Andrew_Linear_Regression_Exercise_1_By_Fida_Mohammad.ipynb

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  "metadata": {
    "colab": {
      "name": "Andrew Linear Regression Exercise 1 By Fida Mohammad.ipynb",
      "provenance": [],
      "collapsed_sections": [],
      "include_colab_link": true
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
    },
    "language_info": {
      "name": "python"
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
      },
      "source": [
        "<a href=\"https://colab.research.google.com/github/hussain0048/Machine-Learning/blob/master/Andrew_Linear_Regression_Exercise_1_By_Fida_Mohammad.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "187O0ncSyJ69"
      },
      "source": [
        "# Machine Learning Exercise 1 - Linear Regression\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "QyYWaUwhyPfl"
      },
      "source": [
        "This notebook covers a Python-based solution for the first programming exercise of the machine learning class on Coursera. Please refer to the exercise text for detailed descriptions and equations.\n",
        "\n",
        "In this exercise we'll implement simple linear regression using gradient descent and apply it to an example problem. We'll also extend our implementation to handle multiple variables and apply it to a slightly more difficult example."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "X3AWV5lEyTDy"
      },
      "source": [
        "## Linear regression with one variable\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "l92hRF5DyYfo"
      },
      "source": [
        "In the first part of the exercise, we're tasked with implementing linear regression with one variable to predict profits for a food truck. Suppose you are the CEO of a restaurant franchise and are considering different cities for opening a new outlet. The chain already has trucks in various cities and you have data for profits and populations from the cities."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "cvuINMwSyZxq"
      },
      "source": [
        "Let's start by importing some libraries and examining the data."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "TOiyAL3Iw1pV"
      },
      "source": [
        "import numpy as np\n",
        "import pandas as pd\n",
        "import matplotlib.pyplot as plt\n",
        "%matplotlib inline"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "ZDeLakSMyi9t",
        "outputId": "d57737b9-a25c-4115-b0f5-f74da8fea2aa"
      },
      "source": [
        "from google.colab import drive\n",
        "drive.mount('/content/drive')"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Mounted at /content/drive\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "gXy9ILjEzYqh"
      },
      "source": [
        "path = '/content/drive/MyDrive/Dataset for Andrew Exercise/ex1data1.txt'"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 206
        },
        "id": "DO4DYpc9zj9j",
        "outputId": "03c12ac3-9f56-4185-8b9a-82faaa6f6d88"
      },
      "source": [
        "data = pd.read_csv(path, header=None, names=['Population', 'Profit'])\n",
        "data.head()"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "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>Population</th>\n",
              "      <th>Profit</th>\n",
              "    </tr>\n",
              "  </thead>\n",
              "  <tbody>\n",
              "    <tr>\n",
              "      <th>0</th>\n",
              "      <td>6.1101</td>\n",
              "      <td>17.5920</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>1</th>\n",
              "      <td>5.5277</td>\n",
              "      <td>9.1302</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>2</th>\n",
              "      <td>8.5186</td>\n",
              "      <td>13.6620</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>3</th>\n",
              "      <td>7.0032</td>\n",
              "      <td>11.8540</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>4</th>\n",
              "      <td>5.8598</td>\n",
              "      <td>6.8233</td>\n",
              "    </tr>\n",
              "  </tbody>\n",
              "</table>\n",
              "</div>"
            ],
            "text/plain": [
              "   Population   Profit\n",
              "0      6.1101  17.5920\n",
              "1      5.5277   9.1302\n",
              "2      8.5186  13.6620\n",
              "3      7.0032  11.8540\n",
              "4      5.8598   6.8233"
            ]
          },
          "metadata": {},
          "execution_count": 6
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 300
        },
        "id": "Ua4EBZ8Vz9OH",
        "outputId": "0e1f2094-7748-4078-80e5-bc22a3708b5e"
      },
      "source": [
        "data.describe()"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "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>Population</th>\n",
              "      <th>Profit</th>\n",
              "    </tr>\n",
              "  </thead>\n",
              "  <tbody>\n",
              "    <tr>\n",
              "      <th>count</th>\n",
              "      <td>97.000000</td>\n",
              "      <td>97.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>mean</th>\n",
              "      <td>8.159800</td>\n",
              "      <td>5.839135</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>std</th>\n",
              "      <td>3.869884</td>\n",
              "      <td>5.510262</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>min</th>\n",
              "      <td>5.026900</td>\n",
              "      <td>-2.680700</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>25%</th>\n",
              "      <td>5.707700</td>\n",
              "      <td>1.986900</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>50%</th>\n",
              "      <td>6.589400</td>\n",
              "      <td>4.562300</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>75%</th>\n",
              "      <td>8.578100</td>\n",
              "      <td>7.046700</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>max</th>\n",
              "      <td>22.203000</td>\n",
              "      <td>24.147000</td>\n",
              "    </tr>\n",
              "  </tbody>\n",
              "</table>\n",
              "</div>"
            ],
            "text/plain": [
              "       Population     Profit\n",
              "count   97.000000  97.000000\n",
              "mean     8.159800   5.839135\n",
              "std      3.869884   5.510262\n",
              "min      5.026900  -2.680700\n",
              "25%      5.707700   1.986900\n",
              "50%      6.589400   4.562300\n",
              "75%      8.578100   7.046700\n",
              "max     22.203000  24.147000"
            ]
          },
          "metadata": {},
          "execution_count": 7
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Q_vJI4I_0FbV"
      },
      "source": [
        "Let's plot it to get a better idea of what the data looks like."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 514
        },
        "id": "YKsb_Ys1z_rP",
        "outputId": "551684e3-f2ea-4c58-fce7-a76b5460bac9"
      },
      "source": [
        "data.plot(kind='scatter', x='Population', y='Profit', figsize=(12,8))"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<matplotlib.axes._subplots.AxesSubplot at 0x7fa805f3ad50>"
            ]
          },
          "metadata": {},
          "execution_count": 8
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 864x576 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "QZY3fqLq0XmJ"
      },
      "source": [
        "Now let's implement linear regression using gradient descent to minimize the cost function.  The equations implemented in the following code samples are detailed in \"ex1.pdf\" in the \"exercises\" folder.\n",
        "\n",
        "First we'll create a function to compute the cost of a given solution (characterized by the parameters theta)."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "2eoGHz4V0Isc"
      },
      "source": [
        "def computeCost(X, y, theta):\n",
        "    inner = np.power(((X * theta.T) - y), 2)\n",
        "    return np.sum(inner) / (2 * len(X))"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "RgG9bFGT0f2w"
      },
      "source": [
        "Let's add a column of ones to the training set so we can use a vectorized solution to computing the cost and gradients."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 423
        },
        "id": "oXBL98ic0e37",
        "outputId": "c2881bb9-c4bb-4ee4-8123-74247eb53d52"
      },
      "source": [
        "data.insert(0, 'Ones', 1)\n",
        "data"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "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>Ones</th>\n",
              "      <th>Population</th>\n",
              "      <th>Profit</th>\n",
              "    </tr>\n",
              "  </thead>\n",
              "  <tbody>\n",
              "    <tr>\n",
              "      <th>0</th>\n",
              "      <td>1</td>\n",
              "      <td>6.1101</td>\n",
              "      <td>17.59200</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>1</th>\n",
              "      <td>1</td>\n",
              "      <td>5.5277</td>\n",
              "      <td>9.13020</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>2</th>\n",
              "      <td>1</td>\n",
              "      <td>8.5186</td>\n",
              "      <td>13.66200</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>3</th>\n",
              "      <td>1</td>\n",
              "      <td>7.0032</td>\n",
              "      <td>11.85400</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>4</th>\n",
              "      <td>1</td>\n",
              "      <td>5.8598</td>\n",
              "      <td>6.82330</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>...</th>\n",
              "      <td>...</td>\n",
              "      <td>...</td>\n",
              "      <td>...</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>92</th>\n",
              "      <td>1</td>\n",
              "      <td>5.8707</td>\n",
              "      <td>7.20290</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>93</th>\n",
              "      <td>1</td>\n",
              "      <td>5.3054</td>\n",
              "      <td>1.98690</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>94</th>\n",
              "      <td>1</td>\n",
              "      <td>8.2934</td>\n",
              "      <td>0.14454</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>95</th>\n",
              "      <td>1</td>\n",
              "      <td>13.3940</td>\n",
              "      <td>9.05510</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>96</th>\n",
              "      <td>1</td>\n",
              "      <td>5.4369</td>\n",
              "      <td>0.61705</td>\n",
              "    </tr>\n",
              "  </tbody>\n",
              "</table>\n",
              "<p>97 rows × 3 columns</p>\n",
              "</div>"
            ],
            "text/plain": [
              "    Ones  Population    Profit\n",
              "0      1      6.1101  17.59200\n",
              "1      1      5.5277   9.13020\n",
              "2      1      8.5186  13.66200\n",
              "3      1      7.0032  11.85400\n",
              "4      1      5.8598   6.82330\n",
              "..   ...         ...       ...\n",
              "92     1      5.8707   7.20290\n",
              "93     1      5.3054   1.98690\n",
              "94     1      8.2934   0.14454\n",
              "95     1     13.3940   9.05510\n",
              "96     1      5.4369   0.61705\n",
              "\n",
              "[97 rows x 3 columns]"
            ]
          },
          "metadata": {},
          "execution_count": 10
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ZZb2pdei0sAq"
      },
      "source": [
        "Now let's do some variable initialization."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "uSyGzneI0mFY"
      },
      "source": [
        "# set X (training data) and y (target variable)\n",
        "cols = data.shape[1]\n",
        "X = data.iloc[:,0:cols-1]\n",
        "y = data.iloc[:,cols-1:cols]"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "7tLLprHJ0yUC"
      },
      "source": [
        "Let's take a look to make sure X (training set) and y (target variable) look correct."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 206
        },
        "id": "LvzXYs550vMy",
        "outputId": "0ea92958-8d10-4369-810f-67ea95bd3f5d"
      },
      "source": [
        "X.head()"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "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>Ones</th>\n",
              "      <th>Population</th>\n",
              "    </tr>\n",
              "  </thead>\n",
              "  <tbody>\n",
              "    <tr>\n",
              "      <th>0</th>\n",
              "      <td>1</td>\n",
              "      <td>6.1101</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>1</th>\n",
              "      <td>1</td>\n",
              "      <td>5.5277</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>2</th>\n",
              "      <td>1</td>\n",
              "      <td>8.5186</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>3</th>\n",
              "      <td>1</td>\n",
              "      <td>7.0032</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>4</th>\n",
              "      <td>1</td>\n",
              "      <td>5.8598</td>\n",
              "    </tr>\n",
              "  </tbody>\n",
              "</table>\n",
              "</div>"
            ],
            "text/plain": [
              "   Ones  Population\n",
              "0     1      6.1101\n",
              "1     1      5.5277\n",
              "2     1      8.5186\n",
              "3     1      7.0032\n",
              "4     1      5.8598"
            ]
          },
          "metadata": {},
          "execution_count": 12
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 206
        },
        "id": "ygnOYR_U01qT",
        "outputId": "554fd584-2977-4049-dadb-bbe644aa398a"
      },
      "source": [
        "y.head()"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "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>Profit</th>\n",
              "    </tr>\n",
              "  </thead>\n",
              "  <tbody>\n",
              "    <tr>\n",
              "      <th>0</th>\n",
              "      <td>17.5920</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>1</th>\n",
              "      <td>9.1302</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>2</th>\n",
              "      <td>13.6620</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>3</th>\n",
              "      <td>11.8540</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>4</th>\n",
              "      <td>6.8233</td>\n",
              "    </tr>\n",
              "  </tbody>\n",
              "</table>\n",
              "</div>"
            ],
            "text/plain": [
              "    Profit\n",
              "0  17.5920\n",
              "1   9.1302\n",
              "2  13.6620\n",
              "3  11.8540\n",
              "4   6.8233"
            ]
          },
          "metadata": {},
          "execution_count": 13
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "vLKdoe4s09i2"
      },
      "source": [
        "The cost function is expecting numpy matrices so we need to convert X and y before we can use them.  We also need to initialize theta."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "RefDLKou05R1"
      },
      "source": [
        "X = np.matrix(X.values)\n",
        "y = np.matrix(y.values)\n",
        "theta = np.matrix(np.array([0,0]))"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "tdAjAXDM1DtB"
      },
      "source": [
        "Here's what theta looks like."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "nu5Ba6hx1Ack",
        "outputId": "1bd35820-e2d6-4b54-df43-2ba585ed9aa2"
      },
      "source": [
        "theta"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "matrix([[0, 0]])"
            ]
          },
          "metadata": {},
          "execution_count": 15
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "mRphGCzV1KZG"
      },
      "source": [
        "Let's take a quick look at the shape of our matrices."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "5SHOcheZ1JJt",
        "outputId": "0b5ff7cc-8a44-4259-bd78-a5aa9de4717f"
      },
      "source": [
        "X.shape, theta.shape, y.shape"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "((97, 2), (1, 2), (97, 1))"
            ]
          },
          "metadata": {},
          "execution_count": 16
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "_0Qse_yn1QvT"
      },
      "source": [
        "Now let's compute the cost for our initial solution (0 values for theta)."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "krxgEHxn1NF7",
        "outputId": "b69a9ecc-10ab-4665-9ad7-9dd9898de250"
      },
      "source": [
        "computeCost(X, y, theta)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "32.072733877455676"
            ]
          },
          "metadata": {},
          "execution_count": 17
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "epsX6tD41UpS"
      },
      "source": [
        "So far so good.  Now we need to define a function to perform gradient descent on the parameters theta using the update rules defined in the text."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "YSmtSIqA1Tfr"
      },
      "source": [
        "def gradientDescent(X, y, theta, alpha, iters):\n",
        "    temp = np.matrix(np.zeros(theta.shape))\n",
        "    parameters = int(theta.ravel().shape[1])\n",
        "    cost = np.zeros(iters)\n",
        "    \n",
        "    for i in range(iters):\n",
        "        error = (X * theta.T) - y\n",
        "        \n",
        "        for j in range(parameters):\n",
        "            term = np.multiply(error, X[:,j])\n",
        "            temp[0,j] = theta[0,j] - ((alpha / len(X)) * np.sum(term))\n",
        "            \n",
        "        theta = temp\n",
        "        cost[i] = computeCost(X, y, theta)\n",
        "        \n",
        "    return theta, cost"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Zeub7trx1ctE"
      },
      "source": [
        "Initialize some additional variables - the learning rate alpha, and the number of iterations to perform."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "MIgAyngS1aCZ"
      },
      "source": [
        "alpha = 0.01\n",
        "iters = 1000"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "7ktDe-6x1eKc"
      },
      "source": [
        "Now let's run the gradient descent algorithm to fit our parameters theta to the training set."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "MKMZJk9I1khg",
        "outputId": "28aec448-7966-4491-fde5-2eeaaede7098"
      },
      "source": [
        "g, cost = gradientDescent(X, y, theta, alpha, iters)\n",
        "g"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "matrix([[-3.24140214,  1.1272942 ]])"
            ]
          },
          "metadata": {},
          "execution_count": 20
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "er5497z21yzV"
      },
      "source": [
        "Finally we can compute the cost (error) of the trained model using our fitted parameters."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "bkkFl3Aw1va8",
        "outputId": "6200f30a-083a-4d7c-8c49-dc9baf7a96cb"
      },
      "source": [
        "computeCost(X, y, g)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "4.515955503078912"
            ]
          },
          "metadata": {},
          "execution_count": 21
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "bRPgUqc513C1"
      },
      "source": [
        "Now let's plot the linear model along with the data to visually see how well it fits."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 530
        },
        "id": "yScC9GKj12Ey",
        "outputId": "8cf61377-884b-4076-d1ce-e9e1d03e1c5c"
      },
      "source": [
        "x = np.linspace(data.Population.min(), data.Population.max(), 100)\n",
        "f = g[0, 0] + (g[0, 1] * x)\n",
        "\n",
        "fig, ax = plt.subplots(figsize=(12,8))\n",
        "ax.plot(x, f, 'r', label='Prediction')\n",
        "ax.scatter(data.Population, data.Profit, label='Traning Data')\n",
        "ax.legend(loc=2)\n",
        "ax.set_xlabel('Population')\n",
        "ax.set_ylabel('Profit')\n",
        "ax.set_title('Predicted Profit vs. Population Size')"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "Text(0.5, 1.0, 'Predicted Profit vs. Population Size')"
            ]
          },
          "metadata": {},
          "execution_count": 22
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 864x576 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "-JKBOewn2ArB"
      },
      "source": [
        "Looks pretty good!  Since the gradient decent function also outputs a vector with the cost at each training iteration, we can plot that as well.  Notice that the cost always decreases - this is an example of a convex optimization problem."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 530
        },
        "id": "MNPcukVb176Y",
        "outputId": "49a71644-2066-4b14-8a70-df45a51794b3"
      },
      "source": [
        "fig, ax = plt.subplots(figsize=(12,8))\n",
        "ax.plot(np.arange(iters), cost, 'r')\n",
        "ax.set_xlabel('Iterations')\n",
        "ax.set_ylabel('Cost')\n",
        "ax.set_title('Error vs. Training Epoch')"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "Text(0.5, 1.0, 'Error vs. Training Epoch')"
            ]
          },
          "metadata": {},
          "execution_count": 23
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 864x576 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "gHFKNRJL2SXw"
      },
      "source": [
        "## Linear regression with multiple variables"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "pMAAMqPp2RnT"
      },
      "source": [
        "Exercise 1 also included a housing price data set with 2 variables (size of the house in square feet and number of bedrooms) and a target (price of the house).  Let's use the techniques we already applied to analyze that data set as well."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 206
        },
        "id": "y_TDq44k2EqO",
        "outputId": "74370a52-9a4b-4bb3-dd64-7654301dfe3e"
      },
      "source": [
        "path = '/content/drive/MyDrive/Dataset for Andrew Exercise/ex1data2.txt'\n",
        "data2 = pd.read_csv(path, header=None, names=['Size', 'Bedrooms', 'Price'])\n",
        "data2.head()"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "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>Size</th>\n",
              "      <th>Bedrooms</th>\n",
              "      <th>Price</th>\n",
              "    </tr>\n",
              "  </thead>\n",
              "  <tbody>\n",
              "    <tr>\n",
              "      <th>0</th>\n",
              "      <td>2104</td>\n",
              "      <td>3</td>\n",
              "      <td>399900</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>1</th>\n",
              "      <td>1600</td>\n",
              "      <td>3</td>\n",
              "      <td>329900</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>2</th>\n",
              "      <td>2400</td>\n",
              "      <td>3</td>\n",
              "      <td>369000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>3</th>\n",
              "      <td>1416</td>\n",
              "      <td>2</td>\n",
              "      <td>232000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>4</th>\n",
              "      <td>3000</td>\n",
              "      <td>4</td>\n",
              "      <td>539900</td>\n",
              "    </tr>\n",
              "  </tbody>\n",
              "</table>\n",
              "</div>"
            ],
            "text/plain": [
              "   Size  Bedrooms   Price\n",
              "0  2104         3  399900\n",
              "1  1600         3  329900\n",
              "2  2400         3  369000\n",
              "3  1416         2  232000\n",
              "4  3000         4  539900"
            ]
          },
          "metadata": {},
          "execution_count": 24
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "7w4rBZTR3Up0"
      },
      "source": [
        "For this task we add another pre-processing step - normalizing the features.  This is very easy with pandas."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "WjeX0lYc3VSL",
        "outputId": "27c1a6ec-29f4-4d1f-bffa-92f42c58877e",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 206
        }
      },
      "source": [
        "data2 = (data2 - data2.mean()) / data2.std()\n",
        "data2.head()"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "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>Size</th>\n",
              "      <th>Bedrooms</th>\n",
              "      <th>Price</th>\n",
              "    </tr>\n",
              "  </thead>\n",
              "  <tbody>\n",
              "    <tr>\n",
              "      <th>0</th>\n",
              "      <td>0.130010</td>\n",
              "      <td>-0.223675</td>\n",
              "      <td>0.475747</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>1</th>\n",
              "      <td>-0.504190</td>\n",
              "      <td>-0.223675</td>\n",
              "      <td>-0.084074</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>2</th>\n",
              "      <td>0.502476</td>\n",
              "      <td>-0.223675</td>\n",
              "      <td>0.228626</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>3</th>\n",
              "      <td>-0.735723</td>\n",
              "      <td>-1.537767</td>\n",
              "      <td>-0.867025</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>4</th>\n",
              "      <td>1.257476</td>\n",
              "      <td>1.090417</td>\n",
              "      <td>1.595389</td>\n",
              "    </tr>\n",
              "  </tbody>\n",
              "</table>\n",
              "</div>"
            ],
            "text/plain": [
              "       Size  Bedrooms     Price\n",
              "0  0.130010 -0.223675  0.475747\n",
              "1 -0.504190 -0.223675 -0.084074\n",
              "2  0.502476 -0.223675  0.228626\n",
              "3 -0.735723 -1.537767 -0.867025\n",
              "4  1.257476  1.090417  1.595389"
            ]
          },
          "metadata": {},
          "execution_count": 25
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "hKtFZoqj3msz"
      },
      "source": [
        "Now let's repeat our pre-processing steps from part 1 and run the linear regression procedure on the new data set."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "VCoOyHQK3j_7",
        "outputId": "8f521fe2-6e52-4e01-9765-52dbaa1d60f4",
        "colab": {
          "base_uri": "https://localhost:8080/"
        }
      },
      "source": [
        "# add ones column\n",
        "data2.insert(0, 'Ones', 1)\n",
        "\n",
        "# set X (training data) and y (target variable)\n",
        "cols = data2.shape[1]\n",
        "X2 = data2.iloc[:,0:cols-1]\n",
        "y2 = data2.iloc[:,cols-1:cols]\n",
        "\n",
        "# convert to matrices and initialize theta\n",
        "X2 = np.matrix(X2.values)\n",
        "y2 = np.matrix(y2.values)\n",
        "theta2 = np.matrix(np.array([0,0,0]))\n",
        "\n",
        "# perform linear regression on the data set\n",
        "g2, cost2 = gradientDescent(X2, y2, theta2, alpha, iters)\n",
        "\n",
        "# get the cost (error) of the model\n",
        "computeCost(X2, y2, g2)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.13070336960771892"
            ]
          },
          "metadata": {},
          "execution_count": 26
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "_HsZVfmT3zbp"
      },
      "source": [
        "We can take a quick look at the training progess for this one as well."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "N2iAyrzS3t3_",
        "outputId": "06493859-f69f-4f9f-a106-3b30693d677a",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 530
        }
      },
      "source": [
        "fig, ax = plt.subplots(figsize=(12,8))\n",
        "ax.plot(np.arange(iters), cost2, 'r')\n",
        "ax.set_xlabel('Iterations')\n",
        "ax.set_ylabel('Cost')\n",
        "ax.set_title('Error vs. Training Epoch')"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "Text(0.5, 1.0, 'Error vs. Training Epoch')"
            ]
          },
          "metadata": {},
          "execution_count": 27
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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pUw3XkiRJXcJwXbWpU+0WIkmS1CUM11Xr7RaSWXclkiRJqpjhumpTp8KaNbBsWd2VSJIkqWKG66r1jnVt1xBJkqSOZ7iuWu8sjT7UKEmS1PEM11VzCnRJkqSuYbiumlOgS5IkdQ3DddUmToTRow3XkiRJXcBwXbURI4rWa7uFSJIkdTzD9XBwCnRJkqSuYLgeDk6BLkmS1BUM18PBbiGSJEldwXA9HKZOLcL1hg11VyJJkqQKGa6Hw9SpsH49PPFE3ZVIkiSpQobr4eAU6JIkSV3BcD0cnAJdkiSpKxiuh4PhWpIkqSsYrofDi15UrB9+uN46JEmSVCnD9XAYPx7GjYPFi+uuRJIkSRUyXA+X6dPhoYfqrkKSJEkVMlwPl2nTbLmWJEnqcIbr4WLLtSRJUsczXA+XadOKcJ1ZdyWSJEmqiOF6uEybBmvWOEujJElSBzNcD5fp04u1XUMkSZI6luF6uEybVqx9qFGSJKljGa6Hiy3XkiRJHc9wPVx6Z2m05VqSJKljGa6Hy5gx0NNjy7UkSVIHM1wPp97h+CRJktSRDNfDyVkaJUmSOprhejg5S6MkSVJHM1wPp2nT4NFHYe3auiuRJElSBQzXw2n69GL680cfrbsSSZIkVaDScB0RR0XE3RGxMCLO7uf46RFxa0TcHBE/i4g55f6ZEfFsuf/miPhilXUOGyeSkSRJ6mijqnrjiBgJnA8cCSwC5kfEvMy8o+G0r2fmF8vzjwHOA44qj92TmftVVV8tesO1/a4lSZI6UpUt1wcBCzPz3sxcA1wEHNt4QmauaNjcHsgK66mfszRKkiR1tCrD9XTgwYbtReW+54mId0fEPcAngTMbDs2KiJsi4tqIOKzCOodPTw+MGmW3EEmSpA5V+wONmXl+Zu4OfBD423L3w8Cumbk/8H7g6xGxQ99rI+K0iFgQEQuWLFkyfEUP1ogRxTTotlxLkiR1pCrD9WJgRsP2LuW+TbkIOA4gM1dn5hPl6xuBe4CX9L0gM7+UmXMzc25PT0/TCq+UE8lIkiR1rCrD9XxgdkTMiogxwAnAvMYTImJ2w+abgd+W+3vKByKJiBcDs4F7K6x1+EyfbriWJEnqUJWNFpKZ6yLiDOAKYCRwQWbeHhHnAAsycx5wRkQcAawFlgKnlpcfDpwTEWuBDcDpmflkVbUOqxkz4Mori/GuI+quRpIkSU1UWbgGyMzLgMv67PtIw+v3bOK6bwHfqrK22syYAU8/DcuWwcSJdVcjSZKkJqr9gcaus+uuxfrBBzd/niRJktqO4Xq4zSif8XzggXrrkCRJUtMZroebLdeSJEkdy3A93KZOhdGjbbmWJEnqQIbr4TZiRDEcny3XkiRJHcdwXYddd7XlWpIkqQMZruswY4Yt15IkSR3IcF2HXXeFRYtg/fq6K5EkSVITGa7rMGMGrFsHjz5adyWSJElqIsN1HXqH47PftSRJUkcxXNehdyIZ+11LkiR1FMN1HWy5liRJ6kiG6zrsuCOMG2fLtSRJUocxXNchwrGuJUmSOpDhui6OdS1JktRxDNd1seVakiSp4xiu6zJjBjz2GKxaVXclkiRJahLDdV16RwxZtKjeOiRJktQ0huu69I51bdcQSZKkjmG4rsvMmcX6vvvqrEKSJElNZLiuy4wZMGIE/O53dVciSZKkJjFc12X06CJgG64lSZI6huG6TrNmGa4lSZI6iOG6ToZrSZKkjmK4rtOsWfDww/Dss3VXIkmSpCYwXNdp1qxiff/99dYhSZKkpjBc16k3XNs1RJIkqSMYrutkuJYkSeoohus6TZ0K22xjuJYkSeoQhus6jRhRzNRouJYkSeoIhuu6ORyfJElSxzBc181wLUmS1DEM13WbNQuWLoXly+uuRJIkSUNkuK6bI4ZIkiR1DMN13WbOLNaGa0mSpLZnuK6bLdeSJEkdw3Bdt0mTYPx4w7UkSVIHGFC4joj/Gsg+DUIEvPjFcO+9dVciSZKkIRpoy/VLGzciYiRwYPPL6VJ77AELF9ZdhSRJkoZos+E6Ij4UEU8B+0bEinJ5CngM+O6wVNgNZs8uuoWsW1d3JZIkSRqCzYbrzPx4Zo4HPpWZO5TL+MzcKTM/NEw1dr499oC1a+GBB+quRJIkSUMw0G4h/xMR2wNExMkRcV5E7FZhXd1l9uxi/dvf1luHJEmShmSg4foLwMqIeDnwAeAe4D8rq6rbGK4lSZI6wkDD9brMTOBY4POZeT4wvrqyuszUqbD99j7UKEmS1OZGDfC8pyLiQ8AfA4dFxAhgdHVldZmIot+1LdeSJEltbaAt18cDq4F3ZuYjwC7ApyqrqhvNnm24liRJanMDCtdloP4asGNEHA2sykz7XDeTw/FJkiS1vYHO0PhHwA3AHwJ/BFwfEW+vsrCuM3t2Eazvv7/uSiRJkjRIA+0W8mHgFZl5amaeAhwE/N2WLoqIoyLi7ohYGBFn93P89Ii4NSJujoifRcSchmMfKq+7OyLeMNAv1LYcMUSSJKntDTRcj8jMxxq2n9jSteUU6ecDbwTmACc2hufS1zPzZZm5H/BJ4Lzy2jnACRTTrh8F/Ev5fp1rjz2KteFakiSpbQ10tJDLI+IK4Bvl9vHAZVu45iBgYWbeCxARF1EM5XdH7wmZuaLh/O2BLF8fC1yUmauB30XEwvL9fjnAetvPlCkwbpzD8UmSJLWxzYbriNgDmJKZfx0RbwNeXR76JcUDjpszHXiwYXsRcHA/n/Fu4P3AGOD3G669rs+107fwee0twhFDJEmS2tyWuoV8FlgBkJnfzsz3Z+b7ge+Ux4YsM8/PzN2BDwJ/uzXXRsRpEbEgIhYsWbKkGeXUy3AtSZLU1rYUrqdk5q19d5b7Zm7h2sXAjIbtXcp9m3IRcNzWXJuZX8rMuZk5t6enZwvltIE99oD77oO1a+uuRJIkSYOwpXA9YTPHtt3CtfOB2RExKyLGUDygOK/xhIiY3bD5ZqC32XYecEJEbBMRs4DZFEMBdraXvKQYju++++quRJIkSYOwpXC9ICL+tO/OiPjfwI2buzAz1wFnAFcAdwIXZ+btEXFORBxTnnZGRNweETdT9Ls+tbz2duBiiocfLwfenZnrt+J7tae99irWd95Zbx2SJEkalMjMTR+MmELRv3oNG8P0XIqHD99aztzYEubOnZsLFiyou4yhWb4cJkyAc8+FD36w7mokSZLUj4i4MTPn9ndss6OFZOajwCsj4veAfcrd38/MHzW5RgHsuCO86EW2XEuSJLWpAY1znZnXANdUXIsA9t7bcC1JktSmBjpDo4ZLb7jeTHcdSZIktSbDdavZe2946il46KG6K5EkSdJWMly3mr33LtZ2DZEkSWo7hutW0xuu77qr3jokSZK01QzXrWbq1GLUEFuuJUmS2o7hutVEOGKIJElSmzJct6K99jJcS5IktSHDdSvae2945BFYtqzuSiRJkrQVDNetyBFDJEmS2pLhuhUZriVJktqS4boVzZoF22wDd9xRdyWSJEnaCobrVjRyZNF6fdttdVciSZKkrWC4blX77gu33lp3FZIkSdoKhutW9bKXwUMPwRNP1F2JJEmSBshw3ar23bdY23otSZLUNgzXreplLyvWt9xSbx2SJEkaMMN1q5o6FSZPNlxLkiS1EcN1q4rwoUZJkqQ2Y7huZS97WTEc34YNdVciSZKkATBct7J994WVK+Hee+uuRJIkSQNguG5lPtQoSZLUVgzXreylLy36XtvvWpIkqS0YrlvZdtvBHnvYci1JktQmDNetzhFDJEmS2obhutXtuy8sXAhPP113JZIkSdoCw3WrO+AAyIRf/7ruSiRJkrQFhutWd8ABxfrGG+utQ5IkSVtkuG5106YVU6H/6ld1VyJJkqQtMFy3gwMPtOVakiSpDRiu28EBB8AddxSzNUqSJKllGa7bwYEHwoYNjnctSZLU4gzX7cCHGiVJktqC4bod7LIL9PT4UKMkSVKLM1y3gwgfapQkSWoDhut2ccABcPvtsGpV3ZVIkiRpEwzX7eLAA2HdOrj11rorkSRJ0iYYrtuFDzVKkiS1PMN1u9htt+Khxuuvr7sSSZIkbYLhul1EwKGHwnXX1V2JJEmSNsFw3U4OOQTuuguWLq27EkmSJPXDcN1ODjmkWN9wQ711SJIkqV+G63Yydy6MGAG//GXdlUiSJKkfhut2Mn487LOP/a4lSZJalOG63RxySDFiyIYNdVciSZKkPgzX7eaQQ2DZMvjNb+quRJIkSX0YrttN70ON9ruWJElqOZWG64g4KiLujoiFEXF2P8ffHxF3RMQtEXF1ROzWcGx9RNxcLvOqrLOt7Lkn7Lij/a4lSZJa0Kiq3jgiRgLnA0cCi4D5ETEvM+9oOO0mYG5mroyIPwc+CRxfHns2M/erqr62NWIEHHywLdeSJEktqMqW64OAhZl5b2auAS4Cjm08ITOvycyV5eZ1wC4V1tM5XvUquO02J5ORJElqMVWG6+nAgw3bi8p9m/Iu4AcN22MjYkFEXBcRx/V3QUScVp6zYMmSJUOvuF0cfjhkws9/XnclkiRJatASDzRGxMnAXOBTDbt3y8y5wEnAZyNi977XZeaXMnNuZs7t6ekZpmpbwMEHw+jR8JOf1F2JJEmSGlQZrhcDMxq2dyn3PU9EHAF8GDgmM1f37s/MxeX6XuDHwP4V1tpett0WDjrIcC1JktRiqgzX84HZETErIsYAJwDPG/UjIvYH/pUiWD/WsH9iRGxTvp4MvApofBBSr3kN3HgjPP103ZVIkiSpVFm4zsx1wBnAFcCdwMWZeXtEnBMRx5SnfQoYB3yzz5B7ewMLIuLXwDXAuX1GGdHhh8O6dQ7JJ0mS1EIqG4oPIDMvAy7rs+8jDa+P2MR1vwBeVmVtbe+VryyG5bv2Wjii3z9GSZIkDbOWeKBRgzB+PBxwgP2uJUmSWojhup295jVw/fWwalXdlUiSJAnDdXs7/HBYvRpuuKHuSiRJkoThur0ddljR7/rqq+uuRJIkSRiu29vEiTB3Llx1Vd2VSJIkCcN1+zvyyKJbyPLldVciSZLU9QzX7e7II2H9evjxj+uuRJIkqesZrtvdoYfC9tvbNUSSJKkFGK7b3ZgxxZB8hmtJkqTaGa47wZFHwm9+Aw88UHclkiRJXc1w3QmOPLJY23otSZJUK8N1J5gzB6ZNM1xLkiTVzHDdCSLgDW+AK6+EdevqrkaSJKlrGa47xdFHw9Kl8Itf1F2JJElS1zJcd4ojj4TRo+F//qfuSiRJkrqW4bpTjB8Pr32t4VqSJKlGhutOcvTRcOedcM89dVciSZLUlQzXneTNby7W3/9+vXVIkiR1KcN1J9l9d9h7b/je9+quRJIkqSsZrjvN0UfDtdfCihV1VyJJktR1DNed5i1vgbVr4Qc/qLsSSZKkrmO47jSvfCVMnQqXXFJ3JZIkSV3HcN1pRo6Et70NLrsMnnmm7mokSZK6iuG6E7397bByJVx+ed2VSJIkdRXDdSc67DDo6bFriCRJ0jAzXHeiUaPgrW8tZmt89tm6q5EkSeoahutO9fa3w9NPw5VX1l2JJElS1zBcd6rXvhYmTYKLL667EkmSpK5huO5Uo0fDH/4hXHpp0YItSZKkyhmuO9nJJxejhnznO3VXIkmS1BUM153sVa+CmTPhwgvrrkSSJKkrGK47WUTRev3DH8LDD9ddjSRJUsczXHe6k0+GDRvgG9+ouxJJkqSOZ7judHvuCa94BfzXf9VdiSRJUsczXHeDU06Bm28uFkmSJFXGcN0NTjoJttkG/u3f6q5EkiSpoxmuu8GkScWY1xdeWAzNJ0mSpEoYrrvFn/4prFjhjI2SJEkVMlx3i8MOKx5utGuIJElSZQzX3SICTjsNfvELuP32uquRJEnqSIbrbnLKKTBmDHzhC3VXIkmS1JEM191k8mQ48UT46ldh2bK6q5EkSeo4hutu8573wDPPwJe/XHclkiRJHcdw3W323x8OPxw+9zlYt67uaiRJkjqK4bobvfe9cP/9MG9e3ZVIkiR1FMN1NzrmGJg5E/7pn+quRJIkqaMYrrvRyJFw5pnws58VQ/NJkiSpKSoN1xFxVETcHRELI+Lsfo6/PyLuiIhbIuLqiNit4dipEfXNjtkAABUeSURBVPHbcjm1yjq70mmnwU47wT/+Y92VSJIkdYzKwnVEjATOB94IzAFOjIg5fU67CZibmfsClwCfLK+dBHwUOBg4CPhoREysqtautP32Rd/r738fbr657mokSZI6QpUt1wcBCzPz3sxcA1wEHNt4QmZek5kry83rgF3K128ArsrMJzNzKXAVcFSFtXand78bxo+Hj3+87kokSZI6QpXhejrwYMP2onLfprwL+MEgr9VgTJwIf/EX8M1vwt13112NJElS22uJBxoj4mRgLvCprbzutIhYEBELlixZUk1xne5974OxY+Gcc+quRJIkqe1VGa4XAzMatncp9z1PRBwBfBg4JjNXb821mfmlzJybmXN7enqaVnhXmTKlmLXxG9+AW26puxpJkqS2VmW4ng/MjohZETEGOAF43qwlEbE/8K8UwfqxhkNXAK+PiInlg4yvL/epCn/917DDDvB3f1d3JZIkSW2tsnCdmeuAMyhC8Z3AxZl5e0ScExHHlKd9ChgHfDMibo6IeeW1TwL/QBHQ5wPnlPtUhUmTioA9bx5cf33d1UiSJLWtyMy6a2iKuXPn5oIFC+ouo309/TS8+MWwzz5w9dUQUXdFkiRJLSkibszMuf0da4kHGtUCxo2Dj3wErrkGvve9uquRJElqS4ZrbfRnfwZ77w0f+ACsXr3l8yVJkvQ8hmttNHo0nHceLFwIn/983dVIkiS1HcO1nu+oo+CNbyzGvX7ssS2fL0mSpOcYrvVCn/kMPPts0T1EkiRJA2a41gvtvTecdRZceCH88Id1VyNJktQ2DNfq34c/DHvsAaefXrRiS5IkaYsM1+rfttvCF78I99wD//f/1l2NJElSWzBca9Ne9zo45RT45CfhllvqrkaSJKnlGa61eZ/5TDE9+sknO/a1JEnSFhiutXmTJ8MFF8Ctt8Lf/V3d1UiSJLU0w7W27M1vLmZv/PSn4dpr665GkiSpZRmuNTCf/jTsvnvRB3vZsrqrkSRJakmGaw3MuHHFuNeLF8O73gWZdVckSZLUcgzXGriDD4ZPfAK+/W0477y6q5EkSWo5hmttnfe/H972NvjgB+EnP6m7GkmSpJZiuNbWiYCvfAVe/GI4/nh4+OG6K5IkSWoZhmttvR12KLqGrFgBxx0HK1fWXZEkSVJLMFxrcPbZB77+dZg/H049FTZsqLsiSZKk2hmuNXjHHlsM0XfJJfC3f1t3NZIkSbUbVXcBanPvex/85jfw8Y/DzJlw2ml1VyRJklQbw7WGJgI+9zl44AE4/XTYccfiQUdJkqQuZLcQDd3o0UXXkFe/Gk4+GS67rO6KJEmSamG4VnNstx1873uw777wB38AP/5x3RVJkiQNO8O1mmfHHeHyy4sxsN/0JvjhD+uuSJIkaVgZrtVcPT1wzTWwxx5w9NHwgx/UXZEkSdKwMVyr+XbeuQjYc+YUk8x85zt1VyRJkjQsDNeqxk47wdVXwwEHFH2wzz+/7ookSZIqZ7hWdSZOLAL2W94CZ5wBZ53lTI6SJKmjGa5Vre22g29/G/78z+FTn4J3vANWraq7KkmSpEoYrlW9kSOLbiHnngsXXQSHHQYPPlh3VZIkSU1nuNbwiIAPfhAuvRTuvhsOPNCxsCVJUscxXGt4HXss3HBD8cDjEUfAeedBZt1VSZIkNYXhWsNvr73g+uvhmGPgAx8oxsN+7LG6q5IkSRoyw7XqscMO8K1vwec/X4wosu++cOWVdVclSZI0JIZr1ScC3v1umD8fJk+GN7wBzjwTnn667sokSZIGxXCt+r3sZUXAPvPMoiV7n33gqqvqrkqSJGmrGa7VGrbdFv75n+GnP4WxY+H1r4d3vhOWLq27MkmSpAEzXKu1vOpVcPPN8KEPwX/+Z/Hw4wUXOLOjJElqC4ZrtZ6xY+Ef/xEWLIDZs+Fd74KDD4Zf/rLuyiRJkjbLcK3Wtd9+RTeRr30NHnoIXvlKOOUUuP/+uiuTJEnql+FarS0CTjqpmNXx7LPh4ovhJS+B977XsbElSVLLMVyrPYwbBx//OPz2t/DHfwyf+xzsvjt89KOwbFnd1UmSJAGGa7WbGTPg3/8dbr8djjoKzjkHdtsN/uZvbMmWJEm1M1yrPe21F3zzm3DTTUXIPvfcImSfeSY88EDd1UmSpC5luFZ7228/+O//hjvvhBNPhC98oegucuKJcN11kFl3hZIkqYsYrtUZ9tyzGA/7nnvgL/8SLrsMDj20GMLvwgth9eq6K5QkSV3AcK3OsuuucN55sHgxnH8+rFhRPAC5665w1lnFqCOSJEkVqTRcR8RREXF3RCyMiLP7OX54RPwqItZFxNv7HFsfETeXy7wq61QHGjcO/uIv4I474Iorilbs884r+mq/+tXwla/AM8/UXaUkSeowlYXriBgJnA+8EZgDnBgRc/qc9gDwv4Cv9/MWz2bmfuVyTFV1qsONGAGvfz1ceiksWgSf+AQsWQLvfCdMnQp/8idF+F63ru5KJUlSB6iy5fogYGFm3puZa4CLgGMbT8jM+zLzFmBDhXVIhalTi64hd91VzPz4h38I3/52MdrItGlFS/dPfwob/J+jJEkanCrD9XTgwYbtReW+gRobEQsi4rqIOK6/EyLitPKcBUuWLBlKreomEUXXkAsugEcfhe98B37v9+CrX4XDDy+G9Hvve+FHP4K1a+uuVpIktZFWfqBxt8ycC5wEfDYidu97QmZ+KTPnZubcnp6e4a9Q7W/sWDjuuGI4v8ceg699DfbfH774RXjd62DnneHkk4tp11esqLtaSZLU4qoM14uBGQ3bu5T7BiQzF5fre4EfA/s3szjpBcaNg5NOgnnz4Iknii4jxx4Ll18Oxx8PkycXXUg++9niQUnH0JYkSX1UGa7nA7MjYlZEjAFOAAY06kdETIyIbcrXk4FXAXdUVqnU1/bbw1vfWnQVefRR+MlP4D3vgd/9Dt73PnjpS4up2N/5TrjoInj88borliRJLSCywta3iHgT8FlgJHBBZn4sIs4BFmTmvIh4BfAdYCKwCngkM18aEa8E/pXiQccRwGcz88ub+6y5c+fmggULKvsu0nPuuw+uuqpYfvhDWLq06Me9337wmtcU/bYPO6xo6ZYkSR0nIm4suy+/8FiV4Xo4Ga5Vi/Xr4cYb4cor4eqriynXV60qjs2ZUwTt3mX61jzPK0mSWpXhWhouq1cXYfsnPymWn/0MnnqqODZjBhx0UDEl+8EHw4EHFt1PJElSWzFcS3VZtw5uuaUI2jfcANdfD/feWxwbMQL22acI2gcdBAccULR2jx1bb82SJGmzDNdSK1myBObPL4L29dcXoXvp0uLYqFHFFO377ff8Zaed6q1ZkiQ9x3AttbJMuOceuPnm5y+LG0au3GUX2HffomW7d9l7b9hhh/rqliSpS20uXI8a7mIk9REBe+xRLG9/+8b9S5bAr39dBO2bboLbbitGJ1mzZuM5u+zy/MC9114wezZMmVK8ryRJGla2XEvtZN26YqztO+8sJrLpXe68E1au3Hje9ttvDOx77FEE7t7XL3pR0d9bkiQNii3XUqcYNaoIyrNnwzHHbNy/YQM88ADcdVfRxWThwmK57bZixsm1azeeu+22sPvusNtu/S9Tphi+JUkaJMO11AlGjICZM4ulr/Xr4cEHNwbu3uW+++DnP4dly55//pgxsOuuxdIbuKdPh2nTNi6TJxvAJUnqh+Fa6nQjR24M3kcc8cLjK1YUrd733//C5fLL4eGHX3jN6NFF95LGwN24TJkCO+9chPBR/mdGktQ9/H89qdvtsEMx3vY++/R/fM0aeOQReOihYlm8eOPrhx4quqL86EcvbAHvNWlSEbR7eor1pl5PngwTJxbBXZKkNmW4lrR5jd1ENmflyqKV+6GH4LHHnr8sWVKs77gDrr0WnniiGIKwP+PHF4F80qQibPe+3tS+iRNhwoTiIU67qkiSama4ltQc221XPCi5++5bPnfduiJg94buxx6Dxx8vJtN58smNy9KlcPvtG7cbH8zsK6II5jvsADvuWKx7l77bffeNH1+E895l220dylCSNCiGa0nDb9Sool/2lCkDvyYTnnnmhQH8ySeLfuMrVsDy5Rtfr1hRnHv//Ru3n356YJ8V8fywvbXL2LHFsu22m389erQhXpI6jOFaUnuIgHHjimXGjMG9x/r18NRTzw/gvcszz7xwefrpF+57/PEXnrNhw+DqGTFi4EF8zJjql1Gjnr+MHGn4l6StZLiW1D1Gjiz6Z0+Y0Lz3zITVqzeG7ZUrYdWqYnn22aG9fuaZovvMs88WXWLWrHnhsrmuMs0wcuTGsD169AsD+ECWLV03cmTxF41WXEcUy4gRW3490POqeO1fgqSWYbiWpKGI2NjKvNNOw//5mZsO3luzrF5dtOyvW/fCZe3a/vdvaVm7tnjfLZ2zYUPx2ZtbD/ZfB7rJQAJ4YxAfyLqqc6t+/ypr6Wtzf7EZjmvq/vzhumZT+//6r/sfZrZGhmtJamcRG7t1dLLMYtlSCG/mev36jZ+bWexvxutmvtdgPrv3z3NL66rOrfr9q6ylr03tH65rqvj8/o638ves+l/vBsFwLUlqfY0tsZLUwvyvlCRJktQkhmtJkiSpSQzXkiRJUpMYriVJkqQmMVxLkiRJTWK4liRJkprEcC1JkiQ1ieFakiRJahLDtSRJktQkhmtJkiSpSQzXkiRJUpMYriVJkqQmMVxLkiRJTWK4liRJkprEcC1JkiQ1ieFakiRJahLDtSRJktQkhmtJkiSpSSIz666hKSJiCXB/TR8/GXi8ps/W8PE+dwfvc+fzHncH73N3qOs+75aZPf0d6JhwXaeIWJCZc+uuQ9XyPncH73Pn8x53B+9zd2jF+2y3EEmSJKlJDNeSJElSkxium+NLdRegYeF97g7e587nPe4O3ufu0HL32T7XkiRJUpPYci1JkiQ1ieF6CCLiqIi4OyIWRsTZddejwYuIGRFxTUTcERG3R8R7yv2TIuKqiPhtuZ5Y7o+I+H/lvb8lIg6o9xtoa0TEyIi4KSL+p9yeFRHXl/fzvyNiTLl/m3J7YXl8Zp11a+AiYkJEXBIRd0XEnRFxqL/nzhMR7yv/m31bRHwjIsb6e25/EXFBRDwWEbc17Nvq329EnFqe/9uIOHW46jdcD1JEjATOB94IzAFOjIg59ValIVgHfCAz5wCHAO8u7+fZwNWZORu4utyG4r7PLpfTgC8Mf8kagvcAdzZsfwL4p8zcA1gKvKvc/y5gabn/n8rz1B7+Gbg8M/cCXk5xv/09d5CImA6cCczNzH2AkcAJ+HvuBF8Fjuqzb6t+vxExCfgocDBwEPDR3kBeNcP14B0ELMzMezNzDXARcGzNNWmQMvPhzPxV+fopiv8jnk5xT/+jPO0/gOPK18cC/5mF64AJEfGiYS5bgxARuwBvBv693A7g94FLylP63ufe+38J8LryfLWwiNgROBz4MkBmrsnMZfh77kSjgG0jYhSwHfAw/p7bXmb+BHiyz+6t/f2+AbgqM5/MzKXAVbwwsFfCcD1404EHG7YXlfvU5sp/KtwfuB6YkpkPl4ceAaaUr73/7euzwFnAhnJ7J2BZZq4rtxvv5XP3uTy+vDxfrW0WsAT4Stn9598jYnv8PXeUzFwMfBp4gCJULwduxN9zp9ra329tv2vDtdQgIsYB3wLem5krGo9lMbSOw+u0sYg4GngsM2+suxZVahRwAPCFzNwfeIaN/4QM+HvuBOU/8R9L8ZepacD2DFPLpOrV6r9fw/XgLQZmNGzvUu5Tm4qI0RTB+muZ+e1y96O9/zxcrh8r93v/29OrgGMi4j6Krly/T9E3d0L5z8rw/Hv53H0uj+8IPDGcBWtQFgGLMvP6cvsSirDt77mzHAH8LjOXZOZa4NsUv3F/z51pa3+/tf2uDdeDNx+YXT6VPIbiIYp5NdekQSr73X0ZuDMzz2s4NA/ofcL4VOC7DftPKZ9SPgRY3vDPVWpRmfmhzNwlM2dS/GZ/lJnvAK4B3l6e1vc+997/t5fnt2xriQqZ+QjwYETsWe56HXAH/p47zQPAIRGxXfnf8N777O+5M23t7/cK4PURMbH8V47Xl/sq5yQyQxARb6LovzkSuCAzP1ZzSRqkiHg18FPgVjb2xf0bin7XFwO7AvcDf5SZT5b/If88xT9BrgT+JDMXDHvhGrSIeC3wV5l5dES8mKIlexJwE3ByZq6OiLHAf1H0wX8SOCEz762rZg1cROxH8dDqGOBe4E8oGpT8PXeQiPg/wPEUIz7dBPxvin61/p7bWER8A3gtMBl4lGLUj0vZyt9vRLyT4v/LAT6WmV8ZlvoN15IkSVJz2C1EkiRJahLDtSRJktQkhmtJkiSpSQzXkiRJUpMYriVJkqQmMVxLUouLiKfL9cyIOKnJ7/03fbZ/0cz3l6RuY7iWpPYxE9iqcN0wU92mPC9cZ+Yrt7ImSVIDw7UktY9zgcMi4uaIeF9EjIyIT0XE/Ii4JSL+DIoJciLipxExj2LGOiLi0oi4MSJuj4jTyn3nAtuW7/e1cl9vK3mU731bRNwaEcc3vPePI+KSiLgrIr5WTuJARJwbEXeUtXx62P90JKkFbKlFQ5LUOs6mnFUSoAzJyzPzFRGxDfDziLiyPPcAYJ/M/F25/c5yNrNtgfkR8a3MPDsizsjM/fr5rLcB+wEvp5glbX5E/KQ8tj/wUuAh4OfAqyLiTuCtwF6ZmRExoenfXpLagC3XktS+Xg+cEhE3A9cDOwGzy2M3NARrgDMj4tfAdcCMhvM25dXANzJzfWY+ClwLvKLhvRdl5gbgZoruKsuBVcCXI+JtFNMQS1LXMVxLUvsK4C8zc79ymZWZvS3Xzzx3UsRrgSOAQzPz5cBNwNghfO7qhtfrgVGZuQ44CLgEOBq4fAjvL0lty3AtSe3jKWB8w/YVwJ9HxGiAiHhJRGzfz3U7Akszc2VE7AUc0nBsbe/1ffwUOL7s190DHA7csKnCImIcsGNmXga8j6I7iSR1HftcS1L7uAVYX3bv+CrwzxRdMn5VPlS4BDiun+suB04v+0XfTdE1pNeXgFsi4leZ+Y6G/d8BDgV+DSRwVmY+Uobz/owHvhsRYyla1N8/uK8oSe0tMrPuGiRJkqSOYLcQSZIkqUkM15IkSVKTGK4lSZKkJjFcS5IkSU1iuJYkSZKaxHAtSZIkNYnhWpIkSWoSw7UkSZLUJP8ffvqpHtFx4DkAAAAASUVORK5CYII=\n",
            "text/plain": [
              "<Figure size 864x576 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "rdDVNJwV36kt"
      },
      "source": [
        "Instead of implementing these algorithms from scratch, we could also use scikit-learn's linear regression function.  Let's apply scikit-learn's linear regressio algorithm to the data from part 1 and see what it comes up with."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "r_q6Z0FP32pz",
        "outputId": "0d8b63e6-f429-44a0-afb0-fdece55bbd98",
        "colab": {
          "base_uri": "https://localhost:8080/"
        }
      },
      "source": [
        "from sklearn import linear_model\n",
        "model = linear_model.LinearRegression()\n",
        "model.fit(X, y)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stderr",
          "text": [
            "/usr/local/lib/python3.7/dist-packages/sklearn/utils/validation.py:590: FutureWarning: np.matrix usage is deprecated in 1.0 and will raise a TypeError in 1.2. Please convert to a numpy array with np.asarray. For more information see: https://numpy.org/doc/stable/reference/generated/numpy.matrix.html\n",
            "  FutureWarning,\n",
            "/usr/local/lib/python3.7/dist-packages/sklearn/utils/validation.py:590: FutureWarning: np.matrix usage is deprecated in 1.0 and will raise a TypeError in 1.2. Please convert to a numpy array with np.asarray. For more information see: https://numpy.org/doc/stable/reference/generated/numpy.matrix.html\n",
            "  FutureWarning,\n"
          ]
        },
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "LinearRegression()"
            ]
          },
          "metadata": {},
          "execution_count": 28
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "-nJ1rRV94DdA"
      },
      "source": [
        "Here's what the scikit-learn model's predictions look like."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "I_Up78sS39AY",
        "outputId": "bbb4e675-f518-47c9-d41e-453dd4b94b9f",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 585
        }
      },
      "source": [
        "x = np.array(X[:, 1].A1)\n",
        "f = model.predict(X).flatten()\n",
        "\n",
        "fig, ax = plt.subplots(figsize=(12,8))\n",
        "ax.plot(x, f, 'r', label='Prediction')\n",
        "ax.scatter(data.Population, data.Profit, label='Traning Data')\n",
        "ax.legend(loc=2)\n",
        "ax.set_xlabel('Population')\n",
        "ax.set_ylabel('Profit')\n",
        "ax.set_title('Predicted Profit vs. Population Size')"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stderr",
          "text": [
            "/usr/local/lib/python3.7/dist-packages/sklearn/utils/validation.py:590: FutureWarning: np.matrix usage is deprecated in 1.0 and will raise a TypeError in 1.2. Please convert to a numpy array with np.asarray. For more information see: https://numpy.org/doc/stable/reference/generated/numpy.matrix.html\n",
            "  FutureWarning,\n"
          ]
        },
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "Text(0.5, 1.0, 'Predicted Profit vs. Population Size')"
            ]
          },
          "metadata": {},
          "execution_count": 29
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 864x576 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "LpMNs1Wk4Ks_"
      },
      "source": [
        "That's it!  Thanks for reading.  In Exercise 2 we'll take a look at logistic regression for classification problems."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "-mljJAPb4Jqk"
      },
      "source": [
        ""
      ],
      "execution_count": null,
      "outputs": []
    }
  ]
}