18 线性回归.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 一元线性回归模型\n",
    "\n",
    "y = a + bx + ε\n",
    "\n",
    "其中，a为模型的截距项，b为模型的斜率项，ε为模型的误差项。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 360x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# 导入第三方模块\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "import seaborn as sns\n",
    "\n",
    "\n",
    "# 导入数据集\n",
    "income = pd.read_csv('Salary_Data.csv')\n",
    "# 绘制散点图\n",
    "sns.lmplot(x = 'YearsExperience', y = 'Salary', data = income, ci = None)\n",
    "# 显示图形\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "使用最小二乘法求解a和b\n",
    "\n",
    "![](https://tva1.sinaimg.cn/large/006y8mN6ly1g916ponxuvj30ly03fweq.jpg)\n",
    "\n",
    "问题转化为求解函数J(a,b)的最小值，使用偏导数的方法求解出参数a和b，进而得到目标函数的最小值。\n",
    "\n",
    "第一步：展开平方项\n",
    "\n",
    "![](https://tva1.sinaimg.cn/large/006y8mN6ly1g916vj2vpej30ro04faag.jpg)\n",
    "\n",
    "第二步：设偏导数为0\n",
    "\n",
    "![](https://tva1.sinaimg.cn/large/006y8mN6ly1g916vv2hhzj30p10853za.jpg)\n",
    "\n",
    "第三步：和公式转换\n",
    "\n",
    "![](https://tva1.sinaimg.cn/large/006y8mN6ly1g916w6zg1hj30mc08e751.jpg)\n",
    "\n",
    "第四步：化解\n",
    "\n",
    "![](https://tva1.sinaimg.cn/large/006y8mN6ly1g916wimhn9j30qs079js5.jpg)\n",
    "\n",
    "第五步：将参数a带入，求解b\n",
    "\n",
    "![](https://tva1.sinaimg.cn/large/006y8mN6ly1g916wwc5xfj30h106maal.jpg)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Intercept          25792.200199\n",
       "YearsExperience     9449.962321\n",
       "dtype: float64"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 使用第三方模块statsmodels的ols函数\n",
    "\n",
    "# 导入第三方模块\n",
    "import statsmodels.api as sm\n",
    "# 利用收入数据集，构建回归模型\n",
    "fit = sm.formula.ols('Salary ~ YearsExperience', data = income).fit()\n",
    "# 返回模型的参数值\n",
    "fit.params"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 多元线性回归模型\n",
    "\n",
    "![](https://tva1.sinaimg.cn/large/006y8mN6ly1g917t56dinj30jc05yt93.jpg)\n",
    "\n",
    "y=β0+β1x1+β2x2+…+βpxn+ε\n",
    "\n",
    "第一步：构建目标函数\n",
    "\n",
    "![](https://tva1.sinaimg.cn/large/006y8mN6ly1g917uc9iifj30f8036dfy.jpg)\n",
    "\n",
    "第二步：展开平方项\n",
    "\n",
    "J(β)=(y-Xβ)'(y-Xβ)\n",
    "\n",
    "=(y'-β'X')(y-Xβ)\n",
    "\n",
    "=(y'y-y'Xβ-β'X'y+β'X'Xβ)\n",
    "\n",
    "第三步：求偏导\n",
    "\n",
    "![](https://tva1.sinaimg.cn/large/006y8mN6ly1g917utrwz4j30jk03k3yq.jpg)\n",
    "\n",
    "第四步：计算偏回归系数的值\n",
    "\n",
    "X'Xβ=X'y\n",
    "\n",
    "β=(X'X)-1X'y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "模型的偏回归系数分别为：\n",
      " Intercept               58581.516503\n",
      "C(State)[T.Florida]       927.394424\n",
      "C(State)[T.New York]     -513.468310\n",
      "RD_Spend                    0.803487\n",
      "Administration             -0.057792\n",
      "Marketing_Spend             0.013779\n",
      "dtype: float64\n",
      "对比预测值和实际值的差异：\n",
      "        Prediction       Real\n",
      "8   150621.345802  152211.77\n",
      "48   55513.218079   35673.41\n",
      "14  150369.022458  132602.65\n",
      "42   74057.015562   71498.49\n",
      "29  103413.378282  101004.64\n",
      "44   67844.850378   65200.33\n",
      "4   173454.059692  166187.94\n",
      "31   99580.888895   97483.56\n",
      "13  128147.138397  134307.35\n",
      "18  130693.433835  124266.90\n"
     ]
    }
   ],
   "source": [
    "# 导入模块\n",
    "from sklearn import model_selection\n",
    "# 导入数据\n",
    "Profit = pd.read_excel('Predict to Profit.xlsx')\n",
    "# 将数据集拆分为训练集和测试集\n",
    "train, test = model_selection.train_test_split(Profit, test_size = 0.2, random_state=1234)\n",
    "# 根据train数据集建模\n",
    "model = sm.formula.ols('Profit ~ RD_Spend + Administration + Marketing_Spend + C(State)', data = train).fit()\n",
    "print('模型的偏回归系数分别为：\\n', model.params)\n",
    "# 删除test数据集中的Profit变量，用剩下的自变量进行预测\n",
    "test_X = test.drop(labels = 'Profit', axis = 1)\n",
    "pred = model.predict(exog = test_X)\n",
    "print('对比预测值和实际值的差异：\\n',pd.DataFrame({'Prediction':pred,'Real':test.Profit}))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "模型的偏回归系数分别为：\n",
      " Intercept          58068.048193\n",
      "RD_Spend               0.803487\n",
      "Administration        -0.057792\n",
      "Marketing_Spend        0.013779\n",
      "Florida             1440.862734\n",
      "California           513.468310\n",
      "dtype: float64\n"
     ]
    }
   ],
   "source": [
    "# 生成由State变量衍生的哑变量\n",
    "dummies = pd.get_dummies(Profit.State)\n",
    "# 将哑变量与原始数据集水平合并\n",
    "Profit_New = pd.concat([Profit,dummies], axis = 1)\n",
    "# 删除State变量和California变量（因为State变量已被分解为哑变量，New York变量需要作为参照组）\n",
    "Profit_New.drop(labels = ['State','New York'], axis = 1, inplace = True)\n",
    "\n",
    "# 拆分数据集Profit_New\n",
    "train, test = model_selection.train_test_split(Profit_New, test_size = 0.2, random_state=1234)\n",
    "# 建模\n",
    "model2 = sm.formula.ols('Profit ~ RD_Spend + Administration + Marketing_Spend + Florida + California', data = train).fit()\n",
    "print('模型的偏回归系数分别为：\\n', model2.params)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Profit=58068.05+0.80RD_Spend-0.06Administation+0.01Marketing_Spend+1440.86Florida+513.47California"
   ]
  }
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