arghyadeep99/linear-regression.ipynb

## linear-regression.ipynb
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  "nbformat_minor": 0,
  "metadata": {
    "colab": {
      "name": "Linear Regression.ipynb",
      "version": "0.3.2",
      "provenance": [],
      "collapsed_sections": [],
      "include_colab_link": true
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
      },
      "source": [
        "<a href=\"https://colab.research.google.com/gist/arghyadeep99/b3ce6d5ee4e0f7e3b5a60f7ffc613aaf/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": "_q9XYSs79_5L",
        "colab_type": "text"
      },
      "source": [
        "# Linear Regression\n",
        "Regression is a statistical way to establish a relationship between a dependent variable and a set of independent variable(s). Linear regression is a technique where we try to fit a linear equation on a graph plot such that the line passes through the data points in a **general** manner. "
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "dlYY6s4o-p7I",
        "colab_type": "text"
      },
      "source": [
        "<figure>\n",
        "<center>\n",
        "<img src='https://drive.google.com/uc?id=1Gh_oGD9OsZK_jq_R1WNcb8qmPVkybyFY'/>\n",
        "  </center>\n",
        "  <center><figcaption><b>A simple example of Linear Regression</b></figcaption></center>\n",
        "</figure>\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "PmrbNBPSBrSF",
        "colab_type": "text"
      },
      "source": [
        "Still confused? Ok well, let's see some examples. I give you some coordinates:\n",
        "\n",
        "**(1,2),(2,4),(4,8),(5,10).**\n",
        "\n",
        "These are pairs of X-Y coordinates. Now, if I say X=10, what is Y? Obviously, Y=20.\n",
        "\n",
        "How could you answer it? Your mind found a pattern in this data. Then your mind tried to fit an equation to relate this pattern. This is how our brain works. Your mind was spontaneously quick in fitting the curve **Y=2X** in this case. \n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Yl5EiOz1FgxF",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "%matplotlib inline\n",
        "import matplotlib.pyplot as plt\n",
        "import seaborn as sns; sns.set()\n",
        "import numpy as np\n",
        "import statsmodels.api as sm\n",
        "import pandas as pd\n",
        "from sklearn.linear_model import LinearRegression"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "1HGEgVMUsKC2",
        "colab_type": "code",
        "outputId": "a8c782e3-67dd-4983-fb35-8ece22de8adc",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 301
        }
      },
      "source": [
        "X,Y=[1,2,3,4,5,6,7,8],[2,4,6,8,10,12,14,16]\n",
        "plt.plot(X,Y,label='linear', color='green')\n",
        "plt.ylabel('Y')\n",
        "plt.xlabel('X')\n",
        "plt.scatter(X,Y)\n",
        "plt.title('Graph for y=2x')\n",
        "plt.legend()\n",
        "plt.show()"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
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gaivvN2pmWRZERGKoOVznYdcBn/35SwyrMVwnZywLIiIjaMxwnRyxLIiImqGwohCLjt8b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\nSIRg7QAAAABJRU5ErkJggg==\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "a4xVlZmNF3oA",
        "colab_type": "text"
      },
      "source": [
        "Let's take one more example:\n",
        "\n",
        "**(1,1),(2,4),(5,25),(8,64).**\n",
        "\n",
        "If X=30, what is Y? It's 900. \n",
        "\n",
        "Again, your mind was quick to fit the equation **Y=X<sup>2</sup>**. "
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "WO7Em4SuGBJ6",
        "colab_type": "code",
        "outputId": "44900ba2-756c-4145-8863-c63579844ae9",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 301
        }
      },
      "source": [
        "X,Y=[1,2,3,4,5,6,7,8],[1,4,9,16,25,36,49,64]\n",
        "plt.plot(X,Y,label='polynomial', color='orange')\n",
        "plt.ylabel('Y')\n",
        "plt.xlabel('X')\n",
        "plt.scatter(X,Y)\n",
        "plt.title('Graph for y=x^2')\n",
        "plt.legend()\n",
        "plt.show()"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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c0tLSSE5OBiA5OZm0tDRyc3NdFUukeTIcBOybRMiOu3EEdCTviv+oKOScXDrd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\nQmCC\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "LvdXh6qnIVx8",
        "colab_type": "code",
        "outputId": "c7a31c4f-ecbe-4db7-b00d-f8c4296b4ef5",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 291
        }
      },
      "source": [
        "#Now we plot both of them together\n",
        "\n",
        "plt.plot([1,2,3,4,5,6,7,8],[2,4,6,8,10,12,14,16],label='linear', color='green')\n",
        "plt.plot([1,2,3,4,5,6,7,8],[1,4,9,16,25,36,49,64],label='polynomial')\n",
        "plt.ylabel('Y')\n",
        "plt.xlabel('X')\n",
        "plt.legend()\n",
        "plt.show()"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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4hOyiGhL6+3HX+BiZEis0QcpCCCtobjGzfnsmG3dm4+7ai0duHsSI/v5qxxLi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\nLIQQQrRLykIIIUS7pCyEEEK06/8DXY+ce7QWfTUAAAAASUVORK5CYII=\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Kk0uu6PPrvRi",
        "colab_type": "text"
      },
      "source": [
        "We can take some random data plotted on a graph and then try to fit a linear equation that passes through it."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "OxFbZkfZr2e6",
        "colab_type": "code",
        "outputId": "5cdfe93a-3456-4cbd-a821-277665699f29",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 274
        }
      },
      "source": [
        "rng = np.random.RandomState(1)\n",
        "x = 10 * rng.rand(50)\n",
        "y = 2 * x - 8 + rng.randn(50)\n",
        "plt.scatter(x, y);"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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xvdN3GwAofJYXkB988EHdd999VjdTsMzYNTz9boOqihLdc9tnqBcASBuriRyWatdwJpP5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\nRDAAAIhgAAAQwQAAIIIBAEDS/wFdJyjKQoE53AAAAABJRU5ErkJggg==\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "mIPI-mfNr_OC",
        "colab_type": "code",
        "outputId": "da8cd224-d23b-4f02-f549-f8b5c9488feb",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 274
        }
      },
      "source": [
        "model = LinearRegression(fit_intercept=True)\n",
        "\n",
        "model.fit(x[:, np.newaxis], y)\n",
        "# What is np.newaxis? Stack Overflow: https://stackoverflow.com/questions/29241056/how-does-numpy-newaxis-work-and-when-to-use-it\n",
        "xfit = np.linspace(0, 10, 1000)\n",
        "yfit = model.predict(xfit[:, np.newaxis])\n",
        "\n",
        "plt.scatter(x, y)\n",
        "plt.plot(xfit, yfit);"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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3MQYBMBiAZxePN1q5c0IZEdmDTWvm7du32/L0DmWgSruorAE7Mwtxu7alzzH9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\nTkSuQmCC\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "B2zsaZnJEI__",
        "colab_type": "text"
      },
      "source": [
        "This is the intuition behind regression. First example was the case for **linear regression**, whereas second example was a case of **polynomial regression**.\n",
        "\n",
        "\n",
        "\n",
        "This is one of the algorithms we try to replicate in machine learning. As we already know by now, machine learning, in layman words, is a way in which we show the machine some outputs and then ask the machine to predict what the input should be, so that the desired output is yielded.\n",
        "\n",
        "\n",
        "\n",
        "\n",
        "Using this **most suitable line** predicted by the machine, it will try to fit or predict input for new/future outputs.\n",
        "\n",
        "\n",
        "\n",
        "\n",
        "In the examples seen above, Y is the *dependent variable* whereas X is the *independent variable*. "
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "jZMcFfPsJOrY",
        "colab_type": "text"
      },
      "source": [
        "We all are aware about the equation of a line:\n",
        "\n",
        "##<center>*y=mx+c*</center>\n",
        "where \n",
        "m=slope, c=intercept\n",
        "\n",
        "In machine learning, we need to adjust these **m and c** to fit the right line. These are known as **_hyper-parameters_**. \n",
        "\n",
        "Tweaking these values will help us predict the generalized line that will pass through **most** of the data and understand a general pattern.\n",
        "\n",
        "<figure>\n",
        "<center>\n",
        "<img src='https://drive.google.com/uc?id=1iAEFeyjWk15I0khdcI1xkT5To75e9QU7'/>\n",
        "  <figcaption><b>y=mx+c</b></figcaption></center>\n",
        "</figure>\n",
        "\n",
        "### Let us take one more example. If you want to find a corelation between a human's weight and height. \n",
        "\n",
        "We can see that the data points are being plotted. Now, the linear regression algorithm will try to fit a straight line that passes through the data points in such a way that the  best line is fit.\n",
        "<figure>\n",
        "<center>\n",
        "<img src='https://drive.google.com/uc?id=1iGh6YwVp4eHuPAryaX4nfZtQtOmLQug-'/>\n",
        "  <figcaption><b>Finding relation between body weight and height using linear regression</b></figcaption></center>\n",
        "</figure>"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "0AVM09dhL813",
        "colab_type": "text"
      },
      "source": [
        "But how do you define best fit? What are the parameters on the basis of which a line is defined as **best-fit**?\n",
        "\n",
        "We see the difference between actual and predicted. At each iteration, we compute the sum of \"error\", i.e. the difference of the predicted value and actual value. and then try to minimise that sum. This is the working of linear regression.\n",
        "\n",
        "_The best fit line is the one with such values of 'm' and 'c' such that it produces least error for all pairs (X,Y)._\n",
        "\n",
        "<figure>\n",
        "<center>\n",
        "<img src='https://drive.google.com/uc?id=1jQfqrkIaZ2jdColAogj-18YkiHSh1KZd'/>\n",
        "  <figcaption><b></b></figcaption></center>\n",
        "</figure>\n",
        "\n",
        "But the question is, for how many values of (m,c) can we check in an infinite space? The answer to that is Gradient Descent, something we will learn in detail in next few hours. I will just leave a simple formula to get your \"neurons\" activated. \n",
        "\n",
        "In this gradient descent method, we will try to predict values of 'm' and 'c'. We will compute \"loss\" function that will tell us how far we are from the destination/minima and then accordingly update our 'm' and 'c' by adjusting it with a derivative term.\n",
        "\n",
        "Let's see our loss function first. Here, we will use \"mean squared error function\".\n",
        "\n",
        "\n",
        "<figure>\n",
        "<center>\n",
        "<img src='https://drive.google.com/uc?id=12_-islxcYpkrd8rz1Lhq4p1kHNGLjmq4'/>\n",
        "  <figcaption><b></b></figcaption></center>\n",
        "</figure>\n",
        "\n",
        "Our goal is to \"minimise\" this loss function so that we are as close as possible to our destination value. We know that to minimise a function, we can use differentiation. Let's see how that works.\n",
        "\n",
        "\n",
        "<figure>\n",
        "<center>\n",
        "<img src='https://drive.google.com/uc?id=18MvnwqpW1wIu3MxeUDWGwqhIP77YgUDe'/>\n",
        "  <figcaption><b></b></figcaption></center>\n",
        "</figure>\n",
        "\n",
        "So, now let's see how our derivative will look like wrt m and c.\n",
        "\n",
        "<figure>\n",
        "<center>\n",
        "<img src='https://drive.google.com/uc?id=1kXHifeCE9FZS-a53qaVq17JrFywcbQSt'/>\n",
        "  <figcaption><b></b></figcaption></center>\n",
        "</figure>\n",
        "\n",
        "\n",
        "<figure>\n",
        "<center>\n",
        "<img src='https://drive.google.com/uc?id=1lihxJkQsR1bbKdk6xL3zwoflctMd7knz'/>\n",
        "  <figcaption><b></b></figcaption></center>\n",
        "</figure>\n",
        "\n",
        "\n",
        "So, now we need to update our existing 'm' and 'c' values in this way:\n",
        "\n",
        "m(t+1) = f(m(t), error(t))\n",
        "\n",
        "\n",
        "c(t+1) = f(c(t), error(t))\n",
        "\n",
        "\n",
        "<figure>\n",
        "<center>\n",
        "<img src='https://drive.google.com/uc?id=1o4An1V6RpC6O0cLQIbOGfYfqjbF-qsfF'/>\n",
        "  <figcaption><b></b></figcaption></center>\n",
        "</figure>\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "r2k_mq1NsMYr",
        "colab_type": "code",
        "outputId": "13710d51-d45b-44a7-8354-4ff8428fad65",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 1000
        }
      },
      "source": [
        "m = 0\n",
        "c = 0\n",
        "\n",
        "L = 0.001  # The learning Rate\n",
        "epochs = 1000  # The number of iterations to perform gradient descent\n",
        "\n",
        "n = float(len(X)) # Number of elements in X\n",
        "# Performing Gradient Descent \n",
        "for i in range(epochs): \n",
        "    Y_pred = m*x + c  # The current predicted value of Y\n",
        "    D_m = (-2/n) * sum(x * (y - Y_pred))  # Derivative wrt m\n",
        "    D_c = (-2/n) * sum(y - Y_pred)  # Derivative wrt c\n",
        "    m = m - L * D_m  # Update m\n",
        "    c = c - L * D_c  # Update c\n",
        "    if i%100==0:\n",
        "      print(\"Epoch {}\".format(i))\n",
        "      plt.scatter(x,y) \n",
        "      plt.plot([min(x), max(x)], [min(Y_pred), max(Y_pred)], color='red')  # regression line\n",
        "      plt.show()\n",
        "    \n",
        "print (m, c)\n"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAAYMAAAEBCAYAAACaHMnBAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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yr29evHgxzp075/Znhw4dQm1tLVJSUqDRaFBSUoJXX30Vu3btglar9bnDkean\nL+xATX1Lp9dTkmLxzq9nt/157ydnsG5rBWrrW9AnKRZ3zUvD1PTBwewqEUUwv8LAWxMmTEBxcTEG\nDhzo1fvM5iY4nd53MxyGS9xtGNPrNLh73ui2YZ+e/M6VwuG6Ay0arxmIzuuOxmsGvLtujUaAydTb\nq89XdM7g/Pnz6NevHwDggw8+gEajaftzpPJ2ZVBPDrYP1LwCEZEniobBE088AbPZDEEQ0Lt3b7zx\nxhvQ6SJ3zrqnK4PcBcbLD97q8XM5r0BESlP0zvzXv/5VyY8POT35Bu/LUlJTgsHtjZ+byogoULgD\nOYB68g3elwNolmQMh17X/q+Km8qIKJAid8wmSK4c8vHkym/wvgz59GRegYjIHwwDP3gqHX2ljt/g\nfR3yuXJTGRFRoDEM/OCpdHQrjYBOyz+7qiPkb40iIiJfMQz80N1qHqeMtrmA1pu6pyEfAH7XKCIi\n8hXDAL5XDfU05HMldzd1d0M+j689yL0ERKSaqF9N1Dru33pTb715Hz5e3e173a3ycae71UKt7Xrz\nOhFRIEV9GPiy1LPVxLH9cfe80W2TvxrB8+92d1PnATVEpKaoHyby9xt5xyGfx9ce9Gm1EA+oISI1\nRf2TQaC/kfu6QazjU4YpweCxEB0RUaBF/ZNBoL+R+7NBjHsJiEgtUR8GSuzu5U2diMJN1IcBwJs3\nEVHUhgF3+xIRXRaVYeBLGWkiokgWlauJ/NlbQEQUiaIyDLjbl4iovagMA+72JSJqLyrDgCeHERG1\nF5UTyDw5jIiovagMA4B7C4iIrhSVw0RERNQew4CIiBgGRETEMCAiIjAMiIgIEbqaqLUIXV2DiGQu\nGyUi6lbEhQGL0BEReS/iholYhI6IyHsRFwYsQkdE5L2IGyYyJRjc3vhNCQYeaENE5EHEPRl4KkJ3\nw3AT/rb1i7agaJ1LOHy8Wo1uEhGFlIgLg4lj++PueaNhSjBAgOuJ4O55o/FZpZlzCUREHkTcMBFw\nuQhdSko8amoaAQBvbSp3+7ucSyAiisAnA094oA0RkWdREwY80IaIyLOIHCZyhwfaEBF5FjVhAPBA\nGyIiT/weJtqwYQPmz5+PMWPG4L333mv3s5aWFqxcuRKzZs3C3LlzUVZW5m9zRESkAL+fDNLS0vDH\nP/4Rb775Zqefvf322+jduzd27tyJ06dP44477sCOHTsQFxfnb7NERBRAfj8ZjBw5EiNGjIBG0/mj\ntm7dipycHADA0KFDcd1112H//v3+NklERAGm6Gqic+fOYeDAgW1/Tk1NRXU1d/wSEYWaboeJFi9e\njHPnzrn92aFDh6DVagPeqY6o3Rf0AAAEr0lEQVRMpt4+vzclJT6APQkf0Xjd0XjNQHRedzReM6Ds\ndXcbBu+//77PHz5gwACcPXsWycnJAICqqipMmDDB588jIiJlKDpMNHfuXOTn5wMATp8+jc8//xyT\nJ09WskkiIvKBIMuy7M8HlJaW4ne/+x0aGhoQExOD2NhYvPPOOxgxYgSam5uRm5uLiooKaDQaPP74\n45g5c2ag+k5ERAHidxgQEVH4i5raRERE5BnDgIiIGAZERMQwICIiMAyIiAgMAyIiQgSHwalTp5CT\nk4M5c+YgJycHp0+fVrtLiqqvr8d9992HOXPmYP78+XjooYdQV1endreC5rXXXsOoUaNw8uRJtbsS\nFKIoYs2aNZg9ezbmz5+Pp59+Wu0uKa6srAyLFi3CwoULsWDBAuzYsUPtLikiLy8P06dP7/TvWfF7\nmhyhVqxYIZeUlMiyLMslJSXyihUrVO6Rsurr6+UPP/yw7c8vvfSS/OSTT6rYo+A5duyYfO+998rT\npk2TT5w4oXZ3guL555+XX3zxRdnpdMqyLMs1NTUq90hZTqdTHj9+fNvfb0VFhTxu3DhZkiSVexZ4\nR44ckc+dO9fp37PS97SIfDIwm80oLy9HZmYmACAzMxPl5eUR/U05MTGxXd2ncePGeSwwGElsNhue\ne+45PPPMM2p3JWgsFgtKSkrw6KOPQhAEAECfPn1U7pXyNBoNGhsbAQCNjY3o27ev29L54W78+PFI\nTU1t91ow7mkReexlVVUV+vXr11ZRVavVom/fvqiqqmormhfJnE4n/vGPf2D69Olqd0Vxr776KhYs\nWIBBgwap3ZWgOXPmDBITE/Haa6/ho48+QlxcHB599FGMHz9e7a4pRhAEvPLKK3jwwQfRq1cvWCwW\ntwdqRapg3NMiL1YJzz//PHr16oU777xT7a4o6ujRozh27BiWL1+udleCSpIknDlzBmPGjEFxcTFW\nrVqFhx9+GE1NTWp3TTEOhwN/+ctfsHbtWpSVleGNN97AypUrYbFY1O5axIjIMEhNTcX58+chSRIA\n1/88Fy5c6PToFYny8vLwzTff4JVXXonIR+grHTlyBJWVlZgxYwamT5+O6upq3HvvvThw4IDaXVNU\namoqdDpd25DBjTfeiKSkJJw6dUrlnimnoqICFy5cQHp6OgAgPT0dsbGxqKysVLlnwRGMe1pE3i1M\nJhPS0tJQWloKwFVZNS0tLeKHiP7whz/g2LFjeP3116HX69XujuLuv/9+HDhwAHv27MGePXvQv39/\nvP3225g0aZLaXVNUcnIyJkyYgIMHDwJwrTIxm824+uqrVe6Zcvr374/q6mp8/fXXAIDKykqYzWYM\nGTJE5Z4FRzDuaRFbtbSyshK5ubloaGhAQkIC8vLyMGzYMLW7pZgvv/wSmZmZGDp0KIxGIwBg0KBB\neP3111XuWfBMnz4df/7znzFy5Ei1u6K4M2fO4KmnnsLFixeh0+mwcuVKZGRkqN0tRW3cuBFvvfVW\n26T5I488EpEl8V944QXs2LEDtbW1SEpKQmJiIjZv3qz4PS1iw4CIiHouIoeJiIjIOwwDIiJiGBAR\nEcOAiIjAMCAiIjAMiIgIDAMiIgLDgIiIAPx/8pYyG8Ye3AAAAAAASUVORK5CYII=\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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Qfv0VovJyoPtoHaBSwTJmHEyZM1HX/69y986v5NyigohCW/iGgSgCq1cj7oUX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\nrkJggg==\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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KFy+j5PIeoS6ZiEgFCoMAMOXnEbf0eWzzZ2G22ynudRUFYyZQeln3gN97x+4s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\nERQGIiIC/H8AeEOx1UvebQAAAABJRU5ErkJggg==\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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f/IJDQQDY3m8g/lCyM5pMRF6MTwYuJr1101IrsHM7BJUaun9PQcGbb0MIDKr2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\niIgIDAMiIgLDgIiIAPx/I6+ZE1Ar4tAAAAAASUVORK5CYII=\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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wZss2DuWXasqniHhMYdCUORxErFqJ9fHJWA7mUnb9DdhT0nC2O9vl2+sqUaFA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\nSsQAAAAASUVORK5CYII=\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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DxNxUTAX5FPz9n+T/698YzaKrXaJCgSAi1VEYNBWGQcj6tVgnTyDop3SKrr4W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\ncXSF16qlAAAAAABJRU5ErkJggg==\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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qbfzZU+WcKUTmkTzUFJfg8W8/wvCTewGlEhVLXkPt5CcAueH/5eZsbG9sUZk1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\nAABJRU5ErkJggg==\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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gW7OGVQJqDf+0VPKRG4nEACAiV2AY2EGVkQ7/+W9j1o7ttUpCVRkkY2kHuFXq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\ngIiIwDAgIiIA/w8S0ll0EKk2GwAAAABJRU5ErkJggg==\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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fdb2WAnvlt2sDORGpDYVBE2L5+Sea/X0YzUbfhBEZRUbSp+Q89yKbj5awLHkv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\nAABJRU5ErkJggg==\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "stream",
          "text": [
            "1.995761650430102 -7.790938274543669\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "CxpX9ucOtzck",
        "colab_type": "text"
      },
      "source": [
        "Let's see the gradient descent in action, how it fits the line.\n",
        "\n",
        "<figure>\n",
        "<center>\n",
        "<img src='https://drive.google.com/uc?id=1kmdiIzpzXHmbp9UCJX6e3_BqIrqKxy5L'/>\n",
        "  <figcaption><b>Simulation of Linear Regression using Gradient Descent</b></figcaption></center>\n",
        "</figure>"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "DwmCCfnZTfLi",
        "colab_type": "text"
      },
      "source": [
        "\n",
        "But for the sake of linear regression, we have mathematical formulas that help determine the best fit line which you guys will learn in 4th semester/already know. \n",
        "\n",
        "\n",
        "Note that we are only dealing with X and Y. If we deal with more independent variables, then we deal with something known as **multi linear regression**.\n",
        "\n",
        "\n",
        "<figure>\n",
        "<center>\n",
        "<img src='https://drive.google.com/uc?id=16c8UEoUNQH4xk7SrGOHn-hXW0o7xasCG'/>\n",
        "  <figcaption><b></b></figcaption></center>\n",
        "</figure>"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ONKN_Pz5Wq6L",
        "colab_type": "text"
      },
      "source": [
        "In real life, we deal with cases where a variable depends on more than one independent variables. This is known as **multi variate linear regression**.\n",
        "\n",
        "\n",
        "\n",
        "Now let's code an example on multi linear regression using a library in Python called **[sklearn](https://scikit-learn.org/stable/)**. \n",
        "\n",
        "\n",
        "\n",
        "In this section, we will discuss the basic of using multi variate linear regression on the diabetes dataset."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "6FhqphIAW0Li",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "#Importing the required libraries\n",
        "\n",
        "import matplotlib.pylab as plt\n",
        "import numpy as np\n",
        "%matplotlib inline\n",
        "from sklearn import datasets"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "OvfthRv3XRif",
        "colab_type": "text"
      },
      "source": [
        "Ten baseline variables, age, sex, body mass index, average blood pressure, and six blood serum measurements were obtained for each of n = 442 diabetes patients, as well as the response of interest, a quantitative measure of disease progression one year after baseline."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "n52PmNWpXTyr",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "diabetes=datasets.load_diabetes() #loading the diabetes dataset into a variable"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "MkOlS8ZjXfLn",
        "colab_type": "code",
        "outputId": "fca65217-8c3b-4af2-9b31-3a2af8b6cacc",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 734
        }
      },
      "source": [
        "print(diabetes.data.shape) # feature shape\n",
        "# 442 patients, 10 variables\n",
        "print(diabetes.target.shape)\n",
        "print(diabetes.feature_names) \n",
        "#s1....s6 indicate 6 blood serum measurements\n",
        "print(diabetes.DESCR)"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "(442, 10)\n",
            "(442,)\n",
            "['age', 'sex', 'bmi', 'bp', 's1', 's2', 's3', 's4', 's5', 's6']\n",
            ".. _diabetes_dataset:\n",
            "\n",
            "Diabetes dataset\n",
            "----------------\n",
            "\n",
            "Ten baseline variables, age, sex, body mass index, average blood\n",
            "pressure, and six blood serum measurements were obtained for each of n =\n",
            "442 diabetes patients, as well as the response of interest, a\n",
            "quantitative measure of disease progression one year after baseline.\n",
            "\n",
            "**Data Set Characteristics:**\n",
            "\n",
            "  :Number of Instances: 442\n",
            "\n",
            "  :Number of Attributes: First 10 columns are numeric predictive values\n",
            "\n",
            "  :Target: Column 11 is a quantitative measure of disease progression one year after baseline\n",
            "\n",
            "  :Attribute Information:\n",
            "      - Age\n",
            "      - Sex\n",
            "      - Body mass index\n",
            "      - Average blood pressure\n",
            "      - S1\n",
            "      - S2\n",
            "      - S3\n",
            "      - S4\n",
            "      - S5\n",
            "      - S6\n",
            "\n",
            "Note: Each of these 10 feature variables have been mean centered and scaled by the standard deviation times `n_samples` (i.e. the sum of squares of each column totals 1).\n",
            "\n",
            "Source URL:\n",
            "https://www4.stat.ncsu.edu/~boos/var.select/diabetes.html\n",
            "\n",
            "For more information see:\n",
            "Bradley Efron, Trevor Hastie, Iain Johnstone and Robert Tibshirani (2004) \"Least Angle Regression,\" Annals of Statistics (with discussion), 407-499.\n",
            "(https://web.stanford.edu/~hastie/Papers/LARS/LeastAngle_2002.pdf)\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "4zbW30u7YCqc",
        "colab_type": "text"
      },
      "source": [
        "Linear Regression assumes the following model:\n",
        "\n",
        "## <center>y=Xβ+c+ϵ </center>\n",
        "\n",
        "X: data \n",
        "\n",
        "β: coefficients \n",
        "\n",
        "c: intercept \n",
        "\n",
        "ϵ: error, cannot be explained by model \n",
        "\n",
        "y: target "
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "7N34uQUGYVp6",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "# Sperate train and test data\n",
        "from sklearn.model_selection import train_test_split\n",
        "X_train, X_test, y_train, y_test = train_test_split(diabetes.data, diabetes.target, test_size=0.2) #80-20 split"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "F0oBmEymYbmG",
        "colab_type": "code",
        "outputId": "d0609d45-31a1-4350-c923-785f4195589a",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 34
        }
      },
      "source": [
        "model = LinearRegression()\n",
        "model.fit(X_train,y_train)\n",
        "model.score(X_test, y_test)"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.6194766672779816"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 65
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "ebE8KtFtZQSq",
        "colab_type": "code",
        "outputId": "a04ed9c0-23c8-43e6-bb22-54dee7589389",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 68
        }
      },
      "source": [
        "model.coef_"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "array([  18.88242783, -242.86752218,  486.33856538,  331.66616316,\n",
              "       -790.72746354,  454.32462054,  112.04669869,  244.10860833,\n",
              "        669.21323554,   67.06270312])"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 66
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "vEIjWCxGbAef",
        "colab_type": "code",
        "outputId": "cb2a30ca-c5f6-4371-c2bf-f2d4a0864ab3",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 34
        }
      },
      "source": [
        "model.intercept_"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "151.759734141401"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 67
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "t_CCpWZ9bEFs",
        "colab_type": "code",
        "outputId": "2ea4d863-b720-4343-f3d0-d358b30d8005",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 408
        }
      },
      "source": [
        "model.predict(X_test)"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "array([ 63.99614541,  60.10110903, 201.61144192, 158.12329435,\n",
              "       105.0981974 , 158.89135164, 163.73279561, 194.92919673,\n",
              "       103.39792751, 149.70960855, 143.95513722, 110.88535919,\n",
              "        82.40501143,  87.55418625, 109.59371078, 215.85278811,\n",
              "       126.88095303, 235.06655432, 126.19576577,  70.53619534,\n",
              "       118.90650439, 161.42298075, 277.10570878, 178.24602755,\n",
              "        75.62071255, 121.86716878, 140.7220144 , 210.99699123,\n",
              "       190.21668417,  79.34342518, 184.09781062, 282.15206582,\n",
              "       159.50903361, 182.8411674 , 159.11714049, 204.50286345,\n",
              "       226.14604274, 126.53151102, 145.36084203, 253.70822103,\n",
              "       180.34138931, 107.25084526, 136.23887777, 197.87279907,\n",
              "       187.13417899, 157.7009701 ,  83.14552259,  54.25821638,\n",
              "       158.12689357, 162.76145062, 258.0576136 , 148.40268022,\n",
              "        61.0175373 , 139.30966036, 102.43282008, 121.34445951,\n",
              "       114.59742667, 225.81191799,  53.41036952, 260.94062338,\n",
              "       118.15344287, 232.11656066, 151.64154348, 191.70009751,\n",
              "       158.57203375,  60.37222056,  72.94031334, 126.41058344,\n",
              "       128.16599158, 159.71663481, 108.97416245,  83.68191227,\n",
              "       176.99837151,  78.24173047, 126.8780034 , 222.26557288,\n",
              "       117.50825903, 225.61527101, 127.99338295, 109.47925339,\n",
              "       106.08661978, 233.13840841, 148.12401299,  88.09698485,\n",
              "       195.03721801, 122.05286845, 195.52544339, 201.40366029,\n",
              "        97.82403515])"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 68
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "h4NsyfvnFbsW",
        "colab_type": "code",
        "outputId": "bc75f4c5-2865-4b63-d40d-ce438922e732",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 595
        }
      },
      "source": [
        "X_train = sm.add_constant(X_train) # adding a constant\n",
        " \n",
        "model = sm.OLS(y_train, X_train).fit()\n",
        "predictions = model.predict(X_train) \n",
        " \n",
        "print_model = model.summary()\n",
        "print(print_model)"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "                            OLS Regression Results                            \n",
            "==============================================================================\n",
            "Dep. Variable:                      y   R-squared:                       0.483\n",
            "Model:                            OLS   Adj. R-squared:                  0.468\n",
            "Method:                 Least Squares   F-statistic:                     31.98\n",
            "Date:                Wed, 04 Sep 2019   Prob (F-statistic):           2.00e-43\n",
            "Time:                        14:53:16   Log-Likelihood:                -1909.7\n",
            "No. Observations:                 353   AIC:                             3841.\n",
            "Df Residuals:                     342   BIC:                             3884.\n",
            "Df Model:                          10                                         \n",
            "Covariance Type:            nonrobust                                         \n",
            "==============================================================================\n",
            "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
            "------------------------------------------------------------------------------\n",
            "const        151.7597      2.932     51.752      0.000     145.992     157.528\n",
            "x1            18.8824     67.701      0.279      0.780    -114.281     152.046\n",
            "x2          -242.8675     71.183     -3.412      0.001    -382.879    -102.856\n",
            "x3           486.3386     75.011      6.484      0.000     338.798     633.879\n",
            "x4           331.6662     75.910      4.369      0.000     182.358     480.974\n",
            "x5          -790.7275    475.582     -1.663      0.097   -1726.161     144.706\n",
            "x6           454.3246    394.953      1.150      0.251    -322.518    1231.168\n",
            "x7           112.0467    234.966      0.477      0.634    -350.114     574.207\n",
            "x8           244.1086    180.298      1.354      0.177    -110.523     598.740\n",
            "x9           669.2132    199.906      3.348      0.001     276.012    1062.414\n",
            "x10           67.0627     74.094      0.905      0.366     -78.674     212.799\n",
            "==============================================================================\n",
            "Omnibus:                        1.029   Durbin-Watson:                   1.933\n",
            "Prob(Omnibus):                  0.598   Jarque-Bera (JB):                1.065\n",
            "Skew:                           0.044   Prob(JB):                        0.587\n",
            "Kurtosis:                       2.745   Cond. No.                         229.\n",
            "==============================================================================\n",
            "\n",
            "Warnings:\n",
            "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "iBkV-FxQS1TY",
        "colab_type": "text"
      },
      "source": [
        "References:\n",
        "\n",
        "1. https://towardsdatascience.com/linear-regression-using-gradient-descent-97a6c8700931\n",
        "2. My Udacity PyTorch Nanodegree Program."
      ]
    }
  ]
}