02. Simple Linear Regression.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Topics\n",
    "1. Simple Linear Regression Basics\n",
    "2. Bivariate(Two Variable) Statistics\n",
    "3. Coefficient of Determination"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Simple Linear Regression Basics\n",
    "### Def Regression: \n",
    "\"Using the relationship between variables to find the best fit line or the regression equation that can be used to make predictions.\"    \n",
    "\n",
    "If we have only one variable then Mean is the best way to predict.  \n",
    "<img src=\"Image/LR-1.JPG\">\n",
    "\n",
    "$\\hat{y}$ = $b_o + b_1$x  \n",
    "here $b_1$ = 0  \n",
    "$\\hat{y}$ = $b_o$ = 10  ( because mean is the best fit, i.e, $b_o$ = 10 )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Distance between best fit line and the observed value is called Residuals or Error.\n",
    "Squaring the residuals\n",
    "<img src=\"Image/LR-2.JPG\">\n",
    "Residual Squared Sum (RSS)of Error or (SSE)- Sum of square of difference between actual value & predicted value.  \n",
    "Sum of Squared Errors(SSE) = 25 + 49 + 1 + 4 + 16 + 25 = 120"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Bivariate(Two Variable) Statistics"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Best fit line will always pass from Centroid\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\hat{y}$ = $b_o + b_1$x\n",
    "\n",
    "$\\hat{y}$ = Mean(Expected) value of y for a given value of x\n",
    "\n",
    "Least Square Criterian: We need to minimize this term $\\sum(y_i - \\hat{y_i})^2$  \n",
    "Using Ordinary Least Square(OLS) Criteria:  \n",
    "- Ordinary least squares  (OLS) is a non-iterative method that fits a model such that the sum-of-squares of differences of observed and predicted values is minimized.\n",
    "\n",
    "$b_o = \\bar{y} - b_1\\bar{x}$  \n",
    "$b_1 = \\frac{\\sum(x_i - \\bar{x})(y_i - \\bar{y})}{(x_i - \\bar{x})^2}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"Image/LR-3.JPG\">\n",
    "\n",
    "When we conducted regression SSE deceased from 120 to 30.075.  \n",
    "So, where does this 89.925 go?\n",
    "89.925 is Sum of Squares due to Regression(SSR)  \n",
    "SST = SSR + SSE  \n",
    "120 = 89.925 + 30.075  \n",
    "Correlation Coefficient: Its the strength of the relationship between the relative movements of two variables. The values range between -1.0 and 1.0. A calculated number greater than 1.0 or less than -1.0 means that there was an error in the correlation measurement."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Coefficient of Determination($r^2$)\n",
    "$r^2 = \\frac{SSR}{SST} = \\frac{89.925}{120}$ = 0.7493 or 74.93%  \n",
    "So, 74.93% of the Total Sum of Squares can be explained by using the estimated regression equation to predict the tip amount. The remainder is error."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.9437054475140341\n"
     ]
    }
   ],
   "source": [
    "# Simple Linear Regression\n",
    "\n",
    "# Importing the libraries\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import pandas as pd\n",
    "\n",
    "# Importing the dataset\n",
    "dataset = pd.read_csv('Data/Regression/Company_Profit.csv')\n",
    "X = dataset.iloc[:, :-1].values\n",
    "y = dataset.iloc[:, 1].values\n",
    "\n",
    "# Splitting the dataset into the Training set and Test set\n",
    "from sklearn.model_selection import train_test_split\n",
    "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size = 0.33, random_state = 0)\n",
    "\n",
    "# Fitting Simple Linear Regression to the Training set\n",
    "from sklearn.linear_model import LinearRegression\n",
    "linear_model = LinearRegression()\n",
    "linear_model.fit(X_train, y_train)\n",
    "\n",
    "#Check R-square on training data\n",
    "print(linear_model.score(X_train, y_train))\n",
    "\n",
    "# Predicting the Test set results\n",
    "y_predict = linear_model.predict(X_test)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Visualising the Training set results\n",
    "plt.scatter(X_train, y_train, color = 'green')\n",
    "plt.plot(X_train, linear_model.predict(X_train), color = 'red')\n",
    "plt.title('Training Set -> Startup_Yrs_Operation Vs Profit')\n",
    "plt.xlabel('Startup_Yrs_Operation')\n",
    "plt.ylabel('Profit')\n",
    "plt.show()\n",
    "\n",
    "# Comapre the Values \n",
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "x_serial = list(range(1, len(y_predict) + 1))\n",
    "plt.scatter(x_serial, y_predict, color = 'red')\n",
    "plt.scatter(x_serial, y_test, color = 'blue')\n",
    "plt.title('y_pred(red) vs y_test(blue)')\n",
    "plt.xlabel('Serial Number')\n",
    "plt.ylabel('Profit')\n",
    "plt.show()\n",
    "\n",
    "# Visualising the Test set results\n",
    "plt.scatter(X_test, y_test, color = 'black')\n",
    "plt.plot(X_train, linear_model.predict(X_train), color = 'red')\n",
    "plt.title('Test Set -> Startup_Yrs_Operation Vs Profit')\n",
    "plt.xlabel('Startup_Yrs_Operation')\n",
    "plt.ylabel('Profit')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Assumptions of Linear Regression\n",
    "#### 1. Linearity\n",
    "- linear regression is sensitive to outlier effects\n",
    "- needs the relationship between the independent and dependent variables to be linear\n",
    "- linearity assumption can best be tested with scatter plots  \n",
    "\n",
    "#### 2. Homoscedasticity\n",
    "- meaning the residuals are equal across the regression line\n",
    "- Heteroscedasticity Test to check - The Goldfeld-Quandt Test \n",
    "<img src=\"Image/homosca.JPG\" width=\"600\" />\n",
    "\n",
    "#### 3. Multivariate Normality\n",
    "- This assumption can best be checked with a histogram or a Q-Q-Plot\n",
    "- Normality can be checked with a goodness of fit test(Kolmogorov-Smirnov)  \n",
    "<img src=\"Image/normal.JPG\" width=\"300\" />\n",
    "\n",
    "#### 4. No Autocorrelation in the Data\n",
    "- when the residuals are not independent from each other.  \n",
    "- in simple terms when the value of y(x+1) is not independent from the value of y(x)\n",
    "- Durbin-Watson test\n",
    "\n",
    "#### 5. Lack of Multicollinearity\n",
    "- Multicollinearity: Model cannot differentiate between the effect of D1 and D2 as these are totally related.\n",
    "- fixed using correlation in data pre processing"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Gradient Descent\n",
    "\n",
    "This is one of the best optimisation algorithms to minimise the errors (difference of actual value and predicted value).  \n",
    "This finds the model parameters iteratively.  \n",
    "\n",
    "Consider an example were we need to find the min. value of a equation: $y = 2 + 3x + 4x^2$\n",
    "\n",
    "For this, take the derivative of above equation w.r.t x and equate it to zero. This will give the point where the value of the equation will be min.\n",
    "\n",
    "The main goal of Gradient descent is to minimize the cost value. i.e. min $J(\\theta_o, \\theta_1)$  \n",
    "Error = $\\sum$ (predicted output - actual output$)^2 $  \n",
    "\n",
    "Cost Function:  \n",
    "$J(\\theta) =  \\frac{1}{2}\\sum \\limits _{i=1}^{m}(h_\\theta(x^{(i)}) - y^{(i)})^2$  \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a href=\"https://www.analyticsvidhya.com/blog/2017/03/introduction-to-gradient-descent-algorithm-along-its-variants/\"> More Read </a> "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Algorithm \n",
    "$h_\\theta(x) = \\theta_0 + \\theta_1x$  \n",
    "\n",
    "$J(\\theta) =  \\frac{1}{2}\\sum \\limits _{i=1}^{m}(h_\\theta(x^{(i)}) - y^{(i)})^2$   \n",
    "Where $J(\\theta)$ is Error or Cost Function  \n",
    "Goal if to minimize $J(\\theta)$  \n",
    "Let $\\theta_0 = 0$ for simplification  \n",
    "\n",
    "$J(\\theta_1) = (\\theta_1x - y)^2$  \n",
    "Now do partial derivative to w.r.t $\\delta m$ to find $\\delta$ change along the curve\n",
    "\n",
    "$m = m - \\frac {\\partial J(m)}{\\partial m}$  \n",
    "\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Questions & Answers\n",
    "**Does Linear Regression need scaling?  \n",
    "A. No  \n",
    "**Why?  \n",
    "A. The model coef_ takes care of scale of the data.   \n",
    "**Does Linear Regression do scaling internally  \n",
    "A. yes,  \n",
    "**Why?  \n",
    "So that coef tell comparable importance of features  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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