Polynomial Regression.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "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": [
    "import operator\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "import statsmodels.api as sm\n",
    "from sklearn.linear_model import LinearRegression\n",
    "from sklearn.metrics import mean_squared_error, r2_score\n",
    "from sklearn.preprocessing import PolynomialFeatures\n",
    "np.random.seed(0)\n",
    "x = 2 - 3 * np.random.normal(0, 1, 20)\n",
    "y = x - 2 * (x ** 2) + 0.5 * (x ** 3) + np.random.normal(-5, 5, 20)\n",
    "plt.scatter(x,y, s=10)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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vzZ68VfCLyC5S5x+hhqZWFjS3UVddRX1N/64HfPopXHABzJnTdW79eqiqCr1GEUkHdf4RaWhqZfKspcx84T0mz1pKQ1NrdnLOnKDL33vvjsE/Y0a2y1fwi0gJKfwjsqC5jY1btgKwcctWlr60Ijh5awbf+172wMmTYdOmIPB1VywRCYmWfSJSV13FIy+vZtyLj/Hfz93bcXK//aChAUaMiKc4EUkdhX8UVq6k/pQTWfH++x3Hb70Vrrqq8M1TRERComWfsLS3w9VXB8F+0EGwPfiPOAJaWoJlne3zIiIRU+dfaosWwTHHwLZtHccffBDGj4+nJhGRTtT5l8Knn8K4cUEXP3p0NvjPPBM+/jjo8hX8IlJG1PkXY86cjjt1tnv+eTjuuMjLERHZWer8e+qDD4I7YXXeonn55dktmgp+ESlz6vx3hjtMmQJXXtlxfODA4BaI2qIpIgmj8O/O3/4GJ52U3amz3W23wU9/qp06IpJYWvbpLHeL5sEHZ4P/yCNhzZrgU4D25otIwqnz327hwmCLpne6XfCsWcFOHhGRCpLuzn/DBjj77KCLr6vLBv93v5vdoqngF5EKlM7O/5FH4Kyzuo7/+c9w7LGRlyMiErX0dP65WzRzg/+KK4K7Zbkr+EUkNSq/8587F844o+PYoEHBFs2amnhqEhGJWeV3/mefnX18223BpRdaWhT8IpJqoYa/mV1mZivNbLmZ/Tpn/FozW5WZOynMGli3Lljy0RZNEZEvhLbsY2bfAcYCh7n7JjPbNzNeA4wDRgADgflmdqC7bw2lkH32CeXXiogkWZid/8XALe6+CcDd12fGxwKz3X2Tu78DrAJGhVhH4jQ0tXLj3Dc63udXRKSEwgz/A4E6M1tsZn8xsyMy44OA1TnHtWTGOjCziWbWaGaNbW1tIZZZXrq90buISIkUFf5mNt/M3sjzbyzBktI+wFHAfwIPm5kB+RbdvcuA+zR3r3X32qqqqmLKTJTON3pf0JyeNz4RiU5Ra/7ufkKhOTO7GHjU3R14ycy2Af0IOv0hOYcOBtYWU0clqauu4pHGFjZu2Uqf3r2oq07PG5+IRCfMff7/C4wB/mxmBwK7Ax8C84AHzew3BCd8q4GXQqwjUepr+jN1/OEsaG6jrrqK+pr+cZckIhUozPCfAcwwszeAzcD5mU8By83sYaAJaAcmhbbTJ6Hqa/or9EUkVKGFv7tvBn5QYO6XwC/D+tsiItK9yv+Gr4iIdKHwFxFJIYW/iEgKKfxFRFJI4S8ikkIKfxGRFFL4i4ikkMJfRCSFFP4iIimk8BcRSSGFv4hICin8RURSSOEvIpJCCn8RkRRS+IuIpJDCX0QkhRT+IiIppPAXEUkhhb+ISAop/EVEUkjhLyKSQqGFv5mNNLMXzexVM2s0s1GZcTOzqWa2ysyWmdk3w6pBRETyC7Pz/zVwk7uPBG7MPAc4BajO/JsI3BNiDSIikkeY4e9A38zjrwBrM4/HAjM98CLwVTPbL8Q6RESkk91C/N1XAM+Y2W0EbzLfzowPAlbnHNeSGVuX+8NmNpHgkwFDhw4NsUwRkfQpKvzNbD4wIM/U9cDxwJXuPsfMzgLuA04ALM/x3mXAfRowDaC2trbLvIiI7Lqiwt/dTyg0Z2YzgcszTx8BpmcetwBDcg4dTHZJSEREIhDmmv9a4NjM4zFAc+bxPOC8zK6fo4CP3X1dvl8gIiLhCHPN/z+AKWa2G/A5mfV74CngVGAV8BlwYYg1iIhIHqGFv7svBL6VZ9yBSWH9XRER2TF9w1dEJIUU/iIiKaTwFxFJIYW/iEgKKfxFRFJI4S8ikkIKfxGRFFL4i4ikkMJfRCSFFP4iIimk8BcRSSGFv4hICin8RURSSOEvIpJCCn8RkRRS+IuIpJDCX0QkhRT+IiIppPAXEUkhhb+ISAop/EVEUqio8Dez75vZcjPbZma1neauNbNVZrbSzE7KGT85M7bKzK4p5u+LiMiuKbbzfwM4E/i/3EEzqwHGASOAk4G7zayXmfUC7gJOAWqA8ZljRUQkQrsV88PuvgLAzDpPjQVmu/sm4B0zWwWMysytcve3Mz83O3NsUzF1iIhIz4S15j8IWJ3zvCUzVmi8CzObaGaNZtbY1tYWUpkiIum0w87fzOYDA/JMXe/ucwv9WJ4xJ/+bjef7Be4+DZgGUFtbm/cYERHZNTsMf3c/YRd+bwswJOf5YGBt5nGhcRERiUhYyz7zgHFmtoeZDQeqgZeAl4FqMxtuZrsTnBSeF1INIiJSQFEnfM3s34E7gCrgSTN71d1PcvflZvYwwYncdmCSu2/N/MylwDNAL2CGuy8v6hWIiEiPmXv5L6fX1tZ6Y2Nj3GWIiCSKmS1x99p8c/qGr4hICin8RURSSOEvIpJCCn8RkRRS+IuIpJDCX0QkhRT+IiIppPAXEUmhor7hmwQNTa0saG6jrrqK+pr+cZcjIlIWKrrzb2hqZfKspcx84T0mz1pKQ1Nr3CWJiJSFig7/Bc1tbNyyFYCNW7ayoFn3BRARgQoP/7rqKvr07gVAn969qKuuirkiEZHyUNFr/vU1/Zk6/nCt+YuIdFLR4Q/BG4BCX0Sko4pe9hERkfwU/iIiKaTwFxFJIYW/iEgKKfxFRFJI4S8ikkKJuIG7mbUB78VdRwH9gA/jLiImeu3ppNeeHPu7e95vtyYi/MuZmTW6e23cdcRBr12vPW0q6bVr2UdEJIUU/iIiKaTwL960uAuIkV57Oum1VwCt+YuIpJA6fxGRFFL4i4ikkMK/RMzsajNzM+sXdy1RMbNbzexvZrbMzB4zs6/GXVPYzOxkM1tpZqvM7Jq464mKmQ0xs+fNbIWZLTezy+OuKWpm1svMlprZE3HXUgoK/xIwsyFAPfB+3LVErAE4xN0PA94Ero25nlCZWS/gLuAUoAYYb2Y18VYVmXbgKnc/GDgKmJSi177d5cCKuIsoFYV/adwO/AxI1dlzd3/W3dszT18EBsdZTwRGAavc/W133wzMBsbGXFMk3H2du7+SebyBIAQHxVtVdMxsMPCvwPS4aykVhX+RzOx0YI27vxZ3LTG7CHg67iJCNghYnfO8hRQF4HZmNgw4HFgcbyWR+i1Bg7ct7kJKpeJv41gKZjYfGJBn6nrgOuDEaCuKTnev3d3nZo65nmBZ4IEoa4uB5RlL1ac9M9sLmANc4e6fxF1PFMzsNGC9uy8xs+PirqdUFP47wd1PyDduZocCw4HXzAyCZY9XzGyUu38QYYmhKfTatzOz84HTgOO98r800gIMyXk+GFgbUy2RM7PeBMH/gLs/Gnc9EToaON3MTgX2BPqa2Z/c/Qcx11UUfcmrhMzsXaDW3ZN01b9dZmYnA78BjnX3trjrCZuZ7UZwYvt4YA3wMnCOuy+PtbAIWNDd3A/83d2viLueuGQ6/6vd/bS4aymW1vylGHcCewMNZvaqmf0u7oLClDm5fSnwDMEJz4fTEPwZRwM/BMZk/l+/mumEJaHU+YuIpJA6fxGRFFL4i4ikkMJfRCSFFP4iIimk8BcRSSGFv4hICin8RURS6P8BwLfbPTV7T4AAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# transforming the data to include another axis\n",
    "x = x[:, np.newaxis]\n",
    "y = y[:, np.newaxis]\n",
    "\n",
    "model = LinearRegression()\n",
    "model.fit(x, y)\n",
    "y_pred = model.predict(x)\n",
    "\n",
    "plt.scatter(x, y, s=10)\n",
    "plt.plot(x, y_pred, color='r')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                            OLS Regression Results                            \n",
      "==============================================================================\n",
      "Dep. Variable:                      y   R-squared:                       0.605\n",
      "Model:                            OLS   Adj. R-squared:                  0.583\n",
      "Method:                 Least Squares   F-statistic:                     27.56\n",
      "Date:                Sun, 23 Feb 2020   Prob (F-statistic):           5.42e-05\n",
      "Time:                        21:01:54   Log-Likelihood:                -84.954\n",
      "No. Observations:                  20   AIC:                             173.9\n",
      "Df Residuals:                      18   BIC:                             175.9\n",
      "Df Model:                           1                                         \n",
      "Covariance Type:            nonrobust                                         \n",
      "==============================================================================\n",
      "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
      "------------------------------------------------------------------------------\n",
      "const        -18.6612      4.015     -4.648      0.000     -27.097     -10.225\n",
      "x1             8.2110      1.564      5.250      0.000       4.925      11.497\n",
      "==============================================================================\n",
      "Omnibus:                       12.845   Durbin-Watson:                   2.532\n",
      "Prob(Omnibus):                  0.002   Jarque-Bera (JB):               11.123\n",
      "Skew:                          -1.376   Prob(JB):                      0.00384\n",
      "Kurtosis:                       5.403   Cond. No.                         2.59\n",
      "==============================================================================\n",
      "\n",
      "Warnings:\n",
      "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n"
     ]
    }
   ],
   "source": [
    "X = x\n",
    "y = y\n",
    "X2 = sm.add_constant(X)\n",
    "est = sm.OLS(y, X2)\n",
    "est2 = est.fit()\n",
    "print(est2.summary())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "###### Degree 2: \n",
    "<p> Underfit, high bias, low variance </p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "11.043997201942759\n",
      "0.8317748036823445\n"
     ]
    },
    {
     "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": [
    "# transforming the data to include another axis\n",
    "x = x[:, np.newaxis]\n",
    "y = y[:, np.newaxis]\n",
    "\n",
    "polynomial_features= PolynomialFeatures(degree=2)\n",
    "x_poly = polynomial_features.fit_transform(x)\n",
    "\n",
    "model = LinearRegression()\n",
    "model.fit(x_poly, y)\n",
    "y_poly_pred = model.predict(x_poly)\n",
    "\n",
    "rmse = np.sqrt(mean_squared_error(y,y_poly_pred))\n",
    "r2 = r2_score(y,y_poly_pred)\n",
    "print(rmse)\n",
    "print(r2)\n",
    "\n",
    "plt.scatter(x, y, s=10)\n",
    "# sort the values of x before line plot\n",
    "sort_axis = operator.itemgetter(0)\n",
    "sorted_zip = sorted(zip(x,y_poly_pred), key=sort_axis)\n",
    "x, y_poly_pred = zip(*sorted_zip)\n",
    "plt.plot(x, y_poly_pred, color='m')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "###### Degree 3:\n",
    "<p> Correct fit with low variance and low bias. </p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "5.749825845681209\n",
      "0.9544017842288466\n"
     ]
    },
    {
     "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": [
    "# transforming the data to include another axis\n",
    "x = x[:, np.newaxis]\n",
    "y = y[:, np.newaxis]\n",
    "\n",
    "polynomial_features= PolynomialFeatures(degree=3)\n",
    "x_poly = polynomial_features.fit_transform(x)\n",
    "\n",
    "model = LinearRegression()\n",
    "model.fit(x_poly, y)\n",
    "y_poly_pred = model.predict(x_poly)\n",
    "\n",
    "rmse = np.sqrt(mean_squared_error(y,y_poly_pred))\n",
    "r2 = r2_score(y,y_poly_pred)\n",
    "print(rmse)\n",
    "print(r2)\n",
    "\n",
    "plt.scatter(x, y, s=10)\n",
    "# sort the values of x before line plot\n",
    "sort_axis = operator.itemgetter(0)\n",
    "sorted_zip = sorted(zip(x,y_poly_pred), key=sort_axis)\n",
    "x, y_poly_pred = zip(*sorted_zip)\n",
    "plt.plot(x, y_poly_pred, color='m')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### degree 20:\n",
    "<p> Degree 20 covers most of the data points. However, this is a case of over fitting of data. Thereby, it will fail to generalize on unseen data.</p>\n",
    "<p> To prevent over-fitting, we can add more training samples so that the algorithm doesn’t learn the noise in the system and can become more generalized.</p>\n",
    "<p> To understand the best fit line, Bias vs Variance Trade off must be understood. </p>\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.855936596068998\n",
      "0.9952492219767469\n"
     ]
    },
    {
     "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": [
    "# transforming the data to include another axis\n",
    "x = x[:, np.newaxis]\n",
    "y = y[:, np.newaxis]\n",
    "\n",
    "polynomial_features= PolynomialFeatures(degree=20)\n",
    "x_poly = polynomial_features.fit_transform(x)\n",
    "\n",
    "model = LinearRegression()\n",
    "model.fit(x_poly, y)\n",
    "y_poly_pred = model.predict(x_poly)\n",
    "\n",
    "rmse = np.sqrt(mean_squared_error(y,y_poly_pred))\n",
    "r2 = r2_score(y,y_poly_pred)\n",
    "print(rmse)\n",
    "print(r2)\n",
    "\n",
    "plt.scatter(x, y, s=10)\n",
    "# sort the values of x before line plot\n",
    "sort_axis = operator.itemgetter(0)\n",
    "sorted_zip = sorted(zip(x,y_poly_pred), key=sort_axis)\n",
    "x, y_poly_pred = zip(*sorted_zip)\n",
    "plt.plot(x, y_poly_pred, color='m')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "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": "3.7.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}