Lab_2_Simple_Linear_Regression.ipynb

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  "metadata": {
    "colab": {
      "name": "Lab 2: Simple Linear Regression.ipynb",
      "provenance": [],
      "collapsed_sections": [],
      "authorship_tag": "ABX9TyM8t2Zfcc1qjwe9k+tmm5/h",
      "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/mirsazzathossain/CSE317-Lab/blob/main/Lab_2_Simple_Linear_Regression.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "aye6_J7499fJ"
      },
      "source": [
        "## **Problem Statement:**\n",
        "Suppose you want to predict the price of a book from the number of pages it has. You collected some data and it is stored in a csv file named bookprice.csv\n",
        "\n",
        "The n = 10 data points (a<sub>k</sub>, b<sub>k</sub>), where k = 1, 2, ..., 10  are given as below, on the right is the scatter plot of the data:\n",
        "\n",
        "<table>\n",
        "  <tr>\n",
        "    <td colspan=\"2\" align=\"center\">Data Points</td>\n",
        "    <td align=\"center\">Scatter Plot</td>\n",
        "  </tr>\n",
        "  <tr>\n",
        "    <td align=\"center\">a</td>\n",
        "    <td align=\"center\">b</td>\n",
        "    <td rowspan=\"11\" align=\"center\">\n",
        "      <img 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\" alt=\"Scatter Plot\" />\n",
        "    </td>\n",
        "  </tr>\n",
        "  <tr>\n",
        "    <td align=\"center\">166</td>\n",
        "    <td align=\"center\">54</td>\n",
        "  </tr>\n",
        "  <tr>\n",
        "    <td align=\"center\">195</td>\n",
        "    <td align=\"center\">82</td>\n",
        "  </tr>\n",
        "  <tr>\n",
        "    <td align=\"center\">200</td>\n",
        "    <td align=\"center\">72</td>\n",
        "  </tr>\n",
        "  <tr>\n",
        "    <td align=\"center\">260</td>\n",
        "    <td align=\"center\">72</td>\n",
        "  </tr>\n",
        "  <tr>\n",
        "    <td align=\"center\">265</td>\n",
        "    <td align=\"center\">90</td>\n",
        "  </tr>\n",
        "  <tr>\n",
        "    <td align=\"center\">335</td>\n",
        "    <td align=\"center\">124</td>\n",
        "  </tr>\n",
        "  <tr>\n",
        "    <td align=\"center\">370</td>\n",
        "    <td align=\"center\">94</td>\n",
        "  </tr>\n",
        "  <tr>\n",
        "    <td align=\"center\">450</td>\n",
        "    <td align=\"center\">118</td>\n",
        "  </tr>\n",
        "  <tr>\n",
        "    <td align=\"center\">517</td>\n",
        "    <td align=\"center\">152</td>\n",
        "  </tr>\n",
        "  <tr>\n",
        "    <td align=\"center\">552</td>\n",
        "    <td align=\"center\">132</td>\n",
        "  </tr>\n",
        "</table>\n",
        "\n",
        "\n",
        "What we want is to get the least square line that fits the data.\n",
        "\n",
        "That is, we want to find the coefficients $c_1$ and $c_2$ of a linear function $f(x) = c_1 + c_2 x$ such that, mean squared error $J = \\frac{1}{n} \\left\\|f(a)-b\\right\\|^2 $ is minimized.\n",
        "\n",
        "Now we know from the lecture that the values of $c_1$ and $c_2$ for which $J$ is minimum is given by,\n",
        "$$c_1 = avg(b) - avg(a)c_2 \\text{ and } c_2 = \\rho std(b)/std(a)$$\n",
        "\n",
        "You are to use the procedure below to find and plot the least square line for the above data.\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "L_m0EBogKK3S"
      },
      "source": [
        "## **Procedure:**"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "crbxWmm56UA1"
      },
      "source": [
        "#### **1. Import necessary packages:**\n",
        "To read the data from csv file we will use a python package named **\"pandas\"**. To start working with pandas we first need to import it. We will also import other necessary packages that will be needed for this lab exercise."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "K4UgfP8N5-PL"
      },
      "source": [
        "import pandas as pd\n",
        "import numpy as np\n",
        "import matplotlib.pyplot as plt"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Mx72zbCvK-xa"
      },
      "source": [
        "#### **2. Upload and load dataset:**\n",
        "At first we have to upload the dataset to google colab to start working with it. Please download the **\"bookprice.csv\"** dataset from piazza resourse or [click here](https://drive.google.com/file/d/1CVAFMl9xaIC9IGq9Fq-B6JHq1tHuhcbI/view?usp=sharing) to download it. Then click on files form sidebar, drag and drop your file to side bar to upload the dataset.\n",
        "\n",
        "Now, use `data = pd.read_csv(\"bookprice.csv\")` to load the data.\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "_v5J2XrWA85K"
      },
      "source": [
        "# Write appropriate code here"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "mCr8VqgRNz0w"
      },
      "source": [
        "#### **3. Seperate the data:**\n",
        "You can use simple indexing (**i.e.** `data['a']`) to seperate the data into a, b. By default this will return a panda dataset. To convert it to numpy array use `to_numpy()` method."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "fSg4SNysCtDV"
      },
      "source": [
        "# Write appropriate code here"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "PABJcZetTvmG"
      },
      "source": [
        "#### **4. Plot the data:**\n",
        "Draw a scatter plot using `plt.scatter(a, b)`"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "l5jww1hNUKYj"
      },
      "source": [
        "# Write appropriate code here"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "jPL1W0X6PYSi"
      },
      "source": [
        "#### **5. Find the value of $c_1$ and $c_2$:**\n",
        "1.   First compute the demean vectors $\\tilde{a}$ and $\\tilde{b}$ \n",
        "2.   Use the demeaned vectors $\\tilde{a}$ and $\\tilde{b}$ to compute $\\rho$, $std(a)$, and $std(b)$\n",
        "3.   Use the equation given in problem statement to find the value of $c_1$ and $c_2$\n",
        "\n",
        "[**Note:** The values you should get are $c_1 = 31.2181$ and $c_2 = 0.2048$]"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "o93HQF9lRHTv"
      },
      "source": [
        "# Write appropriate code here"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "IkjPDxW3R8Cn"
      },
      "source": [
        "#### **6. Generate and fit data:**\n",
        "Use `np.linspace()` method to generate some data (page numbers) and plot the least square fit line (price) on the scatter plot. The graph of the fitted line should look like the following figure:\n",
        "\n",
        "![scatter.png](data:image/png;base64,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)\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "YKj1EapcDXwB"
      },
      "source": [
        "# Write appropriate code here"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "gVefi7w5aDzn"
      },
      "source": [
        "#### **7. Using numpy function: polyfit():**\n",
        "Numpy has its own polynomial fit function called polyfit. Use `np.polyfit(a, b, 1)` to get the coffecients (make sure you understand the order of the coefficients produced).\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "iXa1d62gagjY"
      },
      "source": [
        "# Write appropriate code here"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "x9UPrCd8bCW9"
      },
      "source": [
        "#### **8. Plot the quadratic polynomial:**\n",
        "The Last argument in the `np.polyfit()` function should be 1 if you want a fit a linear function. It can be some other number which will indicate the degree of the polynomial you want to fit. Plot the quadratic polynomial that fits the data.\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "3KiNXeFobvkm"
      },
      "source": [
        "# Write appropriate code here"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "LGGMaVMVcDAf"
      },
      "source": [
        "#### **9. Goodness of fit: $r^2$ statistic:**\n",
        "A commonly used statistic that is related to the error, but is not the same as the error is the goodness-of-fit $r^2$ (r-squared) statistic. The $r^2$ statistic ranges from a value of $0$ for absolutely no relation between\n",
        "the data and the line to a value of $1$ which occurs only if all of the data fall exactly on the line, **i.e.**, no error. In some engineering disciplines, an equation fitted to data is acceptable only if $r^2 > 0.9$. Other\n",
        "engineering disciplines might find an $r^2$ low as $0.7$ acceptable for use.\n",
        "\n",
        "The $r^2$ statistic is computed as follows:\n",
        "\n",
        "\n",
        "$r^2 = 1 - SSE / SST$, where $SSE$ is the sum of squared errors: $(b - \\text{b_pred})^T(b - \\text{b_pred})$ and $SST = \\text{Total sum of squares : } (b - avg(b))^T(b - avg(b))$\n",
        "\n",
        "\n",
        "Find the **SSE** and **SST** values for the given data and the value of $r^2$ statistic.\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "33729c-deQGW"
      },
      "source": [
        "# Write appropriate code here"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "2zLJdD4KeJD8"
      },
      "source": [
        "#### **10. Displaying the equation on the graph:**\n",
        "1.   Create a string that contains the equation. You can insert a number to a string and formate as follow: `eqn = 'y = %.2f + %.2fx' %(c_1, c_2)`\n",
        "2.   Use `plt.text(Xcor, Ycor, 'string'`) to display the equation on the graph.\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "EbzZQ8Pkfq7p"
      },
      "source": [
        "# Write appropriate code here"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "BL0H8sGCl57_"
      },
      "source": [
        "## **Exercise:**\n",
        "A healthy child’s systolic blood pressure p (in millimeters of mercury) and weight w (in pounds) are approximately related by the equation,\n",
        "\n",
        "$$p = \\beta_0 + \\beta_1 \\log(w)$$\n",
        "\n",
        "\n",
        "Use the following experimental data to estimate the systolic blood pressure of a healthy child weighing 100 pounds.\n",
        "\n",
        "<table>\n",
        "  <tr>\n",
        "    <td>w</td>\n",
        "    <td>44</td>\n",
        "    <td>61</td>\n",
        "    <td>81</td>\n",
        "    <td>113</td>\n",
        "    <td>131</td>\n",
        "  </tr>\n",
        "\n",
        "  <tr>\n",
        "    <td>p</td>\n",
        "    <td>91</td>\n",
        "    <td>98</td>\n",
        "    <td>103</td>\n",
        "    <td>110</td>\n",
        "    <td>112</td>\n",
        "  </tr>\n",
        "</table>"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "ahlLlK3lnEFt"
      },
      "source": [
        "# Write appropriate code here"
      ],
      "execution_count": null,
      "outputs": []
    }
  ]
}