IBM-ML240EN-SkillsNetwork_labs_02a_LAB_Transforming_Target.jupyterlite (1).ipynb

{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "# Machine Learning Foundation\n",
        "\n",
        "## Section 2, Part a: Regression Intro: Transforming Target \n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## Learning objectives\n",
        "\n",
        "By the end of this lesson, you will be able to:\n",
        "\n",
        "* Apply transformations to make target variable more normally distributed for regression\n",
        "* Apply inverse transformations to be able to use these in a regression context\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 32,
      "metadata": {},
      "outputs": [],
      "source": [
        "#!pip install -U scikit-learn"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 33,
      "metadata": {
        "trusted": true
      },
      "outputs": [],
      "source": [
        "#import piplite\n",
        "#await piplite.install(['tqdm', 'seaborn', 'skillsnetwork', 'pandas', 'numpy', 'scikit-learn'])"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 34,
      "metadata": {
        "trusted": true
      },
      "outputs": [],
      "source": [
        "import numpy as np\n",
        "import pandas as pd\n",
        "import matplotlib.pyplot as plt\n",
        "%matplotlib inline\n",
        "import skillsnetwork\n",
        "# Surpress warnings:\n",
        "def warn(*args, **kwargs):\n",
        "    pass\n",
        "import warnings\n",
        "warnings.warn = warn"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "In the following cell we load the data and define some useful plotting functions.\n"
      ]
    },
    {
      "attachments": {},
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "**Observation:** \n",
        "I got an error because the function load_boston is no longer available in sktlearn.datasets anymore. So, to solve it, I downloaded manually the data set as 'BostonHousing.csv' to use it.\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 68,
      "metadata": {
        "trusted": true
      },
      "outputs": [],
      "source": [
        "np.random.seed(72018)\n",
        "\n",
        "boston = pd.read_csv('data\\BostonHousing.csv', sep =',')\n",
        "boston.columns = [ x.upper() for x in boston.columns]\n",
        "\n",
        "#def to_2d(array):\n",
        "#    return array.reshape(array.shape[0], -1)\n",
        "\n",
        "#def boston_dataframe(description=False):    #We won't need this function because we have already get the csv file as an DataFrame\n",
        "#    data = boston.data\n",
        "#    target = boston.target\n",
        "#    names = boston.feature_names\n",
        "#    \n",
        "#    target = to_2d(target)\n",
        "#    \n",
        "#    data_all = np.concatenate([data, target], axis=1)\n",
        "#    names_all = np.concatenate([names, np.array(['MEDV'])], axis=0)\n",
        "#    \n",
        "#    if description:\n",
        "#        return pd.DataFrame(data=data_all, columns=names_all), boston.DESCR\n",
        "#    \n",
        "#    else:  \n",
        "#        return pd.DataFrame(data=data_all, columns=names_all)\n",
        "    \n",
        "def plot_exponential_data():\n",
        "    data = np.exp(np.random.normal(size=1000))\n",
        "    plt.hist(data)\n",
        "    plt.show()\n",
        "    return data\n",
        "    \n",
        "def plot_square_normal_data():\n",
        "    data = np.square(np.random.normal(loc=5, size=1000))\n",
        "    plt.hist(data)\n",
        "    plt.show()\n",
        "    return data"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "### Loading in Boston Data\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 36,
      "metadata": {
        "trusted": true
      },
      "outputs": [],
      "source": [
        "boston_data = pd.DataFrame(boston)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 37,
      "metadata": {},
      "outputs": [],
      "source": [
        "boston_data.to_csv('boston_data.csv', index = False)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 38,
      "metadata": {
        "trusted": true
      },
      "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>0</th>\n",
              "      <td>0.00632</td>\n",
              "      <td>18.0</td>\n",
              "      <td>2.31</td>\n",
              "      <td>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</td>\n",
              "      <td>296</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</td>\n",
              "      <td>0.469</td>\n",
              "      <td>6.421</td>\n",
              "      <td>78.9</td>\n",
              "      <td>4.9671</td>\n",
              "      <td>2</td>\n",
              "      <td>242</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</td>\n",
              "      <td>0.469</td>\n",
              "      <td>7.185</td>\n",
              "      <td>61.1</td>\n",
              "      <td>4.9671</td>\n",
              "      <td>2</td>\n",
              "      <td>242</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</td>\n",
              "      <td>0.458</td>\n",
              "      <td>6.998</td>\n",
              "      <td>45.8</td>\n",
              "      <td>6.0622</td>\n",
              "      <td>3</td>\n",
              "      <td>222</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</td>\n",
              "      <td>0.458</td>\n",
              "      <td>7.147</td>\n",
              "      <td>54.2</td>\n",
              "      <td>6.0622</td>\n",
              "      <td>3</td>\n",
              "      <td>222</td>\n",
              "      <td>18.7</td>\n",
              "      <td>396.90</td>\n",
              "      <td>5.33</td>\n",
              "      <td>36.2</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>5</th>\n",
              "      <td>0.02985</td>\n",
              "      <td>0.0</td>\n",
              "      <td>2.18</td>\n",
              "      <td>0</td>\n",
              "      <td>0.458</td>\n",
              "      <td>6.430</td>\n",
              "      <td>58.7</td>\n",
              "      <td>6.0622</td>\n",
              "      <td>3</td>\n",
              "      <td>222</td>\n",
              "      <td>18.7</td>\n",
              "      <td>394.12</td>\n",
              "      <td>5.21</td>\n",
              "      <td>28.7</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>6</th>\n",
              "      <td>0.08829</td>\n",
              "      <td>12.5</td>\n",
              "      <td>7.87</td>\n",
              "      <td>0</td>\n",
              "      <td>0.524</td>\n",
              "      <td>6.012</td>\n",
              "      <td>66.6</td>\n",
              "      <td>5.5605</td>\n",
              "      <td>5</td>\n",
              "      <td>311</td>\n",
              "      <td>15.2</td>\n",
              "      <td>395.60</td>\n",
              "      <td>12.43</td>\n",
              "      <td>22.9</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>7</th>\n",
              "      <td>0.14455</td>\n",
              "      <td>12.5</td>\n",
              "      <td>7.87</td>\n",
              "      <td>0</td>\n",
              "      <td>0.524</td>\n",
              "      <td>6.172</td>\n",
              "      <td>96.1</td>\n",
              "      <td>5.9505</td>\n",
              "      <td>5</td>\n",
              "      <td>311</td>\n",
              "      <td>15.2</td>\n",
              "      <td>396.90</td>\n",
              "      <td>19.15</td>\n",
              "      <td>27.1</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>8</th>\n",
              "      <td>0.21124</td>\n",
              "      <td>12.5</td>\n",
              "      <td>7.87</td>\n",
              "      <td>0</td>\n",
              "      <td>0.524</td>\n",
              "      <td>5.631</td>\n",
              "      <td>100.0</td>\n",
              "      <td>6.0821</td>\n",
              "      <td>5</td>\n",
              "      <td>311</td>\n",
              "      <td>15.2</td>\n",
              "      <td>386.63</td>\n",
              "      <td>29.93</td>\n",
              "      <td>16.5</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>9</th>\n",
              "      <td>0.17004</td>\n",
              "      <td>12.5</td>\n",
              "      <td>7.87</td>\n",
              "      <td>0</td>\n",
              "      <td>0.524</td>\n",
              "      <td>6.004</td>\n",
              "      <td>85.9</td>\n",
              "      <td>6.5921</td>\n",
              "      <td>5</td>\n",
              "      <td>311</td>\n",
              "      <td>15.2</td>\n",
              "      <td>386.71</td>\n",
              "      <td>17.10</td>\n",
              "      <td>18.9</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>10</th>\n",
              "      <td>0.22489</td>\n",
              "      <td>12.5</td>\n",
              "      <td>7.87</td>\n",
              "      <td>0</td>\n",
              "      <td>0.524</td>\n",
              "      <td>6.377</td>\n",
              "      <td>94.3</td>\n",
              "      <td>6.3467</td>\n",
              "      <td>5</td>\n",
              "      <td>311</td>\n",
              "      <td>15.2</td>\n",
              "      <td>392.52</td>\n",
              "      <td>20.45</td>\n",
              "      <td>15.0</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>11</th>\n",
              "      <td>0.11747</td>\n",
              "      <td>12.5</td>\n",
              "      <td>7.87</td>\n",
              "      <td>0</td>\n",
              "      <td>0.524</td>\n",
              "      <td>6.009</td>\n",
              "      <td>82.9</td>\n",
              "      <td>6.2267</td>\n",
              "      <td>5</td>\n",
              "      <td>311</td>\n",
              "      <td>15.2</td>\n",
              "      <td>396.90</td>\n",
              "      <td>13.27</td>\n",
              "      <td>18.9</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>12</th>\n",
              "      <td>0.09378</td>\n",
              "      <td>12.5</td>\n",
              "      <td>7.87</td>\n",
              "      <td>0</td>\n",
              "      <td>0.524</td>\n",
              "      <td>5.889</td>\n",
              "      <td>39.0</td>\n",
              "      <td>5.4509</td>\n",
              "      <td>5</td>\n",
              "      <td>311</td>\n",
              "      <td>15.2</td>\n",
              "      <td>390.50</td>\n",
              "      <td>15.71</td>\n",
              "      <td>21.7</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>13</th>\n",
              "      <td>0.62976</td>\n",
              "      <td>0.0</td>\n",
              "      <td>8.14</td>\n",
              "      <td>0</td>\n",
              "      <td>0.538</td>\n",
              "      <td>5.949</td>\n",
              "      <td>61.8</td>\n",
              "      <td>4.7075</td>\n",
              "      <td>4</td>\n",
              "      <td>307</td>\n",
              "      <td>21.0</td>\n",
              "      <td>396.90</td>\n",
              "      <td>8.26</td>\n",
              "      <td>20.4</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>14</th>\n",
              "      <td>0.63796</td>\n",
              "      <td>0.0</td>\n",
              "      <td>8.14</td>\n",
              "      <td>0</td>\n",
              "      <td>0.538</td>\n",
              "      <td>6.096</td>\n",
              "      <td>84.5</td>\n",
              "      <td>4.4619</td>\n",
              "      <td>4</td>\n",
              "      <td>307</td>\n",
              "      <td>21.0</td>\n",
              "      <td>380.02</td>\n",
              "      <td>10.26</td>\n",
              "      <td>18.2</td>\n",
              "    </tr>\n",
              "  </tbody>\n",
              "</table>\n",
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            ],
            "text/plain": [
              "       CRIM    ZN  INDUS  CHAS    NOX     RM    AGE     DIS  RAD  TAX  \\\n",
              "0   0.00632  18.0   2.31     0  0.538  6.575   65.2  4.0900    1  296   \n",
              "1   0.02731   0.0   7.07     0  0.469  6.421   78.9  4.9671    2  242   \n",
              "2   0.02729   0.0   7.07     0  0.469  7.185   61.1  4.9671    2  242   \n",
              "3   0.03237   0.0   2.18     0  0.458  6.998   45.8  6.0622    3  222   \n",
              "4   0.06905   0.0   2.18     0  0.458  7.147   54.2  6.0622    3  222   \n",
              "5   0.02985   0.0   2.18     0  0.458  6.430   58.7  6.0622    3  222   \n",
              "6   0.08829  12.5   7.87     0  0.524  6.012   66.6  5.5605    5  311   \n",
              "7   0.14455  12.5   7.87     0  0.524  6.172   96.1  5.9505    5  311   \n",
              "8   0.21124  12.5   7.87     0  0.524  5.631  100.0  6.0821    5  311   \n",
              "9   0.17004  12.5   7.87     0  0.524  6.004   85.9  6.5921    5  311   \n",
              "10  0.22489  12.5   7.87     0  0.524  6.377   94.3  6.3467    5  311   \n",
              "11  0.11747  12.5   7.87     0  0.524  6.009   82.9  6.2267    5  311   \n",
              "12  0.09378  12.5   7.87     0  0.524  5.889   39.0  5.4509    5  311   \n",
              "13  0.62976   0.0   8.14     0  0.538  5.949   61.8  4.7075    4  307   \n",
              "14  0.63796   0.0   8.14     0  0.538  6.096   84.5  4.4619    4  307   \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  \n",
              "5      18.7  394.12   5.21  28.7  \n",
              "6      15.2  395.60  12.43  22.9  \n",
              "7      15.2  396.90  19.15  27.1  \n",
              "8      15.2  386.63  29.93  16.5  \n",
              "9      15.2  386.71  17.10  18.9  \n",
              "10     15.2  392.52  20.45  15.0  \n",
              "11     15.2  396.90  13.27  18.9  \n",
              "12     15.2  390.50  15.71  21.7  \n",
              "13     21.0  396.90   8.26  20.4  \n",
              "14     21.0  380.02  10.26  18.2  "
            ]
          },
          "execution_count": 38,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "boston_data.head(15)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "### Determining Normality\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Making our target variable normally distributed often will lead to better results\n",
        "\n",
        "If our target is not normally distributed, we can apply a transformation to it and then fit our regression to predict the transformed values.\n",
        "\n",
        "How can we tell if our target is normally distributed? There are two ways:\n",
        "\n",
        "* Visually\n",
        "* Using a statistical test\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "#### Visually\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Plotting a histogram:\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 39,
      "metadata": {
        "trusted": true
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          },
          "output_type": "display_data"
        }
      ],
      "source": [
        "boston_data.MEDV.hist();"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Does not look normal due to that right tail. Let's try to verify statistically:\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 40,
      "metadata": {
        "trusted": true
      },
      "outputs": [],
      "source": [
        "from scipy.stats.mstats import normaltest # D'Agostino K^2 Test"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Without getting into Bayesian vs. frequentist debates, for the purposes of this lesson, the following will suffice:\n",
        "\n",
        "* This is a statistical test that tests whether a distribution is normally distributed or not. It isn't perfect, but suffice it to say: \n",
        "    * This test outputs a \"p-value\". The _higher_ this p-value is the _closer_ the distribution is to normal.\n",
        "    * Frequentist statisticians would say that you accept that the distribution is normal (more specifically: fail to reject the null hypothesis that it is normal) if p > 0.05.\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 41,
      "metadata": {
        "trusted": true
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "NormaltestResult(statistic=90.9746873700967, pvalue=1.7583188871696095e-20)"
            ]
          },
          "execution_count": 41,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "normaltest(boston_data.MEDV.values)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "p-value _extremely_ low. Our y variable we've been dealing with this whole time was not normally distributed!\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Linear Regression assumes a normally distributed residuals which can be aided by transforming y variable. Let's try some common transformations to try and get y to be normally distributed: \n",
        "\n",
        "* Log\n",
        "* Square root\n",
        "* Box cox\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "### Testing log\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "The log transform can transform data that is significantly skewed right to be more normally distributed:\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 42,
      "metadata": {
        "trusted": true
      },
      "outputs": [
        {
          "data": {
            "image/png": 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",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          },
          "output_type": "display_data"
        }
      ],
      "source": [
        "data = plot_exponential_data()"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 43,
      "metadata": {
        "trusted": true
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          },
          "output_type": "display_data"
        }
      ],
      "source": [
        "plt.hist(np.log(data));"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "**Apply transform to Boston data:**\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 44,
      "metadata": {
        "trusted": true
      },
      "outputs": [],
      "source": [
        "log_medv = np.log(boston_data.MEDV)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 45,
      "metadata": {
        "trusted": true
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          },
          "output_type": "display_data"
        }
      ],
      "source": [
        "log_medv.hist();"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 46,
      "metadata": {
        "trusted": true
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "NormaltestResult(statistic=17.21801696640697, pvalue=0.00018245472768345307)"
            ]
          },
          "execution_count": 46,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "normaltest(log_medv)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Conclusion: closer, but still not normal.\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "### Exercise: \n",
        "\n",
        "The square root transformation is another transformation that can transform non-normally distributed data into normally distributed data:\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 47,
      "metadata": {
        "scrolled": true,
        "trusted": true
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          },
          "output_type": "display_data"
        }
      ],
      "source": [
        "data = plot_square_normal_data()"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Slightly skewed right.\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 48,
      "metadata": {
        "scrolled": true,
        "trusted": true
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          },
          "output_type": "display_data"
        }
      ],
      "source": [
        "plt.hist(np.sqrt(data));"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Apply the square root transformation to the Boston data target and test whether the result is normally distributed.\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 69,
      "metadata": {
        "trusted": true
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "(array([  9.,  18.,  40.,  67., 131., 121.,  48.,  36.,  10.,  26.]),\n",
              " array([2.23606798, 2.71956796, 3.20306794, 3.68656793, 4.17006791,\n",
              "        4.65356789, 5.13706788, 5.62056786, 6.10406784, 6.58756783,\n",
              "        7.07106781]),\n",
              " <BarContainer object of 10 artists>)"
            ]
          },
          "execution_count": 69,
          "metadata": {},
          "output_type": "execute_result"
        },
        {
          "data": {
            "image/png": "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",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          },
          "output_type": "display_data"
        }
      ],
      "source": [
        "sqrt_medv = np.sqrt(boston_data.MEDV)\n",
        "plt.hist(sqrt_medv)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 70,
      "metadata": {
        "trusted": true
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "NormaltestResult(statistic=20.487090826863067, pvalue=3.558645701429252e-05)"
            ]
          },
          "execution_count": 70,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "normaltest(sqrt_medv)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "### Box cox\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "The box cox transformation is a parametrized transformation that tries to get distributions \"as close to a normal distribution as possible\".\n",
        "\n",
        "It is defined as:\n",
        "\n",
        "$$ \\text{boxcox}(y_i) = \\frac{y_i^{\\lambda} - 1}{\\lambda} $$\n",
        "\n",
        "You can think of as a generalization of the square root function: the square root function uses the exponent of 0.5, but box cox lets its exponent vary so it can find the best one.\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 52,
      "metadata": {
        "trusted": true
      },
      "outputs": [],
      "source": [
        "from scipy.stats import boxcox"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 53,
      "metadata": {
        "trusted": true
      },
      "outputs": [],
      "source": [
        "bc_result = boxcox(boston_data.MEDV)\n",
        "boxcox_medv = bc_result[0]\n",
        "lam = bc_result[1]"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 54,
      "metadata": {
        "scrolled": true,
        "trusted": true
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "0.2166209012915364"
            ]
          },
          "execution_count": 54,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "lam"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 55,
      "metadata": {
        "trusted": true
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          },
          "output_type": "display_data"
        }
      ],
      "source": [
        "boston_data['MEDV'].hist();"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 56,
      "metadata": {
        "trusted": true
      },
      "outputs": [
        {
          "data": {
            "image/png": 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s7JZfCPwR8NhYi5ogVXVdVW2rqmkWXpnxhar60zGXNTGSnNPdqEA33fBa4OHxVjU5qupp4Kkkr+iadgFju5mjtbtlLgbeDDzUzSsDvLuqPjO+kibKFuBg92UqzwFurypv99NqbQY+tTCGYgPwb1X1ufGWNHHeBtza3SnzOPDWcRXi6wckqUFNTctIkhYY7pLUIMNdkhpkuEtSgwx3SWqQ4S5JDTLcJalB/wPrwItf581YKwAAAABJRU5ErkJggg==",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          },
          "output_type": "display_data"
        }
      ],
      "source": [
        "plt.hist(boxcox_medv);"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 57,
      "metadata": {
        "scrolled": true,
        "trusted": true
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "NormaltestResult(statistic=4.513528775533045, pvalue=0.1046886692817602)"
            ]
          },
          "execution_count": 57,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "normaltest(boxcox_medv)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Significantly more normally distributed (according to p value) than the other two distributions - above 0.05, even!\n",
        "\n",
        "Now that we have a normally distributed y-variable, let's try a regression!\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "### Testing regression:\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 58,
      "metadata": {
        "trusted": true
      },
      "outputs": [],
      "source": [
        "from sklearn.linear_model import LinearRegression\n",
        "from sklearn.metrics import r2_score\n",
        "from sklearn.model_selection import train_test_split\n",
        "from sklearn.preprocessing import (StandardScaler, \n",
        "                                   PolynomialFeatures)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 59,
      "metadata": {
        "trusted": true
      },
      "outputs": [],
      "source": [
        "lr = LinearRegression()"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "**Reload clean version of `boston_data`:**\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Same steps as before.\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "**Create X and y**\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 74,
      "metadata": {
        "trusted": true
      },
      "outputs": [],
      "source": [
        "y_col = \"MEDV\"\n",
        "\n",
        "X = boston_data.drop(y_col, axis=1)\n",
        "y = boston_data[y_col]"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "**Create Polynomial Features**\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 75,
      "metadata": {
        "trusted": true
      },
      "outputs": [],
      "source": [
        "pf = PolynomialFeatures(degree=2, include_bias=False)\n",
        "X_pf = pf.fit_transform(X)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "**Train test split**\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 76,
      "metadata": {
        "trusted": true
      },
      "outputs": [],
      "source": [
        "X_train, X_test, y_train, y_test = train_test_split(X_pf, y, test_size=0.3, \n",
        "                                                    random_state=72018)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "**Fit `StandardScaler` on `X_train` as before**\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 77,
      "metadata": {
        "trusted": true
      },
      "outputs": [],
      "source": [
        "s = StandardScaler()\n",
        "X_train_s = s.fit_transform(X_train)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "**Discuss: what transformation do we need to apply next?**\n",
        "\n",
        "Apply the appropriate transformation.\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "trusted": true
      },
      "outputs": [],
      "source": [
        "pass # your code here"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "trusted": true
      },
      "outputs": [],
      "source": []
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "As before, we'll now:\n",
        "\n",
        "1. Fit regression\n",
        "1. Transform testing data\n",
        "1. Predict on testing data\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 78,
      "metadata": {
        "trusted": true
      },
      "outputs": [
        {
          "ename": "NameError",
          "evalue": "name 'y_train_bc' is not defined",
          "output_type": "error",
          "traceback": [
            "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
            "\u001b[1;31mNameError\u001b[0m                                 Traceback (most recent call last)",
            "\u001b[1;32mc:\\Users\\jhony\\OneDrive\\Imagens\\2023\\Estudos independentes\\computação estudos\\Data Science\\Machine-Learning\\Supervised_Machine_Learning\\IBM-ML240EN-SkillsNetwork_labs_02a_LAB_Transforming_Target.jupyterlite (1).ipynb Cell 66\u001b[0m in \u001b[0;36m<cell line: 1>\u001b[1;34m()\u001b[0m\n\u001b[1;32m----> <a href='vscode-notebook-cell:/c%3A/Users/jhony/OneDrive/Imagens/2023/Estudos%20independentes/computa%C3%A7%C3%A3o%20estudos/Data%20Science/Machine-Learning/Supervised_Machine_Learning/IBM-ML240EN-SkillsNetwork_labs_02a_LAB_Transforming_Target.jupyterlite%20%281%29.ipynb#Y122sZmlsZQ%3D%3D?line=0'>1</a>\u001b[0m y_train_bc\u001b[39m.\u001b[39mshape\n",
            "\u001b[1;31mNameError\u001b[0m: name 'y_train_bc' is not defined"
          ]
        }
      ],
      "source": [
        "y_train_bc.shape"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "trusted": true
      },
      "outputs": [],
      "source": [
        "lr.fit(X_train_s, y_train_bc)\n",
        "X_test_s = s.transform(X_test)\n",
        "y_pred_bc = lr.predict(X_test_s)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "### Discussion\n",
        "\n",
        "* Are we done?\n",
        "* What did we predict?\n",
        "* How would you interpret these predictions?\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "#### Inverse transform\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Every transformation has an inverse transformation. The inverse transformation of $f(x) = \\sqrt{x}$ is $f^{-1}(x) = x^2$, for example. Box cox has an inverse transformation as well: notice that we have to pass in the lambda value that we found from before:\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "trusted": true
      },
      "outputs": [],
      "source": [
        "from scipy.special import inv_boxcox"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "trusted": true
      },
      "outputs": [],
      "source": [
        "# code from above\n",
        "bc_result = boxcox(boston_data.MEDV)\n",
        "boxcox_medv = bc_result[0]\n",
        "lam = bc_result[1]"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "trusted": true
      },
      "outputs": [],
      "source": [
        "inv_boxcox(boxcox_medv, lam)[:10]"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "trusted": true
      },
      "outputs": [],
      "source": [
        "boston_data['MEDV'].values[:10]"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Exactly the same, as we would hope!\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "### Exercise:\n",
        "\n",
        "1. Apply the appropriate inverse transformation to `y_pred_bc`.\n",
        "2. Calculate the $R^2$ using the result of this inverse transformation and `y_test`.  \n",
        "\n",
        "**Hint:** Should be two lines of code.\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "trusted": true
      },
      "outputs": [],
      "source": [
        "pass # your code here"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "trusted": true
      },
      "outputs": [],
      "source": []
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## LAB Exercise: \n",
        "\n",
        "### Determine the R^2 of a LinearRegression without the box cox transformation. Is it higher or lower?\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "trusted": true
      },
      "outputs": [],
      "source": [
        "### BEGIN SOLUTION\n",
        "lr = LinearRegression()\n",
        "lr.fit(X_train_s,y_train)\n",
        "lr_pred = lr.predict(X_test_s)\n",
        "r2_score(lr_pred,y_test)\n",
        "### END SOLUTION"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "---\n",
        "### Machine Learning Foundation (C) 2020 IBM Corporation\n"
      ]
    }
  ],
  "metadata": {
    "kernelspec": {
      "display_name": "Python 3",
      "language": "python",
      "name": "python3"
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      "codemirror_mode": {
        "name": "ipython",
        "version": 3
      },
      "file_extension": ".py",
      "mimetype": "text/x-python",
      "name": "python",
      "nbconvert_exporter": "python",
      "pygments_lexer": "ipython3",
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