IANS_LINEAR_REGRESSION.ipynb

{
  "nbformat": 4,
  "nbformat_minor": 0,
  "metadata": {
    "colab": {
      "name": "IANS_LINEAR_REGRESSION.ipynb",
      "provenance": [],
      "authorship_tag": "ABX9TyNtc0/yWwpYqaaET4JSzlRX",
      "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/rgantonio/danknight_webnotes/blob/main/IANS_LINEAR_REGRESSION.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "# In a Nutshell: Linear Regression: \n",
        "\n"
      ],
      "metadata": {
        "id": "oUHxOrAVZANX"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Linear regression is nothing more than a curve fitting model. Simple linear regression assumes that a data set can be modelled using a simple line. That means for a given data set, we estimate the trend to be in the form of:\n",
        "\n",
        "$$ y =\\beta_0 + \\beta_1 x $$\n",
        "\n",
        "Let's take a simple example. First let's import some simple packages."
      ],
      "metadata": {
        "id": "MYuGGijxPSFV"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "####################################\n",
        "# Importing packages for simulation\n",
        "####################################\n",
        "import numpy as np\n",
        "import pandas as pd\n",
        "import matplotlib.pyplot as plt\n",
        "import seaborn as sns"
      ],
      "metadata": {
        "id": "gCOpq6v4qofJ"
      },
      "execution_count": 2,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "Let's say we have an array for different y values and x values. Our goal is to determine the coefficients $\\beta_0$ and $\\beta_1$ that creates a line that best fits the data set."
      ],
      "metadata": {
        "id": "4KNBbra8xhy5"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "####################################\n",
        "# X and y data\n",
        "####################################\n",
        "x = np.array([ 0, 1, 3, 4, 6, 7, 9, 10, 11, 12, 13, 14, 17, 18, 19, 21, 22, 24, 25, 26, 28, 31, 33, 34, 35, 36, 38, 39])\n",
        "y = np.array([-65,  35, 132,  90,  82, 116,  48, 172, 221, 166, 147, 127, 150, 294, 192, 289, 237, 329, 360, 256, 355, 342, 457, 317, 290, 391, 437, 513])"
      ],
      "metadata": {
        "id": "WsXlPqH0x3gb"
      },
      "execution_count": 12,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "Let's take a quick look at the plot."
      ],
      "metadata": {
        "id": "7q9f1KOm0Bfr"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "####################################\n",
        "# Initial plot\n",
        "####################################\n",
        "fig = plt.figure(figsize=(8,4),dpi=150)\n",
        "plt.scatter(x,y)\n",
        "plt.ylabel('y')\n",
        "plt.xlabel('x')"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 579
        },
        "id": "JgTArREa0EYy",
        "outputId": "5cc5e656-57a7-444e-938f-a85281f5b303"
      },
      "execution_count": 34,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "Text(0.5, 0, 'x')"
            ]
          },
          "metadata": {},
          "execution_count": 34
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 1200x600 with 1 Axes>"
            ],
            "image/png": "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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "At a glance there could be a reliable trend. The question is, what is the best linear equation that can represent the trend. We could do trial and error guesses because it's just a simple linear trend. But can we do something to systematically calculate the coefficients? Answer is **YES**."
      ],
      "metadata": {
        "id": "EpAipYG11CNp"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "# Residual Sum of Squares (RSS)\n",
        "\n"
      ],
      "metadata": {
        "id": "wfOL_Rwj1v2J"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "These problems are optimization problems. Basically, our goal is to find a linear equation:\n",
        "\n",
        "$$ \\hat{y} = \\beta_0 + \\beta_1 x $$\n",
        "\n",
        " Such that the error difference between our model (i.e., the line equation) and the actual data points is minimized! Essentially, we have a minimization problem. We write $\\hat{y}$ to indicate that this is the predicted values based on the linear equation. We let $y$ be the actual data set values.\n",
        "\n",
        "There are several ways to quantify errors. For example, we can get the actual error difference:\n",
        "\n",
        "$$e = \\hat{y} - y $$\n",
        "\n",
        "Or the absolute difference:\n",
        "\n",
        "$$|e| = |\\hat{y} - y| $$\n",
        "\n",
        "But the most popular one is the residual sum of square errors (RSS). RSS is essentially:\n",
        "\n",
        "$$ RSS = e_1^2 + e_2^2 + e_3^2 + ... + e_n^2 $$\n",
        "\n",
        "Where the subscripts pertain to the data point. Let's focus on RSS. Our minimization problem is essentially to pick the coeffcients $\\beta_0$ and $\\beta_1$ such that it minimizes the RSS. Expanding the equation results in:\n",
        "\n",
        "$$ \\min RSS = \\left(y_1 - (\\beta_0 + \\beta_1 x_1) \\right)^2 + \\left(y_1 - (\\beta_0 + \\beta_1 x_1) \\right)^2 + \\left(y_1 - (\\beta_0 + \\beta_1 x_1) \\right)^2...$$\n",
        "\n",
        "Compressing it results in:\n",
        "\n",
        "$$ \\min RSS = \\sum_{i=1}^n \\left(y_i - (\\beta_0 + \\beta_1 x_i) \\right)^2$$\n",
        "\n",
        "If you recall from calculus, minimization or maximization means we get the partial derivative of $\\beta_0$ and $\\beta_1$ of RSS and equate it to 0. We would have:\n",
        "\n",
        "$$\\frac{\\partial RSS}{\\partial \\beta_0} = -2 \\sum \\left( y_i - (\\beta_0 + \\beta_1 x_i)\\right)$$\n",
        "\n",
        "$$\\frac{\\partial RSS}{\\partial \\beta_1} = -2 \\sum \\left( y_i - (\\beta_0 + \\beta_1 x_i)\\right)(x_i)$$\n",
        "\n",
        "Do some research on the math but $\\beta_0$ and $\\beta_1$ is deterministic from these two working equations since we have 2 unknowns and 2 equations. You should arrive at:\n",
        "\n",
        "$$\\beta_0 = \\mu_y - \\beta_1 \\mu_x $$\n",
        "\n",
        "$$\\beta_1 =  \\frac{\\sum(x_i - \\mu_x)(y_i - \\mu_y)}{\\sum(x_i-\\mu_x)^2} $$\n",
        "\n",
        "Let's try it out. First let's check the means of $x$ and $y$."
      ],
      "metadata": {
        "id": "kslJGFOfPPym"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "####################################\n",
        "# Calculating terms\n",
        "####################################\n",
        "mean_x = x.mean()\n",
        "mean_y = y.mean()\n",
        "\n",
        "print(\"Mean of x: %f\" % mean_x)\n",
        "print(\"Mean of y: %f\" % mean_y)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "FzwqItGw1vV6",
        "outputId": "f3f607c5-8cad-4f9e-daf9-70b847f12a74"
      },
      "execution_count": 17,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Mean of x: 19.142857\n",
            "Mean of y: 231.428571\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Let's calculate $\\beta_1$:"
      ],
      "metadata": {
        "id": "J5LP0raP6gFq"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "####################################\n",
        "# Calculating beta_1\n",
        "####################################\n",
        "\n",
        "# Numerator part\n",
        "num_part = 0\n",
        "\n",
        "for i in range(len(x)):\n",
        "  num_part += (x[i]-mean_x)*(y[i]-mean_y)\n",
        "\n",
        "# Denominator part\n",
        "den_part = 0\n",
        "\n",
        "for i in range(len(x)):\n",
        "  den_part += (x[i]-mean_x)**2\n",
        "\n",
        "beta_1 = num_part/den_part\n",
        "\n",
        "print(\"Beta_1 is: %f\" % beta_1)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "yuhlRSh71c-7",
        "outputId": "8bcab72e-b72d-48b9-b65c-7c548d36734c"
      },
      "execution_count": 18,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Beta_1 is: 10.701175\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Let's calculate $\\beta_0$:"
      ],
      "metadata": {
        "id": "0c9Pf0_56ivq"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "####################################\n",
        "# Calculating beta_0\n",
        "####################################\n",
        "beta_0 = mean_y - beta_1*mean_x\n",
        "\n",
        "print(\"Beta_0 is: %f\" % beta_0)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "qwm-lO_46l1t",
        "outputId": "26c9e103-bd89-4271-ee62-fef4a60b8c21"
      },
      "execution_count": 19,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Beta_0 is: 26.577499\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Since we have:\n",
        "\n",
        "$$\\beta_0 = 26.58 $$\n",
        "$$\\beta_1 = 10.70 $$\n",
        "\n",
        "Our working equation would be:\n",
        "\n",
        "$$ \\hat{y} = 26.58 + 10.70 x $$\n",
        "\n",
        "Let's try to predict values and compare it to the earlier values:"
      ],
      "metadata": {
        "id": "yrYoHqm_640i"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "####################################\n",
        "# Predicting values\n",
        "####################################\n",
        "y_pred = beta_0 + beta_1*x\n",
        "\n",
        "print(y_pred)\n",
        "print(y)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "eysXtLWs7IaL",
        "outputId": "eea31409-6a69-4aa6-9e9e-a9c327275bc8"
      },
      "execution_count": 22,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "[ 26.57749872  37.27867416  58.68102504  69.38220048  90.78455136\n",
            " 101.4857268  122.88807768 133.58925312 144.29042856 154.991604\n",
            " 165.69277944 176.39395488 208.4974812  219.19865664 229.89983208\n",
            " 251.30218296 262.0033584  283.40570928 294.10688472 304.80806016\n",
            " 326.21041104 358.31393736 379.71628824 390.41746368 401.11863912\n",
            " 411.81981456 433.22216544 443.92334088]\n",
            "[-65  35 132  90  82 116  48 172 221 166 147 127 150 294 192 289 237 329\n",
            " 360 256 355 342 457 317 290 391 437 513]\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "At a glance the data does not look perfect but if we plot the linear equation we should see a nice trend."
      ],
      "metadata": {
        "id": "5v1p8y3u7eRz"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "####################################\n",
        "# Comparison plot\n",
        "####################################\n",
        "\n",
        "fig = plt.figure(figsize=(8,4),dpi=150)\n",
        "plt.scatter(x,y)\n",
        "plt.plot(x,y_pred,color='orange',lw=2)\n",
        "plt.legend(['predicted data','original data'])\n",
        "plt.ylabel('y')\n",
        "plt.xlabel('x')\n"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 579
        },
        "id": "sAODNrgQ7lOp",
        "outputId": "b9536aa7-f198-4764-8a78-9218f0dd3475"
      },
      "execution_count": 33,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "Text(0.5, 0, 'x')"
            ]
          },
          "metadata": {},
          "execution_count": 33
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 1200x600 with 1 Axes>"
            ],
            "image/png": "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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "# Multivariate Linear Regression\n",
        "\n"
      ],
      "metadata": {
        "id": "BHrUMxpl9EDZ"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "What we have just introduced is the essence of linear regression. It's just a curve fitting function! Our example only handles a single feature, but in practice we should be able to handle multiple ones. In general, a linear regerssion predictive model is in the form of:\n",
        "\n",
        "$$ \\hat{y} = \\sum_{i=1}^n \\beta_i x_i $$\n",
        "\n",
        "It is essentially a linear combination. Our goal is still the same, what are the appropriate $\\beta_i$ coefficients?\n",
        "\n",
        "Unforunately, solving for $\\beta_i$ is not as straightforward as what we have demonstrated. It is possible to generate hand calculations but that is a gruesome way to approach it. Instead, we can use computers to help us.\n",
        "\n",
        "We'll borrow the idea from what we have learned. Essentially, we have:\n",
        "\n",
        "1. Write down the working model.\n",
        "2. Write a cost function. In our case it was the RSS.\n",
        "3. Do optimization to minimize RSS and find the appropriate $\\beta_i$.\n",
        "\n",
        "Simple right? For a multivariate linear regression, we still use the RSS cost function but it's in the form of:\n",
        "\n",
        "$$J(\\beta) = \\frac{1}{2m} \\sum_{j=1}^m \\left(y^j-\\sum_{i=0}^n \\beta_ix_i^j \\right)^2 $$\n",
        "\n",
        "Take note that the $j$ terms here pertains to the entry number while the $i$ terms pertains to the maltivariable term. Our goal is to minimize the cost function $J(\\beta)$. We also do a partial derivative on each $\\beta_i$ term. The partial derivative for each term is called the gradient descent:\n",
        "\n",
        "$$ \\nabla J = \\begin{bmatrix}\n",
        " \\frac{\\partial J}{\\partial \\beta_0 } \\\\ \n",
        " \\frac{\\partial J}{\\partial \\beta_1 } \\\\\n",
        " \\frac{\\partial J}{\\partial \\beta_2 } \\\\\n",
        " ... \\\\\n",
        " \\frac{\\partial J}{\\partial \\beta_n }\\\\\n",
        "\\end{bmatrix} $$\n",
        "\n",
        "Then each term would have the form:\n",
        "\n",
        "$$\\frac{\\partial J}{\\partial \\beta_k } = -\\frac{1}{m} \\sum_{j=1}^m \\left( y^j - \\sum_{i=0}^n \\beta_i x_i^j \\right)x_k^j$$\n",
        "\n",
        "This is the working equation. Previously, we took the derivative and equate it to zero to find the optimal $beta$ coefficients. Here, the gradient descent function is that same derivative term but this time, we find the $beta$ terms and update it for every step. For each $j$ iteration, we allow the machine to calculate the next step until we reach a local minimum. This is the numerical method of what we just did earlier! You might ask, how do we update $\\beta_k$ then? Well:\n",
        "\n",
        "$$\\beta_0 := \\beta_0 - \\alpha \\left( \\frac{1}{m} \\sum_{j=1}^m \\left( y^j - \\sum_{i=0}^n \\beta_i x_i^j \\right)x_k^j\\right) $$\n",
        "\n",
        "$$\\beta_1 := \\beta_1 - \\alpha \\left( \\frac{1}{m} \\sum_{j=1}^m \\left( y^j - \\sum_{i=0}^n \\beta_i x_i^j \\right)x_k^j\\right) $$\n",
        "\n",
        "$$... $$\n",
        "\n",
        "$$\\beta_n := \\beta_n - \\alpha \\left( \\frac{1}{m} \\sum_{j=1}^m \\left( y^j - \\sum_{i=0}^n \\beta_i x_i^j \\right)x_k^j\\right) $$\n",
        "\n",
        "The $\\alpha$ term here is a learning rate. The learning rate tells us how \"heavy\" each step should be. The iterations can last up to infinity but there are two cases where we stop the gradient descent: (1) we reach the maximum desired iterations and (2) we reach the desired tolerable error. This completes the entire algorithm. We'll explore this next time :)"
      ],
      "metadata": {
        "id": "XRCehZPkAULG"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "# Polynomial Regression\n",
        "\n"
      ],
      "metadata": {
        "id": "ygRCGWfdMz8V"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Sometimes some some features may have a synergistic effect. For example, $x_1$ and $x_2$ combined may have a very nice contribution to the overall trend. That is, instead of just looking solely at feature $x_1$ and $x_2$ separately, we may be interested in $x_1\\cdot x_2$ combination! This are, of course, hidden in the data set which may not be seen immediately. We know these to be non-linear effects but that shouldn't stop us from doing good linear regression.\n",
        "\n",
        "In this way, let's say we are interested in the combination of $x_1$ and $x_2$ features, instead of restricting ourselves to a linear regression:\n",
        "\n",
        "$$\\hat{y} = \\beta_0 + \\beta_1 x_1 + \\beta_2 x_2$$\n",
        "\n",
        "We can create a **polynomial regression** which captures non-linear effects better. However, in polynomial regression we apply multiple permutations! For example:\n",
        "\n",
        "$$\\hat{y} = \\beta_0 + \\beta_1 x_1 + \\beta_2 x_2 + \\beta_3 x_1^2 + \\beta_4 x_2^2 + \\beta_5 x_1x_2 $$\n",
        "\n",
        "Observe that we placed all combinations even starting from the isolated linear components (i.e., $\\beta_1 x_1$). This is the essence of polynomial regression. How do we find the optimal $\\beta_i$ coefficients? We just use the gradient descent method."
      ],
      "metadata": {
        "id": "thI2PySdObWW"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "# Metrics for Performance\n"
      ],
      "metadata": {
        "id": "c-Ath98sRa3W"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "\n",
        "Of course, just like any system, we need metrics of performance to measure how good our model (or design) is. Linear (or polynomial) regression problems are used for predicting values. That is how close is our model to the actual data. The outputs are meant for predictions and not categories. That means, the outputs or the model measures real numbers.\n",
        "\n",
        "For linear regression models we have the following metrics:\n",
        "\n",
        "1. **Mean absolute error (MAE)**:\n",
        "\n",
        "$$ P = \\frac{1}{n} \\sum_{i=1}^n |y_i - \\hat{y}_i| $$\n",
        "\n",
        "2. **Mean squared error (MSE)**:\n",
        "\n",
        "$$ P = \\frac{1}{n} \\sum_{i=1}^n \\left(y_i - \\hat{y}_i \\right)^2$$\n",
        "\n",
        "3. **Root mean squared error (RMSE)**:\n",
        "\n",
        "$$ P = \\sqrt{\\frac{1}{n} \\sum_{i=1}^n \\left(y_i - \\hat{y}_i \\right)^2}$$\n",
        "\n",
        "These metrics are important for evaluating how good our linear regression model is."
      ],
      "metadata": {
        "id": "hrNxUR8VR4XW"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "# Regularization"
      ],
      "metadata": {
        "id": "Vq2PB35ZS0he"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Regularization is the method used for feature selection or reduction of overfitting. The term **overfitting** occurs when a model hits all the trained data perfectly. However, when we start testing it to unseen data, it performs poorly. We say that the model is overfitted to the trained data if this happens. To reduce overfitting we apply regularization. Regularization is like extra tuning for the cost function. What it basically does is to \"lessen\" the effects of overfitting based on the $\\beta_i$ parameters. You may need to research or review this even deeper. There are three types of regularization methods for linear regression:\n",
        "\n",
        "1. **L1 Regularization: Lasso Regression **\n",
        "\n",
        "$$ J = \\sum_{i=1}^n \\left(y_i - \\beta_0 - \\sum_{j=1}^p \\beta_jx_j \\right)^2 + \\lambda \\sum_{j=1}^p |B_j| $$\n",
        "\n",
        "Lasso regression is a simple absolute value tuning parameter with some $\\lambda$ hyperparameter. What it does is to suppress drastic changes in the trained $\\beta_i$ parameters. The advantage of lasso regression is that it can prune out features that it computes to be \"useless\". For example, sometimes a feature my not really have any significant information to the prediction.\n",
        "\n",
        "2. **L2 Regularization: Ridge Regression **\n",
        "\n",
        "$$ J = \\sum_{i=1}^n \\left(y_i - \\beta_0 - \\sum_{j=1}^p \\beta_jx_j \\right)^2 + \\lambda \\sum_{j=1}^p B_j^2 $$\n",
        "\n",
        "Ridge regression is a squared hyperparameter. The only difference with lasso is that this time, in ridge regression, the larger errors are penalized greatly. In otherwords, because of that square relationship, for every large cost, it is also greatly penalized. The only cons for this is that it cannot prune features. It is possible that the $\\beta_i$ can be close to 0. You might need to investigate the model further if this happens.\n",
        "\n",
        "3. **Combination of L1 and L2 Regression: Elastic Net**\n",
        "\n",
        "$$ J =  \\frac{1}{2n} \\sum_{i=1}^n \\left(y_i - x_i^j\\beta_j \\right)^2 + \\lambda \\left(\\frac{1-\\alpha}{2}\\sum_{j=1}^m \\beta_j^2 + \\alpha \\sum_{j=1}^m |\\beta_j| \\right)$$\n",
        "\n",
        "The elastic net regression combines both the lasso and ridge regression into one. The only trick though is that we specifiy the $\\alpha$ pamater as the weighting on how much lasso or how much ridge do we want. This provides a more sophisticated model but also combines both benefits of L1 and L2 regressions."
      ],
      "metadata": {
        "id": "TbwhfeXdS2IJ"
      }
    }
  ]
}