    "y_test= test['CLV']"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##Exploratory data analysis\n",
    "\n",
    "We start by computing the correlations between the response and the predictors as a reference for our analysis."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>CLV</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>CLV</th>\n",
       "      <td>1.000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Acq_Expense</th>\n",
       "      <td>-0.425</td>\n",
       "    </tr>\n",
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       "      <th>Ret_Expense</th>\n",
       "      <td>0.411</td>\n",
       "    </tr>\n",
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       "      <th>First_Purchase</th>\n",
       "      <td>0.558</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Revenue</th>\n",
       "      <td>0.123</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Employees</th>\n",
       "      <td>0.543</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Crossbuy</th>\n",
       "      <td>0.358</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Frequency</th>\n",
       "      <td>-0.078</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Industry</th>\n",
       "      <td>0.127</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
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      ],
      "text/plain": [
       "                  CLV\n",
       "CLV             1.000\n",
       "Acq_Expense    -0.425\n",
       "Ret_Expense     0.411\n",
       "First_Purchase  0.558\n",
       "Revenue         0.123\n",
       "Employees       0.543\n",
       "Crossbuy        0.358\n",
       "Frequency      -0.078\n",
       "Industry        0.127"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "train.corr().round(3)[['CLV']]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To explore possible nonlinearities in the data, we build scatter plots for all the numerical predictors. The scatter plots include local regression fits for visualisation. Nonlinear patters are apparent for several predictors, in particular the acquisition and retention expenses, customer revenue, and buying frequency. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
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h0DUwMj7+fiaQt/TMvDd9UW4m62pKzgb4mhJqy/JcvcjNhbh9zDxuPXNjzHXA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\nHqdgLiIi4nEK5iIiIh73/wG2wTv+hr3Y0wAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x261b9df2908>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(figsize=(8,5))\n",
    "sns.regplot(train['Acq_Expense'], train['CLV'],  scatter_kws = {'s': 25}, lowess=True, color=blue)\n",
    "sns.despine()\n",
    "fig.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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tmsnVp05nRGHMwxKKiEiGNM493+xt7uA/nn2X/3x+M80diZ7tEyuK+Pw5s/jE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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x261b9dfd9b0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(figsize=(8,5))\n",
    "sns.regplot(train['Ret_Expense'], train['CLV'],  scatter_kws = {'s': 25}, lowess=True, color=blue)\n",
    "sns.despine()\n",
    "fig.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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LuqpC6qvnUV9dSF1VIdVFeZNOE9PPID46T+NyXZ+5eIiXf7Fm2lyXTs19fpxe1HtxiINt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\ncQpzERERj/v/w8jt/tMtmdoAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x261b9d5eb38>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(figsize=(8,5))\n",
    "sns.regplot(train['First_Purchase'], train['CLV'],  scatter_kws = {'s': 25}, lowess=True, color=blue)\n",
    "sns.despine()\n",
    "fig.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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3LfwpVN3s0sIElmKxND+BBUB48Z6AIAhdZR6WCSwIN3qgVo73BARB6CrzsExg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\nYQ4AgM8R5gAA+BxhDgCAzxHmAAD43P8HVWa3k6m3RXgAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x261b9d6e630>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(figsize=(8,5))\n",
    "sns.regplot(train['Revenue'], train['CLV'],  scatter_kws = {'s': 25}, lowess=True, color=blue)\n",
    "sns.despine()\n",
    "fig.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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W6brydxWR5II0nkXBPEs6BobYd8qb2733dC8Hp1kOFa4NVNsUz7pX1JTwU3/x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\nTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x261ba401a20>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(figsize=(8,5))\n",
    "sns.regplot(train['Employees'], train['CLV'],  scatter_kws = {'s': 25}, lowess=True, color=blue)\n",
    "sns.despine()\n",
    "fig.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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VHOVL6ip3qVwuF3QbZmWM6ZR0sLu7Wx0dHUE3BwAAP6QW+wIycwAAIo5gDgBA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\nYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x261ba4414e0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(figsize=(8,5))\n",
    "sns.regplot(train['Frequency'], train['CLV'],  scatter_kws = {'s': 25}, lowess=True, color=blue)\n",
    "sns.despine()\n",
    "fig.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##Regression Splines\n",
    "\n",
    "We use the [<TT>patsy</TT>](http://patsy.readthedocs.io/en/latest/index.html) library to construct spline expansions. Our implementation is based on two core functions: [<TT>dmatrix</TT>](http://patsy.readthedocs.io/en/latest/API-reference.html#patsy.dmatrix), the general design matrix constructor, and [<TT>bs</TT>](http://patsy.readthedocs.io/en/latest/API-reference.html#spline-regression), the building block for regression splines.\n",
    "\n",
    "The next cell builds a linear basis expansion for a cubic spline (degree=3) with four degrees of freedom (df=4, therefore using one knot) for the retention expense. The function generates B-splines rather than the step functions we have seen in the lectures. B-splines generate an equivalent model, but lead to more numerically stable computations. "
   ]
  },
  {
   "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>bs(predictor, degree=3, df=4)[0]</th>\n",
       "      <th>bs(predictor, degree=3, df=4)[1]</th>\n",
       "      <th>bs(predictor, degree=3, df=4)[2]</th>\n",
       "      <th>bs(predictor, degree=3, df=4)[3]</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Customer</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>126</th>\n",
       "      <td>0.318851</td>\n",
       "      <td>0.489163</td>\n",
       "      <td>0.191170</td>\n",
       "      <td>0.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>103</th>\n",
       "      <td>0.391723</td>\n",
       "      <td>0.460276</td>\n",
       "      <td>0.142328</td>\n",
       "      <td>0.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>330</th>\n",
       "      <td>0.031471</td>\n",
       "      <td>0.251704</td>\n",
       "      <td>0.595972</td>\n",
       "      <td>0.120853</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>0.009401</td>\n",
       "      <td>0.131247</td>\n",
       "      <td>0.569199</td>\n",
       "      <td>0.290153</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>216</th>\n",
       "      <td>0.164663</td>\n",
       "      <td>0.477499</td>\n",
       "      <td>0.356010</td>\n",
       "      <td>0.001828</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "          bs(predictor, degree=3, df=4)[0]  bs(predictor, degree=3, df=4)[1]  \\\n",
       "Customer                                                                       \n",
       "126                               0.318851                          0.489163   \n",
       "103                               0.391723                          0.460276   \n",
       "330                               0.031471                          0.251704   \n",
       "17                                0.009401                          0.131247   \n",
       "216                               0.164663                          0.477499   \n",
       "\n",
       "          bs(predictor, degree=3, df=4)[2]  bs(predictor, degree=3, df=4)[3]  \n",
       "Customer                                                                      \n",
       "126                               0.191170                          0.000000  \n",
       "103                               0.142328                          0.000000  \n",
       "330                               0.595972                          0.120853  \n",
       "17                                0.569199                          0.290153  \n",
       "216                               0.356010                          0.001828  "
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from patsy import dmatrix\n",
    "\n",
    "X = dmatrix('bs(predictor, degree=3, df=4) - 1', {'predictor': train['Ret_Expense']}, return_type='dataframe')\n",
    "# The -1 excludes the intercept\n",
    "X.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Given the design matrix, we estimate the model by OLS. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "LinearRegression(copy_X=True, fit_intercept=True, n_jobs=1, normalize=False)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sklearn.linear_model import LinearRegression\n",
    "ols = LinearRegression()\n",
    "ols.fit(X, y_train)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "from patsy import build_design_matrices\n",
    "\n",
    "a=train['Ret_Expense'].min()\n",
    "b=train['Ret_Expense'].max()\n",
    "x=np.linspace(a, b)\n",
    "\n",
    "X_predict = build_design_matrices([X.design_info], {'predictor': x})[0] # remember that X is our original design matrix\n",
    "y_fitted=ols.predict(X_predict)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, we plot the regression fit. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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wRdJhz549PPXUU0ycOJGVK1cyevRo7r777qxI7ND4ee5/BB4CXnX3dwDc/UXgxRTFJiIZ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\nSMQouYuIiESMkruIiEjEKLmLiIhEzP8Ph2GR9Rcd7RYAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x261ba8b15c0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(figsize=(8,5))\n",
    "sns.regplot(train['Ret_Expense'], y_train,  scatter_kws = {'s': 25}, fit_reg=False, color=blue)\n",
    "ax.plot(x, y_fitted, color='Black', alpha=0.7)\n",
    "ax.set_xlabel('Retention expense')\n",
    "ax.set_ylabel('Customer Lifetime Value')\n",
    "ax.set_title('Cubic spline (df = 4)', fontsize=13, fontweight='bold')\n",
    "\n",
    "sns.despine()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The next repeats all these steps for a linear spline with one knot. You should experiment by changing the first two lines to try different models (say, by adding more knots) and predictors."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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hjY2NHHLIIUyZMoWLLrpIC89IrMWpNqxBmvHRYXJ39x+a2VjgbeBw4GZ3nxN6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\nlNxFREQS5v8BQSQp5v16Qj4AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x261ba8fc080>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "predictor = 'Ret_Expense'\n",
    "X = dmatrix('bs(variable, degree=1, df=2) - 1', {'variable': train[predictor]}, return_type='dataframe')\n",
    "\n",
    "ols = LinearRegression()\n",
    "ols.fit(X, y_train)\n",
    "\n",
    "a=train[predictor].min()\n",
    "b=train[predictor].max()\n",
    "x=np.linspace(a, b)\n",
    "\n",
    "X_predict = build_design_matrices([X.design_info], {'variable': x})[0] # remember that X is our original design matrix\n",
    "y_fitted=ols.predict(X_predict)\n",
    "\n",
    "fig, ax = plt.subplots(figsize=(8,5))\n",
    "sns.regplot(train[predictor], y_train,  scatter_kws = {'s': 25}, fit_reg=False, color=blue)\n",
    "ax.plot(x, y_fitted, color='Black', alpha=0.7)\n",
    "ax.set_ylabel('Customer Lifetime Value')\n",
    "ax.set_title('Regression spline', fontsize=13, fontweight='bold')\n",
    "\n",
    "sns.despine()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##Natural Splines\n",
    "\n",
    "Estimating a natural cubic spline requires only a straightforward modification. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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vHHTQQRQVFSW0HXV4y0w66U0NHefs0l5v+TtpPYnnASWhRSRtSlVyico/cV1d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\nGCV3ERGRiFFyFxERiZj/D4APyiBgnfZiAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x261ba4cf550>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "predictor = 'Ret_Expense'\n",
    "X = dmatrix('cr(variable, df=4) - 1', {'variable': train[predictor]}, return_type='dataframe')\n",
    "\n",
    "ols = LinearRegression()\n",
    "ols.fit(X, y_train)\n",
    "\n",
    "a=train[predictor].min()\n",
    "b=train[predictor].max()\n",
    "x=np.linspace(a, b)\n",
    "\n",
    "X_predict = build_design_matrices([X.design_info], {'variable': x})[0] # remember that X is our original design matrix\n",
    "y_fitted=ols.predict(X_predict)\n",
    "\n",
    "fig, ax = plt.subplots(figsize=(8,5))\n",
    "sns.regplot(train[predictor], y_train,  scatter_kws = {'s': 25}, fit_reg=False, color=blue)\n",
    "ax.plot(x, y_fitted, color='Black', alpha=0.7)\n",
    "ax.set_ylabel('Customer Lifetime Value')\n",
    "ax.set_title('Natural cubic spline (df=4)', fontsize=13, fontweight='bold')\n",
    "\n",
    "sns.despine()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##Generalised additive model\n",
    "\n",
    "We implement a generalised regression based on naturals splines in the <TT>statlearning</TT> module. To fit the model, we build a dictionary to indicate the number of natural spline basis functions that we request for selected predictors.  All predictors not included in the dictionary are assumed to be linear.   "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "from statlearning import generalised_additive_regression\n",
    "\n",
    "spline_dfs={'Ret_Expense': 4, 'Acq_Expense': 3, 'Frequency': 3, 'Employees': 4}\n",
    "\n",
    "gam = generalised_additive_regression()\n",
    "gam.fit(X_train, y_train, spline_dfs)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We use a backward selection algorithm to find the best specification by cross validation. First, we build a dictionary to specify the maximum degrees of freedom that we are going to consider for each predictor. Because the sample size is small, it is important to not set these numbers high as the selected model may overfit even if we are using cross validation. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{'Acq_Expense': 3, 'Ret_Expense': 3, 'First_Purchase': 2, 'Revenue': 2, 'Employees': 2, 'Crossbuy': 2, 'Frequency': 2}\n"
     ]
    }
   ],
   "source": [
    "max_dfs = {} # initialising dictionary\n",
    "\n",
    "# setting a maximum of two degrees of freedom for each numerical predictor\n",
    "\n",
    "for predictor in  list(X_train.columns[:-1]): # looping over the list of numerical predictors\n",
    "    max_dfs[predictor] = 2 \n",
    "\n",
    "# allowing the acquisition and retention expenses specifically to have higher maximum degrees of freedom \n",
    "\n",
    "max_dfs['Acq_Expense'] = 3\n",
    "max_dfs['Ret_Expense'] = 3\n",
    "\n",
    "print(max_dfs)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The following syntax selects and fits the best specification according to leave one out cross validation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Selected degrees of freedom (backward algorithm): \n",
      "\n",
      "Acq_Expense       3\n",
      "Ret_Expense       3\n",
      "First_Purchase    1\n",
      "Revenue           1\n",
      "Employees         0\n",
      "Crossbuy          1\n",
      "Frequency         2\n",
      "Industry          1\n",
      "dtype: int32\n"
     ]
    }
   ],
   "source": [
    "gam = generalised_additive_regression()\n",
    "gam.backward_selection(X_train, y_train, max_dfs, max_dfs_model=20)\n",
    "\n",
    "gam.info()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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ddyY4W5fUuzCbs0b05bFX1/D8u5uYbKVhRxIRkSQXzzD+bwgmwbkCuBTIBO5IZKiu7soJ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\nRssAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x261ba8fc3c8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "gam.plot_cv()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In certain cases, we may simply want to build the design matrices for the training and test sets.  The module supports this as follows."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "from statlearning import gam_design_matrix, gam_design_matrix_test\n",
    "X_train_gam = gam_design_matrix(X_train, spline_dfs)\n",
    "X_train_gam, X_test_gam = gam_design_matrix_test(X_train, X_test, spline_dfs)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that we need to build the training and test matrices together because we to specify lower and upper bounds for the predictors. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##Model evaluation\n",
    "\n",
    "The test results show that accounting for nonlinear effects leads to substantial improvements in predictive accuracy relative to a linear regression. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "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>Test RMSE</th>\n",
       "      <th>SE</th>\n",
       "      <th>Test R2</th>\n",
       "      <th>SE</th>\n",
       "      <th>Test MAE</th>\n",
       "      <th>SE</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>OLS</th>\n",
       "      <td>0.457</td>\n",
       "      <td>0.037</td>\n",
       "      <td>0.730</td>\n",
       "      <td>0.095</td>\n",
       "      <td>0.399</td>\n",
       "      <td>0.035</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>GAM</th>\n",
       "      <td>0.367</td>\n",
       "      <td>0.035</td>\n",
       "      <td>0.826</td>\n",
       "      <td>0.064</td>\n",
       "      <td>0.306</td>\n",
       "      <td>0.032</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "     Test RMSE     SE  Test R2     SE  Test MAE     SE\n",
       "OLS      0.457  0.037    0.730  0.095     0.399  0.035\n",
       "GAM      0.367  0.035    0.826  0.064     0.306  0.032"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Fitting the two models\n",
    "ols = LinearRegression()\n",
    "ols.fit(X_train, y_train)\n",
    "\n",
    "columns=['Test RMSE', 'SE', 'Test R2', 'SE','Test MAE','SE']\n",
    "rows=['OLS', 'GAM']\n",
    "results=pd.DataFrame(0.0, columns=columns, index=rows) \n",
    "\n",
    "from statlearning import rmse_jack, r2_jack, mae\n",
    "methods=[ols, gam]\n",
    "\n",
    "for i, method in enumerate(methods):\n",
    "\n",
    "    y_pred=method.predict(X_test)\n",
    "    \n",
    "    results.iloc[i,0], results.iloc[i,1] = rmse_jack(y_test, y_pred)\n",
    "    results.iloc[i,2], results.iloc[i,3] = r2_jack(y_test, y_pred)\n",
    "    results.iloc[i,4], results.iloc[i,5] = mae(y_test, y_pred)\n",
    "\n",
    "results.round(3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##To do\n",
    "\n",
    "For future versions of the tutorial: \n",
    "\n",
    "1. Implement smoothing splines.  \n",
    "\n",
    "2. Plot the individual regression functions based on the estimated GAM. \n",
    "\n",
    "##Exercise\n",
    "\n",
    "Using the setting of Tutorial 10 (classification models customer acquisition), build a logistic regression model based on regression splines and investigate whether it leads to improved accuracy for the test data. "
   ]
  }
 ],
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