linear-regression-by-hand-vs-libraries-in-python.ipynb

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        "<a href=\"https://colab.research.google.com/github/burrittresearch/linear-regression-by-hand-vs-libraries-in-python/blob/main/linear-regression-by-hand-vs-libraries-in-python.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
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      "source": [
        "# Linear Regression by Hand vs Libraries in Python\n",
        "This notebook creates a linear regression  model and related statistics by hand using the underlying mathematical formulas and a sample dataset. It then creates another linear regression model and related statistics using python libraries and the same sample dataset. Both methods include calculating the equation of a line $y = mx+b$, the line's slope ($m$) and y-intercept ($b$), the coefficient of determination R-squared ($r^2$), and Pearson's correlation coefficient ($r$). The results are of the two modeling methods are then compared and summarized.\n",
        "\n",
        "### About Least Squares Method\n",
        "Both linear regression models use the Least Squares Method (LSM), which attempts to find the line ($y = mx+b$) of best fit to given data by reducing the sum of squares (residuals) between the the observed and fitted data. LSM is one of the most widely used regression methods.\n",
        "\n",
        "### About $r^2$ and $r$\n",
        "R-squared ($r^2$) or the coefficient of determination measures how well $x$, an independent variable, can explain the variance in $y$, the dependent variable. Values range from 0 to 1, where 0 means $x$ explains nothing about $y$ and where 1 means $x$ explains everything about $y$.   \n",
        "\n",
        "R ($r$) or Pearson's correlation coefficient measures the strength and direction of the linear relationship between $x$ and $y$. Values range from -1 to 1, where -1 means $x$ and $y$ have a perfectly negative relationship and 1 means $x$ and $y$ have a perfectly positive relationship.\n",
        "\n",
        "### Equations Used for Hand Calculations\n",
        "- line\n",
        "$\n",
        "y = mx+b\n",
        "$\n",
        "\n",
        "- slope\n",
        "$\n",
        "m = \\frac{\\sum(x-\\bar{x})(y-\\bar{y})}\n",
        "{\\sum(x-\\bar{x})}\n",
        "$\n",
        "\n",
        "- y-intercept\n",
        "$\n",
        "b = \\bar{y}-a\\bar{x}\n",
        "$\n",
        "\n",
        "- R-squared\n",
        "$\n",
        "r^2 = \\frac{\\sum(\\hat{y}-\\bar{y})}\n",
        "{\\sum(y-\\bar{y})}\n",
        "$\n",
        "\n",
        "- R\n",
        "$\n",
        "r = \\frac{\\sum(x-\\bar{x})(y-\\bar{y})}\n",
        "{\\sqrt{\\sum(x-\\bar{x})^2(y-\\bar{y})^2}}\n",
        "$\n",
        "\n",
        "\n"
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      "metadata": {
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    {
      "cell_type": "markdown",
      "source": [
        "# Project Workflow\n",
        "* Define the Problem\n",
        "* Linear Regression by Hand\n",
        "* Linear Regression Using Python Libraries\n",
        "* Compare Results"
      ],
      "metadata": {
        "id": "RMAYdMm4pgju"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "# Define the Problem\n",
        "Build a linear regression model and related statistics by hand and by using python libraries, and then compare the results."
      ],
      "metadata": {
        "id": "PtDuQJptpgtS"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "# Linear Regression by Hand"
      ],
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        "id": "T0o548djqKfr"
      }
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      "execution_count": 1,
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      "source": [
        "# Import libraries\n",
        "import pandas as pd\n",
        "import matplotlib.pyplot as plt\n",
        "import numpy as np\n",
        "import seaborn as sns\n",
        "from sklearn.linear_model import LinearRegression\n",
        "from sklearn.metrics import r2_score\n"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# Process data\n",
        "x = np.array([1, 2, 3, 4, 5])\n",
        "y = np.array([2, 4, 5, 4, 5])"
      ],
      "metadata": {
        "id": "JyMOdMKNMtTX"
      },
      "execution_count": 2,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "# Create dataframe\n",
        "\n",
        "df_least = pd.DataFrame()"
      ],
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      "execution_count": 3,
      "outputs": []
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        "# Add key columns for manual calculation to dataframe\n",
        "xbar = np.mean(x)\n",
        "ybar = np.mean(y)\n",
        "\n",
        "df_least['x'] = x\n",
        "df_least['y'] = y\n",
        "df_least['x_xbar'] = x - xbar\n",
        "df_least['y_ybar'] = y - ybar\n",
        "df_least['(x_xbar)(y_ybar)'] = df_least['x_xbar'] * df_least['y_ybar']\n",
        "df_least['(x_xbar)^2'] = df_least['x_xbar'] ** 2\n",
        "df_least['(y_ybar)^2'] = df_least['y_ybar'] ** 2\n",
        "\n",
        "df_least"
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              "summary": "{\n  \"name\": \"df_least\",\n  \"rows\": 5,\n  \"fields\": [\n    {\n      \"column\": \"x\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 1,\n        \"min\": 1,\n        \"max\": 5,\n        \"num_unique_values\": 5,\n        \"samples\": [\n          2,\n          5,\n          3\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"y\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 1,\n        \"min\": 2,\n        \"max\": 5,\n        \"num_unique_values\": 3,\n        \"samples\": [\n          2,\n          4,\n          5\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"x_xbar\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 1.5811388300841898,\n        \"min\": -2.0,\n        \"max\": 2.0,\n        \"num_unique_values\": 5,\n        \"samples\": [\n          -1.0,\n          2.0,\n          0.0\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"y_ybar\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 1.224744871391589,\n        \"min\": -2.0,\n        \"max\": 1.0,\n        \"num_unique_values\": 3,\n        \"samples\": [\n          -2.0,\n          0.0,\n          1.0\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"(x_xbar)(y_ybar)\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 1.7888543819998317,\n        \"min\": 0.0,\n        \"max\": 4.0,\n        \"num_unique_values\": 3,\n        \"samples\": [\n          4.0,\n          -0.0,\n          2.0\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"(x_xbar)^2\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 1.8708286933869707,\n        \"min\": 0.0,\n        \"max\": 4.0,\n        \"num_unique_values\": 3,\n        \"samples\": [\n          4.0,\n          1.0,\n          0.0\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"(y_ybar)^2\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 1.6431676725154982,\n        \"min\": 0.0,\n        \"max\": 4.0,\n        \"num_unique_values\": 3,\n        \"samples\": [\n          4.0,\n          0.0,\n          1.0\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    }\n  ]\n}"
            }
          },
          "metadata": {},
          "execution_count": 4
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# Calculate slope and y-intercept\n",
        "m = df_least['(x_xbar)(y_ybar)'].sum() / df_least['(x_xbar)^2'].sum()\n",
        "b = ybar - (m * xbar)\n",
        "m, b"
      ],
      "metadata": {
        "id": "IFPVQLjZEVKF",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "f05efc66-41f1-4bca-d784-2c914f722db6"
      },
      "execution_count": 5,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "(0.6, 2.2)"
            ]
          },
          "metadata": {},
          "execution_count": 5
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# Run yhat test\n",
        "yhat_test = (m * df_least['x'][1:2]) + b\n",
        "yhat_test"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "3zPNiq1xI7mg",
        "outputId": "ddb36a46-e068-4873-c63b-e5d3e5a06f1e"
      },
      "execution_count": 6,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "1    3.4\n",
              "Name: x, dtype: float64"
            ]
          },
          "metadata": {},
          "execution_count": 6
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# Add addition columnns for r² calculation\n",
        "df_least['yhat'] = (m * df_least['x']) + b\n",
        "df_least['yhat_ybar'] = df_least['yhat'] - ybar\n",
        "df_least['(yhat_ybar)^2'] = df_least['yhat_ybar'] ** 2\n",
        "df_least"
      ],
      "metadata": {
        "id": "dpnH0hkeEU23",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 206
        },
        "outputId": "949e6e0a-b365-43f3-e5ea-4e3d5b35c56c"
      },
      "execution_count": 7,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "   x  y  x_xbar  y_ybar  (x_xbar)(y_ybar)  (x_xbar)^2  (y_ybar)^2  yhat  \\\n",
              "0  1  2    -2.0    -2.0               4.0         4.0         4.0   2.8   \n",
              "1  2  4    -1.0     0.0              -0.0         1.0         0.0   3.4   \n",
              "2  3  5     0.0     1.0               0.0         0.0         1.0   4.0   \n",
              "3  4  4     1.0     0.0               0.0         1.0         0.0   4.6   \n",
              "4  5  5     2.0     1.0               2.0         4.0         1.0   5.2   \n",
              "\n",
              "   yhat_ybar  (yhat_ybar)^2  \n",
              "0       -1.2           1.44  \n",
              "1       -0.6           0.36  \n",
              "2        0.0           0.00  \n",
              "3        0.6           0.36  \n",
              "4        1.2           1.44  "
            ],
            "text/html": [
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              "<table border=\"1\" class=\"dataframe\">\n",
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              "      <td>-0.6</td>\n",
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              "      <th>2</th>\n",
              "      <td>3</td>\n",
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              "      <td>1.0</td>\n",
              "      <td>0.0</td>\n",
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              "      <td>4.0</td>\n",
              "      <td>0.0</td>\n",
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              "    <tr>\n",
              "      <th>3</th>\n",
              "      <td>4</td>\n",
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              "      <td>1.0</td>\n",
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              "      <td>0.0</td>\n",
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              "      <td>0.0</td>\n",
              "      <td>4.6</td>\n",
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              "      <td>0.36</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>4</th>\n",
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              "\n",
              "    </div>\n",
              "  </div>\n"
            ],
            "application/vnd.google.colaboratory.intrinsic+json": {
              "type": "dataframe",
              "variable_name": "df_least",
              "summary": "{\n  \"name\": \"df_least\",\n  \"rows\": 5,\n  \"fields\": [\n    {\n      \"column\": \"x\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 1,\n        \"min\": 1,\n        \"max\": 5,\n        \"num_unique_values\": 5,\n        \"samples\": [\n          2,\n          5,\n          3\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"y\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 1,\n        \"min\": 2,\n        \"max\": 5,\n        \"num_unique_values\": 3,\n        \"samples\": [\n          2,\n          4,\n          5\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"x_xbar\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 1.5811388300841898,\n        \"min\": -2.0,\n        \"max\": 2.0,\n        \"num_unique_values\": 5,\n        \"samples\": [\n          -1.0,\n          2.0,\n          0.0\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"y_ybar\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 1.224744871391589,\n        \"min\": -2.0,\n        \"max\": 1.0,\n        \"num_unique_values\": 3,\n        \"samples\": [\n          -2.0,\n          0.0,\n          1.0\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"(x_xbar)(y_ybar)\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 1.7888543819998317,\n        \"min\": 0.0,\n        \"max\": 4.0,\n        \"num_unique_values\": 3,\n        \"samples\": [\n          4.0,\n          -0.0,\n          2.0\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"(x_xbar)^2\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 1.8708286933869707,\n        \"min\": 0.0,\n        \"max\": 4.0,\n        \"num_unique_values\": 3,\n        \"samples\": [\n          4.0,\n          1.0,\n          0.0\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"(y_ybar)^2\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 1.6431676725154982,\n        \"min\": 0.0,\n        \"max\": 4.0,\n        \"num_unique_values\": 3,\n        \"samples\": [\n          4.0,\n          0.0,\n          1.0\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"yhat\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 0.9486832980505137,\n        \"min\": 2.8000000000000003,\n        \"max\": 5.2,\n        \"num_unique_values\": 5,\n        \"samples\": [\n          3.4000000000000004,\n          5.2,\n          4.0\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"yhat_ybar\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 0.9486832980505137,\n        \"min\": -1.1999999999999997,\n        \"max\": 1.2000000000000002,\n        \"num_unique_values\": 5,\n        \"samples\": [\n          -0.5999999999999996,\n          1.2000000000000002,\n          0.0\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"(yhat_ybar)^2\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 0.6734983296193094,\n        \"min\": 0.0,\n        \"max\": 1.4400000000000004,\n        \"num_unique_values\": 4,\n        \"samples\": [\n          0.3599999999999996,\n          1.4400000000000004,\n          1.4399999999999993\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    }\n  ]\n}"
            }
          },
          "metadata": {},
          "execution_count": 7
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# Calculate r², the coefficient of determination\n",
        "r_sq = df_least['(yhat_ybar)^2'].sum() / df_least['(y_ybar)^2'].sum()\n",
        "r_sq"
      ],
      "metadata": {
        "id": "dwvEDYxQbYDO",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "5b1aa72d-aa6e-42bf-cee1-19764da855c1"
      },
      "execution_count": 8,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.5999999999999998"
            ]
          },
          "metadata": {},
          "execution_count": 8
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# Calculate r, Pearson's correlation coefficient\n",
        "r = (df_least['(x_xbar)(y_ybar)'].sum() /\n",
        "        np.sqrt(df_least['(x_xbar)^2'].sum() * df_least['(y_ybar)^2'].sum())\n",
        ")\n",
        "r"
      ],
      "metadata": {
        "id": "LFZEn7OxbYZb",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "490afc89-2f13-4711-ea78-a702b6f474f8"
      },
      "execution_count": 9,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.7745966692414834"
            ]
          },
          "metadata": {},
          "execution_count": 9
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "# Linear Regression using Python Libraries\n"
      ],
      "metadata": {
        "id": "kGE0p3zbHoA4"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# Process data\n",
        "x = x.reshape(-1, 1)\n",
        "y = y.reshape(-1, 1)"
      ],
      "metadata": {
        "id": "h6rrNiz6_TS1"
      },
      "execution_count": 10,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "# Create model and add results to dataframe\n",
        "lr = LinearRegression()\n",
        "lr_model = lr.fit(x, y)\n",
        "lr_model\n",
        "yhat_libr = lr_model.predict(x)\n",
        "\n",
        "df_least['yhat_libr'] = yhat_libr\n",
        "df_least"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 206
        },
        "id": "7fPuYHuZH-2q",
        "outputId": "f5020ce1-e06a-444c-bdb4-6962feb7c678"
      },
      "execution_count": 11,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "   x  y  x_xbar  y_ybar  (x_xbar)(y_ybar)  (x_xbar)^2  (y_ybar)^2  yhat  \\\n",
              "0  1  2    -2.0    -2.0               4.0         4.0         4.0   2.8   \n",
              "1  2  4    -1.0     0.0              -0.0         1.0         0.0   3.4   \n",
              "2  3  5     0.0     1.0               0.0         0.0         1.0   4.0   \n",
              "3  4  4     1.0     0.0               0.0         1.0         0.0   4.6   \n",
              "4  5  5     2.0     1.0               2.0         4.0         1.0   5.2   \n",
              "\n",
              "   yhat_ybar  (yhat_ybar)^2  yhat_libr  \n",
              "0       -1.2           1.44        2.8  \n",
              "1       -0.6           0.36        3.4  \n",
              "2        0.0           0.00        4.0  \n",
              "3        0.6           0.36        4.6  \n",
              "4        1.2           1.44        5.2  "
            ],
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              "\n",
              "      [theme=dark] .colab-df-generate:hover {\n",
              "        background-color: #434B5C;\n",
              "        box-shadow: 0px 1px 3px 1px rgba(0, 0, 0, 0.15);\n",
              "        filter: drop-shadow(0px 1px 2px rgba(0, 0, 0, 0.3));\n",
              "        fill: #FFFFFF;\n",
              "      }\n",
              "    </style>\n",
              "    <button class=\"colab-df-generate\" onclick=\"generateWithVariable('df_least')\"\n",
              "            title=\"Generate code using this dataframe.\"\n",
              "            style=\"display:none;\">\n",
              "\n",
              "  <svg xmlns=\"http://www.w3.org/2000/svg\" height=\"24px\"viewBox=\"0 0 24 24\"\n",
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              "\n",
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            "application/vnd.google.colaboratory.intrinsic+json": {
              "type": "dataframe",
              "variable_name": "df_least",
              "summary": "{\n  \"name\": \"df_least\",\n  \"rows\": 5,\n  \"fields\": [\n    {\n      \"column\": \"x\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 1,\n        \"min\": 1,\n        \"max\": 5,\n        \"num_unique_values\": 5,\n        \"samples\": [\n          2,\n          5,\n          3\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"y\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 1,\n        \"min\": 2,\n        \"max\": 5,\n        \"num_unique_values\": 3,\n        \"samples\": [\n          2,\n          4,\n          5\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"x_xbar\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 1.5811388300841898,\n        \"min\": -2.0,\n        \"max\": 2.0,\n        \"num_unique_values\": 5,\n        \"samples\": [\n          -1.0,\n          2.0,\n          0.0\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"y_ybar\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 1.224744871391589,\n        \"min\": -2.0,\n        \"max\": 1.0,\n        \"num_unique_values\": 3,\n        \"samples\": [\n          -2.0,\n          0.0,\n          1.0\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"(x_xbar)(y_ybar)\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 1.7888543819998317,\n        \"min\": 0.0,\n        \"max\": 4.0,\n        \"num_unique_values\": 3,\n        \"samples\": [\n          4.0,\n          -0.0,\n          2.0\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"(x_xbar)^2\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 1.8708286933869707,\n        \"min\": 0.0,\n        \"max\": 4.0,\n        \"num_unique_values\": 3,\n        \"samples\": [\n          4.0,\n          1.0,\n          0.0\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"(y_ybar)^2\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 1.6431676725154982,\n        \"min\": 0.0,\n        \"max\": 4.0,\n        \"num_unique_values\": 3,\n        \"samples\": [\n          4.0,\n          0.0,\n          1.0\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"yhat\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 0.9486832980505137,\n        \"min\": 2.8000000000000003,\n        \"max\": 5.2,\n        \"num_unique_values\": 5,\n        \"samples\": [\n          3.4000000000000004,\n          5.2,\n          4.0\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"yhat_ybar\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 0.9486832980505137,\n        \"min\": -1.1999999999999997,\n        \"max\": 1.2000000000000002,\n        \"num_unique_values\": 5,\n        \"samples\": [\n          -0.5999999999999996,\n          1.2000000000000002,\n          0.0\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"(yhat_ybar)^2\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 0.6734983296193094,\n        \"min\": 0.0,\n        \"max\": 1.4400000000000004,\n        \"num_unique_values\": 4,\n        \"samples\": [\n          0.3599999999999996,\n          1.4400000000000004,\n          1.4399999999999993\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"yhat_libr\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 0.9486832980505137,\n        \"min\": 2.8000000000000003,\n        \"max\": 5.2,\n        \"num_unique_values\": 5,\n        \"samples\": [\n          3.4000000000000004,\n          5.2,\n          4.0\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    }\n  ]\n}"
            }
          },
          "metadata": {},
          "execution_count": 11
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# Calculate slope and y-intercept\n",
        "m_libr = lr_model.coef_.item()\n",
        "b_libr = lr_model.intercept_.item()\n",
        "\n",
        "m_libr, b_libr"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "L7bxMMCID4RR",
        "outputId": "9fa348eb-514f-4f61-c48d-2d7545996136"
      },
      "execution_count": 12,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "(0.6, 2.2)"
            ]
          },
          "metadata": {},
          "execution_count": 12
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# Calculate r², the coefficient of determination\n",
        "r_sq_libr = r2_score(y, df_least['yhat'])\n",
        "r_sq_libr"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "swDO-d2dLC8t",
        "outputId": "5069a083-5313-42be-a85a-69b66a3d7b08"
      },
      "execution_count": 13,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.6000000000000001"
            ]
          },
          "metadata": {},
          "execution_count": 13
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# Calculate r, Pearson's correlation coefficient\n",
        "r_libr = df_least['x'].corr(df_least['y'], method='pearson')\n",
        "r_libr"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "quYIvUmzOBa0",
        "outputId": "82fd9a59-a0ed-484a-ce70-5ccb1f0205c2"
      },
      "execution_count": 14,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.7745966692414834"
            ]
          },
          "metadata": {},
          "execution_count": 14
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "# Compare Results"
      ],
      "metadata": {
        "id": "ju9a50R1fzoZ"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# Create comparison dataframe\n",
        "stats = [m, b, r_sq, r]\n",
        "stats_libr = [m_libr, b_libr, r_sq_libr, r_libr]\n",
        "df_stats = pd.DataFrame([stats, stats_libr], columns=['m', 'b', 'r^2', 'r'], index=['By Hand', 'Using Libraries'])\n",
        "df_stats"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 125
        },
        "id": "RCR3dgKoD4M4",
        "outputId": "6df03647-03ed-4258-a5da-c05d145c7369"
      },
      "execution_count": 15,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "                   m    b  r^2         r\n",
              "By Hand          0.6  2.2  0.6  0.774597\n",
              "Using Libraries  0.6  2.2  0.6  0.774597"
            ],
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              "      <th>By Hand</th>\n",
              "      <td>0.6</td>\n",
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              "      <td>0.6</td>\n",
              "      <td>0.774597</td>\n",
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              "      <th>Using Libraries</th>\n",
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              "    <button class=\"colab-df-generate\" onclick=\"generateWithVariable('df_stats')\"\n",
              "            title=\"Generate code using this dataframe.\"\n",
              "            style=\"display:none;\">\n",
              "\n",
              "  <svg xmlns=\"http://www.w3.org/2000/svg\" height=\"24px\"viewBox=\"0 0 24 24\"\n",
              "       width=\"24px\">\n",
              "    <path d=\"M7,19H8.4L18.45,9,17,7.55,7,17.6ZM5,21V16.75L18.45,3.32a2,2,0,0,1,2.83,0l1.4,1.43a1.91,1.91,0,0,1,.58,1.4,1.91,1.91,0,0,1-.58,1.4L9.25,21ZM18.45,9,17,7.55Zm-12,3A5.31,5.31,0,0,0,4.9,8.1,5.31,5.31,0,0,0,1,6.5,5.31,5.31,0,0,0,4.9,4.9,5.31,5.31,0,0,0,6.5,1,5.31,5.31,0,0,0,8.1,4.9,5.31,5.31,0,0,0,12,6.5,5.46,5.46,0,0,0,6.5,12Z\"/>\n",
              "  </svg>\n",
              "    </button>\n",
              "    <script>\n",
              "      (() => {\n",
              "      const buttonEl =\n",
              "        document.querySelector('#id_d977dfab-c23b-4b6c-8bcb-99d1ec0d72b5 button.colab-df-generate');\n",
              "      buttonEl.style.display =\n",
              "        google.colab.kernel.accessAllowed ? 'block' : 'none';\n",
              "\n",
              "      buttonEl.onclick = () => {\n",
              "        google.colab.notebook.generateWithVariable('df_stats');\n",
              "      }\n",
              "      })();\n",
              "    </script>\n",
              "  </div>\n",
              "\n",
              "    </div>\n",
              "  </div>\n"
            ],
            "application/vnd.google.colaboratory.intrinsic+json": {
              "type": "dataframe",
              "variable_name": "df_stats",
              "summary": "{\n  \"name\": \"df_stats\",\n  \"rows\": 2,\n  \"fields\": [\n    {\n      \"column\": \"m\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 0.0,\n        \"min\": 0.6,\n        \"max\": 0.6,\n        \"num_unique_values\": 1,\n        \"samples\": [\n          0.6\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"b\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 0.0,\n        \"min\": 2.2,\n        \"max\": 2.2,\n        \"num_unique_values\": 1,\n        \"samples\": [\n          2.2\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"r^2\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 2.482534153247273e-16,\n        \"min\": 0.5999999999999998,\n        \"max\": 0.6000000000000001,\n        \"num_unique_values\": 2,\n        \"samples\": [\n          0.6000000000000001\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"r\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 0.0,\n        \"min\": 0.7745966692414834,\n        \"max\": 0.7745966692414834,\n        \"num_unique_values\": 1,\n        \"samples\": [\n          0.7745966692414834\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    }\n  ]\n}"
            }
          },
          "metadata": {},
          "execution_count": 15
        }
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 16,
      "metadata": {
        "id": "gI3uji6PZ6Qe",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 443
        },
        "outputId": "08443f7c-c85e-4eac-f97b-cde210902685"
      },
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 800x400 with 2 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {}
        }
      ],
      "source": [
        "# Plot the results\n",
        "plt.style.use('fivethirtyeight')\n",
        "plt.rcParams['font.size'] = 10\n",
        "fig = plt.figure(figsize=(8, 4), facecolor='0.85')\n",
        "plt.subplots_adjust(top=0.8)\n",
        "\n",
        "ax1 = fig.add_subplot(1, 2, 1)\n",
        "ax2 = fig.add_subplot(1, 2, 2)\n",
        "\n",
        "for ax in fig.get_axes():\n",
        "  ax.scatter(x, y, label='y')\n",
        "\n",
        "  ax1.plot(x, df_least['yhat'], color='green', label='yhat')\n",
        "  ax1.set_title('By Hand')\n",
        "  ax1.annotate(f'yhat = {round(m, 2)}x + {round(b, 2)}', xy=(1, 1), bbox=dict(boxstyle=\"round\", facecolor='0.85'))\n",
        "\n",
        "  ax2.plot(x, df_least['yhat_libr'], color='black', label='yhat_libr')\n",
        "  ax2.set_title('Using Libraries')\n",
        "  ax2.annotate(f'yhat_libr = {round(m_libr, 2)}x + {round(b_libr, 2)}', xy=(1, 1), bbox=dict(boxstyle=\"round\", facecolor='0.85'))\n",
        "\n",
        "  ax.set(xlabel='x', ylabel='y', xlim=(0), ylim=(0))\n",
        "  ax.label_outer()\n",
        "  ax.legend()\n",
        "\n",
        "fig.suptitle('Linear Regression', fontsize=plt.rcParams['font.size']*1.75)\n",
        "\n",
        "plt.show()"
      ]
    }
  ],
  "metadata": {
    "colab": {
      "provenance": [],
      "toc_visible": true,
      "include_colab_link": true
    },
    "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": "3.8.6"
    },
    "toc-autonumbering": true,
    "toc-showcode": false,
    "toc-showmarkdowntxt": false
  },
  "nbformat": 4,
  "nbformat_minor": 0
}