c8_1多元线性回归.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 目标使 : $\\sum_{i=1}^m (y^{(i)} - \\hat y^{(i)})^2$ 尽可能的小\n",
    "### 因为 $\\hat y = X_b\\cdot\\theta$\n",
    "### 上式也可写成 : $(y-X_b\\cdot\\theta)^T(y-X_b\\cdot\\theta)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 求上式的最小,用凸优化什么的高阶技巧,得到如下公式: \n",
    "### $\\theta = (X_b^T X_b)^{-1} X_b^Ty$\n",
    "\n",
    "$X_b$指的是带有节距 $x_0 = 1$ 项"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 用自定义的类实现多元线性回归"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from sklearn import datasets"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "boston = datasets.load_boston()\n",
    "\n",
    "X = boston.data\n",
    "y = boston.target\n",
    "\n",
    "X = X[y < 50.0]\n",
    "y = y[y < 50.0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "from c2_model_selection import train_test_split\n",
    "\n",
    "X_train, y_train, X_test, y_test = train_test_split(X, y, seed=666)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "LinearRegression()"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from c8_LinearRegression import LinearRegression\n",
    "\n",
    "reg = LinearRegression()\n",
    "reg.fit_normal(X_train, y_train)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-1.18919477e-01,  3.63991462e-02, -3.56494193e-02,  5.66737830e-02,\n",
       "       -1.16195486e+01,  3.42022185e+00, -2.31470282e-02, -1.19509560e+00,\n",
       "        2.59339091e-01, -1.40112724e-02, -8.36521175e-01,  7.92283639e-03,\n",
       "       -3.81966137e-01])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "reg.coef_"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "34.16143549624071"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "reg.interception_"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.8129802602658509"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "reg.score(X_test, y_test)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Scikit-learn 中的回归问题"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from sklearn import datasets"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "boston = datasets.load_boston()\n",
    "\n",
    "X = boston.data\n",
    "y = boston.target\n",
    "\n",
    "X = X[y < 50.0]\n",
    "y = y[y < 50.0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sklearn.model_selection import train_test_split"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=666)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### scikit-learn 中的线性回归"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sklearn.linear_model import LinearRegression\n",
    "\n",
    "lin_reg = LinearRegression()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "LinearRegression(copy_X=True, fit_intercept=True, n_jobs=1, normalize=False)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lin_reg.fit(X_train, y_train)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-1.14235739e-01,  3.12783163e-02, -4.30926281e-02, -9.16425531e-02,\n",
       "       -1.09940036e+01,  3.49155727e+00, -1.40778005e-02, -1.06270960e+00,\n",
       "        2.45307516e-01, -1.23179738e-02, -8.80618320e-01,  8.43243544e-03,\n",
       "       -3.99667727e-01])"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lin_reg.coef_"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "32.64566083965407"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lin_reg.intercept_"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.8008916199519134"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lin_reg.score(X_test, y_test)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# KNN Regressor"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.602674505080953"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sklearn.neighbors import KNeighborsRegressor\n",
    "\n",
    "knn_reg = KNeighborsRegressor()\n",
    "knn_reg.fit(X_train, y_train)\n",
    "knn_reg.score(X_test, y_test)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 使用网格搜索优化 KNN Regressor的超参数 "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Fitting 3 folds for each of 60 candidates, totalling 180 fits\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "[Parallel(n_jobs=-1)]: Done 180 out of 180 | elapsed:    3.3s finished\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "GridSearchCV(cv=None, error_score='raise',\n",
       "       estimator=KNeighborsRegressor(algorithm='auto', leaf_size=30, metric='minkowski',\n",
       "          metric_params=None, n_jobs=1, n_neighbors=5, p=2,\n",
       "          weights='uniform'),\n",
       "       fit_params=None, iid=True, n_jobs=-1,\n",
       "       param_grid=[{'weights': ['uniform'], 'n_neighbors': [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]}, {'weights': ['distance'], 'n_neighbors': [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], 'p': [1, 2, 3, 4, 5]}],\n",
       "       pre_dispatch='2*n_jobs', refit=True, return_train_score='warn',\n",
       "       scoring=None, verbose=1)"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sklearn.model_selection import GridSearchCV\n",
    "\n",
    "param_grid = [\n",
    "    {\n",
    "        'weights':['uniform'],\n",
    "        'n_neighbors':[i for i in range(1,11)]\n",
    "    },\n",
    "    {\n",
    "        'weights':['distance'],\n",
    "        'n_neighbors':[i for i in range(1,11)],\n",
    "        'p':[i for i in range(1,6)]\n",
    "    }\n",
    "]\n",
    "\n",
    "knn_reg = KNeighborsRegressor()\n",
    "grid_search = GridSearchCV(knn_reg, param_grid=param_grid, n_jobs=-1, verbose=1)\n",
    "grid_search.fit(X_train, y_train)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'n_neighbors': 6, 'p': 1, 'weights': 'distance'}"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "grid_search.best_params_"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.6060327991735741"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "grid_search.best_score_   # 这个是基于crossfold的结果"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.7354244906092771"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "grid_search.best_estimator_.score(X_test, y_test)  \n",
    "# 这个是和我们前面得到的结果一样的打分\n",
    "# 可以看出比 0.60 没有优化超参数的结果要好, 但是不如LinearRegressor模型好"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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