Using plot.ly an interactive 3D graph of log likelihood function of a sample set (x,y) is drawn and its maximum found, only to prove that, in fact that happens to provide us, the best fit regression line.

Supplementary 2 - Interactive Visualization of Simple Regression Model

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Simple Regression Model"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Samples Setup\n",
    "\n",
    "Below are the samples. \n",
    "\n",
    "X|Y|\n",
    "---|---|\n",
    "2.2|14\n",
    "2.7|23\n",
    "3|13\n",
    "3.55|22\n",
    "4|15\n",
    "4.5|20\n",
    "4.75|28\n",
    "5.5|23"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-10-25T13:48:44.477635Z",
     "start_time": "2018-10-25T13:48:44.323721Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "bhat_0:9.850824800910123, bhat_1:2.622298065984073\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x18323acbc18>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "x_i = [2.2, 2.7, 3, 3.55, 4, 4.5, 4.75, 5.5]   # a sample set\n",
    "y_i = [14, 23, 13, 22, 15, 20, 28, 23] \n",
    "# x_i = np.random.normal(2, 1.5, 100)\n",
    "\n",
    "x_bar = sum(x_i)/len(x_i)\n",
    "y_bar = sum(y_i)/len(y_i)\n",
    "num,dem = 0,0\n",
    "for i, j in zip(x_i, y_i):\n",
    "    num += (i - x_bar)*(j - y_bar)\n",
    "    dem += (i - x_bar)**2\n",
    "bhat_1 = num/dem\n",
    "bhat_0 = y_bar - bhat_1*x_bar\n",
    "print('bhat_0:{}, bhat_1:{}'.format(bhat_0, bhat_1))\n",
    "    \n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "# construct the line\n",
    "x = np.linspace(1.5, 5.5, 10)\n",
    "y = 9.85 + 2.62*x\n",
    "plt.plot(x,y, color='red')\n",
    "plt.scatter(x_i, y_i)\n",
    "plt.xlim([1,6])\n",
    "plt.ylim([5,40])\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('y')\n",
    "plt.title('Our sample set')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So as per formula, we should be getting around, \n",
    "\n",
    "$$\n",
    "y = 9.85 + 2.62x\n",
    "$$\n",
    "\n",
    "Let us verify visually via graph below by plotting log likelihood function and taking the maximum point out of that surface directly. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Graph Setup"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-10-25T09:37:56.721057Z",
     "start_time": "2018-10-25T09:37:53.653774Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<script>requirejs.config({paths: { 'plotly': ['https://cdn.plot.ly/plotly-latest.min']},});if(!window.Plotly) {{require(['plotly'],function(plotly) {window.Plotly=plotly;});}}</script>"
      ],
      "text/vnd.plotly.v1+html": [
       "<script>requirejs.config({paths: { 'plotly': ['https://cdn.plot.ly/plotly-latest.min']},});if(!window.Plotly) {{require(['plotly'],function(plotly) {window.Plotly=plotly;});}}</script>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "from math import log, pi\n",
    "\n",
    "# plotly offline init\n",
    "from plotly.offline import init_notebook_mode, plot, iplot\n",
    "import plotly.graph_objs as go\n",
    "init_notebook_mode(connected=True)   # this is needed for plot.ly to operate offline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Handling data format"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The log likelihood function of given sample set $(x_i, y_i)$ of size $m$ is given by, \n",
    "\n",
    "$$\n",
    "ln( L(\\beta_0, \\beta_1) ) = -\\dfrac{m}{2}\\big(ln(2\\pi\\sigma^2)\\big) - \\dfrac{ \\sum_{i=1}^{m}[y_i - (\\beta_0 + \\beta_1x_i)]^2}{2\\sigma^2} \n",
    "$$\n",
    "\n",
    "We need to find out how the surface plot of RHS looks like in above equation as its a $f(\\beta_0, \\beta_1)$ and others are constants. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-10-25T09:38:45.323238Z",
     "start_time": "2018-10-25T09:37:56.723058Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>b0</th>\n",
       "      <th>b1</th>\n",
       "      <th>L</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>-40.0</td>\n",
       "      <td>-40.0</td>\n",
       "      <td>-742857.806331</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>-40.0</td>\n",
       "      <td>-39.5</td>\n",
       "      <td>-729440.603831</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
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       "      <td>-716146.196331</td>\n",
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      "text/plain": [
       "     b0    b1              L\n",
       "0 -40.0 -40.0 -742857.806331\n",
       "1 -40.0 -39.5 -729440.603831\n",
       "2 -40.0 -39.0 -716146.196331\n",
       "3 -40.0 -38.5 -702974.583831\n",
       "4 -40.0 -38.0 -689925.766331"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b0 = np.arange(-40,40,0.5)   \n",
    "b1 = np.arange(-40,40,0.5)   \n",
    "\n",
    "sigma = 0.5  # assuming but one could try any value\n",
    "\n",
    "columns = ['b0', 'b1', 'L']\n",
    "df = pd.DataFrame(columns=columns)\n",
    "\n",
    "Z = []\n",
    "for each_b0 in b0:\n",
    "    for each_b1 in b1:        \n",
    "        s_1 = sum([ (j - (each_b0 + each_b1*i))**2 for i,j in zip(x_i,y_i) ])  # i and j are x_i and y_i indexings\n",
    "        L = - (m/2)*log(2*pi*(sigma**2)) - s_1/(2*(sigma**2))\n",
    "        df.loc[len(df)] = [each_b0, each_b1, L]\n",
    "        Z.append(L)\n",
    "\n",
    "Z = np.array(Z).reshape((len(b0),len(b1)))        \n",
    "df.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Plotting the graph"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-10-25T09:59:45.450474Z",
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