{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "view-in-github"
   },
   "source": [
    "<a href=\"https://colab.research.google.com/github/smart-stats/ds4bio_book/blob/main/book/regression_examples.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a> "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Linear models: a classic example"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "colab": {},
    "colab_type": "code",
    "id": "oIimJROu2Hc_"
   },
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "import numpy as np\n",
    "from sklearn.linear_model import LinearRegression\n",
    "import matplotlib.pyplot as plt\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 201
    },
    "colab_type": "code",
    "id": "vsVJc6lS2Qov",
    "outputId": "fc0b2fe3-eccd-4f2a-819f-084bc8e2602d"
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Region</th>\n",
       "      <th>Fertility</th>\n",
       "      <th>Agriculture</th>\n",
       "      <th>Examination</th>\n",
       "      <th>Education</th>\n",
       "      <th>Catholic</th>\n",
       "      <th>Infant.Mortality</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>Courtelary</td>\n",
       "      <td>80.2</td>\n",
       "      <td>17.0</td>\n",
       "      <td>15</td>\n",
       "      <td>12</td>\n",
       "      <td>9.96</td>\n",
       "      <td>22.2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>Delemont</td>\n",
       "      <td>83.1</td>\n",
       "      <td>45.1</td>\n",
       "      <td>6</td>\n",
       "      <td>9</td>\n",
       "      <td>84.84</td>\n",
       "      <td>22.2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>Franches-Mnt</td>\n",
       "      <td>92.5</td>\n",
       "      <td>39.7</td>\n",
       "      <td>5</td>\n",
       "      <td>5</td>\n",
       "      <td>93.40</td>\n",
       "      <td>20.2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>Moutier</td>\n",
       "      <td>85.8</td>\n",
       "      <td>36.5</td>\n",
       "      <td>12</td>\n",
       "      <td>7</td>\n",
       "      <td>33.77</td>\n",
       "      <td>20.3</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>Neuveville</td>\n",
       "      <td>76.9</td>\n",
       "      <td>43.5</td>\n",
       "      <td>17</td>\n",
       "      <td>15</td>\n",
       "      <td>5.16</td>\n",
       "      <td>20.6</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "         Region  Fertility  Agriculture  Examination  Education  Catholic  \\\n",
       "0    Courtelary       80.2         17.0           15         12      9.96   \n",
       "1      Delemont       83.1         45.1            6          9     84.84   \n",
       "2  Franches-Mnt       92.5         39.7            5          5     93.40   \n",
       "3       Moutier       85.8         36.5           12          7     33.77   \n",
       "4    Neuveville       76.9         43.5           17         15      5.16   \n",
       "\n",
       "   Infant.Mortality  \n",
       "0              22.2  \n",
       "1              22.2  \n",
       "2              20.2  \n",
       "3              20.3  \n",
       "4              20.6  "
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dat = pd.read_csv(\"https://raw.githubusercontent.com/bcaffo/ds4bme_intro/master/data/swiss.csv\")\n",
    "dat.head()\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 54
    },
    "colab_type": "code",
    "id": "T34j2o8u2cH_",
    "outputId": "61309a3a-8ff6-489d-9dfd-c1ea40be99d8"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[66.9151816789687,\n",
       " array([-0.17211397, -0.25800824, -0.87094006,  0.10411533,  1.07704814])]"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y = dat.Fertility\n",
    "x = dat.drop(['Region', 'Fertility'], axis=1)\n",
    "fit = LinearRegression().fit(x, y)\n",
    "yhat = fit.predict(x)\n",
    "[fit.intercept_, fit.coef_]\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "colab": {},
    "colab_type": "code",
    "id": "Dwova9_d5BKe"
   },
   "outputs": [],
   "source": [
    "x2 = x\n",
    "x2['Test'] = x2.Agriculture + x2.Examination\n",
    "fit2 = LinearRegression().fit(x2, y)\n",
    "yhat2 = fit2.predict(x2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 283
    },
    "colab_type": "code",
    "id": "RAIz1jJs6YFg",
    "outputId": "de9beb6e-d3e6-4033-eba9-27689bac6ff4"
   },
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(yhat, yhat2);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 283
    },
    "colab_type": "code",
    "id": "626Z0Ql66qWx",
    "outputId": "12617691-af4e-47d3-d080-10223101ec81"
   },
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x3 = x2.drop(['Agriculture'], axis = 1)\n",
    "fit3 = LinearRegression().fit(x3, y)\n",
    "yhat3 = fit3.predict(x3)\n",
    "plt.plot(yhat, yhat3);\n"
   ]
  }
 ],
 "metadata": {
  "colab": {
   "authorship_tag": "ABX9TyPD6YfE7eh7Lu8TLxXjNkFR",
   "include_colab_link": true,
   "name": "regression_examples",
   "provenance": []
  },
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.10.13"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}