2.2-multiple-linear-regression-for-car-price-prediction.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Introduction to Multiple Linear Regression\n",
    "\n",
    "在简单线性回归的代码部分，我们分析了120个二手车价格(price)和车龄(ageOfCar)之间的关系，得到了可以通过车龄来预测价格的线性模型。但是，实际的二手车市场中，影响二手车价格的因素远远不止车龄。与价格相关的还有因素诸如：行驶里程，车辆是否损坏等等。\n",
    "\n",
    "所以如果我们能够将更多的信息加入之前的模型，那么模型的预测能力应该可以进一步提升。所以在本章中，我们会利用上一节介绍的多元线性回归（Multiple Linear Regression）方法，在之前的二手车数据中加入行驶里程（kilometer）和是否车辆有损坏（isDamaged）这两个predictor，来建立一个更准确的二手车价格预测方法。\n",
    "\n",
    "在本章的教程中，你会学到：\n",
    "* 如何对predictor进行筛选\n",
    "* 如何建立哑变量（dummy variable）\n",
    "* 如何使用箱线图（boxplot）\n",
    "* 如何建立多元线性模型\n",
    "* 如何比较不同的多元线性模型的结果\n",
    "\n",
    "## 1.准备工作\n",
    "\n",
    "首先我们来引入Python库。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# imports\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import statsmodels.api as sm\n",
    "\n",
    "# this allows plots to appear directly in the notebook\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2.数据探索\n",
    "\n",
    "接着，我们来读取新的二手车数据，看一看数据中包含的值。\n",
    "\n",
    "**(1) 读入数据**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "auto_df = pd.read_csv('./data/auto_multiple.csv', index_col=0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**(2) 显示数据**\n",
    "\n",
    "用`head()`打印出数据表格（dataframe）的前5行。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "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>name</th>\n",
       "      <th>price</th>\n",
       "      <th>kilometer</th>\n",
       "      <th>vehicleType</th>\n",
       "      <th>ageOfCar</th>\n",
       "      <th>isDamaged</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>Volkswagen_Golf_1.4</td>\n",
       "      <td>1599</td>\n",
       "      <td>150000</td>\n",
       "      <td>limousine</td>\n",
       "      <td>18</td>\n",
       "      <td>no</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>Volkswagen_Golf_1.4</td>\n",
       "      <td>10300</td>\n",
       "      <td>50000</td>\n",
       "      <td>limousine</td>\n",
       "      <td>5</td>\n",
       "      <td>no</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>Volkswagen_Golf_1.4</td>\n",
       "      <td>1600</td>\n",
       "      <td>150000</td>\n",
       "      <td>limousine</td>\n",
       "      <td>14</td>\n",
       "      <td>no</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>Volkswagen_Golf_1.4</td>\n",
       "      <td>7300</td>\n",
       "      <td>70000</td>\n",
       "      <td>limousine</td>\n",
       "      <td>7</td>\n",
       "      <td>no</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>Volkswagen_Golf_1.4</td>\n",
       "      <td>599</td>\n",
       "      <td>125000</td>\n",
       "      <td>limousine</td>\n",
       "      <td>18</td>\n",
       "      <td>no</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                  name  price  kilometer vehicleType  ageOfCar isDamaged\n",
       "0  Volkswagen_Golf_1.4   1599     150000   limousine        18        no\n",
       "1  Volkswagen_Golf_1.4  10300      50000   limousine         5        no\n",
       "2  Volkswagen_Golf_1.4   1600     150000   limousine        14        no\n",
       "3  Volkswagen_Golf_1.4   7300      70000   limousine         7        no\n",
       "4  Volkswagen_Golf_1.4    599     125000   limousine        18        no"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "auto_df.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**(3) 数据字典**\n",
    "- ageOfCar: 车辆的使用年数。\n",
    "- name: 车的型号。\n",
    "- kilometer: 行驶公里数。\n",
    "- vehicleType: 车辆类别。\n",
    "- isDamaged: 有没有事故记录。\n",
    "- price: 二手车的出售价格。\n",
    "\n",
    "以上六个feature中，price是outcome，其它5项的是predictor。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**(4) 数据量**\n",
    "\n",
    "用pandas的`shape()`函数查看数据表格（dataframe）的行数和列数，我们看到这个数据表中有160个observation。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(160, 6)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "auto_df.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**(5) 数据质量检测**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "首先我们来进一步的看看每一项数据的内容，第一件事情我们要了解的是数据库中有没有空值。如果有空值，会在训练模型的部分引发错误，所以我们需要这一步处理空值。\n",
    "\n",
    "Pandas数据表有一个自带函数`isna()`，用于检测每一个feature的每一个值是否为空，然后返回一个同样尺寸的数据表（dataframe）。如果某个数据为空，则新的数据表对应位置的值为True，反之则为False。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "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>name</th>\n",
       "      <th>price</th>\n",
       "      <th>kilometer</th>\n",
       "      <th>vehicleType</th>\n",
       "      <th>ageOfCar</th>\n",
       "      <th>isDamaged</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>False</td>\n",
       "      <td>False</td>\n",
       "      <td>False</td>\n",
       "      <td>False</td>\n",
       "      <td>False</td>\n",
       "      <td>False</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>False</td>\n",
       "      <td>False</td>\n",
       "      <td>False</td>\n",
       "      <td>False</td>\n",
       "      <td>False</td>\n",
       "      <td>False</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>False</td>\n",
       "      <td>False</td>\n",
       "      <td>False</td>\n",
       "      <td>False</td>\n",
       "      <td>False</td>\n",
       "      <td>False</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>False</td>\n",
       "      <td>False</td>\n",
       "      <td>False</td>\n",
       "      <td>False</td>\n",
       "      <td>False</td>\n",
       "      <td>False</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>False</td>\n",
       "      <td>False</td>\n",
       "      <td>False</td>\n",
       "      <td>False</td>\n",
       "      <td>False</td>\n",
       "      <td>False</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "    name  price  kilometer  vehicleType  ageOfCar  isDamaged\n",
       "0  False  False      False        False     False      False\n",
       "1  False  False      False        False     False      False\n",
       "2  False  False      False        False     False      False\n",
       "3  False  False      False        False     False      False\n",
       "4  False  False      False        False     False      False"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "auto_check_null = auto_df.isna()\n",
    "auto_check_null.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "我们可以用`sum()`函数将这些True/False值相加，True会自动变为1，False会自动转为0。加完后的数字，就代表每一个feature中间空值的数量。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "name           0\n",
       "price          0\n",
       "kilometer      0\n",
       "vehicleType    0\n",
       "ageOfCar       0\n",
       "isDamaged      0\n",
       "dtype: int64"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "auto_check_null.sum()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "我们可以看到，没有空值，所以我们可以继续下面的分析。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3.数据可视化"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "接下来，我们对数据进行可视化，用图表将每个predictor与outcome的关系体现出来，看看能否发现出一些规律。\n",
    "\n",
    "5个predictor包含连续变量（continuous variable）和分类变量（categorical variable）。其中：\n",
    "* **连续变量:** 行驶公里数（kilometer）, 车龄（ageOfCar）\n",
    "* **分类变量:** 车辆类别（vehicleType）, 车辆损坏状态（isDamaged）, 车辆名称（name）\n",
    "\n",
    "对于连续变量，我们可以通过绘制散点图，来观察它们与价格（price）之间是否存在相关性。\n",
    "\n",
    "而对于分类变量，我们主要观察两点：\n",
    "1. 包含哪些类别，对应observation的数量是多少。\n",
    "2. 每个类别中价格的分布情况。\n",
    "\n",
    "\n",
    "我们先来看连续变量：\n",
    "\n",
    "**(1) 公里数（kilometer）**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.axes._subplots.AxesSubplot at 0x121f7f0b8>"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAZUAAAEWCAYAAACufwpNAAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADl0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uIDMuMC4yLCBodHRwOi8vbWF0cGxvdGxpYi5vcmcvOIA7rQAAIABJREFUeJzt3X+cXXV95/HXe35kMiRAhiSNkAkkNGgNLiDOCmyspWAB0UJXFHHdGhFLW6m/LT+sj6Vb292CVSu1BalSo2uBSKyhrkoRsLarIAMmwYApEYKZ8CsMSWBCmMxkPvvH/Q65M87N3Jmc+/O8n4/HzZzzOd9zz/fce3M/93y/33OOIgIzM7MstNS6AmZm1jycVMzMLDNOKmZmlhknFTMzy4yTipmZZcZJxczMMuOkYk1J0pGSBiS1pvnvS3pvrevVSNLrd3St62GNxUnFGpakzZLeUDR/gaTtkn4jIn4REbMjYm+N6xiSltayDqkei1NdBtJjs6TL97dOev0eqVYdrTm01boCZlmQtAL4DPCmiPhhreuTBUltETGc8dPOiYhhSacAd0haGxHfrcJ2LSd8pGINT9LvA58GzhxNKEW/zH/ph5OkFkmfkPSYpKclfUXSoePWu1DSlnTk8weS/rOk9ZJ2SPr8uOd7j6SHUtnbJB2V4j9IRdalo4O3p/ibJa1Nz/VDSccVPddmSZdJWg/smqj+WYiIHwEbgFel7YakSyQ9DDxcFFuapjslfTq9Zjsl/bukzrTs5LQfOyStk3RqJepsDSIi/PCjIR/AZmA18BRw/Lhli4EA2tL894H3pun3AJuAo4HZwDeAr45b7zpgJnAG8CLwTeBXgIXA08BvpPLnpud6JYUj/08APyyqRwBLi+ZfndY/CWgFVqT96Cjap7XAIqBzGq+JSsRfej0AAcuBF4DTi+p5O3DY6HaL6w78bXoNF6Z6/xegI833A2dT+JH6W2l+fq0/H37U5lHzCvjhx3Qf6Qv4OWAN0DJu2f6Syh3A+4rKvgIYSl+4o+stLFreD7y9aH418KE0/R3goqJlLenL+qg0Pz6pXAt8clxdNxYlqc3AeybZ7zcD9wDbUrI7G+gCfhv48xLrjO7XDmA78BDwgaLlAZw2bp0AlqZ92s24xJ3KXEZKyEWx24AVtf58+FGbh5u/rNH9IfBy4IuSVOY6RwCPFc0/RiGhLCiKPVU0vXuC+dlp+ijgc6npZwfwLIUjgYUltn0U8NHR8mmdRalOo7ZMUv+3Ae8EjgRWAR8H/oPCEdiXJll3XkR0RcQrI+KacctKbXcehaO2n0+w7CjgbeP253XA4ZPUw5qUO+qt0T0FnA78K/B3FJLMZB6n8GU46khgOD1X9xS3vwX4i4j42hTL/8V+ykx26fALI2IkTf9jemSh1HafodAE+KvAunHLtlA4Uvm9jOpgDc5HKtbwIuJxConlLEmfLWOVG4EPS1oiaTbwv4CbY3ojnq4DrpB0LICkQyW9rWj5UxT6bkb9PfAHkk5SwSxJb5J0cLkbLEooVZG2dwPwGUlHSGqVdIqkDuD/AL8t6cwUnynpVElTTc7WJJxUrClExC+A04C3SvrfkxS/Afgq8APgUQq/wt8/ze3+E3AVcJOk54CfAm8sKvKnwMrUNHR+RPQCvwd8nkLfxibg3dPZdpV9DHgAuJdCE99VFPqxtlAYrPBxCn08W4A/xt8tuaUI36TLzMyy4V8TZmaWGScVMzPLjJOKmZllxknFzMwyk7vzVObNmxeLFy+udTXMzBrGfffd90xEzC+nbO6SyuLFi+nt7a11NczMGoakxyYvVeDmLzMzy4yTipmZZcZJxczMMuOkYmZmmXFSMTOzzDip5FD/wCDrtuygf2Cw1lUxsyaTuyHFebdm7VYuW72e9pYWhkZGuPq84zjnhFL3kzIzmxofqeRI/8Agl61ez4tDIzw/OMyLQyNcunq9j1jMLDNOKjnSt3037S1j3/L2lhb6tu+uUY3MrNk4qeRId1cnQyNjbxo4NDJCd1dnjWpkZs3GSSVH5s7u4OrzjmNmewsHd7Qxs72Fq887jrmzO2pdNTNrEhXrqJd0A/Bm4OmIeFWKHQbcDCwGNgPnR8R2SQI+B5wNvAC8OyLuT+usAD6RnvbPI2Jlir8G+DLQCXwb+GD4NpaTOueEhSxfOo++7bvp7up0QjGzTFXySOXLwFnjYpcDd0TEMcAdaR4K9/Q+Jj0uBq6Fl5LQlcBJwGuBKyV1pXWupXCv79H1xm/LSpg7u4PjF81xQjGzzFUsqUTED4Bnx4XPBVam6ZXA7xTFvxIFdwNzJB0OnAncHhHPRsR24HbgrLTskIi4Ox2dfKXouczMrEaq3aeyICKeSNNPAgvS9EJgS1G5vhTbX7xvgviEJF0sqVdS77Zt2w5sD8zMrKSaddSnI4yq9IFExPUR0RMRPfPnl3WfGTMzm4ZqJ5WnUtMV6e/TKb4VWFRUrjvF9hfvniBuZmY1VO2kciuwIk2vANYUxd+lgpOBnamZ7DbgDEldqYP+DOC2tOw5SSenkWPvKnouMzOrkUoOKb4ROBWYJ6mPwiiuvwRWSboIeAw4PxX/NoXhxJsoDCm+ECAinpX0SeDeVO7PImK08/997BtS/J30MDOzGlLeTu3o6ekJ36PezKx8ku6LiJ5yyvqMejMzy4yTipmZZcZJxczMMuOkYmZmmXFSMTOzzDipmJlZZpxUzMwsM04qZmaWGScVMzPLjJOKmZllxknFzMwy46RiZmaZcVIxM7PMOKmYmVlmnFTMzCwzTipmZpYZJxUzM8uMk4qZmWXGScXMzDLjpGJmZplxUjEzs8w4qZiZWWacVMzMLDNOKmZmlhknFTMzy4yTipmZZcZJxczMMuOkYmZmmXFSMTOzzDipmJlZZpxUzMwsMzVJKpI+LGmDpJ9KulHSTElLJN0jaZOkmyXNSGU70vymtHxx0fNckeIbJZ1Zi30xM7N9qp5UJC0EPgD0RMSrgFbgAuAq4LMRsRTYDlyUVrkI2J7in03lkLQsrXcscBbwd5Jaq7kvZmY2Vq2av9qATkltwEHAE8BpwC1p+Urgd9L0uWmetPx0SUrxmyJiMCIeBTYBr61S/c3MbAJVTyoRsRX4K+AXFJLJTuA+YEdEDKdifcDCNL0Q2JLWHU7l5xbHJ1jHzMxqoBbNX10UjjKWAEcAsyg0X1VymxdL6pXUu23btkpuysws12rR/PUG4NGI2BYRQ8A3gOXAnNQcBtANbE3TW4FFAGn5oUB/cXyCdcaIiOsjoicieubPn5/1/piZWVKLpPIL4GRJB6W+kdOBB4G7gLemMiuANWn61jRPWn5nRESKX5BGhy0BjgF+XKV9MDOzCbRNXiRbEXGPpFuA+4Fh4CfA9cD/BW6S9Ocp9qW0ypeAr0raBDxLYcQXEbFB0ioKCWkYuCQi9lZ1Z8wy0j8wSN/23XR3dTJ3dketq2M2bSr86M+Pnp6e6O3trXU1zF6yZu1WLlu9nvaWFoZGRrj6vOM45wSPObH6Iem+iOgpp6zPqDerof6BQS5bvZ4Xh0Z4fnCYF4dGuHT1evoHBmtdNbNpcVIxq6G+7btpbxn737C9pYW+7btrVCOzA+OkYlZD3V2dDI2MjIkNjYzQ3dVZoxqZHRgnFbMamju7g6vPO46Z7S0c3NHGzPYWrj7vOHfWW8Oq+ugvqz2PNKov55ywkOVL5/k9sabgpJIzHmlUn+bO7nAysabg5q8cabaRRv0Dg6zbsqNh62/WjHykkiOjI41eZF/H8OhIo0b7lewjLrP65COVHGmWkUbNdsRl1kycVHJkdKRRR1sLB81opaOtMUca+dwOs/rlpJIzMfpv7JtrNM1yxGXWjJxUcmS02WhwOHhhaC+Dw9GQzUY+t8OsfrmjPkeaqaPe53aY1ScnlRxptmYjn9thVn/c/JUjbjYys0rzkUrOuNnIzCrJSSWH3GxkZpXi5i8zM8uMk4qZmWXGScXMzDLjpGJmZplxUjEzs8w4qZiZWWacVMzMLDNOKmZmlhknFTMzy4yTipmZZcZJxczMMuOkYmZmmXFSMTOzzDipmJlZZpxUzMwsMzVJKpLmSLpF0s8kPSTpFEmHSbpd0sPpb1cqK0nXSNokab2kE4ueZ0Uq/7CkFbXYFzMz26dWRyqfA74bEb8GHA88BFwO3BERxwB3pHmANwLHpMfFwLUAkg4DrgROAl4LXDmaiBpV/8Ag67bsoH9gsNZVMTOblqrf+VHSocDrgXcDRMQeYI+kc4FTU7GVwPeBy4Bzga9ERAB3p6Ocw1PZ2yPi2fS8twNnATdWa1+ytGbtVi5bvZ72lhaGRka4+rzjOOeEhbWulpnZlNTiSGUJsA34B0k/kfRFSbOABRHxRCrzJLAgTS8EthSt35dipeK/RNLFknol9W7bti3DXclG/8Agl61ez4tDIzw/OMyLQyNcunq9j1jMrOHUIqm0AScC10bEq4Fd7GvqAiAdlURWG4yI6yOiJyJ65s+fn9XTZqZv+27aW8a+Fe0tLfRt312jGpmZTU/ZSUXSUZLekKY7JR08zW32AX0RcU+av4VCknkqNWuR/j6dlm8FFhWt351ipeINp7urk6GRkTGxoZERurs6a1QjM7PpKSupSPo9Cl/+X0ihbuCb09lgRDwJbJH0ihQ6HXgQuBUYHcG1AliTpm8F3pVGgZ0M7EzNZLcBZ0jqSh30Z6RYw5k7u4OrzzuOme0tHNzRxsz2Fq4+7zjmzu6oddXMzKak3I76SyiMsLoHICIelvQrB7Dd9wNfkzQDeAS4kEKCWyXpIuAx4PxU9tvA2cAm4IVUloh4VtIngXtTuT8b7bRvROecsJDlS+fRt3033V2dTihm1pDKTSqDEbFHEgCS2jiAPo+IWAv0TLDo9AnKBoWkNtHz3ADcMN161Ju5szucTMysoZXbp/Kvkj4OdEr6LeDrwD9XrlpmZtaIyk0ql1MYBvwA8PsUmqQ+UalKmZllzScXV0e5zV+dwA0R8fcAklpT7IVKVczMLCs+ubh6yj1SuYNCEhnVCXwv++qYmWXLJxdXV7lJZWZEDIzOpOmDKlMlM7Ps+OTi6io3qewad3Xg1wB+R8ys7vnk4uoqN6l8CPi6pH+T9O/AzcAfVa5aZmbZ8MnF1VVWR31E3Cvp14DRs+A3RsRQ5aplZpYdn1xcPftNKpJOi4g7Jb1l3KKXSyIivlHBupmZZcYnF1fHZEcqvwHcCfz2BMsCcFIxM7OX7DepRMSVklqA70TEqirVyczMGtSkHfURMQJcWoW6mJlZgyt39Nf3JH1M0iJJh40+KlozMzNrOOVepuXtFPpQ3jcufnS21TEzs6z1DwxWbeRbuUllGYWE8joKyeXfgOsqVSkzM8tGta97Vm7z10rglcA1wN9QSDIrK1UpMzM7cLW47lm5RyqviohlRfN3SXqwEhUyM7Ns9G3fTYyMvZ9ijAR923dXrBms3COV+9P94QGQdBLQW5EamZlZJmbNaGVw79ikMrg3mDWjtWLbLPdI5TXADyX9Is0fCWyU9ACFO/4eV5HamZnZtO3as5f2VjFUlFjaW8WuPXsrts1yk8pZFauBmZlVxKwZrWMSCsBQPRypRMRjFauBmZlVxK49e+lo1ZgmsI4KH6mU26diZmYNprurE7VoTEwtqui9ZJxUzMyaVC3uJVNun4qZmTWgat9LxknFzKzJVfNeMm7+MjOzzDipmJk1uf6BQdZt2VHRy7OMcvOXmVkTq9cLSpqZWYOpxQUlnVTMzJpU3/bdtLeM/Zpvb2mhb/vuim3TScXMrEl1d3UyNDIyJjY0MtKcJz9KapX0E0nfSvNLJN0jaZOkmyXNSPGONL8pLV9c9BxXpPhGSWfWZk/MzOpT3k5+/CDwEHBImr8K+GxE3CTpOuAi4Nr0d3tELJV0QSr3dknLgAuAY4EjgO9JenlEVO6iNmZmDabaJz/W5EhFUjfwJuCLaV7AacAtqchK4HfS9Lnsu8vkLcDpqfy5wE0RMRgRjwKbgNdWZw/MrNFUc1htntXqSOWvgUuBg9P8XGBHRAyn+T5gdMzbQmALQEQMS9qZyi8E7i56zuJ1xpB0MXAxwJFHHpndXphZQ6j2sNp60vRDiiW9GXg6Iu6r1jYj4vqI6ImInvnz51drs2ZWB2oxrLZe5GVI8XLgHEmbgZsoNHt9DpgjafTIqRvYmqa3AosA0vJDgf7i+ATrmJkBtRlWWy9yMaQ4Iq6IiO6IWEyho/3OiHgncBfw1lRsBbAmTd+a5knL74yISPEL0uiwJcAxwI+rtBtm1iBqMay2XuRqSPEELgM+ImkThT6TL6X4l4C5Kf4R4HKAiNgArAIeBL4LXOKRX2Y2Xi2G1daLWuy7Cj/686Onpyd6e3trXQ0zq7L+gcGqDautNwe675Lui4iecsr6gpJmlgvVvKdInjmpmE0iz79wrfGtWbuVj61aCwgIPn3+CRUdUuykYrYfeT6/wRpf/8AgH7xpbZordHV84Ka1LF86r2I/kOqpo96sruT5/IZmlMcz6v9lw5NTimfBRypmJYyO8X+RfUMyR8f4uxmsseT1iPORZwamFM+Cj1TMSsjz+Q3NJM9HnGcue9mU4llwUjErIc/nNzSTPJ9R37NkLr++dO6Y2K8vnUvPkrkl1jhwbv6yhlWNUVnVvmy4ZS/vR5xffe/J9D7azw8efobXHzOvogkFnFSsQVWzjdznNzS20SPOS8d9XvL0ni6ZP5v2ttaqJFInFWs4xW3ko53ol65eX9FhktbY8nzEuWbtVi69ZT2tLWLvSPCptzbZpe/NDlSe28ht+ubO7uD4RXNylVD6Bwb52NfXMTg8wgt79jI4PMJHv76u6S59b3ZA8t5GblauDY/vZGjv2Os7Du0NNjy+s2LbdFKxhuNRWWbl0hTjB859KtaQ8txGblauY484JF3xax+leKU4qVjD8qgss8m1tWpME1hba+WOUsDNX2ZmTatv+25mtrWOic1sa22u2wmbmVl15P12wmZmlqFaDGpxn4qZWROr9qAWJxUzsyZXzUEtbv4yM2ty1bxBmY9UzMyaWOHaX+toVQt7Y4RPvfV4X/vLzA5cHm+nWyyP+98/MMhHV61lcDh4YWgvg8PBR1atrehr4CMVsxzI6+10R+V1/zc8/hzDY0cUMzxSiL/+5fMrsk0fqZg1uTzfThfyvf/P7R6aUjwLTipmTS7vtwrI9/7HFOMHzknFrMnl/VYB3V2d7B4aHhPbPTSci/0/pHPGlOJZcFIxa3K+VQDEuEu9j59vVkccOnNK8Sy4o96sDvQPDFb0jOc83ypgw+M72Tsytrln70jhRlWvf/mv1KhW1fH4zhdLxpcuOLgi23RSMauxao1MyuutAp7bPTyleHNxn4pZrlRzZFIez9PIuyMOnbjfqFQ8Cz5SMauh0ZFJL7KvI310ZFKWRxV5PU8D4JDO9inFm8muPXtpH3eTrvZWsWvP3opts+pHKpIWSbpL0oOSNkj6YIofJul2SQ+nv10pLknXSNokab2kE4uea0Uq/7CkFdXeF7MDVY2RWXk+TwNq01ldL2bNaB2TUACG9gazZrSWWOPA1aL5axj4aEQsA04GLpG0DLgcuCMijgHuSPMAbwSOSY+LgWuhkISAK4GTgNcCV44mIts/N4PUj2qMzMr3eRqFX+tt477p2lqo6K/1erFrz95fGucmKrvvVW/+iogngCfS9POSHgIWAucCp6ZiK4HvA5el+FciIoC7Jc2RdHgqe3tEPAsg6XbgLODGqu1MA8pzM0i9qvTIrLyfpzJrRuuElyqp5K/1evHM8y/+Upd8pHil1LSjXtJi4NXAPcCClHAAngQWpOmFwJai1fpSrFR8ou1cLKlXUu+2bdsyq3+jyXszSD2bO7uD4xfNqcjorLyfp7Jrz146Wsf+Xu+ocL9CvVjXt3NK8SzUrKNe0mxgNfChiHhO2vemR0RIymzMW0RcD1wP0NPTU7mxdHWuWp3CVn/yfJ5Kd1cnahEU9S2oRbk4Unv9MfO45s5NE8YrpSZHKpLaKSSUr0XEN1L4qdSsRfr7dIpvBRYVrd6dYqXiVkLem0HyrpJHQ/Vs7uwOzu/pHhM7v6c7d69DtdRi9JeALwEPRcRnihbdCoyO4FoBrCmKvyuNAjsZ2JmayW4DzpDUlTroz0gxKyHvzSCWT/0Dg6zq7RsTW9Xbl4tm38/cvnFK8SzUovlrOfC7wAOS1qbYx4G/BFZJugh4DDg/Lfs2cDawCXgBuBAgIp6V9Eng3lTuz0Y77a20PDeDWD7ludn38R0Tj/ArFc9CLUZ//TuUvJrb6ROUD+CSEs91A3BDdrXLh7xersPyKc/Nvq9bOo/NP+6bMF4pvkyLmTW10WbfjjZxUHsrHW3KTbNvZ8fEVw0oFc+Ck0qZfMKgWeMqjPtSaiPJx2XvAX7+1HNTimfB1/4qg08YNGtco+dnDRadAXnp6vUsXzqv6Y9W7t+yY0rxLPhIZRI+YdCsseX5MjWDJU7wLBXPgpPKJPL8gTRrBnnuqG9rnfhSNKXiWXBSmUSeP5AHwn1QVi/2ddS3cNCMVjra8nN+Vs9RE19jt1Q8C+5TmcToB/LScX0qefhATpf7oKzexOi/ISp518N6M2fWxKO8SsWz4KRSBp8wWL7iPqjRk83y0ilq9WlfR30Ahb6EvHwmtzw7cTN9qXgW3PxVprxeN2mq3Adl9SbPn8lXHn7IlOJZcFKxTLkPyupNnj+Tm/sHphTPgpOKZcoXrbR6k+fP5MNP7ZpSPAvuU7HMNVsfVP/AYFPsS7Psx3Q022eyXCcv6eKb65+cMF4pTipWEc1y0cpqjWSr9Be+R+Q1z2dyKt7Sc+SESeUtPUdWbJtu/jIroVpXU1izdivLr7qT//7Fe1h+1Z3cujbbe835qhD5dd+jz0wpngUnFbMSqjFqqBpf+Hke/VQsjyfkrln/xJTiWXDzl1kJ1Rg1VI0bSOV59NOovDb/lboecyWv0+wjFbMSqjFqqBpf+Hke/QT5bv7bXeLCkaXiWfCRitl+VHrUULUuA5TX0U+Q79sJj4xMfEmaUvEsOKmYTaLSo4aq9YWfx9FPkO/mv/a2iRujSsWz4OYvszrgywBVztzZHZz/mu4xsfN7unPxWr/yZSUu01IingUnFTNrav0Dg9x475YxsRt/vCUXfSqHdE7cGFUqngUnFTNrahse38nQ3rF9CEN7gw2P76xRjarn6ecnTpyl4llwUjGzpvbc7uEpxZvJzPaJv+JLxbPgpGJmTa0WTUD1ov+FoSnFs+CkYmZN7dgjDqW9dezpfu2t4tgjDq1RjapnVvvEibNUPAtOKmbW1ObO7uDTbzt+zD3qP/2243Mx+uvC5YunFM9C8x//mVnu5fXkz9OXvYxXLJjFxqL7p7xiwSxOX/ayim3TScXMciGvJ3/e9uFTuePBJ/mXB5/ijGULKppQwEnFzKzpnb7sZRVPJqPcp2JmZplxUjEzs8w0fFKRdJakjZI2Sbq81vUxM8uzhk4qklqBvwXeCCwD3iFpWW1rZWaWXw2dVIDXApsi4pGI2APcBJxb4zqZmeVWo4/+WggUX360DzhpfCFJFwMXp9kBSRurULfpmgc8U+tKZMT7Un+aZT/A+1JNR5VbsNGTSlki4nrg+lrXoxySeiOip9b1yIL3pf40y36A96VeNXrz11ZgUdF8d4qZmVkNNHpSuRc4RtISSTOAC4Bba1wnM7Pcaujmr4gYlvRHwG1AK3BDRGyocbUOVEM005XJ+1J/mmU/wPtSlxQRk5cyMzMrQ6M3f5mZWR1xUjEzs8w4qVSQpFZJP5H0rTS/RNI96ZIyN6fBBUjqSPOb0vLFRc9xRYpvlHRmUbxql6eRNEfSLZJ+JukhSadIOkzS7ZIeTn+7UllJuibVa72kE4ueZ0Uq/7CkFUXx10h6IK1zjSRNVI+M9uXDkjZI+qmkGyXNbJT3RdINkp6W9NOiWMXfh1LbyHg/PpU+X+sl/ZOkOUXLpvRaT+f9zHJfipZ9VFJImpfm6/Y9yVRE+FGhB/AR4B+Bb6X5VcAFafo64A/T9PuA69L0BcDNaXoZsA7oAJYAP6cwIKE1TR8NzEhlllVwP1YC703TM4A5wNXA5Sl2OXBVmj4b+A4g4GTgnhQ/DHgk/e1K011p2Y9TWaV131ih/VgIPAp0Fr0f726U9wV4PXAi8NOiWMXfh1LbyHg/zgDa0vRVRfsx5dd6qu9n1vuS4osoDCB6DJhX7+9Jpv/Pal2BZn1QOGfmDuA04FvpQ/FM0X+cU4Db0vRtwClpui2VE3AFcEXRc96W1ntp3RQfUy7j/TiUwhexxsU3Aoen6cOBjWn6C8A7xpcD3gF8oSj+hRQ7HPhZUXxMuYz3ZfQKDIel1/lbwJmN9L4Aixn7ZVzx96HUNrLcj3HL/ivwtYlew8le6+n8P6vEvgC3AMcDm9mXVOr6Pcnq4eavyvlr4FJgJM3PBXZExHCa76PwJQdFl5tJy3em8hNdhmbhfuKVsATYBvyDCk15X5Q0C1gQEU+kMk8CC8bvS5l1Xpimx8czFxFbgb8CfgE8QeF1vo/GfF9GVeN9KLWNSnkPhV/lMPX9mM7/s0xJOhfYGhHrxi1q5PekbE4qFSDpzcDTEXFfreuSgTYKh/fXRsSrgV0UDrdfEoWfS3U/Nj21O59LIVEeAcwCzqpppTJUjfeh0tuQ9CfAMPC1Sm2jkiQdBHwc+B/V2ma9/f9zUqmM5cA5kjZTuHLyacDngDmSRk84Lb6kzEuXm0nLDwX6KX0ZmmpenqYP6IuIe9L8LRSSzFOSDk91Phx4Oi2fap23punx8Up4A/BoRGyLiCHgGxTeq0Z8X0ZV430otY1MSXo38GbgnemLkknqO1G8n6m/n1n6VQo/Wtal///dwP2SXjaNfan5ezIttW5/a/YHcCr7Ouq/ztgOxPel6UsY24G4Kk0fy9hOykcodFC2pekl7OukPLaC+/BvwCvS9J8Cn0qP4o7Cq9P0mxjbGfnjFD+MQt9MV3o8ChyWlo3vjDy7QvtxErABOChtayXw/kZ6X/jlPpWKvw+ltpHxfpwFPAjMH1duyq/1VN/PrPdl3LLN7OtTqev3JLPPaK0r0OyPCMQgAAACtElEQVQPxiaVo9OHZFP64Hek+Mw0vyktP7po/T+hMMplI0WjoiiMJPmPtOxPKrwPJwC9wHrgm+mDP5fCQISHge8V/ScQhRun/Rx4AOgpep73pH3cBFxYFO8BfprW+TwZdJ7uZ1/+J/CztL2vpi+rhnhfgBsp9AUNUTiCvKga70OpbWS8H5so9CusTY/rpvtaT+f9zHJfxi3fzL6kUrfvSZYPX6bFzMwy4z4VMzPLjJOKmZllxknFzMwy46RiZmaZcVIxM7PMOKmYTYGkxeOvSCupR9I1afrdkj5f4Tp8KJ25bVZ3nFTMDlBE9EbEB6q4yQ9ROIGzbJJaK1QXszGcVMymSdLR6SKbf6x0z5xxyxdLujPdO+MOSUem+JclXSvpbkmPSDo13ZfjIUlfLlr/DEk/knS/pK9Lmi3pAxSuW3aXpLtKlUvxzZKuknQ/8LZqvCZmTipm0yDpFcBqCvdjubdEsb8BVkbEcRQukHhN0bIuCpdl/zBwK/BZCpck+U+STkg3dvoE8IaIOJHCFQ0+EhHXAI8DvxkRv1mqXNF2+iPixIi4KYv9NptM2+RFzGyc+cAa4C0R8aCkU0uUOwV4S5r+KoUbK43654gISQ8AT0XEAwCSNlC4llQ3hRtU/b90s78ZwI8m2MbJk5S7eao7Z3YgnFTMpm4nhXuyvI7CRRCnYzD9HSmaHp1vA/YCt0fEOyZ5Hk1Sbtc062c2LW7+Mpu6PRTuTvguSf9tP+V+SOFquADvpHC153LdDSyXtBRA0ixJL0/LngcOLqOcWdU5qZhNQ0TsonDvjw8Dh5Qo9n7gQknrgd8FPjiF599Gob/mxrT+j4BfS4uvB74r6a5JyplVna9SbGZmmfGRipmZZcZJxczMMuOkYmZmmXFSMTOzzDipmJlZZpxUzMwsM04qZmaWmf8PeummYk4/CCgAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "auto_df.plot(kind='scatter', x='kilometer', y='price', title='Kilometer ~ Price')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "行驶公里数（kilometer）和价格（price）之间，呈现出来一个微弱的线性关系。并且从散点图中我们能看到，kilometer的值非常的分散，都是以“万”为单位的数值，颗粒度非常大，这可能对模型的帮助会比较有限。我们在后续建模的时候，会进一步评估行驶公里数作为predictor对结果的贡献。\n",
    "\n",
    "**(2) 车龄（ageOfCar）**\n",
    "\n",
    "再来看看车龄（ageOfCar）和价格（price）之间的关系："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.axes._subplots.AxesSubplot at 0x122107c18>"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAZYAAAEWCAYAAABFSLFOAAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADl0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uIDMuMC4yLCBodHRwOi8vbWF0cGxvdGxpYi5vcmcvOIA7rQAAIABJREFUeJzt3XuYnHV99/H3dw/ZhCQkyyaNkA0JmogGhQhbCA21PGA5FRMoB/GyTUQqT69KhVJN0PqABZ5WUPTBp0+xKKnBWiASJdhiEYnVooAuuATDoSzHJIQQNiFkQ7LZ7H6fP+7fktlhZrOz+9u575n9vK5rr8z87sN855D5zu94m7sjIiISS03aAYiISHVRYhERkaiUWEREJColFhERiUqJRUREolJiERGRqJRYRIbBzBaY2TNm1mlmZ6UdT0zhOb0z7Tik8iixiBRhZs1m9l0z6zCznWb2KzM7M2+3q4F/cPcJ7n5XOO5UM/u5me0wsy1m9jMzW1j+Z7CPmc0yMw/JotPMXjCzKwY6Jjyn58oVo1QPJRaRAszsIOABYA9wBDAF+Brwr2Z2bs6uM4F1OcedC3wPuBVoBqYBVwIfHkIMdUONfwCT3X0C8FHgSjM7rUyPK6OIEotUNDO7wsyeDbWDJ8zs7JxttWZ2g5m9ZmbPm9kl4Vd7Xdg+ycxuMbNNZrbRzK41s9pw+F8BncBF7v6Ku+9y99uA/w3cYIlngXcCPwy1gAbgq8A17v4td9/u7r3u/jN3/2R4zHeZ2ZpQC3ot1Igm58T8gpktM7O1wM6R+pJ39wdJEuL7wuO6mX3KzJ4Bnskpmx1ujwuv5Ytmtt3MHjCzcWHbfDP7pZm9bmaPmdmJIxGzVA4lFql0zwK/D0wC/hb4FzM7OGz7JHA6MA84GsjvA/k2sBeYDXwAOAX4s7DtD4FV7t6bd8xK4FDg3e7+LuAl4MOhFnAYMAO4c4B4Dfh74BDgvWH/L+bt81Hgj0hqF3sHONfbT25mg9nHzBaQ1MR+k7PpLOA4YG6Bw74CHAP8HnAQsBToNbPpwL8D14byzwCrzGxqKXFLdVFikYrm7t9z95dDzeAOkl/bx4bN5wM3uvsGd98GfKnvODObBpwBXObuO939VZKmrgvCLlOATQUeclPO9nxNefsUirfd3e9z9y5330JSw/mDvN2+7u7r3X1XoXOY2Zlm9nDov7nLzM4ws0Yz+zBwTbHHDl4DtgLfAq5w9/tztv29u2/Nf1wzqwE+AVzq7hvdvcfdf+nuXcCfAPe4+z3hPbgPaCV5bWWUUluqVDQzWwxcDswKRRPY96V/CLA+Z/fc2zOBemBTzo/8mpx9XgMO5u0OztmeryNnn+eLxDsNuJGkljUxPOa2vN3W5x+X5zzgY8BG4Gzg88DhJH1Cl+/n2CkD1IKKPe4UYCxJ7TDfTOC8kNT61AM/3U8cUsVUY5GKZWYzgW8ClwBN7j4Z+C1JcxMkNYfmnENm5NxeD3SRfNFODn8HuvsRYftPgD8Ov9ZznR+O/e8CIT0dtp0zQNh/Bzjwfnc/kOQXf37z1f6WHL8w1Hx2ufu/uvsJ7j7V3c9294IJbZCKPe5rwG7gXQW2rQe+k/MaTnb38e7+pQL7yiihxCKVbDzJl+EWADO7kNAZHawELjWz6aGDfFnfBnffBPyYpCP+QDOrCR3rfc1SXyPpt7nFzN5hZmPN7KPA3wCf9QLXmwhllwP/y8wuzDnvCWZ2c9htIsmggO2hf+KzpT7pAv0+Iyo83nLgq2Z2SBgUcXwYrPAvwIfDEOva8DqdaGbNA59VqpkSi1Qsd38CuAF4ENgMvB/4Rc4u3yRJHmtJOqnvIems7wnbFwNjgCdImqPuJDR1uXsHcAJJE9ATJM1clwN/GvpyisV0J/ARkj6Jl0Nc1wKrwy5/SzKQYDtJp/f3h/j0y+0zwOPAr0n6aK4Datx9PbCIpDluC0kN5rPou2VUM13oS0YLMzsd+Ia7z0w7FpFqpl8VUrXC3IszzKwuNDtdBfwg7bhEqp1qLFK1zOwA4GfAe4BdJE1Pl7r7G6kGJlLllFhERCQqNYWJiEhUo26C5JQpU3zWrFlphyEiUjEeeeSR19x90Mv0jLrEMmvWLFpbW9MOQ0SkYpjZi6Xsr6YwERGJSolFRESiUmIREZGolFhERCQqJRYREYlKiWWQOjq7eGz963R0dqUdiohIpo264cZDsbptI8tWraW+pobu3l6uP+dIFs6bnnZYIiKZpBrLfnR0drFs1Vp2d/eyo2svu7t7WbpqrWouIiJFKLHsx4Ztu6iv6f8y1dfUsGFbwcuRi4iMekos+9HcOI7u3v4X7Ovu7aW5cVxKEYmIZJsSy340TWjg+nOOZGx9DRMb6hhbX8P15xxJ04SGtEMTEcmkEeu8N7PlwJnAq+7+vlB2EHAHMAt4ATjf3beZmQE3AmcAbwIfd/dHwzFLgC+E017r7itC+THAt4FxJJecvbTQdchjWDhvOgtmT2HDtl00N45TUhERGcBI1li+DZyWV3YFcL+7zwHuD/cBTgfmhL+LgZvgrUR0FXAccCxwlZk1hmNuAj6Zc1z+Y0XVNKGBo2ZMVlIREdmPEUss7v5zYGte8SJgRbi9Ajgrp/xWTzwETDazg4FTgfvcfau7bwPuA04L2w5094dCLeXWnHOJiEiKyt3HMs3dN4XbrwDTwu3pwPqc/TaEsoHKNxQoL8jMLjazVjNr3bJly/CegYiIDCi1zvtQ0yjLdZHd/WZ3b3H3lqlTB32tGhERGYJyJ5bNoRmL8O+roXwjMCNnv+ZQNlB5c4FyERFJWbkTy93AknB7CbA6p3yxJeYD20OT2b3AKWbWGDrtTwHuDdveMLP5YUTZ4pxziYhIikZyuPFtwInAFDPbQDK660vASjO7CHgROD/sfg/JUON2kuHGFwK4+1Yzuwb4ddjvanfvGxDwF+wbbvyj8CciIimzEZr6kVktLS2ua96LiAyemT3i7i2D3V8z70VEJColFhERiUqJRUREolJiERGRqJRYREQkKiUWERGJSolFRESiUmIREZGolFhERCQqJRYREYlKiUVERKJSYhERkaiUWEREJColFhERiUqJRUREolJiERGRqJRYREQkKiUWERGJSolFRESiUmIREZGolFhERCQqJRYREYlKiUVERKJSYhERkaiUWEREJColFhERiUqJRUREolJiERGRqJRYREQkKiUWERGJSolFRESiSiWxmNlfmdk6M/utmd1mZmPN7DAze9jM2s3sDjMbE/ZtCPfbw/ZZOef5XCh/2sxOTeO5iIhIf2VPLGY2Hfg00OLu7wNqgQuA64CvuftsYBtwUTjkImBbKP9a2A8zmxuOOwI4DfhHM6st53MREZG3S6sprA4YZ2Z1wAHAJuAk4M6wfQVwVri9KNwnbD/ZzCyU3+7uXe7+PNAOHFum+EVEpIiyJxZ33wh8BXiJJKFsBx4BXnf3vWG3DcD0cHs6sD4cuzfs35RbXuAYERFJSRpNYY0ktY3DgEOA8SRNWSP5mBebWauZtW7ZsmUkH0pEZNRLoynsQ8Dz7r7F3buB7wMLgMmhaQygGdgYbm8EZgCE7ZOAjtzyAsf04+43u3uLu7dMnTo19vMREZEcaSSWl4D5ZnZA6Cs5GXgC+ClwbthnCbA63L473CdsX+PuHsovCKPGDgPmAL8q03MQEZEi6va/S1zu/rCZ3Qk8CuwFfgPcDPw7cLuZXRvKbgmH3AJ8x8zaga0kI8Fw93VmtpIkKe0FPuXuPWV9Minq6Oxiw7ZdNDeOo2lCQ9rhiIi8xZIf/6NHS0uLt7a2ph3GsKxu28iyVWupr6mhu7eX6885koXzNG5BREaGmT3i7i2D3V8z7ytMR2cXy1atZXd3Lzu69rK7u5elq9bS0dmVdmgiIoASS8XZsG0X9TX937b6mho2bNuVUkQiIv0psVSY5sZxdPf29ivr7u2luXFcShGJiPSnxFJhmiY0cP05RzK2voaJDXWMra/h+nOOVAe+iGRG2UeFyfAtnDeduQcfSNv615k3YzKzp01MOyQRkbcosVQgjQoTkSxTU1iF0agwEck6JZYKU2z0l0aFiUhWKLFUmPFjatnd3X9U2O7uXsaP0aVoRCQblFjKrKOzi8fWvz7kpqude3poqLV+ZQ21xs49o2Y1GxHJOHXel1GMTvfmxnFYjUHPvqV4rMY0j0VEMkM1ljKJ1emueSwiknWqsZRJ31Isu9nXP9K3FEupSWHhvOksmD1FqxuLSCYpsZRJ7KVYmiY0KKGISCapKaxM1IQlIqOFaixlpCYsERkNlFjKTE1YIlLt1BQmIiJRKbGIiEhUSiwiIhKVEouIiESlxCIiIlEpsYiISFRKLCIiEpUSi4iIRKXEIiIiUSmxiIhIVEosIiISlRKLiIhEpcQiIiJRKbGIiEhUSiwiIhJVKonFzCab2Z1m9pSZPWlmx5vZQWZ2n5k9E/5tDPuamX3dzNrNbK2ZHZ1zniVh/2fMbEkaz0VERPpLq8ZyI/Af7v4e4CjgSeAK4H53nwPcH+4DnA7MCX8XAzcBmNlBwFXAccCxwFV9yWg06Ojs4rH1r9PR2ZV2KCIi/ZT9CpJmNgn4IPBxAHffA+wxs0XAiWG3FcB/AsuARcCt7u7AQ6G2c3DY9z533xrOex9wGnBbuZ5LWla3bWTZqrXU19TQ3dvL9eccycJ509MOS0QESKfGchiwBfhnM/uNmX3LzMYD09x9U9jnFWBauD0dWJ9z/IZQVqz8bczsYjNrNbPWLVu2RHwq5dfR2cWyVWvZ3d3Ljq697O7uZemqtaq5iEhmpJFY6oCjgZvc/QPATvY1ewEQaice6wHd/WZ3b3H3lqlTp8Y6bSo2bNtFfU3/t62+poYN23alFJGISH+DTixmNtPMPhRujzOziUN8zA3ABnd/ONy/kyTRbA5NXIR/Xw3bNwIzco5vDmXFyqtac+M4unt7+5V19/bS3DgupYhERPobVGIxs0+SJIB/CkXNwF1DeUB3fwVYb2aHh6KTgSeAu4G+kV1LgNXh9t3A4jA6bD6wPTSZ3QucYmaNodP+lFBW1ZomNHD9OUcytr6GiQ11jK2v4fpzjqRpQkPaoYmIAIPvvP8UycirhwHc/Rkz+51hPO5fAt81szHAc8CFJElupZldBLwInB/2vQc4A2gH3gz74u5bzewa4Ndhv6v7OvKr3cJ501kwewobtu2iuXGckoqIZMpgE0uXu+8xMwDMrI5h9IG4exvQUmDTyQX2dZLEVug8y4HlQ42jkjVNaFBCEZFMGmwfy8/M7PPAODP7Q+B7wA9HLiwREalUg00sV5AMEX4c+J8kzVNfGKmgRIZDk0dF0jXYprBxwHJ3/yaAmdWGsjdHKjCRodDkUZH0DbbGcj9JIukzDvhJ/HBEhk6TR0WyYbCJZay7d/bdCbcPGJmQRIZGk0dFsmGwiWVn3qrCxwD63yqZosmjItkw2MRyGfA9M/svM3sAuAO4ZOTCEimdJo+KZMOgOu/d/ddm9h6gb7b80+7ePXJhiQyNJo+KpG/AxGJmJ7n7GjP747xN7zYz3P37IxibyJBo8qhIuvZXY/kDYA3w4QLbHFBiERGRfgZMLO5+lZnVAD9y95VliklERCrYfjvv3b0XWFqGWEREpAoMdlTYT8zsM2Y2w8wO6vsb0chERKQiDXZJl4+Q9Kn8RV75O+OGI5Woo7NLo7BE5C2DTSxzSZLKCSQJ5r+Ab4xUUFI5tDaXiOQbbFPYCuC9wNeB/0uSaFaMVFBSGbK6NpdWNxZJ12BrLO9z97k5939qZk+MREBSOfrW5trNvmVU+tbmSqtJTDUokfQNtsbyaLjePABmdhzQOjIhSaXI2tpcsWtQqvmIDM1gayzHAL80s5fC/UOBp83scZKrBx85ItFJpvWtzbU0r4aQVm0lZg1KNR+RoRtsYjltRKOQipWltbli1aByaz59SWrpqrUsmD1Fo95EBmGwi1C+ONKBSOXKytpcsWpQWew7Eqkkg62xiFSEGDWorPUdiVSawXbei1SMpgkNHDVj8pBrF7qui8jwqMYiUkCW+o5EKo0Si0gRWek7Eqk0agoTEZGolFgkMzQhUaQ6qClMMkETEkWqh2oskrqsLmYpIkOjxCKp65uQmKtvQqKIVB4lFkmdJiSKVJfUEouZ1ZrZb8zs38L9w8zsYTNrN7M7zGxMKG8I99vD9lk55/hcKH/azE5N55nIcGlCokh1SbPz/lLgSeDAcP864GvufruZfQO4CLgp/LvN3Web2QVhv4+Y2VzgAuAI4BDgJ2b2bnfvKfcTkeHThESR6pFKjcXMmoE/Ar4V7htwEnBn2GUFcFa4vYh9V6u8Ezg57L8IuN3du9z9eaAdOLY8z0BGwnCXYunTvnkHd7aup33zjkiRiUgp0qqx/B9gKTAx3G8CXnf3veH+BqBvrOl0YD2Au+81s+1h/+nAQznnzD2mHzO7GLgY4NBDD433LCRzrrzrcW596KW37i8+/lCuXvT+FCMSGX3KXmMxszOBV939kXI9prvf7O4t7t4yderUcj2slFn75h39kgrArQ++pJqLSJml0RS2AFhoZi8At5M0gd0ITDazvhpUM7Ax3N4IzAAI2ycBHbnlBY6RUaht/esllYvIyCh7YnH3z7l7s7vPIul8X+PuHwN+CpwbdlsCrA637w73CdvXuLuH8gvCqLHDgDnAr8r0NCSD5s2YXFK5iIyMLM1jWQZcbmbtJH0ot4TyW4CmUH45cAWAu68DVgJPAP8BfEojwka32dMmsvj4/n1oi48/lNnTJhY5QkRGgiU//kePlpYWb21tTTsMGUHtm3fQtv515s2YrKQiEoGZPeLuLYPdP0s1FpEoGsePYc60iTSOH5N2KCKjklY3HuU6OruqalLi6raNLL3zMWqthh7v5cvnHlU1qyTHeq+q7T2X7FFiGcWqban6js4u/nplG3t7AZLutstXtrFg9pSK/wKN9V5V23su2aSmsFGqGpeqX/fyGyGp7LO3NykfiqxceCzWe1WN77lkk2oso1TfUvW72fdN3LdUfeX+ui82EKX0ASpZ+mUf672qzvdcskg1llGqGpeqP+KQSdTXWr+y+lrjiEMmlXSerP2yj/VeVeN7LtmkxDJKVeNS9U0TGrjhvKNoqKvhgDG1NNTVcMN5R5X8nLJ24bFY71U1vueSTZrHMspV4wih4T6njs4uFly3ht3d+37dj62v4RfLTkr1NarWUWFZi0fertR5LOpjGeWaJjRU3X/m4T6nvl/2n80ZtpyFX/ax3qssvedZ6suSeNQUJlUnxmiupB5vYOFfiS5rfVkSj2osUlVi/ALu+8Lryhm7vHTV2qqYD5MlGqVWvVRjkaoR6xdw1jrvq5VGqVUvJRapGrESQrV/4WVl4qdGqVUvNYVJ1YiVEPq+8JbmNalVwxde1jrLF86bzoLZUzQqrMpouLFUlbvbNr4tIQz1i7PahsFmdRi1ZJ+GG8uoFvMXcJaG5cagznIpFyUWqTrVlhBiqfa+I8kOdd6LjBLqLJdyUY1FZBRRZ7mUgxKLyCijpkIZaWoKExllsjKPRaqXaiwio0jW5rHEUm1DwyudEotIhYhxOYC+JW/6hhwPZw20rCzjX63JspIpsYiMsBhfwDG+PGPOY4n1ZT7c88ROlhKH+lhERtDqto0suG4Nf/Kth1lw3RrubttY8jliLa4Zax5LrHhinEcLhmaTEotIEcPt5M7aasux5rHEiifGeTTpM5vUFCZSQKymJ+/tvxaf93rJTU8xvzxjzGOJFU+M81TzgqGVTIlFJE+sdvvxY2rp6umfWLp6nPFjakuKJ/alkmNcuvn8Y5q59aGX3io7v6W55HPGSgqa9Jk9SiwieWJ1cr+8vXCTzsvbdzF72sSSYnLAHXpw0l6QvKOzi5WPbOhXtrJ1A5ee/O7UksK2nXt4ZvMOxo+pVWLJACUWkTzxmp6sxPLCOjq7+OuVbSRXSk6yyuUr21Ib+RR7leTh1qCuvOvxfrWnxccfytWL3j/k88nwqfNeJE+sTu4jDjmQurz/YXU1SXkp1r38Rkgq++ztTcqHYriDEmJ3mA8nnvbNO/olFYBbH3yJ9s07hhSLxKEai0gBMZpomiY08NXz5/XrG/nyuUcN4VzF2r5KbxOLMSghZof5cONpW/960fJSmxslnrInFjObAdwKTCP5n3Gzu99oZgcBdwCzgBeA8919m5kZcCNwBvAm8HF3fzScawnwhXDqa919RTmfi1S3GIs1xkhQRxwyifpaoztnIEB9rXHEIZNKOk/MyYQL503nkElj+fkzr/HBOVNoOayppONjxTOr6YCSyqU80mgK2wv8tbvPBeYDnzKzucAVwP3uPge4P9wHOB2YE/4uBm4CCInoKuA44FjgKjNrLOcTERmMpgkNHDVj8rBGcd1w3lE01NVwwJhaGupquOG80ms+xeaHDGUy4ZV3Pc65//QQX1/Tzrn/9BBXrn685HPEiKe+rpbavC6rWkvK0zTaF/ose43F3TcBm8LtHWb2JDAdWAScGHZbAfwnsCyU3+ruDjxkZpPN7OCw733uvhXAzO4DTgNuK9uTESmTGDWf8WNq+13vHmB3d2/Jw5+L9Wssnj+rpOanGPE0N46jvq6Gnpzz1NfVpDpBUmuXpdx5b2azgA8ADwPTQtIBeIWkqQySpLM+57ANoaxYeaHHudjMWs2sdcuWLdHil+qWtV+dw6357NzTQ0Pez/uGWmPnnp6SzjNQv0a548naVTFjrbZQ6VLrvDezCcAq4DJ3fyPpSkm4u5tZtNH67n4zcDNAS0tLyrMApBJU46/O5sZxWI1BTl+N1VjJv+7nzZhcUvlIx7Nw3nTmHnwgbetfZ96Myal22sceil2pUqmxmFk9SVL5rrt/PxRvDk1chH9fDeUbgRk5hzeHsmLlIsNSrb86s/brvmlCAy0z+3eL/u7MxiGttHzmPzzA3/7wCc78hweGtNBnLFq7LFH2xBJGed0CPOnuX83ZdDewJNxeAqzOKV9sifnA9tBkdi9wipk1hk77U0KZyLBU84q5C+dN5xfLTuJf/uw4frHspCHVwm554LmSyotp37yDB9o7+pX9V3tHSXNQsvYjIGvJOy1pNIUtAP4UeNzM2kLZ54EvASvN7CLgReD8sO0ekqHG7STDjS8EcPetZnYN8Ouw39V9Hfkiw1HtvzqHO4z6hY6dJZUXE2MOSqyFPmPS2mXpjAp7gOJrWpxcYH8HPlXkXMuB5fGiE8nuirlZuWLjR1pm8OBz2wqWl6LxgPqSyguJtdBnbDHmQFUyzbwXKSBrvzqzcsVGgLOOnsFVd69j++59o7cmja3lrKNLSyzb3uwuqbyQmAt9SjxaK0wk47J0xca+8xSqJZR6nvx11PZXXsjGIv1excqlPJRYRAqIcUnhWAbqRyj1PKWUj3Q8P3ny1ZLKC3mmSEd/sXIpDyUWkTxZG2kUqx8h1sz7WPG82PFmSeWFvLqj8HtSrFzKQ4lFJE/Whhvv3NNTcD2sUmfMx5p5v3NPD2Pr+78+Y+trSj7PAWMKf/0UKy9kZpHFJouVS3kosYjkydpw4+69PeRVEOjxpLwUb810zzGUme7NjeP6rbQM0N3jJZ/nQ3PfUVJ5IWd/oLmkcikPJRaRPFmb5PZCkaahYuXFxHpe23buoSevj6Wn19m2c09J53nnlPEllRcye9pEDp/Wf//Dp43XiLCUabixSAFZGm4ca20uiHMdlVgX13ro+Y6i5ScPstbSvnkHT2/uPzHz6c07ad+8Q8klRaqxiBQx3NWEY5k9bSKLjz+0X9ni4w8d0hdnjOuoxLq41u9MHFtSeSGxVlqWuFRjEakAVy96P4vnzxrWCr6xrqPSd3Gt3G6WoVxcq9jyG8XKC4kxF0biU2IRqRCzp00cVvPOA+2vFS0v5byxLq61uciQ4GLlhTz3WuF+pmLlUh7K6yIVYrgXHptSpEmvWHkxfYMAxtRCQ20NY2oZ0iCA+YcdVFJ5IR+cM6WkcikP1VhEKsDqto189nttGDU4vXzlvHklr/F1/LuaMCB3PJeF8lK1vrCVZNpKUmtpfXFryfFMKdKXUqy8kJbDmjh82vh+HfiHTxs/pEEJEo9qLCIZ19HZxWW3t7GnB7p6etnTA5fe3jakmktd3gTJ/PuDUayvppTrqAA8v6Xw/sXKC+no7Hpbs9dzr71Z8Rdlq3RKLCIZ9+CzHeRfT9tDeSk2bNvF2LwO9rF1tSWvKPBA+5aSyot58LnCl08qVl7Iupe3F5ysue7l7SXF0me4zY2SUGIRybjXOneXVF5Mc+M4dufN1t+9t6fkTveGIqO/ipUXc/w7C/elFCsv5Dcvvv26MAOVDyRLC49WOiUWkYw7YfbUksoHklw3r/j9wZg4tvCFuIqVF5O/IOb+yguJNSosawuPVjolFpGMizVBcsO2XdTV9v8vX1db+uKaB44rnECKlRez5qnNJZUX8rszC68+UKy8mFiXApCERoWJVIAYEyRjLZt/xCEHUlcDe3NOVVeTlJdib5GKSbHyQsaOKfwVVqy8mKxe4jiWWJe1HiwlFpEKMdwJkn3L5ud+gQ5l2fymCQ189fx5fPbOx6i1Gnq8ly+fe1TJX1gnHT6VNU+/vcP/pMMH38QXax21vksT5K8mUOprk0WxLmtdCiUWkVHirWXzc749h7JsPsRZpHP77sLXti9WXsjsaRN5x4FjeOWNfSsrH3zgmJITcKxLE8Q23JpGbt/R7jDnaOmqtSyYPWVEay7qYxEZJWJfDmD4i3QOf7Ww1uc7+iUVgE1v7KG1yMrJxcS6NEGfGMOWY4xSS+uidaqxiIwiWbocwKlHvIMv//i/C5YP1g/aXi5aXsrs+8YDCg88KFY+kBhNT7FqGmldtE41FpFRppouB5B/qeX9lRfz1CuFZ/sXKy8m1rDlWDWNtC5apxqLiKRmuKPdjp55EMt/+WLB8jT0JYS+WgbsSwhp1TTSqKWqxiIiqZo9bSLntswY0oi3voU1cw1lYc1izW+lNMtBvISQvf6w0qjGIiIVq2lCAzdeMI/P5K38XOoXaF+z3K0P7ltccyiTUPsSwtK8PpahfKFnqT+sVDaUJR0qWUultYMXAAAIE0lEQVRLi7e2tqYdhohEFGsCYPvmHcOahBo7nqwws0fcvWWw+6vGIiIVr2lCQ5Qv8OFOQo0dT6VSH4uIiESlxCIiIlFVfGIxs9PM7GkzazezK9KOR0RktKvoxGJmtcD/A04H5gIfNbO56UYlIjK6VXRiAY4F2t39OXffA9wOLEo5JhGRUa3SR4VNB9bn3N8AHJe/k5ldDFwc7naa2dNDfLwpwGtDPDYtlRZzpcULirlcKi3mSosXisc8s5STVHpiGRR3vxm4ebjnMbPWUsZyZ0GlxVxp8YJiLpdKi7nS4oV4MVd6U9hGYEbO/eZQJiIiKan0xPJrYI6ZHWZmY4ALgLtTjklEZFSr6KYwd99rZpcA9wK1wHJ3XzeCDzns5rQUVFrMlRYvKOZyqbSYKy1eiBTzqFsrTERERlalN4WJiEjGKLGIiEhUSiwFmNkLZva4mbWZ2dvW2LfE18MyMmvN7Og04gyxHB7i7Pt7w8wuy9vnRDPbnrPPlSnEudzMXjWz3+aUHWRm95nZM+HfxiLHLgn7PGNmS1KO+ctm9lR4339gZpOLHDvgZ6jMMX/RzDbmvP9nFDm27MsjFYn3jpxYXzCztiLHpvUazzCzn5rZE2a2zswuDeWZ/TwPEPPIfJ7dXX95f8ALwJQBtp8B/IjkYnXzgYfTjjnEVQu8AszMKz8R+LeUY/sgcDTw25yy64Erwu0rgOsKHHcQ8Fz4tzHcbkwx5lOAunD7ukIxD+YzVOaYvwh8ZhCfnWeBdwJjgMeAuWnEm7f9BuDKjL3GBwNHh9sTgf8mWVIqs5/nAWIekc+zaixDswi41RMPAZPN7OC0gwJOBp5197dfBDxl7v5zYGte8SJgRbi9AjirwKGnAve5+1Z33wbcB5w2YoHmKBSzu//Y3feGuw+RzJ3KjCKv82CksjzSQPGamQHnA7eNdBylcPdN7v5ouL0DeJJkFZDMfp6LxTxSn2cllsIc+LGZPRKWg8lXaCmZ6WWJbGAXUPw/4fFm9piZ/cjMjihnUAOY5u6bwu1XgGkF9snqaw3wCZKaayH7+wyV2yWhuWN5kSaaLL7Ovw9sdvdnimxP/TU2s1nAB4CHqZDPc17MuaJ9nit6HssIOsHdN5rZ7wD3mdlT4ZdVZoUJoguBzxXY/ChJ81hnaF+/C5hTzvj2x93dzCpm7LuZ/Q2wF/hukV2y9Bm6CbiG5MvhGpLmpU+kFEspPsrAtZVUX2MzmwCsAi5z9zeSClYiq5/n/JhzyqN+nlVjKcDdN4Z/XwV+QNJMkCuLS8mcDjzq7pvzN7j7G+7eGW7fA9Sb2ZRyB1jA5r4mxPDvqwX2ydxrbWYfB84EPuahATrfID5DZePum929x917gW8WiSVTr7OZ1QF/DNxRbJ80X2Mzqyf5gv6uu38/FGf681wk5hH5PCux5DGz8WY2se82SefWb/N2uxtYbIn5wPacKnBaiv66M7N3hPZqzOxYkve9o4yxFXM30DcqZgmwusA+9wKnmFljaMI5JZSlwsxOA5YCC939zSL7DOYzVDZ5/X9nF4kla8sjfQh4yt03FNqY5msc/i/dAjzp7l/N2ZTZz3OxmEfs8zzSoxEq7Y9kVMxj4W8d8Deh/M+BPw+3jeQCY88CjwMtKcc8niRRTMopy433kvBcHiPpoPu9FGK8DdgEdJO0K18ENAH3A88APwEOCvu2AN/KOfYTQHv4uzDlmNtJ2sjbwt83wr6HAPcM9BlKMebvhM/pWpIvv4PzYw73zyAZLfRsuWIuFG8o/3bf5zdn36y8xieQNCuuzfkcnJHlz/MAMY/I51lLuoiISFRqChMRkaiUWEREJColFhERiUqJRUREolJiERGRqJRYRFJiZmeFZVaeDCvHnpWz7T1hJdnfmNm7wlyk283s2bCsxj1m9u404xcpRku6iKTAzI4CvgL8obs/b2aHkSyV8Zy7ryVZwPBOd782TG77JbDC3S/IOX4aybyT/T2WkVwttnekno9ILs1jESmRmd1FsizHWOBGd7/ZzC4ClgGvk0wk63L3S8xsKvAN4NBw+GXu/gsz+w7wU3dfnnPei0gucXAbsBzoIUkc1wBfdPcPFohlAskM70agHviCu68OCw3eS7LQ4DHAGZ7BVa+lOimxiJTIzA5y961mNo5kKZRTgV+QXFdkB7AGeCwkln8F/tHdHzCzQ4F73f29ZvYoyazrx3LOexTwz+5+tJl9Eeh096+Y2aeBw9z9rwrEUgcc4MkiiFNIVlaYA8wkudbH73lyaQeRslFTmEjpPm1mZ4fbM4A/BX7m7lsBzOx7QF//x4eAuTkr3x4YahmxGPB3ZvZBoJdkCfa+5dpfVFKRNCixiJTAzE4kSRbHu/ubZvafwFPAe4scUgPMd/fdeed5gqSJ6rGc4mNI1mLKtw44t8j5PwZMBY5x924ze4GkiQ5g5/6ej8hI0KgwkdJMAraFpPIekktTjwf+IKxYWweck7P/j4G/7LtjZvPCza8Anwt9IX0XX/o8ybVS8q0BGnIvsGRmR5rZ74d4Xg1J5X+QNIGJpEqJRaQ0/wHUmdmTwJdI+jQ2An8H/Iqkr+UFYHvY/9NASxhW/ATJqtO4extJZ/8Pzewp4IfA0lDejycdoWcDHwrDjdcBf09ylcLvhvM/DiwmqT2JpEqd9yIRmNkET67QWUdyIaTl7v6DtOMSSYNqLCJxfNHM2kgugPQ8yeWfRUYl1VhERCQq1VhERCQqJRYREYlKiUVERKJSYhERkaiUWEREJKr/D687kvKSpgnFAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "auto_df.plot(kind='scatter', x='ageOfCar', y='price', title='ageOfCar ~ Price')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "从上图中我们看出，车龄（ageOfCar）和价格（price）之间有一个明确的近似线性关系，随着车龄的增加，价格按照一个大致的比例降低。\n",
    "\n",
    "接下来，我们来观察分类变量。\n",
    "\n",
    "**(3) 型号（name）**\n",
    "\n",
    "对于categorical variable，第一件事情是要知道有哪几个值。Numpy有一个去除重复值的方法，叫做`unique()`。传入一个Series，`unique()`就会给出Series中有哪些值。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array(['Volkswagen_Golf_1.4'], dtype=object)"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.unique(auto_df['name'])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "我们发现，汽车型号（name）这个feature中只有1个值，也就是说，所有的车都是大众的高尔夫（Volkswagen Golf）。如果feature只有唯一值，那么它对预测就起不到作用，所以我们建模的时候不会使用型号（name）。\n",
    "\n",
    "**(4) 车辆类别（vehicleType）**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array(['coupe', 'kleinwagen', 'limousine'], dtype=object)"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.unique(auto_df['vehicleType'])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "有趣的事情发生了。同样是大众高尔夫，车辆类别却出现了三类：双门轿车（coupe），小轿车（kleinwagen），豪车（limousine）。由于这些车辆类别是由卖家自己选择的，所以这能够反映出车辆在卖家心里的地位。我们来看一看三种类别的数量占比："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.axes._subplots.AxesSubplot at 0x1221d12e8>"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "auto_df['vehicleType'].value_counts().plot(kind='bar')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "我们看到，绝大部分的observation中，车辆类别（vehicleType）都是豪华车。所以其能给模型带来的信息量也不多，我们决定不将其纳入建模的范畴。\n",
    "\n",
    "**(5) 事故记录（isDamaged）**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.axes._subplots.AxesSubplot at 0x122307400>"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXoAAAEDCAYAAAA7jc+ZAAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADl0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uIDMuMC4yLCBodHRwOi8vbWF0cGxvdGxpYi5vcmcvOIA7rQAADClJREFUeJzt3X+s3fVdx/Hna1TUTbPCuDaspZYMnCEqQm4IhsWYMRVkETRIWIxrCLGJmYLO6KrRkPgXGBVnYoh1ZdZk2YZILBnEBSuLMUb0Fokb4EJlMIqFXiI/pltkdW//uN8l16Zw2/M9l+/2vs9HcnPO93u+53zff9w+++3nnnObqkKS1Nebph5AkrS+DL0kNWfoJak5Qy9JzRl6SWrO0EtSc4Zekpoz9JLUnKGXpOY2TT0AwFlnnVU7duyYegxJ+qZy8ODBF6pqYa3jviFCv2PHDpaWlqYeQ5K+qSR5+mSOc+lGkpoz9JLUnKGXpOYMvSQ1Z+glqbk1Q5/kziRHk3xu1b4zkzyQ5Inh9oxhf5L8UZJDSf41ycXrObwkaW0nc0X/Z8AVx+3bDRyoqvOBA8M2wJXA+cPXLuCO+YwpSZrVmqGvqr8D/vO43VcD+4b7+4BrVu3/81rxj8DmJGfPa1hJ0qmb9QNTW6rqyHD/OWDLcH8r8Myq4w4P+45wnCS7WLnqZ/v27TOO8cbasfu+qUdo5albr5p6BGlDGP3D2Fr538VP+X8Yr6o9VbVYVYsLC2t+gleSNKNZQ//815dkhtujw/5ngXNWHbdt2CdJmsisob8X2Dnc3wnsX7X//cO7by4FXl61xCNJmsCaa/RJPg78CHBWksPALcCtwF1JbgSeBq4bDr8f+AngEPBl4IZ1mFmSdArWDH1Vve81Hrr8BMcW8IGxQ0mS5sdPxkpSc4Zekpoz9JLUnKGXpOYMvSQ1Z+glqTlDL0nNGXpJas7QS1Jzhl6SmjP0ktScoZek5gy9JDVn6CWpOUMvSc0ZeklqztBLUnOGXpKaM/SS1Jyhl6TmDL0kNWfoJak5Qy9JzRl6SWrO0EtSc4Zekpoz9JLUnKGXpOYMvSQ1Z+glqTlDL0nNGXpJam5U6JP8SpJHk3wuyceTfFuSc5M8lORQkk8mOX1ew0qSTt3MoU+yFbgJWKyq7wNOA64HbgNur6rzgBeBG+cxqCRpNmOXbjYB355kE/Bm4AjwbuDu4fF9wDUjzyFJGmHm0FfVs8DvAV9kJfAvAweBl6rq2HDYYWDr2CElSbMbs3RzBnA1cC7wduAtwBWn8PxdSZaSLC0vL886hiRpDWOWbt4DfKGqlqvqq8A9wGXA5mEpB2Ab8OyJnlxVe6pqsaoWFxYWRowhSXo9Y0L/ReDSJG9OEuBy4DHgQeDa4ZidwP5xI0qSxhizRv8QKz90fRj47PBae4APAR9Mcgh4G7B3DnNKkma0ae1DXltV3QLcctzuJ4FLxryuJGl+/GSsJDVn6CWpOUMvSc0ZeklqztBLUnOGXpKaM/SS1Jyhl6TmDL0kNWfoJak5Qy9JzRl6SWrO0EtSc4Zekpoz9JLUnKGXpOYMvSQ1Z+glqTlDL0nNGXpJas7QS1Jzhl6SmjP0ktScoZek5gy9JDVn6CWpOUMvSc0ZeklqztBLUnOGXpKaM/SS1Jyhl6TmDL0kNTcq9Ek2J7k7yb8leTzJDyU5M8kDSZ4Ybs+Y17CSpFM39or+w8BfV9X3AhcCjwO7gQNVdT5wYNiWJE1k5tAneSvww8BegKp6tapeAq4G9g2H7QOuGTukJGl2m0Y891xgGfhokguBg8DNwJaqOjIc8xyw5URPTrIL2AWwffv2EWNI2rH7vqlHaOWpW6+aeoS5GrN0swm4GLijqi4C/pvjlmmqqoA60ZOrak9VLVbV4sLCwogxJEmvZ0zoDwOHq+qhYftuVsL/fJKzAYbbo+NGlCSNMXPoq+o54Jkk7xx2XQ48BtwL7Bz27QT2j5pQkjTKmDV6gF8CPpbkdOBJ4AZW/vK4K8mNwNPAdSPPIUkaYVToq+oRYPEED10+5nUlSfPjJ2MlqTlDL0nNGXpJas7QS1Jzhl6SmjP0ktScoZek5gy9JDVn6CWpOUMvSc0ZeklqztBLUnOGXpKaM/SS1Jyhl6TmDL0kNWfoJak5Qy9JzRl6SWrO0EtSc4Zekpoz9JLUnKGXpOYMvSQ1Z+glqTlDL0nNGXpJas7QS1Jzhl6SmjP0ktScoZek5gy9JDU3OvRJTkvyL0k+NWyfm+ShJIeSfDLJ6ePHlCTNah5X9DcDj6/avg24varOA14EbpzDOSRJMxoV+iTbgKuAjwzbAd4N3D0csg+4Zsw5JEnjjL2i/0Pg14GvDdtvA16qqmPD9mFg68hzSJJGmDn0Sd4LHK2qgzM+f1eSpSRLy8vLs44hSVrDmCv6y4CfTPIU8AlWlmw+DGxOsmk4Zhvw7ImeXFV7qmqxqhYXFhZGjCFJej0zh76qfqOqtlXVDuB64G+r6meBB4Frh8N2AvtHTylJmtl6vI/+Q8AHkxxiZc1+7zqcQ5J0kjatfcjaquozwGeG+08Cl8zjdSVJ4/nJWElqztBLUnOGXpKaM/SS1Jyhl6TmDL0kNWfoJak5Qy9JzRl6SWrO0EtSc4Zekpoz9JLUnKGXpOYMvSQ1Z+glqTlDL0nNGXpJas7QS1Jzhl6SmjP0ktScoZek5gy9JDVn6CWpOUMvSc0ZeklqztBLUnOGXpKaM/SS1Jyhl6TmDL0kNWfoJak5Qy9JzRl6SWpu5tAnOSfJg0keS/JokpuH/WcmeSDJE8PtGfMbV5J0qsZc0R8DfrWqLgAuBT6Q5AJgN3Cgqs4HDgzbkqSJzBz6qjpSVQ8P978EPA5sBa4G9g2H7QOuGTukJGl2c1mjT7IDuAh4CNhSVUeGh54DtrzGc3YlWUqytLy8PI8xJEknMDr0Sb4D+Evgl6vqldWPVVUBdaLnVdWeqlqsqsWFhYWxY0iSXsOo0Cf5FlYi/7GqumfY/XySs4fHzwaOjhtRkjTGmHfdBNgLPF5Vf7DqoXuBncP9ncD+2ceTJI21acRzLwN+DvhskkeGfb8J3ArcleRG4GngunEjSpLGmDn0VfX3QF7j4ctnfV1J0nz5yVhJas7QS1Jzhl6SmjP0ktScoZek5gy9JDVn6CWpOUMvSc0ZeklqztBLUnOGXpKaM/SS1Jyhl6TmDL0kNWfoJak5Qy9JzRl6SWrO0EtSc4Zekpoz9JLUnKGXpOYMvSQ1Z+glqTlDL0nNGXpJas7QS1Jzhl6SmjP0ktScoZek5gy9JDVn6CWpOUMvSc2tS+iTXJHk80kOJdm9HueQJJ2cuYc+yWnAHwNXAhcA70tywbzPI0k6OetxRX8JcKiqnqyqV4FPAFevw3kkSSdhPUK/FXhm1fbhYZ8kaQKbpjpxkl3ArmHzv5J8fqpZGjoLeGHqIdaS26aeQBPwe3O+vvtkDlqP0D8LnLNqe9uw7/+pqj3AnnU4/4aXZKmqFqeeQzqe35vTWI+lm38Gzk9ybpLTgeuBe9fhPJKkkzD3K/qqOpbkF4FPA6cBd1bVo/M+jyTp5KzLGn1V3Q/cvx6vrZPikpi+Ufm9OYFU1dQzSJLWkb8CQZKaM/SS1Jyhl6TmDL2kdZPkZ5J853D/t5Lck+TiqefaaAx9E0nemuT2JEvD1+8neevUc2nD++2q+lKSdwHvAfYCd0w804Zj6Pu4E3gFuG74egX46KQTSfC/w+1VwJ6qug84fcJ5NiTfXtlEkkeq6gfX2ie9kZJ8ipVfgfKjwMXAV4B/qqoLJx1sg/GKvo+vDP88BiDJZaz8oZKmdB0rn5L/8ap6CTgT+LVpR9p4JvvtlZq7XwD2rVqXfxHYOeE8ElX15SRHgXcBTwDHhlu9gVy6aSLJtwLXAu8ANgMvA1VVvzPpYNrQktwCLALvrKrvSfJ24C+q6rKJR9tQvKLvYz/wEvAwJ/i10NJEfgq4iJXvS6rqP77+dku9cQx9H9uq6oqph5CO82pVVZICSPKWqQfaiPxhbB//kOT7px5COs5dSf4E2Jzk54G/Af504pk2HNfom0jyGHAe8AXgf4Cwskb/A5MOpg0tyU3AEeASVr4nP11VD0w71cbj0k0fV049gHQC3wXcxMoa/Z2sXNHrDeYVvaR1lSTAjwE3sPIOnLuAvVX175MOtoG4Ri9pXdXK1eRzw9cx4Azg7iS/O+lgG4hX9JLWTZKbgfcDLwAfAf6qqr6a5E3AE1X1jkkH3CBco5e0ns4Efrqqnl69s6q+luS9E8204XhFL0nNuUYvSc0ZeklqztBLUnOGXpKaM/SS1Nz/Ac7IfuUmUh9+AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "auto_df['isDamaged'].value_counts().plot(kind='bar')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "我们可以看到，事故记录（isDamaged）是分布比较均衡的一个predictor，而且由于只有两个值（yes / no），我们可以将它转换为dummy variable，保存为新的变量名`isDamagedDummy`。\n",
    "\n",
    "首先，我们在auto_df数据表中建立名为isDamagedDummyd的feature，全部赋值为0。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "auto_df['isDamagedDummy'] = 0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "接着，我们利用数据表自带的定位函数`loc()`，传入两个定位参数：`auto_df['isDamaged'] == 'yes'`筛选出所有'isDamaged'值为'yes'的行，`isDamagedDummy`是新的变量名。接着，将`loc()` 函数定位的值修改为1。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "auto_df.loc[auto_df['isDamaged'] == 'yes', 'isDamagedDummy'] = 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "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>0</th>\n",
       "      <th>1</th>\n",
       "      <th>2</th>\n",
       "      <th>3</th>\n",
       "      <th>4</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>name</th>\n",
       "      <td>Volkswagen_Golf_1.4</td>\n",
       "      <td>Volkswagen_Golf_1.4</td>\n",
       "      <td>Volkswagen_Golf_1.4</td>\n",
       "      <td>Volkswagen_Golf_1.4</td>\n",
       "      <td>Volkswagen_Golf_1.4</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>price</th>\n",
       "      <td>1599</td>\n",
       "      <td>10300</td>\n",
       "      <td>1600</td>\n",
       "      <td>7300</td>\n",
       "      <td>599</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>kilometer</th>\n",
       "      <td>150000</td>\n",
       "      <td>50000</td>\n",
       "      <td>150000</td>\n",
       "      <td>70000</td>\n",
       "      <td>125000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>vehicleType</th>\n",
       "      <td>limousine</td>\n",
       "      <td>limousine</td>\n",
       "      <td>limousine</td>\n",
       "      <td>limousine</td>\n",
       "      <td>limousine</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>ageOfCar</th>\n",
       "      <td>18</td>\n",
       "      <td>5</td>\n",
       "      <td>14</td>\n",
       "      <td>7</td>\n",
       "      <td>18</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>isDamaged</th>\n",
       "      <td>no</td>\n",
       "      <td>no</td>\n",
       "      <td>no</td>\n",
       "      <td>no</td>\n",
       "      <td>no</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>isDamagedDummy</th>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                                  0                    1                    2  \\\n",
       "name            Volkswagen_Golf_1.4  Volkswagen_Golf_1.4  Volkswagen_Golf_1.4   \n",
       "price                          1599                10300                 1600   \n",
       "kilometer                    150000                50000               150000   \n",
       "vehicleType               limousine            limousine            limousine   \n",
       "ageOfCar                         18                    5                   14   \n",
       "isDamaged                        no                   no                   no   \n",
       "isDamagedDummy                    0                    0                    0   \n",
       "\n",
       "                                  3                    4  \n",
       "name            Volkswagen_Golf_1.4  Volkswagen_Golf_1.4  \n",
       "price                          7300                  599  \n",
       "kilometer                     70000               125000  \n",
       "vehicleType               limousine            limousine  \n",
       "ageOfCar                          7                   18  \n",
       "isDamaged                        no                   no  \n",
       "isDamagedDummy                    0                    0  "
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "auto_df.head().T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "这样我们的dummy variable就创建好了，我们会用`isDamagedDummy`替代`isDamaged`来建模。\n",
    "\n",
    "接下来，我们观察一下，基于车辆损坏情况的价格分布。\n",
    "\n",
    "不同于车龄和价格，车辆损坏状态是一个非0即1的哑变量（dummy variable）。散点图主要用于体现连续变量（continuous variable）之间的关系，所以不再适用于价格（price）与车辆损坏状态（isDamaged）之间的关系表达。\n",
    "\n",
    "此处我们引入一种新的图表类型，叫做箱线图（boxplot），可以用于体现连续变量基于不同哑变量的分布规律。对于箱线图（boxplot）的解释见下图：\n",
    "\n",
    "![Boxplot Explanation](img/boxplot.png)\n",
    "\n",
    "接下来，我们就用pandas自带的boxplot()函数，来可视化车辆损坏状态（isDamaged）和价格（price）之间的关系："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.axes._subplots.AxesSubplot at 0x11efad4a8>"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "auto_df.boxplot(column=['price'], by='isDamaged')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "可以看到，对于有损坏的车辆，价格要低于没有损坏的车辆。然后有一些价格非常高的值，这可能是个别车况非常好的二手车，或者是个别卖家定价远高于市场水平的情况。（备注：所有数据都对应同一款车型）"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 4.建立模型（Multiple Linear Model）\n",
    "\n",
    "**(1) 数据预处理**\n",
    "\n",
    "和simple linear regression类似，我们会得到包含3个predictor的线性模型，它的公式是：\n",
    "\n",
    "$y = \\beta_0 + \\beta_1x_1 + \\beta_2x_2 + \\beta_3x_3$\n",
    "\n",
    "这里的$x_1, x_2, x_3$就是我们的三个predictor: 车龄（ageOfCar），公里数（kilometer）和事故记录（isDamagedDummy）。\n",
    "\n",
    "接下来，我们用Statsmodels来建立Multiple Linear Regression的模型。\n",
    "\n",
    "我们将'ageOfCar', 'isDamagedDummy', 'kilometer'三个predictor赋值给X，并且加入常数1。\n",
    "\n",
    "然后将'price'赋值给Y，作为outcome。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/usr/local/Cellar/python3/3.6.0/Frameworks/Python.framework/Versions/3.6/lib/python3.6/site-packages/numpy/core/fromnumeric.py:2389: FutureWarning: Method .ptp is deprecated and will be removed in a future version. Use numpy.ptp instead.\n",
      "  return ptp(axis=axis, out=out, **kwargs)\n"
     ]
    }
   ],
   "source": [
    "X = auto_df[['ageOfCar', 'isDamagedDummy', 'kilometer']]\n",
    "X = sm.add_constant(X)\n",
    "Y = auto_df['price']"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "我们把X和Y用`head()`函数打印出前5行，验证格式正确。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "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>const</th>\n",
       "      <th>ageOfCar</th>\n",
       "      <th>isDamagedDummy</th>\n",
       "      <th>kilometer</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1.0</td>\n",
       "      <td>18</td>\n",
       "      <td>0</td>\n",
       "      <td>150000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>1.0</td>\n",
       "      <td>5</td>\n",
       "      <td>0</td>\n",
       "      <td>50000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>1.0</td>\n",
       "      <td>14</td>\n",
       "      <td>0</td>\n",
       "      <td>150000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>1.0</td>\n",
       "      <td>7</td>\n",
       "      <td>0</td>\n",
       "      <td>70000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>1.0</td>\n",
       "      <td>18</td>\n",
       "      <td>0</td>\n",
       "      <td>125000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   const  ageOfCar  isDamagedDummy  kilometer\n",
       "0    1.0        18               0     150000\n",
       "1    1.0         5               0      50000\n",
       "2    1.0        14               0     150000\n",
       "3    1.0         7               0      70000\n",
       "4    1.0        18               0     125000"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0     1599\n",
       "1    10300\n",
       "2     1600\n",
       "3     7300\n",
       "4      599\n",
       "Name: price, dtype: int64"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Y.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**(2) 训练模型**\n",
    "\n",
    "接下来我们来训练模型，找到最佳系数$\\beta_0, \\beta_1, \\beta_2, \\beta_3$。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "model1 = sm.OLS(Y, X).fit()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 5.评估模型\n",
    "\n",
    "接下来我们看看模型的效果如何，使用`summary()`方法给出模型的参数。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>          <td>price</td>      <th>  R-squared:         </th> <td>   0.789</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>       <th>  Adj. R-squared:    </th> <td>   0.785</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>  <th>  F-statistic:       </th> <td>   194.5</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Fri, 02 Aug 2019</td> <th>  Prob (F-statistic):</th> <td>1.70e-52</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>23:48:46</td>     <th>  Log-Likelihood:    </th> <td> -1273.2</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>   160</td>      <th>  AIC:               </th> <td>   2554.</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>   156</td>      <th>  BIC:               </th> <td>   2567.</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Model:</th>              <td>     3</td>      <th>                     </th>     <td> </td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Covariance Type:</th>      <td>nonrobust</td>    <th>                     </th>     <td> </td>   \n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "         <td></td>           <th>coef</th>     <th>std err</th>      <th>t</th>      <th>P>|t|</th>  <th>[0.025</th>    <th>0.975]</th>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>const</th>          <td> 1.009e+04</td> <td>  364.504</td> <td>   27.685</td> <td> 0.000</td> <td> 9371.344</td> <td> 1.08e+04</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>ageOfCar</th>       <td> -380.9576</td> <td>   20.540</td> <td>  -18.547</td> <td> 0.000</td> <td> -421.530</td> <td> -340.385</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>isDamagedDummy</th> <td> -618.4489</td> <td>  114.427</td> <td>   -5.405</td> <td> 0.000</td> <td> -844.475</td> <td> -392.423</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>kilometer</th>      <td>   -0.0143</td> <td>    0.002</td> <td>   -6.076</td> <td> 0.000</td> <td>   -0.019</td> <td>   -0.010</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td>11.845</td> <th>  Durbin-Watson:     </th> <td>   2.144</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.003</td> <th>  Jarque-Bera (JB):  </th> <td>  28.522</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td> 0.159</td> <th>  Prob(JB):          </th> <td>6.41e-07</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td> 5.044</td> <th>  Cond. No.          </th> <td>9.32e+05</td>\n",
       "</tr>\n",
       "</table><br/><br/>Warnings:<br/>[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.<br/>[2] The condition number is large, 9.32e+05. This might indicate that there are<br/>strong multicollinearity or other numerical problems."
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                            OLS Regression Results                            \n",
       "==============================================================================\n",
       "Dep. Variable:                  price   R-squared:                       0.789\n",
       "Model:                            OLS   Adj. R-squared:                  0.785\n",
       "Method:                 Least Squares   F-statistic:                     194.5\n",
       "Date:                Fri, 02 Aug 2019   Prob (F-statistic):           1.70e-52\n",
       "Time:                        23:48:46   Log-Likelihood:                -1273.2\n",
       "No. Observations:                 160   AIC:                             2554.\n",
       "Df Residuals:                     156   BIC:                             2567.\n",
       "Df Model:                           3                                         \n",
       "Covariance Type:            nonrobust                                         \n",
       "==================================================================================\n",
       "                     coef    std err          t      P>|t|      [0.025      0.975]\n",
       "----------------------------------------------------------------------------------\n",
       "const           1.009e+04    364.504     27.685      0.000    9371.344    1.08e+04\n",
       "ageOfCar        -380.9576     20.540    -18.547      0.000    -421.530    -340.385\n",
       "isDamagedDummy  -618.4489    114.427     -5.405      0.000    -844.475    -392.423\n",
       "kilometer         -0.0143      0.002     -6.076      0.000      -0.019      -0.010\n",
       "==============================================================================\n",
       "Omnibus:                       11.845   Durbin-Watson:                   2.144\n",
       "Prob(Omnibus):                  0.003   Jarque-Bera (JB):               28.522\n",
       "Skew:                           0.159   Prob(JB):                     6.41e-07\n",
       "Kurtosis:                       5.044   Cond. No.                     9.32e+05\n",
       "==============================================================================\n",
       "\n",
       "Warnings:\n",
       "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
       "[2] The condition number is large, 9.32e+05. This might indicate that there are\n",
       "strong multicollinearity or other numerical problems.\n",
       "\"\"\""
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "model1.summary()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "比起之前仅用车龄（ageOfCar）作为predictor的simple linear regression，这个模型的R值（R-squared）值从0.692提升到0.789。说明加入新的数据（公里数，损坏记录）后，我们有了更好的模型。\n",
    "\n",
    "带入系数，模型可以写作：$price = 10090 - 380*ageOfCar -618 * isDamagedDummy - 0.0143*kilometer$\n",
    "\n",
    "这个公式表示，在其他条件相同的情况下：\n",
    "* 车龄每增加1年，价格预期会下跌380美元；\n",
    "* 有损坏记录的车，价格会比没有损坏记录的车平均便宜618元；\n",
    "* 每多开1km，价格预期下降0.0143美元；\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 6.Predictor的选择\n",
    "\n",
    "虽然以上模型中，三个predictor的P值都约等于0，意味着它们都对于预测结果有显著意义。但由于我们之前对于kilometer和price的线性关系有疑问，不妨试一试如果剔除kilometer，我们的模型效果会提升还是下降。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>          <td>price</td>      <th>  R-squared:         </th> <td>   0.739</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>       <th>  Adj. R-squared:    </th> <td>   0.736</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>  <th>  F-statistic:       </th> <td>   222.5</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Fri, 02 Aug 2019</td> <th>  Prob (F-statistic):</th> <td>1.53e-46</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>23:48:51</td>     <th>  Log-Likelihood:    </th> <td> -1290.2</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>   160</td>      <th>  AIC:               </th> <td>   2586.</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>   157</td>      <th>  BIC:               </th> <td>   2596.</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Model:</th>              <td>     2</td>      <th>                     </th>     <td> </td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Covariance Type:</th>      <td>nonrobust</td>    <th>                     </th>     <td> </td>   \n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "         <td></td>           <th>coef</th>     <th>std err</th>      <th>t</th>      <th>P>|t|</th>  <th>[0.025</th>    <th>0.975]</th>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>const</th>          <td> 8896.8688</td> <td>  340.241</td> <td>   26.149</td> <td> 0.000</td> <td> 8224.829</td> <td> 9568.908</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>ageOfCar</th>       <td> -431.1245</td> <td>   20.848</td> <td>  -20.680</td> <td> 0.000</td> <td> -472.303</td> <td> -389.946</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>isDamagedDummy</th> <td> -594.2428</td> <td>  126.764</td> <td>   -4.688</td> <td> 0.000</td> <td> -844.627</td> <td> -343.859</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td>26.621</td> <th>  Durbin-Watson:     </th> <td>   2.062</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.000</td> <th>  Jarque-Bera (JB):  </th> <td>  53.364</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td> 0.754</td> <th>  Prob(JB):          </th> <td>2.58e-12</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td> 5.393</td> <th>  Cond. No.          </th> <td>    89.8</td>\n",
       "</tr>\n",
       "</table><br/><br/>Warnings:<br/>[1] Standard Errors assume that the covariance matrix of the errors is correctly specified."
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                            OLS Regression Results                            \n",
       "==============================================================================\n",
       "Dep. Variable:                  price   R-squared:                       0.739\n",
       "Model:                            OLS   Adj. R-squared:                  0.736\n",
       "Method:                 Least Squares   F-statistic:                     222.5\n",
       "Date:                Fri, 02 Aug 2019   Prob (F-statistic):           1.53e-46\n",
       "Time:                        23:48:51   Log-Likelihood:                -1290.2\n",
       "No. Observations:                 160   AIC:                             2586.\n",
       "Df Residuals:                     157   BIC:                             2596.\n",
       "Df Model:                           2                                         \n",
       "Covariance Type:            nonrobust                                         \n",
       "==================================================================================\n",
       "                     coef    std err          t      P>|t|      [0.025      0.975]\n",
       "----------------------------------------------------------------------------------\n",
       "const           8896.8688    340.241     26.149      0.000    8224.829    9568.908\n",
       "ageOfCar        -431.1245     20.848    -20.680      0.000    -472.303    -389.946\n",
       "isDamagedDummy  -594.2428    126.764     -4.688      0.000    -844.627    -343.859\n",
       "==============================================================================\n",
       "Omnibus:                       26.621   Durbin-Watson:                   2.062\n",
       "Prob(Omnibus):                  0.000   Jarque-Bera (JB):               53.364\n",
       "Skew:                           0.754   Prob(JB):                     2.58e-12\n",
       "Kurtosis:                       5.393   Cond. No.                         89.8\n",
       "==============================================================================\n",
       "\n",
       "Warnings:\n",
       "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
       "\"\"\""
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X = auto_df[['ageOfCar', 'isDamagedDummy']]\n",
    "X = sm.add_constant(X)\n",
    "Y = auto_df['price']\n",
    "model2 = sm.OLS(Y, X).fit()\n",
    "model2.summary()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "结果是，剔除公里数（kilometer）后，R值下降了5%。这说明公里数对于价格是一个有意义的predictor。\n",
    "\n",
    "R值下面有一个Adj. R-squared，翻译成**修正后的R值**，通常小于R值。下面我们来解释一下其含义：\n",
    "\n",
    "首先，加入更多的predictor不会让结果变得更糟，因为模型可以通过将系数设为0来排除新的predictor对结果的影响。\n",
    "\n",
    "但是新加入predictor会让模型变得更复杂，从而更容易使得模型过拟合。我们希望越简单的模型越好，即predictor越少越好。修正后的R值（Adj. R-squared）实际上就是在R值的基础上，根据predictor的数量加入了一定的“惩罚”。Predictor的数量越多，惩罚越大。想更深入了解的同学可以参考这一篇文章：[under-penalizes complexity](http://scott.fortmann-roe.com/docs/MeasuringError.html).\n",
    "\n",
    "用于比较当两个模型表现的时候，使用修正后的R值进行比较会给我们更全面的评估结果。在我们的模型中，移除公里数后，修正后的R值也下降了约5%。所以这进一步说明公里数对于价格是一个有意义的predictor。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 7.模型的应用"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**(1) 预测**\n",
    "\n",
    "根据上述分析，model1是更好的模型。我们来利用其`predict()`函数对一个车龄10年，没有损坏记录，行驶了11.9万公里的车进行价格预测："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4585.8560290237365"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 建立用于预测的数据，数组中四个值分别代表：截距常量，ageOfCar，isDamagedDummy，kilometer\n",
    "new_X = [1, 10, 0, 119000]\n",
    "# 使用model1的predictor函数对新的车辆进行预测\n",
    "new_predictions = model1.predict(new_X)\n",
    "#结果以数组形式输出，此处的[0]是将数组中的数字取出。\n",
    "new_predictions[0] "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "所以，我们预计一个车龄10年，没有损坏记录，行驶了11.9万公里的车价格约为4586美元。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**(2) 模型的应用范围**\n",
    "\n",
    "基于训练数据的分布，我们的价格预测模型可以应用于车龄6-20年，行驶里程数为30000至150000公里之间的二手车。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Summary & Next\n",
    "\n",
    "本章我们利用多元线性回归（Multiple Linear Regression）建立了一个更准确的二手车价格预测模型，同时涵盖了以下的知识要点：\n",
    "\n",
    "* 如何对predictor进行筛选\n",
    "* 如何建立哑变量（dummy variable）\n",
    "* 如何使用箱线图（boxplot）\n",
    "* 如何建立多元线性模型\n",
    "* 如何比较不同的多元线性模型的结果\n",
    "\n",
    "至此，我们完成了线性回归部分的学习。利用这几章的知识，同学们也已经具备解决一些实际问题的能力了。接下来，我们会继续一起探索扩展能力的边界。\n",
    "\n",
    "下一章中，我们会介绍逻辑回归（Logistic Regression），这个分析方法常用于解决医疗领域的诊断（分类）问题。\n",
    "\n",
    "**下一章：**[3.1 逻辑回归（Multiple Linear Regression）介绍](https://github.com/RyC37/data-science-tutorial-for-beginners/blob/master/3.1-logistic-regression.ipynb)"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "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": "3.6.0"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}