Polynomial_Regression.ipynb

{
  "nbformat": 4,
  "nbformat_minor": 0,
  "metadata": {
    "colab": {
      "name": "Polynomial_Regression.ipynb",
      "provenance": [],
      "collapsed_sections": [],
      "authorship_tag": "ABX9TyNUw9TkjideQvq6kswklFla"
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
    },
    "language_info": {
      "name": "python"
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "5t0CFixgmOUf"
      },
      "source": [
        "##Implementation of Polynomial Regression:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 265
        },
        "id": "zqDBo0WxzbFf",
        "outputId": "860631d8-25bf-4750-8905-72ebf5b4d8cf"
      },
      "source": [
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "\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(-3, 3, 20)\n",
        "plt.scatter(x,y, s=10)\n",
        "plt.show()"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "OvW7QISUzvtY"
      },
      "source": [
        "Now, first let's apply Linear Regression to the model. \n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 301
        },
        "id": "WMeZ3WQjz361",
        "outputId": "b3cfc996-ad13-46a9-cf75-77c6e907ea4a"
      },
      "source": [
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "\n",
        "from sklearn.linear_model import LinearRegression\n",
        "from sklearn.metrics import mean_squared_error, r2_score\n",
        "\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(-3, 3, 20)\n",
        "\n",
        "# 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",
        "rmse = np.sqrt(mean_squared_error(y,y_pred))\n",
        "r2 = r2_score(y,y_pred)\n",
        "print(\"RMSE =\",rmse)\n",
        "print(\"R2 score =\",r2)\n",
        "\n",
        "plt.scatter(x, y, s=10)\n",
        "plt.plot(x, y_pred, color='r')\n",
        "plt.show()\n",
        "\n",
        "##The plot we get is:"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "RMSE = 15.908242501429998\n",
            "R2 score = 0.6386750054827146\n"
          ],
          "name": "stdout"
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ayjpE4TR0rzH"
      },
      "source": [
        "We can see that the data is unable to capture the patterns in the curve, so this is a case of under-fitting. Let's try and add an ${x^2}$ to this model to increase it's complexity. "
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 301
        },
        "id": "HJuMhwvT3RBI",
        "outputId": "d3cd6750-7990-47ba-cd82-d5a41154524a"
      },
      "source": [
        "import operator\n",
        "\n",
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "\n",
        "from sklearn.linear_model import LinearRegression\n",
        "from sklearn.metrics import mean_squared_error, r2_score\n",
        "from sklearn.preprocessing import PolynomialFeatures\n",
        "\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(-3, 3, 20)\n",
        "\n",
        "# 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 = \",rmse)\n",
        "print(\"R2 score = \",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()"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "RMSE =  10.120437473614711\n",
            "R2 score =  0.8537647164420812\n"
          ],
          "name": "stdout"
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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0xaMU3FLAgq8vIC5Zk62IeEXhL2F34IkD1H20Dhd0lP2kjJwrwttWEpEzU/hL2AT7hy7aavpmE5kVmZQ+Ukrq2WrziEQDhb+ERc+eUJtn61EKbi1gwdfU5hGJJgp/mXIHHjtA7Y21AJT9tIycD6nNIxJtFP4yZYJ9Qeo/V0/T3U1kviXU5ilRm0ckGin8ZUr07Omh+qpqOl/qpPC2Qs7+17OJS1KbRyRaKfxl0lp/0krdX9ZhcUbZ42Xk/InaPCLRTuEvExbsC1L/2Xqa7mkic02ozTNfbR6RWKDwlwnp3t1NzZ/V0PlyJ4WfLuTsf1GbRySWKPxl3FofbaXuY3VYglH+ZDmzPzjb65JEZJwU/jJmgd4A9Z+uZ9+395H51kzKHikjZV6K12WJyAQo/GVMund1U3NVDZ3bOin8TCFn/7PaPCKxTOEvZ9T6SCt1H6/DEo3yp8uZfbnaPCKxTuEvpxToDVD/qXr23bePrLdlUfpwKSnFavOITAcKfxlV985uqq+qpquyi6LPF1HylRLiEtXmEZkuFP5ykpaHWti5fieWZCz/2XJmvX+W1yWJyBRT+MuwQE+A3bftZv/G/WSdH2rzFKnNIzIdKfwFCLV5PlxN1/Yuir5QRMk/qc0jMp0p/IUjW46wY90OiIPlP1/OrPepzSMy3Sn8fa71J628+hevkjI/hRW/WKFbMIv4hL7X+1jDfzRQc1UNmedmcs7/naPgF/ERHfn7kHOO1770Gm989Q1mf2g2y364jPjUeK/LEpEIUvj7jHOO3bftpmlDE3M/MZfF31qMxZvXZYlIhIW97WNmnzEzZ2azQ4/NzDaY2W4z225m54S7Bhnigo6dn9xJ04YmCm8rZPG3FfwifhXW8DezIuA9wBsjFl8GLAr9rAe+Hc4aZIgLOGpvrGX/d/ZTfHsxC+5agJmCX8Svwn3k/+/A5wE3Ytk64AE35AVghpnNDXMdvhYcCFLz5zW0/KCF+V+eT8mdJQp+EZ8LW/ib2TqgyTlXecKqAqBhxOPG0LITn7/ezLaa2dYDBw6Eq8xpL9gXpOaqGg48coCzv3Y28/9uvoJfRCZ3wtfMNgN5o6y6A/gSQy2fCXHObQQ2AlRUVLgzbC6jCPQEqL6imkO/OMTCDQspvKXQ65JEJEpMKvydc5eMttzMlgMlQGXoKLMQeNnM1gBNQNGIzQtDy2QKBboCVH2wiiO/PsLijYvJ/3i+1yWJSBQJS9vHOVflnJvjnJvvnJvPUGvnHOdcM/AUcF1o1M95QLtzbn846vCrwY5Btl+6nSPPHWHpD5Yq+EXkJF6M838GeB+wG+gGPupBDdPWwOEBtl+6nc6XOyl9qJQ5V83xuiQRiUIRCf/Q0f+x3x1wUyTe12/62/rZvnY7XTVdlP2kjNnrNN2iiIxOV/hOE33NfVReUklvfS/lT5Yz61LdmVNETk3hPw30NvZSeXElfY19LP/5cma+e6bXJYlIlFP4x7ie13uofHclA20DrHx2JdkXZHtdkojEAIV/DOve1U3lxZUEjgZYuXklWWuyvC5JRGKEwj9GddV0UXlJJW7AsfLXK8lclel1SSISQxT+MaizspPKtZVYvLHquVWkl6V7XZKIxBjN5BVjOrZ2sO1d24hLjmPVbxT8IjIxCv8Y0v77diovriQhO4FVv11F2uI0r0sSkRiltk8EbappYcuuA1y4KIe1pbnjeu7h5w5TdXkVyfnJrPyflaQUpYSpShHxAx35R8immhZufegVHnh+L7c+9AqbalrG/NxDzx6i6rIqUualsOo3qxT8IjJpCv8I2bLrAD0DAQB6BgJs2TW2OQranm6j6gNVpC5JZdVzq0iemxzOMkXEJxT+EXLhohxSE+MBSE2M58JFOWd8TutPWqn+UDUZKzNY9b+rSMpJCneZIuIT6vlHyNrSXDZcs3rMPf/mHzZTe30tWedlseKZFSRk659KRKaOEiWC1pbmjulE7/7/2k/dx+uY8c4ZlD9VTkKG/plEZGqp7RNlmn/YTN3H6pj5npks//lyBb+IhIXCP4q0PdlG7Q21BNek8fhfxvG/r7V5XZKITFMK/yhx+H8OU31VNcGyFD590SG+/9Ib4x4SKiIyVgr/KND+QjtV66pIW5zG7z6XxpG48Q8JFREZD4W/xzq3d1J1WRVJeUmseHYFbzsnd9xDQkVExktnEz3UvaubyvdUEpcex8rNK0mem8zaueMbEioiMhEKf4/0NvRSeUklBGDlcytJnZ86vG6sQ0JFRCZK4e+B/tZ+KtdWMnhkkFW/XkX6Ut2WWUQiS+EfYQNHBtj+3u30vdHHimdXkHmOZuASkchT+EdQoCtA1eVVdFV3Uf5UOTPePsPrkkTEpxT+ERIcDFJ9VTUdz3dQ+kgpsy6d5XVJIuJjYR3qaWa3mFmtmVWb2ddGLL/dzHabWZ2ZvTecNUSL+s/Uc+iZQyy+dzFzrpzjdTki4nNhO/I3s3cB64CVzrk+M5sTWl4KXA2UAfnAZjNb7JwLhKsWrzXd10TThiYKP1VI/ifyvS5HRCSsR/6fBL7qnOsDcM61hpavAx52zvU5514DdgNrwliHpw5tPsSum3dx1vvPYsHXF3hdjogIEN7wXwxcaGZ/MLPfmNlbQssLgIYR2zWGlk07XbVdVF9ZTfqydEp/VIrFm9cliYgAk2z7mNlmIG+UVXeEXvss4DzgLcCjZnb2OF57PbAeoLi4eDJlemLg4ABVl1cRlxRH+dPlJGTp3LqIRI9JJZJz7pJTrTOzTwKPOecc8KKZBYHZQBNQNGLTwtCyE197I7ARoKKiwk2mzkgL9gfZccUO+hr6WPXrVcddvSsiEg3C2fZ5AngXgJktBpKANuAp4GozSzazEmAR8GIY64go5xw7/3on7b9pZ+n9S8k+P9vrkkREThLOXsT9wP1mtgPoB64PfQuoNrNHgRpgELhpOo30abyrkeb/ambe384j9891fx4RiU5hC3/nXD/wF6dYdydwZ7je2yttT7dR/7l6cq7MYf4/zve6HBGRU9L9/KdI5/ZOXr32VTLPzWTpD5ZicRrZIyLRS+E/Bfpb+qn6QBXx2fGUP1lOfFq81yWJiJyWxh9OUqA3wI4/2cFA2wCrt6wmOT/Z65JERM5I4T8Jzjnqbqyj44UOyn5aptszi0jMUNtnEvZ+ZS+tD7VS8s8l5HxIc+2KSOxQ+E9Q66OtvP73r5N7XS7FX4y9K5BFxN8U/hPQ8WIHtdfXknVBFks2LsFMI3tEJLYo/Mept6GXHet2kJSXRPnj5cQl608oIrFHJ3zHYbBzkB0f3EGgK8DKzStJyknyuiQRkQlR+I+RCzpqP1JL5/ZOlv98Oell6V6XJCIyYQr/MdrzpT20PdHGwm8u1Py7IhLz1LAeg/3f30/DvzaQ/1f5FNwyLeedERGfUfifwZEtR9i5ficzLp7Bwg0LNbJHRKYFhf9p9OzpYcef7iClJIWyH5cRl6g/l4hMD0qzUxhsH6Tq8ioIwvKfLSdxZqLXJYmITBmd8B1FcDBI9VXV9OzqYcWmFaQtSvO6JBGRKaXwH0X9p+o5/OxhFn93MTPfOdPrckREppzaPidoureJpnuaKPx0Ifkfy/e6HBGRsFD4j3Bo0yF23bqLWZfPYsHXFnhdjohI2Cj8Q7pqu6j+cDXppeks+9EyLF5DOkVk+lL4AwMHB6i6vIq45DiWP72chEydChGR6c33KRfsD7LjQzvoa+xj1XOrSJmX4nVJIiJh5+vwd86x85M7af9tO8seXEb2edlelyQiEhG+bvs0/FsDzfc3M+/v5pF7ba7X5YiIRIxvw7/tqTb2fH4POR/OYf4/zPe6HBGRiApb+JvZKjN7wcy2mdlWM1sTWm5mtsHMdpvZdjM7J1w1nEpnZSc119aQeW4mS7+/FIvTyB4R8ZdwHvl/DfhH59wq4O9DjwEuAxaFftYD3w5jDSfpa+6j6gNVJMxIoPypcuLT4iP59iIiUSGc4e+ArNDv2cC+0O/rgAfckBeAGWY2N4x1DAv0BNixbgcDBwdY/vRykucmR+JtRUSiTjhH+9wG/MrMvsHQh8z5oeUFQMOI7RpDy/aHsRacc9TdWMfRF49S9lgZmaszw/l2IiJRbVLhb2abgbxRVt0BXAx8yjn3UzO7Cvgv4JJxvPZ6htpCFBcXT6ZMAPZ+eS+tD7dS8i8l5PxpzqRfT0QklplzLjwvbNYOzHDOORua/qrdOZdlZt8BnnPOPRTarg54p3PulEf+FRUVbuvWrROupfWRVmquriH3+lyWfm+pZuMSEV8ws5eccxWjrQtnz38fcFHo93cDu0K/PwVcFxr1cx5DHwpha/l0vNhB7Q21ZF+YzZLvLFHwi4gQ3p7/x4FvmlkC0EuohQM8A7wP2A10Ax8NVwG9jb1UfbCKpLlJlD1WRlyyby9rEBE5TtjC3zn3O+DcUZY74KZwve9ICTMSOOs9Z1H8xWKSZidF4i1FRGLCtL63T0JGAsseWOZ1GSIiUUd9EBERH1L4i4j4kMJfRMSHFP4iIj6k8BcR8SGFv4iIDyn8RUR8SOEvIuJDCn8RER9S+IuI+JDCX0TEhxT+IiI+pPAXEfEhhb+IiA8p/EVEfEjhLyLiQwp/EREfUviLiPiQwl9ExIcU/iIiPqTwFxHxIYW/iIgPKfxFRHxI4S8i4kOTCn8z+7CZVZtZ0MwqTlh3u5ntNrM6M3vviOWXhpbtNrMvTub9RURkYiZ75L8D+BDw25ELzawUuBooAy4F7jWzeDOLB74FXAaUAteEthURkQhKmMyTnXOvApjZiavWAQ875/qA18xsN7AmtG63c25P6HkPh7atmUwdIiIyPuHq+RcADSMeN4aWnWq5iIhE0BmP/M1sM5A3yqo7nHNPTn1Jw++7HlgPUFxcHK63ERHxpTOGv3Pukgm8bhNQNOJxYWgZp1l+4vtuBDYCVFRUuAnUICIipxCuts9TwNVmlmxmJcAi4EXgj8AiMysxsySGTgo/FaYaRETkFCZ1wtfM/hS4G8gBfm5m25xz73XOVZvZowydyB0EbnLOBULPuRn4FRAP3O+cq57UHoiIyLiZc9HfUamoqHBbt271ugwRkZhiZi855ypGW6crfEVEfEjhLyLiQwp/EREfUviLiPiQwl9ExIcmNdQzFmyqaWHLrgNcuCiHtaW5XpcjIhIVpvWR/6aaFm596BUeeH4vtz70CptqWrwuSUQkKkzr8N+y6wA9AwEAegYCbNl1wOOKRESiw7QO/wsX5ZCaGA9AamI8Fy7K8bgiEZHoMK17/mtLc9lwzWr1/EVETjCtwx+GPgAU+iIix5vWbR8RERmdwl9ExIcU/iIiPqTwFxHxIYW/iIgPKfxFRHwoJmbyMrMDwF6v6xjFbKDN6yI85Of91777U6zt+zzn3KhXt8ZE+EcrM9t6qinS/MDP+699177HOrV9RER8SOEvIuJDCv/J2eh1AR7z8/5r3/1p2uy7ev4iIj6kI38RER9S+IuI+JDCf4qY2WfMzJnZbK9riRQz+7qZ1ZrZdjN73MxmeF1TuJnZpWZWZ2a7zeyLXtcTKWZWZGa/NrMaM6s2s7/xuqZIM7N4M3vFzH7mdS1TQeE/BcysCHgP8IbXtUTYJqDcObcC2Anc7nE9YWVm8cC3gMuAUuAaMyv1tqqIGQQ+45wrBc4DbvLRvh/zN8CrXhcxVRT+U+Pfgc8Dvjp77px71jk3GHr4AlDoZT0RsAbY7Zzb45zrBx4G1nlcU0Q45/Y7514O/X6UoRAs8LaqyDGzQuD9wH96XctUUfhPkpmtA5qcc5Ve1+KxG4FfeF1EmBUADSMeN+KjADzGzOYDq4E/eFtJRP0HQwd4Qa8LmSrTfhrHqWBmm4G8UVbdAXyJoZbPtHS6fXfOPRna5g6G2gIPRrI2iTwzywB+CtzmnOvwup5IMLPLgVbn3Etm9k6v65kqCv8xcM5dMtpyM1sOlACVZgZDbY+XzWyNc645giWGzan2/RgzuwG4HLjYTf+LRpqAohGPC0PLfMHMEhkK/gedc495XU8EXQB80MzeB6QAWWb2Q+fcX3hc16ToIq8pZGavAxXOuVi669+EmdmlwF3ARc65A17XE25mlsDQie2LGQr9PwLXOueqPVMEA6MAAAB+SURBVC0sAmzo6OYHwCHn3G1e1+OV0JH/Z51zl3tdy2Sp5y+TcQ+QCWwys21mdp/XBYVT6OT2zcCvGDrh+agfgj/kAuAjwLtD/9bbQkfCEqN05C8i4kM68hcR8SGFv4iIDyn8RUR8SOEvIuJDCn8RER9S+IuI+JDCX0TEh/4f2/MzbZiq11MAAAAASUVORK5CYII=\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "dYzbugsBkvA9"
      },
      "source": [
        ""
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Z9V2d37w4ggv"
      },
      "source": [
        "Now, let's again try and add an ${x^4}$ feature too."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 301
        },
        "id": "vT3aOzC_4ZqU",
        "outputId": "246f6493-4f76-4980-9e2a-0ab2f06afa61"
      },
      "source": [
        "import operator\n",
        "\n",
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "\n",
        "from sklearn.linear_model import LinearRegression\n",
        "from sklearn.metrics import mean_squared_error, r2_score\n",
        "from sklearn.preprocessing import PolynomialFeatures\n",
        "\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(-3, 3, 20)\n",
        "\n",
        "# transforming the data to include another axis\n",
        "x = x[:, np.newaxis]\n",
        "y = y[:, np.newaxis]\n",
        "\n",
        "polynomial_features= PolynomialFeatures(degree=4)\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 = \",rmse)\n",
        "print(\"R2 score = \",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()"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "RMSE =  3.3362527533373756\n",
            "R2 score =  0.9841082585798676\n"
          ],
          "name": "stdout"
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "FsokoKiV5MOu"
      },
      "source": [
        "Let's increase the power to 10 now. "
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 301
        },
        "id": "NEnBIrdf5JTO",
        "outputId": "a39ae2af-a297-4257-eb3a-fc247640e453"
      },
      "source": [
        "import operator\n",
        "\n",
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "\n",
        "from sklearn.linear_model import LinearRegression\n",
        "from sklearn.metrics import mean_squared_error, r2_score\n",
        "from sklearn.preprocessing import PolynomialFeatures\n",
        "\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(-3, 3, 20)\n",
        "\n",
        "# transforming the data to include another axis\n",
        "x = x[:, np.newaxis]\n",
        "y = y[:, np.newaxis]\n",
        "\n",
        "polynomial_features= PolynomialFeatures(degree=10)\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 = \",rmse)\n",
        "print(\"R2 score = \",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()"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "RMSE =  2.0243974275604706\n",
            "R2 score =  0.9941487971704481\n"
          ],
          "name": "stdout"
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "cMaS49d55tNK"
      },
      "source": [
        "## taking an even higher degree of 20, we get:"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 301
        },
        "id": "LT3F-Ju65td0",
        "outputId": "6da6bad3-c51e-4dd3-dc62-b6344aa0d5f3"
      },
      "source": [
        "import operator\n",
        "\n",
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "\n",
        "from sklearn.linear_model import LinearRegression\n",
        "from sklearn.metrics import mean_squared_error, r2_score\n",
        "from sklearn.preprocessing import PolynomialFeatures\n",
        "\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(-3, 3, 20)\n",
        "\n",
        "# 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 = \",rmse)\n",
        "print(\"R2 score = \",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()"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "RMSE =  1.1131604486554747\n",
            "R2 score =  0.9982308317622455\n"
          ],
          "name": "stdout"
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "sjTJuHo16PEf"
      },
      "source": [
        "So, what we observe is that as the degree keeps on increasing, the RMSE value (which indicates the bias) decreases, and the R2 score (which indicates the variance) increases. So, the variance increases and the bias decreases as the degree increases (which eventually leads to over-fitting). When the variance is low at the start, and the bias is high, it is the case of under-fitting. Hence, there is a clear trade-off between the bias and variance for a polynomial regression model. "
      ]
    }
  ]
}