machine-learning/Linear_Regression.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "6iHSJ1olpeh9"
   },
   "source": [
    "# Overview\n",
    "\n",
    "In statistics, linear regression is a linear approach to modelling the relationship between a scalar response (or dependent variable) and one or more explanatory variables (or independent variables).\n",
    "\n",
    "Linear regression is a common algorithm for making inferences and predictions. \n",
    "\n",
    "The goal with linear regression is to minimize the vertical distance between all the data points and a line of best fit. The distance between each data point (its y-value) and the fitted line is called a ***residual***.\n",
    "\n",
    "References:\n",
    "- [Linear Regression - Wikipedia](https://en.wikipedia.org/wiki/Linear_regression)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "jupyter": {
     "source_hidden": true
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "RMSE: 7.8533843501953315\n",
      "R-squared: 0.8880683003040512\n",
      "\n"
     ]
    },
    {
     "data": {
      "image/png": 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Gl0Zn8ya7QL8RsOXe+eZSi2KBlSRJa5YSPPtHGHdBdsFWRSv43NnZo1VV3unUwlTkHUCSJBW5OW/A778II0/Lyuvmn4HvTYTDf9Rs5fVnE6c1y9+j0mCBlaQSM3TCVLY9fwxDJ0zNO4rKXV0dPH4L3Lw/TJsAlVXZ3QW+ex9stkeT//X1S+uw+6b6Na+PuYRAkkrI0AlTGXzvFICP/xzYa4c8I6lczZ6WvSHB9Iezeav9od9w2Lh5vt6GTpjKsPum8s22/9nm17yWs8BKUomoX16X8we6Gl3dMpg0Au6/EpYugtbts7Ou+54KFc3zi9vlX+srKyl+zQsssJJUElZWXpfzB7oazbsvwsj+8NZT2bztoXDCUNhgm2aLsLqv9eX8mpcFVpKK3Cd+oKdEq7QUgKXRCiIAf6BrPS1dAg8NgQevg7paqOoMx1wJe37r46+x5tCQ8rqcX/MtW+4XcUXEFhHx+4iYHRELI+K5iNin3v6IiMsi4u3C/vER4VerpBZhxR/ordJSfjj9l/xw+i8/LrLLDb53ihe5aO3NfAZuORwmXpWV1x2Pg/6PwV7fbtbyCjCkgeV1XY9X+ci1wEbEBsDDQC1wHLArcDbwYb3DzgUGAj8A9gMWAOMioi2SVOb8ga4mU7sIxl8Ct/SCWc9Duw3hi7fC1/4EnbvlEumso3b8xLyUCn6/aC9+v2gvlq6ksqx4vFqOvJcQnAe8kVI6ud6215Z/EBEBnAlckVIaWdj2bWAWcCLw5xWfMCKqgPo3pevU6KklqZmcddSODf6V6vLjpTWa8SiMHACzC2fsd/sCHHctdNwk11jLlwP852s+WErlSo8ddNSOLh9owfJeQnAC8GRE/DUi3o2IZyLi1Hr7twU2A8Yv35BSqgEeAw5YxXNeANTUe7zZJMklqRkM7LUDgxpYSv2BrjVaPB/Gnge/OjYrrx27wn//Ab7869zL63IN+Zr3a115F9jtgB8CU4FjgJ8BQyPiO4X9mxX+nLXC582qt29FVwPV9R5bNmZgSWpu/kBXo5h2P/zsAHjs/4AEPb+ZrXXdpU/eyT5ldV/zfq0L8l9CUAE8mVL6UWF+JiJ2J1vvetu6PGFKaTGwePkczbwAXZKawvIf2EPHTf7UPn+ga7UW1cA9F8HTv83m6q2g702wfa98c63Bp5cT+LWu/8i7wL4NvLDCtheBLxY+fqfwZ9fCsdSbn23SZJJUZAb22oFUu5ilv/7PNn+ga7VeHgujz4J5hR+h+54KR14MVaVxecjyr+0h907hLL/WVU/eBfZhYKcVtu0ITC98/BpZie1FobBGRGeyuxH8rHkiSlLxOO2w7RlaKLCnH7G9P9C1cgtmw93nwXN/zeYNe8AJw2Cbg/LNtQ4G9trBr3N9St4FdgjwSET8CPgL8Fnge4UHKaUUETcCF0XEVLJCezkwE/hHHoElqVicdtj2eUdQsUkJJv8d7joHPnofogIOGACH/what8s7ndRoci2wKaUnIuLzZBde/YSsoJ6ZUvpDvcOuBToAvwC6AA8Bx6aUFjVzXEmSite8d2DM2fDS6GzedFfoNxy22DvfXFITyPsMLCml0cDo1exPZOX2J80WSpKkUpESPPtHGHdBdsFWRSv43P/C586GVm3yTic1idwLrCRJWkdzZsCoM2HahGzevCf0GwGb7Z5nKqnJWWAlSSo1dXXw5K3ZW8EumQ+VVXD4BXDA6VDpj3aVP7/KJUkqJbOnwZ2nw/SHs3mr/bO1rht7pb5aDgusJEmloG4ZTBoB918JSxdB6w7ZPV33PRUq8n5jTal5WWAlSSp2774II/vDW09l87aHwglDYYNtco0l5cUCK0lSsVpWCw8NgQeuhbpaqKqGY66APb8FvlW6WjALrCRJxWjmMzByAMx6Ppt3PA76DIbO3fLNJRUBC6wkScWkdhE8cA08PBTSMmi3IRx/Hez+Rc+6SgUWWEmSisWMR7OzrrOnZvNuX4DjroWOm+SbSyoyFlhJkvK2ZAFMuAwe+zmQoGNX6D0YdumTdzKpKFlgJUnK06sT4c6BMGd6Nvf8ZnahVrsNco0lFTMLrCRJeVhUA/dcBE//Npurt4K+N8H2vfLNJZUAC6wkSc3t5bth9Jkw7+1s3vfU7E0JqjrlGksqFRZYSZKay4LZcPd58Nxfs3nDHnDCMNjmoHxzSSXGAitJUlNLCSb/He46Bz56H6ICDhgAh/8IWrfLO51UciywkiQ1pXnvwJiz4aXR2bzprtBvOGyxd765pBJmgZUkqSmkBM/+EcZdkF2wVdEKPve/8LmzoVWbvNNJJc0CK0lSY5szA0adCdMmZPPmPaHfCNhs9zxTSWXDAitJUmOpq4Mnb4Xxl8CS+VBZla1zPWAAVPojV2os/r9JkqTGMHsa3Hk6TH84m7faP1vruvEO+eaSypAFVpKk9VG3DCaNgPuvhKWLoHUHOPIS2Pe7UFGRdzqpLFlgJUlaV7NegJH9YebT2bzdYdB3KGywda6xpHJngZUkaW0tXQIPDYEHr4O6WqiqhmOuhD2/CRF5p5PKngVWkqS1MfMZ+Ed/eHdyNu90PPQeDJ03zzeX1IJYYCVJaojahTDxGnhkGKRl0H4jOO5a2P2LnnWVmpkFVpKkNZnxaLbWdfYr2bz7F7Py2mHjfHNJLZQFVpKkVVk8HyZcBo//AkjQcTPoMwR2Pj7vZFKLZoGVJGllpt0PowZm76oF2QVaR18J7brkGkuSBVaSpE9aOAfuuQie+V02V3eHvjfC9r3yTCWpHgusJEnLvTwWRp8F897O5n1PhSMvhqpO+eaS9AkWWEmSFsyGu8+D5/6azRv2yN4GdusD880laaUssJKklislmPx3uOsc+Oh9iAo4YAAc/iNo3S7vdJJWwQIrSWqZ5r0DY86Gl0Zn86a7Zmddt9g731yS1sgCK0lqWVKCZ/8I4y6ARTVQ0Qo+97/wubOhVZu800lqAAusJKnlmDMDRp0J0yZk8+Y9od8I2Gz3PFNJWksWWElS+aurgydvhfGXwJL5UFmVrXM9YABU+qNQKjX+v1aSVN5mT4M7T4fpD2fzVvtna1033iHfXJLWmQVWklSe6pbBpBFw/5WwdBG07gBHXgL7fhcqKvJOJ2k9WGAlSeVn1gswsj/MfDqbtzsM+g6FDbbONZakxmGBlSSVj6VL4KEh8OB1UFcLVdVwzBWw57cgIu90khqJBVaSVB5mPgMjB8Cs57N5p+Oh92DovHm+uSQ1OhcBSZJyM3TCVLY9fwxDJ0xd9yepXQj3Xgy39MrKa/uN4Iu3wlf/aHmVypRnYCVJuRg6YSqD750C8PGfA3ut5Z0BZjyarXWd/Uo27/5FOO5a6LBxY0aVVGRyPQMbEZdERFrh8VK9/W0jYkREzI6I+RFxe0R0zTOzJGn91S+vyw2+d0rDz8Qung93nQu/OjYrrx03y864fulXllepBSiGM7CTgSPrzUvrfTwE6A18GagBhgN3AAc1WzpJUqNaWXldrkFnYqfdD6MGZu+qBbDnN+HoK6DdBo0dVVKRKoYCuzSl9M6KGyOiGjgF+HpK6b7CtpOBFyNi/5TSo82cU5K0nlYsr20Lfy6qd8wqS+zCOXDPRfDM77K5ujv0vRG279VUcSUVqWIosDtExEyy71+TgAtSSjOAvYHWwPjlB6aUXoqIGcABwEoLbERUAVX1NnVqquCSpIZbWXkdT2cAjmTu6kvsy2Nh9Fkw7+1s/uz3oNfFUNWxGZJLKjZ534XgMeAk4Fjgh8C2wD8johOwGbAkpTRnhc+ZVdi3KheQLTdY/nizcSNLktbFkFUsG1jt8Qtmw99OgT99NSuvG/aAk8fC8ddZXqUWLNcCm1Iam1L6a0rp3ymlccDxQBfgK+vxtFcD1fUeW653UEnSejvrqB3X4ujEzT1fhxGfhef/BlEBB50BP3wYtj6wqSJKKhHFsITgYymlORExBdgeuBdoExFdVjgL2xX41JrZes+xGFi8fA7feUWSisLy5QCruoBruU35kD9t8Rd6vPRAYcOu0G84bLF3U0eUVCLyXkLwCRHREegBvA08BdQCvert3wnoTrZWVpJUYgb22oFBqzwTm/hy5UT+2eE8esx+ACpaw2EXwPcesLxK+oRcz8BGxPXAKGA60A24FFgG/CmlVBMRtwKDI+IDYC4wDJjkHQgkqXQtPxN7c70zsd14n0ta38Ihlc9lPwW67Qn9RkDX3XJKKamY5b2EYEvgT8BGwHvAQ8D+KaX3CvvPAuqA28nuLDAOOC2HnJKkRjSw1w5ULquD+94FYGSbH9OpogZatYXDfwT794fKvH9ESSpWuX53SCl9dQ37FwH9Cw9JUhn54e7BzPuyjzvEIuh+AJwwHDbePt9gkopeUa2BlSS1AMuWwsM3wa313oTx6CvhpLssr5IaxN/PSJKaz6zJMLI/zHyGSNBt55vguOuITQ8G7xojqYEssJKkprd0CTw0GB68HupqoaqaOPYqouc3LK6S1poFVpLUtN56GkYOgHcnZ/NOx0PvwdB583xzSSpZFlhJUtOoXQgTr4ZHhkGqg/YbwXHXwu5f9KyrpPVigZUkNb7pk+DOATD7lWze/YtZee2wcb65JJUFC6wkqfEsng8TLoPHfwEk6LgZ9BkMO/fOO5mkMmKBlSQ1jmn3w6iBMGdGNu/5TTj6Cmi3Qb65JJUdC6wkaf0snAP3XATP/C6bq7tD3xth+155ppJUxiywkqR199JdMGYQzHs7m/c9FY68GKo65ZtLUlmzwEqS1t6C2TD2XHj+b9m8YQ/oNxy2PjDfXJJaBAusJKnhUoLJd8Bd58JH70NUwIGnw2EXQOt2eaeT1EJYYCVJDTPvHRg9CF4ek82b7pqddd1i73xzSWpxLLCSpNVLCZ79I4y7ABbVQEUrOOQcOHgQtGqTdzpJLZAFVpK0anNmwKgzYNp92dxtT+g3Arrulm8uSS2aBVaS9Gl1dfDkrTD+ElgyH1q1zda5HjAAKv3RISlffheSJH3S+6/AnafDjEeyufsBcMJw2Hj7fHNJUoEFVpKUWbYUHh0B918FSxdB6w5w5CWw73ehoiLvdJL0MQusJAlmvQAj+8PMp7N5u8Oh702wwdb55pKklbDASlJLtnQJPDQYHrwe6mqhqhqOvQp6fgMi8k4nSStlgZWkluqtp2HkAHh3cjbv1Bt63wCdN883lyStgQVWklqa2oUw8Wp4ZBikOmi/ERx/Hez2Bc+6SioJFlhJakmmT4I7B8DsV7J59y/BcT+FDhvnm0uS1oIFVpJagsXzYcKl8PgtQIKOm0GfIbDz8Xknk6S1ZoGVpHI37X4YNTB7Vy2APb8FR18B7brkGkuS1pUFVpLK1cI5cM9F8Mzvsrm6O5xwE/Q4ItdYkrS+LLCSVI5eugvGDIJ5bwMBn/0e9PoJVHXMO5kkrTcLrCSVkwXvw9jz4Pm/ZfNG22dvA7v1AfnmkqRGZIGVpHKQEky+A+46Bz6aDVEBBw6Ew86H1u3yTidJjcoCK0mlbu7bMOZseHlMNm+6G/QbDlvslW8uSWoiFlhJKlUpwdO/g3EXwuIaqGgNh/wvHDwIWrXJO50kNRkLrCSVqj99FWZMzD7uthf0GwFdd801kiQ1BwusJJWSurr/fPz6g9CmLRx+Iex/GlT6LV1Sy+B3O0kqFe+/AnecDlRm85b7wxeGwcbb5xpLkpqbBVaSit2ypfDoCLj/KliyBDgo2/7N26F9+1yjSVIe1rrARkQlcBbwFaA78IkrBVJKGzZONEkSsybDyP4w8xkAWvU4jIGnXQddutOqqirncJKUj3U5A3sx8F3gBuAK4EpgG+BE4LLGCiZJLdrSJfDQYHjweqirhapqOPYqouc3aB2RdzpJytW6FNhvAKemlMZExCXAn1JK0yLi38D+wNDGDChJLc5bT8PIAfDu5GzeqTf0vgE6b55vLkkqEutSYDcDnit8PB+oLnw8Gri8MUJJUotUuxAmXg2PDINUB+03guOuhd2/CJ51laSPrUuBfRPYHJgBTAOOBp4G9gUWN140SWpBpk+COwfA7FeyefcvwXE/hQ4b55tLkorQuhTYvwO9gMeAYcDvI+IUsgu6hjRiNkkqf4vnw4RL4fFbgAQdN4M+Q2Dn4/NOJklFa60LbErp/Hof/7+ImA4cCExNKY1qzHCSVNam3QejzoA5M7J5z2/C0VdCuy65xpKkYrcut9E6BHgkpbQUIKX0KPBoRLSKiENSSg82dkhJKisL58A9F8Izv8/m6u5wwk3Q44hcY0lSqViXJQT3k62BfXeF7dWFfZXrG0qSytZLd8GYQTDv7Wz+7Peh10+gqmO+uSSphFSsw+cEkFayfSNgwboGiYjzIyJFxI31trWNiBERMTsi5kfE7RHRdV3/DknKzYL34W+nwJ+/lpXXjbaHk++G46+1vErSWmrwGdiIuKPwYQJ+ExH17zhQCfwX8Mi6hIiIfYHvA/9eYdcQoDfwZaAGGA7cwcfvoyhJRS4leP52GHsufDQbogIOPB0OuwBat8s7nSSVpLVZQlBT+DOAecDCevuWAI8Ct6xtgIjoCPwBOBW4qN72auAU4OsppfsK204GXoyI/QtrbyWpeM19O1su8PJd2bzpbtBvOGyxV765JKnENbjAppROBoiI14HrU0rrvFxgBSOAMSml8RFxUb3tewOtgfH1MrwUETOAA8gK86dERBVQ/w3COzVSTklqmJSyC7TGXQiLa6CiNRzyv3DwIGjVJu90klTy1uU2Wpc21l8eEV8F9iJ7E4QVbQYsSSnNWWH7rMK+VbkAuLhRAkrS2vpwenZrrFfvz+Zue0G/EdB113xzSVIZWZe7EBARXwK+QvbmBZ84nZBSatDvxiJiK+Am4KiU0qJ1ybEKVwOD682dyN49TJKaTl0dPPFLGH8J1C6AVm3h8Ath/9Ogcp2+1UqSVmGt70IQEQOBX5OdCd0TeByYDWwHjF2Lp9ob2BR4OiKWRsRS4FBgYOHjWUCbiOiywud1Bd5Z1ZOmlBanlOYuf5Ct15WkpvP+K/Cb42HsOVl57X4g/OBhOGig5VWSmsC6fGc9DfheSulPEXEScG1K6dWIuAzYcC2eZwKwxwrbfg28BPwUeAOoJXvb2tsBImInsrO+k9YhtyQ1rmVLYdJwmHg1LF0EbTrCkZfAPqdAxbrcpVCS1BDrUmC785/bZS3kPxdJ/Y7swqoBDXmSlNI84Pn62yJiATA7pfR8Yb4VGBwRHwBzgWHAJO9AICl3sybDyP4w85ls7nEE9L0JunTPN5cktQDrUmDfITvTOh2YAewP/AvYluwWW43pLKCO7AxsFTCO7AywJOVj6RL45w3Zo64W2lbDMVdBz29ANPa3QEnSyqxLgb0POAF4huxX/kMKF3XtQ/YmA+sspXTYCvMioH/hIUn5eutpGDkA3p2czTv1hj6DodPqbowiSWps61Jgv0fh4q+U0oiImA0cCNwJ/LwRs0lScahdmK1zfWQYpDpov3H2FrC7fcGzrpKUg3W5D2wd2a/1l89/Bv7cmKEkqWhMn5Stdf1gWjbv8WU49qfQYaN8c0lSC7bWBTYidgD6AdsACXgV+EdK6bXGjSZJOVo8HyZcCo/fAiTotDn0GQI7HZd3Mklq8daqwEbEBcBlZEsI3iW7aGsT4KcR8aOU0vWNH1GSmtm0++DOM6BmRjbv+S04+gpo1yXXWJKkTIMLbEQcDlwBXA7clFL6sLB9Q+BM4JqIeDyl9GBTBJWkJrdwDtxzITzz+2zu0h36DoUeh+caS5L0SZFSatiBEf8PmJNS+v4q9v8C6JRS+loj5ltvEdEZqKmpqaFz5855x5FUrF4aA6MHwfx3gIDPfg96/QSqOuadTJLK3ty5c6murgaoLryT6mqtzRKCzwLfWs3+3wG/XYvnk6T8LXgfxp4Lz9+ezRttDycMh60PyDeXJGmV1qbAdgVeX83+1wBvhiipNKSUldax58JHsyEq4MCBcNj50Lpd3ukkSauxNgW2LbBkNftrgTbrF0eSmsHct2HMIHj5rmzedDfoNxy22CvfXJKkBlnb22h9NyLmr2Jfp/UNI0lNKqXsAq1xF8LiGqhoDYecAwefBa3897cklYq1KbAzgFMbcIwkFZ8Pp8OoM+DV+7O5217QbwR03TXfXJKktdbgAptS2qYJc0hS06irgyd+CeMvgdoF0KotHH4h7H8aVK7Lu2lLkvLmd29J5ev9V+DOATBjUjZ3PxBOGAYbb59vLknSerHASio/y5bCpOFw/1WwbDG07gBHXQr7nAIVFXmnkyStJwuspPIyazKM7A8zn8nmHkdA35uyd9WSJJWFtXkr2W4ppZlNGUaS1tnSJfDPG7JHXS20rYZjroKe34CIvNNJkhrR2pyBnRwR/VNKf2yyNJK0Lt56CkaeDu9OzuadekPvG6Dz5vnmkiQ1ibUpsBcCP4+IzwPfTyl90ESZJKlhahdm61wnDYdUB+03huOvhd2+4FlXSSpjDb6aIaV0M/BfwEbACxHRt8lSSdKaTH8EfnYQPDI0K697fBn6Pw67f9HyKkllbq0u4kopvQYcEREDgDsi4kVg6QrH+F6MkprO4vkw4VJ4/BYgQafNoc8Q2Om4vJNJkprJWt+FICK2Br4AfAiMZIUCK0lNZtp9cOcZUFN40789vwVHXwHtuuQaS5LUvNaqwEbEqcANwHhgt5TSe02SSpLqWzgH7rkQnvl9NnfpDn2HQo/Dc40lScrH2txG627gs8CAlNJvmy6SJNXz0hgYPQjmvwMEfPZ70OsnUNUx72SSpJyszRnYSuC/UkpvNlUYSfrYgvdh7Lnw/O3ZvNH20G8EdN8/31ySpNw1uMCmlI5qyiCSBEBKWWkdey58NBuiEg48HQ47H1q3yzudJKkI+FaykorH3LdhzCB4+a5s7ro79BsO3fbMN5ckqahYYCXlL6XsAq1xF8LiGqhoDYeeCwedCa3a5J1OklRkLLCS8vXhdBh1Brx6fzZ32ytb69p113xzSZKKlgVWUj7q6uCJX8L4S6B2AbRqC4dfCPufBpV+a5IkrZo/JSQ1v/dfgTsHwIxJ2dz9wGyt60Y98s0lSSoJFlhJzWfZUpg0HO6/CpYthjYd4chLYJ9ToKIi73SSpBJhgZXUPGZNhpH9YeYz2dzjCOh7U/auWpIkrQULrKSmtXQJ/POG7FFXC22r4ZiroOc3ICLvdJKkEmSBldR03noKRg6Ad1/I5p16Q5/B0GmzfHNJkkqaBVZS46tdmK1znTQcUh203wiOvw52+4JnXSVJ680CK6lxTX8kO+v6wbRs3v1LcNxPocPG+eaSJJUNC6ykxrF4Hoy/FJ64JZs7bgZ9hsDOx+ebS5JUdiywktbfKxOyd9OqeSOb9/wWHH0FtOuSayxJUnmywEpaJyklltbMhvGX0eq527KlrV26Q9+h0OPwvONJksqYBVbSOln6/F38YkQ74ES+t+n/o/V+34FeP4GqjnlHkySVOd/6RtLaWfA+/PXk7FHw992Hw/HXWl4lSc3CAiupYVKC5/4Gw/eFyXdQR+XHuy58qj1DJ0zNMZwkqSVxCYGkNZs7E0YPgiljAXivww58d/ZJHF3vkMH3TgFgYK8dcggoSWpJLLCSVi0leOZ3MO4iWFwDFa15dMv/4VtTDoLU6hMFFiyxkqTmkesSgoj4YUT8OyLmFh6TIuK4evvbRsSIiJgdEfMj4vaI6JpnZqnF+PB1+N2JcOfpWXntthe/7/k7vjrlUGpX82/fwfdOcTmBJKlJ5b0G9k3gfGBvYB/gPmBkROxW2D8E6At8GTgU6AbckUNOqeWoq4PHfg43HwivToRWbeGoyxm23c1c9Ehdg57CEitJakqRUso7wydExAfAOcDfgPeAr6eU/lbYtzPwInBASunRBj5fZ6CmpqaGzp07N1FqqUy8PzU74zpjUjZ3PxD6DYeNerDt+WOo/92idYIza9oBcGP1Qmrjk08VwGvX9G6W2JKk0jZ37lyqq6sBqlNKc9d0fNGsgY2ISrIzrR2ASWRnZVsD45cfk1J6KSJmAAcAKy2wEVEFVNXb1KmpMktlY9lSmDQM7r8ali2GNh3hyEtgn1OgIvtFzVlH7fjxGleAWrLiuvzjFZ111I5Nn1uS1CLlXmAjYg+ywtoWmA98PqX0QkT0BJaklOas8CmzgM1W85QXABc3QVSpPM2aDP84Dd5+Npt7HAF9b8reVaue5RdmfVxiY+XFFWDQUTt6IZckqcnkXmCBl4GeQDXwJeC2iDh0PZ7vamBwvbkT2VpbSfUtXQL/vCF71NVC22o45mro+XWy94X9tE+V2JWwvEqSmlruBTaltAR4pTA+FRH7AmcA/w9oExFdVjgL2xV4ZzXPtxhYvHyOVfwgllq0t56CkQPg3Reyeec+0PsG6LS6X25kVldiLa+SpOaQe4FdiQqyNaxPkf2GshdwO0BE7AR0J1tyIGlt1S6E+6+CScMh1UH7jeH462C3z6/yrOvKrKzEWl4lSc0l1wIbEVcDY4EZZL/q/zpwGHBMSqkmIm4FBhfuTDAXGAZMaugdCCTVM/2R7KzrB9OyeY8vw7E/hQ4brdPTLS+rQ+6dwlmWV0lSM8r1NlqFgtoL2ByoAf4N/DSldG9hf1vgBuBrZGdlxwGnpZRWuYRgJX+Ht9FSy7Z4Hoy/FJ64JZs7bQ59hsBOx63+8yRJaiZrexutorsPbGOzwKpFe2UCjDoDat7I5r2+DUddDu265BpLkqT6SvY+sJIa0cIPYdxF8Ozvs7lLd+g7FHocnm8uSZIagQVWKjcvjYHRg2D+O0DAft+HI34MVR3zTiZJUqOwwErlYsH7cNc5MPmObN5oh+xtYLvvn28uSZIamQVWKnUpwfO3w9hz4aPZEJVw4Olw2AXQum3e6SRJanQWWKmUzZ2ZLReYMjabu+6enXXttme+uSRJakIWWKkUpQTP/C67UGtxDVS0hkPPhYPOhFZt8k4nSVKTssBKpebD17NbY706MZu77QX9RkDXXfNMJUlSs7HASqWiri57M4Lxl0LtAmjVFg6/EPY/DSr9v7IkqeXwp55UCt6fCneeDjMmZXP3A7O1rhv1yDeXJEk5sMBKxWzZUpg0DO6/GpYthjYd4chLYJ9ToKIi73SSJOXCAisVq3eeh5H94e1ns7nHEdD3puxdtSRJasEssFKxWboE/nk9/PMGqFsKbavhmKuh59chIu90kiTlzgIrFZM3n8rOur73Yjbv3Ad63wCdNss3lyRJRcQCKxWD2oVw/5UwaQSkOmi/MRx/Hez2ec+6SpK0AguslLfXH4Y7B8AHr2bzHl+GY38KHTbKN5ckSUXKAivlZfE8GH8JPPHLbO60OfQZAjsdl2ssSZKKnQVWysMr42HUmVDzRjbv9W046nJo1yXPVJIklQQLrNScFn4I4y6EZ/+QzV26Q9+h0OPwfHNJklRCLLBSc3lxNIwZBPNnAQH7fR+O+DFUdcw7mSRJJcUCKzW1Be/DXefA5DuyeaMdsreB7b5/vrkkSSpRFlipqaQEz9+eldeFH0BUwkED4dDzoXXbvNNJklSyLLBSU5g7E0YPgiljs7nr7tlZ12575ptLkqQyYIGVGlNK8MzvYNxFsLgGKlrDoefCQWdCqzZ5p5MkqSxYYKXG8uF0GDUQXp2YzVvsDScMh6675hpLkqRyY4GV1lddHTxxC4y/FGoXQKu2cMRFsP9pUFGZdzpJksqOBVZaH+9PhTtPhxmTsnnrg+CEYbBRj3xzSZJUxiyw0rpYthQmDYP7r4Zli6FNRzjqUtj7f6CiIu90kiSVNQustLbeeR5G9oe3n83mHr2g703QZatcY0mS1FJYYKWGWroE/nk9/PMGqFsKbavh2GvgM1+DiLzTSZLUYlhgpYZ46ykYOQDefSGbd+4DvW+ATpvlm0uSpBbIAiutTu1CuP9KmDQCUh203xh6Xw+7nuhZV0mScmKBlVZl+iPZWdcPpmXzHl/Jlgx02CjfXJIktXAWWGlFi+dl93R94pZs7tQN+gyBnY7NN5ckSQIssNInvTIBRp0BNW9k817fhqOvyC7YkiRJRcECKwEs/BDGXQTP/j6bu2wNJwyF7Q7LNZYkSfo0C6z04mgYMwjmzwIC9vsB9PoxtOmQdzJJkrQSFli1XPPfg7HnwOS/Z/NGO0C/4dB9/3xzSZKk1bLAquVJCZ77G4w9FxZ+AFEJBw2EQ8+H1m3zTidJktbAAquWZe5MGH0WTLk7m7vunp117bZnvrkkSVKDWWDVMqQET/8W7vkxLK6BitZw6Llw0JnQqk3e6SRJ0lqoyDuA1BiGTpjKtuePYeiEqZ/e+eHr8Nt+MGpgVl677QXffzArsJZXSZJKjmdgVfKGTpjK4HunAHz858BeO0BdHTz+C5hwKdR+BK3awuEXwv6nQaVf+pIklSp/iquk1S+vyw2+dwobfPQ633r3OnjjsWzj1gfBCcNgox45pJQkSY3JAquStbLyWskyvlc5hq88eTtELbTpCEddCnv/D1S4YkaSpHKQ60/0iLggIp6IiHkR8W5E/CMidlrhmLYRMSIiZkfE/Ii4PSK65pVZxWFl5XWXmM4/2vyY81r/maqo5YFl/8WvP/Mn2Pe7lldJkspI3mdgDwVGAE8UslwF3BMRu6aUFhSOGQL0Br4M1ADDgTuAg5o/rorBiuW1DbX0b/UPTqu8k9axjJrUnstqv83tdZ+Df85jXtup2ZpYSZJUFiKllHeGj0XEJsC7wKEppQcjohp4D/h6SulvhWN2Bl4EDkgpPdqA5+wM1NTU1NC5c+cmTK/msu35Y1j+VfuZeIXrWv+cHSveAuDuZfvy49qTeI8NPj4+gNeu6d38QSVJUoPMnTuX6upqgOqU0tw1HZ/3GdgVVRf+/KDw595Aa2D88gNSSi9FxAzgAOBTBTYiqoCqeps6NU1U5eWso3bk5nuf46xWf+O7lXdRGYn3U2d+UnsSd9XtR1ZZP3m8JEkqH0VTYCOiArgReDil9Hxh82bAkpTSnBUOn1XYtzIXABc3RUYVh4E93uXbT/yELoveAODvyw7istpv8SGfPsM+6KgdXT4gSVKZKZoCS7YWdnfg4PV8nquBwfXmTsCb6/mcKgaL58H4S+CJX9IFmN9mUwbO/zb31e210sMtr5IklaeiuDQ7IoYDfYDDU0r1y+Y7QJuI6LLCp3Qt7PuUlNLilNLc5Q9gXlNkVjN7ZTzcfAA88cts3us7dBz0JD17fXWlh1teJUkqX7megY2IAIYBnwcOSym9tsIhTwG1QC/g9sLn7AR0ByY1Y1TlZeGHMO5CePYP2dxlazhhKGx3GAADe2XLpuvflcDyKklSect7CcEI4OtAP2BeRCxf11qTUlqYUqqJiFuBwRHxATCXrPBOasgdCFTiXhwNYwbB/FlAwH4/gF4/hjYdPnHY8rI65N4pnGV5lSSp7OV6G62IWNVffnJK6TeFY9oCNwBfI7u7wDjgtJTSSpcQrOTv8DZapWbB+3DXOTD5jmzeaAfoNwK675dvLkmS1CRK6jZaKaVowDGLgP6Fh8pZSvDc32DsubDwA4hKOOgMOPQ8aN0273SSJKlI5L2EQMrMnQmjB8GUsdncdffsrGu3nrnGkiRJxccCq3ylBE//Fu75MSyugYrWcOi5cNCZ0KpN3ukkSVIRssAqPx++DqPOgFcnZvMWe2dnXTfdJc9UkiSpyFlg1fzq6uCJW7I3Jaj9CFq1hSMugv1Pg4rKvNNJkqQiZ4FV83p/KowcAG8U7oK29cHZfV036pFvLkmSVDIssGoey5bCI0Nh4jWwbDG06QhHXQZ7nwwVRfGGcJIkqURYYNX03nkeRvaHt5/N5u2PhD43Qpet8kwlSZJKlAVWTWfpYnjwenhoMNQthbZd4Nir4TNfg1jjLYAlSZJWygKrpvHmU9lZ1/dezOad+0DvwdCpa765JElSybPAqnEt+QgmXgWTRkCqgw6bwPHXwa4netZVkiQ1CgusGs/rD8OdA+CDV7P5v/4bjr0G2m+Yby5JklRWLLBaf4vnwb0Xw5O3ZnOnbtD3RtjxmFxjSZKk8mSB1fp5ZTyMOhNq3sjmvb4DR18ObatzjSVJksqXBVbr5qMPYNyF8K8/ZnOXrbM3JNjusFxjSZKk8meB1dp7cRSMORvmzwIC9vsB9PoxtOmQdzJJktQCWGDVcPPfg7HnwOS/Z/NGO0C/EdB9v3xzSZKkFsUCqzVLCZ77G4w9FxZ+AFEJBw2EQ8+H1m3zTidJkloYC6xWb+5MGH0WTLk7m7vuAf2GQ7eeucaSJEktlwVWK5cSPP1buOciWDwXKlrDoefBwWdCZeu800mSpBbMAqtP+/B1uHMgvPZANm+xd7bWddNdco0lSZIEFljVV7cMHv8FTLgMaj+CVu3giItg/x9CRWXe6SRJkgALrJZ7b0r2NrBvPJbNWx+c3dd1ox755pIkSVqBBbalW7YUHhkKE6+BZYuhTUc46jLY+2SoqMg7nSRJ0qdYYFuyd56Dkf3h7X9l8/ZHQp8boctWucaSJElaHQtsS7R0MTx4PTw0GOqWQtsucOzV8JmvQUTe6SRJklbLAtvSvPlUdtb1vRezeZe+cPwN0KlrvrkkSZIayALbUiz5CCZeBZNGQKqDDpvA8dfDbifmnUySJGmtWGBbgtcfzu4w8MGr2fxf/w3HXgPtN8w3lyRJ0jqwwJazxfNg/CXwxC+zuVM36Hsj7HhMnqkkSZLWiwW2XL0yHkadCTVvZPNe34GjL4e21bnGkiRJWl8W2HKz8EMYdyE8+4ds7rI1nDAMtjs031ySJEmNxAJbTl4cDWMGwfxZQGRvAXvERdCmQ97JJEmSGo0FthzMfw/GngOT/57NG+8IJwyH7vvlm0uSJKkJWGBLWUrw3N9g7Lmw8AOISjj4TDjkXGjdNu90kiRJTcICW6rmzoTRZ8GUu7O56x5w4gjY/DP55pIkSWpiFthSkxI8/Vu45yJYPBcq28Ch58JBZ0Jl67zTSZIkNTkLbCn58HW4cyC89kA2b7EP9BsBm+6cayxJkqTmZIEtBXXL4PFfwITLoPYjaNUuu7vA/j+Eisq800mSJDUrC2yxe29K9jawbzyWzVsfDCcMhY165JtLkiQpJxbYYrVsKTwyFCZeA8sWQ5tOcNSlsPfJUFGRdzpJkqTcWGCL0TvPwcj+8Pa/snn7o6DvjVC9Za6xJEmSioEFtpgsXQwPXg8PDYa6pdC2Cxx7DXzmqxCRdzpJkqSiYIEtFm8+mZ11fe+lbN6lLxx/A3Tqmm8uSZKkImOBzduSj+D+K+HRmyHVQYdN4PjrYbcT804mSZJUlHK9GigiDomIURExMyJSRJy4wv6IiMsi4u2IWBgR4yNih5ziNr7XH4L/OwgmDc/K63/9N/R/3PIqSZK0Gnlfzt4B+BfQfxX7zwUGAj8A9gMWAOMiom3zxGsii+fB6EHwm97wwavQqRt8/S/whV9A+w3zTidJklTUcl1CkFIaC4wFiBUuUopsw5nAFSmlkYVt3wZmAScCf17Zc0ZEFVBVb1OnRo69fqaOh1FnwNw3s3nvk+Coy6Btda6xJEmSSkUxr4HdFtgMGL98Q0qpJiIeAw5gFQUWuAC4uOnjraWPPoBxF8K//pjNXbaGE4bBdofmm0uSJKnEFHOB3azw56wVts+qt29lrgYG15s7AW82Yq619+KobMnAgneByN4C9oiLoE2HXGNJkiSVomIusOskpbQYWLx8XnFpQrOa/y7cdQ688I9s3nhHOGE4dN8vv0ySJEklrpgL7DuFP7sCb9fb3hV4ttnTrI2U4Lm/wthzYeGHEJVw8JlwyLnQurSvP5MkScpbMRfY18hKbC8KhTUiOpPdjeBn+cVatZQSCz94FcaeR7tX7iUAuu4B/YZDt545p5MkSSoPuRbYiOgIbF9v07YR0RP4IKU0IyJuBC6KiKlkhfZyYCbwj2aO2iAL332B/e7+KgCPtaqi/SHnwEFnQmXrfINJkiSVkbzPwO4D3F9vXn7x1W3AScC1ZPeK/QXQBXgIODaltKj5Iq6FDbb5z8f/c49nXSVJkppA3veBnQis8iqrlFICflJ4FL/6F4xtsmN+OSRJkspY3u/EJUmSJK0VC6wkSZJKigVWkiRJJcUCK0mSpJJigZUkSVJJscBKkiSppFhgJUmSVFIssJIkSSopFlhJkiSVFAusJEmSSooFVpIkSSXFAitJkqSSYoGVJElSSbHASpIkqaRYYCVJklRSLLCSJEkqKRZYSZIklRQLrCRJkkqKBVaSJEklxQIrSZKkkmKBlSRJUkmxwEqSJKmkWGAlSZJUUiywkiRJKikWWEmSJJUUC6wkSZJKigVWkiRJJcUCK0mSpJJigZUkSVJJscBKkiSppFhgJUmSVFIssJIkSSopFlhJkiSVFAusJEmSSooFVpIkSSXFAttEfjZxWt4RJEmSypIFthHVL63D7nuFoROm5phGkiSpPFlgG8nQCVMZdt8rn9g2+N4pllhJkqRGZoFtBEMnTGXwvVNWus8SK0mS1LgssOtpdeV1OUusJElS47HAroeGlNflLLGSJEmNwwK7HoY0sLyu6/GSJEn6NAvsejjrqB0/uSG1Zt5LlzHvpcsgtV7z8ZIkSVprJVFgI6J/RLweEYsi4rGI+GzemQAG9tqBQZ8opQGpTfYgPnHsoKN2ZGCvHZo1nyRJUjkq+gIbEf8NDAYuBfYC/gWMi4hNcw1W8OkS+2mWV0mSpMZT9AUWGATcklL6dUrpBeAHwEfA/+Qb6z9WV2Itr5IkSY2rqAtsRLQB9gbGL9+WUqorzAes4nOqIqLz8gfQqTmyrqzEWl4lSZIaX1EXWGBjoBKYtcL2WcBmq/icC4Caeo83myzdCpaX2MDyKkmS1FQipZR3hlWKiG7AW8CBKaVJ9bZfCxyaUtpvJZ9TBVTV29QJeLOmpobOnTs3dWRJkiStpblz51JdXQ1QnVKau6bjWzV9pPXyPrAM6LrC9q7AOyv7hJTSYmDx8jkiVnaYJEmSSlRRLyFIKS0BngJ6Ld8WERWFedKqPk+SJEnlq9jPwEJ2C63bIuJJ4HHgTKAD8Os8Q0mSJCkfRV9gU0r/LyI2AS4ju3DrWeDYlNKKF3ZJkiSpBSj6AguQUhoODM87hyRJkvJX1GtgJUmSpBVZYCVJklRSLLCSJEkqKRZYSZIklRQLrCRJkkqKBVaSJEklxQIrSZKkklIS94FtDHPnzs07giRJklZibXtapJSaKEpxiIgtgDfzziFJkqQ12jKl9NaaDmoJBTaAbsC8RnrKTmSFeMtGfM6Wzte08fmaNg1f18bna9o0fF0bn69p41vxNe0EzEwNKKdlv4Sg8CKssck3VNaHAZiXUnJdQiPwNW18vqZNw9e18fmaNg1f18bna9r4VvKaNvh19SIuSZIklRQLrCRJkkqKBXbtLQYuLfypxuFr2vh8TZuGr2vj8zVtGr6ujc/XtPGt82ta9hdxSZIkqbx4BlaSJEklxQIrSZKkkmKBlSRJUkmxwEqSJKmkWGAbKCIOiYhRETEzIlJEnJh3plIXERdExBMRMS8i3o2If0TETnnnKmUR8cOI+HdEzC08JkXEcXnnKicRcX7he8CNeWcpZRFxSeF1rP94Ke9cpS4itoiI30fE7IhYGBHPRcQ+eecqZRHx+kq+VlNEjMg7W6mKiMqIuDwiXit8nU6LiB9HvXc2WJOyfyeuRtQB+BfwK+COnLOUi0OBEcATZF+LVwH3RMSuKaUFuSYrXW8C5wNTgQC+A4yMiD1TSpNzTVYGImJf4PvAv/POUiYmA0fWm5fmFaQcRMQGwMPA/cBxwHvADsCHeeYqA/sClfXm3YF7gb/mE6csnAf8kOxn1GRgH+DXQA0wtCFPYIFtoJTSWGAsfOKtz7QeUkrH1p8j4iTgXWBv4ME8MpW6lNKoFTZdGBE/BPYn+yahdRQRHYE/AKcCF+Ucp1wsTSm9k3eIMnIe8EZK6eR6217LK0y5SCm9V3+OiPOBacAD+SQqCwcCI1NKYwrz6xHxNeCzDX0ClxComFQX/vwg1xRlovArmq+S/fZgUt55ysAIYExKaXzeQcrIDoVlWa9GxB8ionvegUrcCcCTEfHXwrKsZyLi1LxDlZOIaAN8E/hV8kb66+MRoFdE7AgQEZ8BDqZworAhPAOrohARFcCNwMMppedzjlPSImIPssLaFpgPfD6l9EK+qUpb4R8Ce5H9KlGN4zHgJOBlYHPgYuCfEbF7SmlensFK2HZkv5YdTLYka19gaEQsSSndlmuy8nEi0AX4Ta4pSt81QGfgpYhYRrZE48KU0h8a+gQWWBWLEWTrig7OO0gZeBnoSXZG+0vAbRFxqCV23UTEVsBNwFEppUV55ykXhWVZy/07Ih4DpgNfAW7NJ1XJqwCeTCn9qDA/ExG7Az8ALLCN4xRgbEppZt5BStxXgG8AXydb3tYTuDEiZjb0H1sWWOUuIoYDfYBDUkpv5p2n1KWUlgCvFManChcenUF28ZHW3t7ApsDT9da/VwKHRMQAoCqltCyvcOUipTQnIqYA2+edpYS9Daz4D9UXgS/mkKXsRMTWZBcdfiHvLGXgOuCalNKfC/Nzhdf3Ahr4jy0LrHJTuF3GMODzwGEpJS82aBoVQFXeIUrYBGCPFbb9GngJ+KnltXEULpLrAfwu7ywl7GFgxVsR7kh2Zlvr72SyC43HrOlArVF7oG6FbctYi2uzLLANVPjmWv/MwLYR0RP4IKU0I59UJW8E2a8P+gHzImKzwvaalNLC/GKVroi4mmwR/AygE9nrexhwTI6xSlphPeYn1mVHxAJgtuu1111EXA+MIitX3YBLyX6A/SnPXCVuCPBIRPwI+AvZFd3fKzy0HgrXaZwM3JZS8nZv628U2V1yZpAtIdgTGER2q9IGCS+ia5iIOIzs3norui2ldFKzhikTEbGqL76TU0q/ac4s5SIibgV6kV0UU0N2v9KfppTuzTVYmYmIicCzKaUzc45SsiLiz8AhwEZk9yt9iOwijmm5BitxEdEHuJrs/q+vAYNTSrfkm6r0RcTRwDhgp5TSlLzzlLqI6ARcTvYb2E2BmWT/eL2ssAxuzc9hgZUkSVIp8T6wkiRJKikWWEmSJJUUC6wkSZJKigVWkiRJJcUCK0mSpJJigZUkSVJJscBKkiSppFhgJUmSVFIssJIkSSopFlhJagIRURkRj0TEHStsr46INyLiytV87sSISIXH4oh4KyJGRcQX1iHHJRHx7Dr8J0hS0bLASlITSCktA04Cjo2Ib9TbNQz4ALh0DU9xC7A50AP4IvAC8OeI+EXjp5Wk0mKBlaQmklKaApwPDIuIzSOiH/BV4NsppSVr+PSPUkrvpJTeTCk9mlI6D/g+cGpEHLn8oIj4aURMiYiPIuLViLg8IloX9p0EXAx8pt4Z3ZMK+wZFxHMRsaBwRvjmiOjY6C+CJDUBC6wkNa1hwL+A3wG/AC5LKf1rHZ/rNuBDoP5SgnlkZ3p3Bc4ATgXOKuz7f8ANwGSys7mbF7YB1AEDgd2A7wBHANeuYy5JalaRUso7gySVtYjYGXgReA7YK6W0dA3HTwSeTSmduZJ9jwIfpJSOX8Xn/i/w1ZTSPoX5EuDElFLPNfydXwL+L6W08Zr+eyQpb63yDiBJLcD/AB8B2wJbAq+vx3MF8PGZh4j4b7IzqT2AjmTf1+eu8UmyZQgXADsDnQuf1zYi2qeUPlqPfJLU5FxCIElNKCIOJPuVfh/gceDWiIh1fK5KYAfgtcJ8APAH4K7C8+8JXAm0WcPzbAOMBv5NdoHY3kD/wu7Vfq4kFQPPwEpSE4mI9sBvgJ+llO6PiNfIlhH8APjZOjzld4ANgNsL84HA9JTSx7fkioitV/icJUDlCtv2JjuBcXZKqa7weV9ZhzySlAsLrCQ1navJfuV/PkBK6fXCGtXrI2JsSun11Xxu+4jYjOz79JbA58nO5P4spXR/4ZipQPeI+CrwBNC7cFx9rwPbRkRP4E2yi75eAVoDp0fEKOAgslItSSXBi7gkqQlExKHABOCwlNJDK+wbR1ZMj0wr+SZcuIjr0MK4BJgNPAX8KqX09xWOvZZsjW0VMAZ4FLgkpdSlsL+KbJlBL6ALcHJK6TcRcRZwTmHbg4VjfgtskFKas17/8ZLUxCywkiRJKilexCVJkqSSYoGVJElSSbHASpIkqaRYYCVJklRSLLCSJEkqKRZYSZIklRQLrCRJkkqKBVaSJEklxQIrSZKkkmKBlSRJUkmxwEqSJKmk/H9fzaP0AXmZFwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 800x600 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# https://stackoverflow.com/questions/53779773/python-linear-regression-best-fit-line-with-residuals\n",
    "import numpy, scipy, matplotlib\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy.optimize import curve_fit\n",
    "\n",
    "xData = numpy.array([1.1, 2.2, 3.3, 4.4, 5.0, 6.6, 7.7])\n",
    "yData = numpy.array([1.1, 20.2, 30.3, 60.4, 50.0, 60.6, 70.7])\n",
    "\n",
    "\n",
    "def func(x, a, b): # simple linear example\n",
    "    return a * x + b\n",
    "\n",
    "\n",
    "initialParameters = numpy.array([1.0, 1.0])\n",
    "\n",
    "# curve fit the test data\n",
    "fittedParameters, pcov = curve_fit(func, xData, yData, initialParameters)\n",
    "\n",
    "modelPredictions = func(xData, *fittedParameters) \n",
    "\n",
    "absError = modelPredictions - yData\n",
    "\n",
    "SE = numpy.square(absError) # squared errors\n",
    "MSE = numpy.mean(SE) # mean squared errors\n",
    "RMSE = numpy.sqrt(MSE) # Root Mean Squared Error, RMSE\n",
    "Rsquared = 1.0 - (numpy.var(absError) / numpy.var(yData))\n",
    "print('RMSE:', RMSE)\n",
    "print('R-squared:', Rsquared)\n",
    "\n",
    "print()\n",
    "\n",
    "\n",
    "##########################################################\n",
    "# graphics output section\n",
    "def ModelAndScatterPlot(graphWidth, graphHeight):\n",
    "    f = plt.figure(figsize=(graphWidth/100.0, graphHeight/100.0), dpi=100)\n",
    "    axes = f.add_subplot(111)\n",
    "\n",
    "    # first the raw data as a scatter plot\n",
    "    axes.plot(xData, yData,  'D')\n",
    "\n",
    "    # create data for the fitted equation plot\n",
    "    xModel = numpy.linspace(min(xData), max(xData))\n",
    "    yModel = func(xModel, *fittedParameters)\n",
    "\n",
    "    # now the model as a line plot\n",
    "    axes.plot(xModel, yModel)\n",
    "\n",
    "    # now add individual line for each point\n",
    "    for i in range(len(xData)):\n",
    "        lineXdata = (xData[i], xData[i]) # same X\n",
    "        lineYdata = (yData[i], modelPredictions[i]) # different Y\n",
    "        plt.plot(lineXdata, lineYdata)\n",
    "\n",
    "    axes.set_xlabel('X Data') # X axis data label\n",
    "    axes.set_ylabel('Y Data') # Y axis data label\n",
    "\n",
    "    plt.show()\n",
    "    plt.close('all') # clean up after using pyplot\n",
    "\n",
    "graphWidth = 800\n",
    "graphHeight = 600\n",
    "ModelAndScatterPlot(graphWidth, graphHeight)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Process\n",
    "\n",
    "1. Get sample data\n",
    "2. Split it into a training and test set\n",
    "3. Design a model that predicts outcomes for the training set\n",
    "4. Test the model against the test set\n",
    "5. Use the model to predict future \"real-world\" values"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Variables\n",
    "\n",
    "The dependent (predicted) variable is often written as $Y$, and the independent (predictors) variables are often represented as $x_1, x_2, x_3, ..., x_n$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Simple Linear Regression\n",
    "\n",
    "Simple Linear Regression is also known as **Classic Linear Regression** or **Least Squares Method**.\n",
    "\n",
    "Simple linear regression for a population (hence in Greek) is mathematically written as:\n",
    "$$\n",
    "y = \\beta_0 + \\beta_1 x_1 + \\epsilon\n",
    "$$\n",
    "\n",
    "Where:\n",
    "- $y$ - dependent variable (predicted value / output)\n",
    "- $\\beta_0$ - constant (minimum value of $y$ even if other terms are 0)\n",
    "- $\\beta_1$ - independent variable coefficient (quantifies effect of $x_1$ on $y$\n",
    "- $x_1$ - independent variable (predictor / input)\n",
    "- $\\epsilon$ - error (how far off the observed income is from the prediction)\n",
    "\n",
    "The $\\epsilon$ is also called the residual. The sum of all residuals across all observations is 0.\n",
    "\n",
    "For a sample, we similarly write the equation as:\n",
    "\n",
    "$$\n",
    "\\hat{y} = b_0 + b_1 x_1\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Methods for Minimizing Distance\n",
    "\n",
    "There are several methods for minimizing the distance between your regression line and the data points:\n",
    "\n",
    "- Least Squares Method (Sum of Squared Errors)\n",
    "- Sum of Absolute Errors"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Least Squares Method"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This method fits the regression line by minimizing the sum of the squares of the residuals. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Regression Evaluation Metrics\n",
    "\n",
    "There are three primary ways to evaluate your regression line:\n",
    "- Mean Absolute Error\n",
    "- Mean Squared Error\n",
    "- Root Mean Squared Error"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Mean Absolute Error\n",
    "\n",
    "**Mean Absolute Error** (MAE) is the mean of the absolute value of the errors:\n",
    "\n",
    "$$\\frac 1n\\sum_{i=1}^n|y_i-\\hat{y}_i|$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Mean Squared Error\n",
    "\n",
    "\n",
    "**Mean Squared Error** (MSE) is the mean of the squared errors:\n",
    "\n",
    "$$\\frac 1n\\sum_{i=1}^n(y_i-\\hat{y}_i)^2$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Root Mean Squared Error\n",
    "\n",
    "**Root Mean Squared Error** (RMSE) is the square root of the mean of the squared errors:\n",
    "\n",
    "$$\\sqrt{\\frac 1n\\sum_{i=1}^n(y_i-\\hat{y}_i)^2}$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Comparing\n",
    "\n",
    "Comparing these metrics:\n",
    "\n",
    "- **MAE** is the easiest to understand, because it's the average error.\n",
    "- **MSE** is more popular than MAE, because MSE \"punishes\" larger errors, which tends to be useful in the real world.\n",
    "- **RMSE** is even more popular than MSE, because RMSE is interpretable in the \"y\" units.\n",
    "\n",
    "All of these are **loss functions**, because we want to minimize them."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Example"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "import seaborn as sns\n",
    "from sklearn.model_selection import train_test_split\n",
    "from sklearn.linear_model import LinearRegression\n",
    "\n",
    "%matplotlib inline\n",
    "plt.rcParams['figure.figsize'] = [9.5, 6]\n",
    "\n",
    "df = pd.read_csv( '../data/USA_housing.csv' )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "[Expore the USA_Housing data](../data/USA_Housing.ipynb)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Split the Data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Index(['Avg. Area Income', 'Avg. Area House Age', 'Avg. Area Number of Rooms',\n",
       "       'Avg. Area Number of Bedrooms', 'Area Population', 'Price', 'Address'],\n",
       "      dtype='object')"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.columns"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Create a new dataframe with just the columns we want as inputs\n",
    "X = df[ [ 'Avg. Area Income', 'Avg. Area House Age', 'Avg. Area Number of Rooms', 'Avg. Area Number of Bedrooms', 'Area Population' ] ]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [],
   "source": [
    "# The column we want to predict\n",
    "y = df[ 'Price' ]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [],
   "source": [
    "X_train, X_test, y_train, y_test = train_test_split( X, y, test_size=0.4, random_state=101)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Run Linear Regression"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "LinearRegression()"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lm = LinearRegression()\n",
    "lm.fit( X_train, y_train )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-2640159.796853739\n"
     ]
    }
   ],
   "source": [
    "# Print the intercept for our line\n",
    "print( lm.intercept_ )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[2.15282755e+01 1.64883282e+05 1.22368678e+05 2.23380186e+03\n",
      " 1.51504200e+01]\n"
     ]
    }
   ],
   "source": [
    "# Print the coefficients for each feature\n",
    "print( lm.coef_ )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Coeff</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>Avg. Area Income</th>\n",
       "      <td>21.528276</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Avg. Area House Age</th>\n",
       "      <td>164883.282027</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Avg. Area Number of Rooms</th>\n",
       "      <td>122368.678027</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Avg. Area Number of Bedrooms</th>\n",
       "      <td>2233.801864</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Area Population</th>\n",
       "      <td>15.150420</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                                      Coeff\n",
       "Avg. Area Income                  21.528276\n",
       "Avg. Area House Age           164883.282027\n",
       "Avg. Area Number of Rooms     122368.678027\n",
       "Avg. Area Number of Bedrooms    2233.801864\n",
       "Area Population                   15.150420"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Create a dataframe out of our coefficients (cdf = coefficients data frame)\n",
    "cdf = pd.DataFrame( data=lm.coef_, index=X.columns, columns=[ 'Coeff' ] )\n",
    "cdf"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Make Predictions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1260960.70567629,  827588.75560301, 1742421.24254363, ...,\n",
       "        372191.40626868, 1365217.15140901, 1914519.54178955])"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "predictions = lm.predict( X_test )\n",
    "predictions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x13a800a90>"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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EanftBNrDsRQADKg0w+7iunKiaj6CtSJp+c9NmloctGnc01dmn9fdj5/tSkBkkqaePJf5dZ0cEcVlv9A/BDcAMKDSZlaintdOrUgU/zWTE2Ud3FVe08rdyiuYDnzwej166qWudTk5hR+bhfHvr1wq6oE7tredVUnKfqE/CG4AYEClzayMma0qEvZ3Ox05dlEHd5VVLhVlqn/QH9xVr7XxH/+Vm65JDFQ2jXuruqUefS48YCmYNd/nyKd26MSF+Z63b/uc3g3kOjkuSsp+oT9YvwAAPZZmVUHa67TWe0QpegUd3FXW0TOVVc/3v37iwnzodOCk+TFewXTVxg16o1rTNSGbtVs9dGhn8/d6y/SzfQtufMGW83aE/QyKXqGjbBDSiVu/QHADAD3U7Q/DsG6px557OdME3nYH4G1qDNdLu18pqFurFLrBHxaYFGhGBaXdClaRDcENAORE1PLITjMIQb3IiHQ6MC8vWgO7qECTDE3+xAU31NwAQA/1ogB1ves9/FqVPBfNlkvFxPbusIxVsNMpWJ90zxPn6IoaIAQ3ANBDvShAzdLllNaYBf/tdPfjZzVmSaXG/eGNmRbfWYrNKpVLxcjs1qsL1TXTmaOO+fIc4I0yghsA6KFejeIfSxl3pP0Q8OtjVpxUra3EfuD3QzBLU1txsbuoyo26mEJEcHZDqZhqhpD/XOQPE4oBoIeyjuIPFquWGosm36jWIl83O1fR1JPnUs97WZFkVt/zNMguvXU51fOKXkF7t27Wl596PjQ48wPNux8/m+pa7IfKJ4IbAOixtKP4W4tYg9mIqNUKR45dTB3Y+Jyrf1CnyVTk1WItbDXnan7GJiorUzBrFghH7boqmGnFObqico5jKQDIodm5SmgRa1BYQWs7NSD+lN6oY5phUDDT1P4tuu+Z85G1OCvONYOVqOPDr9+5Qy/MHNDJ6X0ENjlGcAMAOeNnbNLUtLQGM1lrQIJTelcG/WwqxrJzmnryXGwtTvB7NzlR1gN3bF81vZm278HBsRQA5EzaYlZp9Qfy7FxFb11eSv0+wWMY/1pZ59Z4Y5J/IlT0xlRNcTyUeM2CSU5tDQeMUjCLPa4Lq59hk/fgIrgBgJxJe7TktzzfMv2sSm1MCw4ew0jS3q2b9ciplzLdazCWWcpY6xN5zWWncW9MV1+5ITbT0mqsURjdehdp6onIygwXjqUAIGeijpbGTCoVPZnq/5bVi4yd6v/Omum4puit+vWJC/Nt3nFdNzMti7UVvXl5SVdtDJ/XY9Kqx8a9Mf21K+u/n1LR06Zxb9VxUtxAv3KpSGAzZMjcAEDOTO3fEroQ85qip3s/sU2TE2XtmTmuhWr6rEaYhWpNH/jf/0TV2oquKXodX6/bastOS8vhGRcn6a133n1ssbbS7JhaqNZU9Ap6MLCkU1Joi7w3ZrRzDyGCGwDIGf8D+fDT51cFHJcWa832725Nxg0GBHnUbi7I7yTzv5f+v+975nzzqKtU9HT49m1kbYYQizMBIKfilmxKg7+0cr2ZpBdmDvT7NrBO4hZnkrkBgBwITiL2B8TFLdl88NDO0KMrvIvVCKOL4AYA+qx1ErE/fbg07oV2C90QKIBtPbpCHasRRhvBDYCRFJYp6VftRdhcm/qvnUyr605aP7QvL3U+V2YQbRr3NL5xQ/Pnt3frZp24MJ+Lnyf6j+AGwMiJypT4ehH0BIOrqMrH1oF4JungrvKq5ZujeCxV9ArNrjEpX4Eq8oGCYgAjJ6pQd9O4p7drK6sChqJX6GjAW9gHr6S262WCGYthXJZQbmRhHnvuZS07p4KZbvsbm/ST16qhwUtroCp1/jPDYIgrKCa4ATBybpl+NlNgUC4VdXJ6X+b3ifrgvWLD2MjXyRS9MUnWcVAS11HWzs8MgyMuuGFCMYCRk7WLpt2ZMlG1NHGBTXP68IApl4ra875rUz3XGzM9cMcHu7KYMq6jDKOLmhsAIydsAnBcRqXdluKsH7BFb0zvLLmBzeqcf/WvEp9TbjlW6vS4LyoDRxv4aCO4ATByggW5SbUwnbQUZ92y3Y2N2v2S9PvsZh1M2HFf63vRBj7aCG4AjKTJiXLkB223Om+idkSNmoJZpsAmqfsprkusNTOE0URwAwABcUFPlLgP4yu9sZEObrJmbOLa9P1rRB33mUQRMSQR3ABAR6I+jE+/+LqOnqmMdGDTzmLKqCLse544J6ke4EQd91FnAx/dUgDQgfueOR/6YfzYcy+PdGAjSVddsSFTxiaqrVuSlp3Tl596XrNzFU3t36KiV1j1uEnau3Vzp7eMIUFwAwAt/A/aW6af1Z6Z45qdq0Q+L2z3k1T/MB51abvF/OxXUlFytbasI8cuanKirIO7yrLAY07S0TOVyJ8VRgvBDQAEBD9ond49Zgr70Dxy7GLvb3CApD0myrJGwg+YTlyYX9MG7gc/AMENAARE1XyEfWgyKC5e2mOiLN9HP2BieB/iENwAQECWD00KWOM9/mcvxx4T+cd/aQ/wgvNror73/EwgEdwAwCpZPjTDClvxrtqK0+Gnz4c+llRn49fTFKz+X63rGcK+9wzvg49WcAAICBu8Z6rX3uyZOR66OuDLT/35QE8XXk+tqyT8mUBxxcNpBvFFTZlmeB+kFMGNmX1D0scl/cw598shj39Y0v8t6YXGl55yzn2ti/cIAB1LmnrrC35oVhaqMql5bBKcYXPiwnzzWoMY2HgFU225tx1dSWsTpGyD+NoZuIjRkOZY6g8lfTThOd91zu1s/ENgAyBXsnRASfUPzZPT+1QuFUM7ch459dKqaw2ibgQ25VJRn7vtpubRUZhN4+9uOE/TFUXNDLohMbhxzn1H0us9uBcAWBdZOqB8s3OVgQ1cesGvb7l/crv+8oGP6aFDO+UV1gY5lxZrzVlBaTqZGMSHbuhWQfGvmtk5M/sTM9vWpWsCQFdkbRv2Mz2I1rovanKirCOf2qFyI/MSDHP8TNk1RU9JTlyY7/atYgR1I7j5vqT3Oud2SPrXkmajnmhmXzSz02Z2en6e/wED6I2sbcNZhsqNonKpGFmvFHecZ6bE7jLm1KAbOg5unHM/d8692fjvb0nyzOy6iOc+7Jzb7ZzbvXkzqUcAvZG1bZgP2Ghx37ek/VALizU9cMf2ZnYnDDU36IaOW8HN7K9L+v+cc87MPqR6wPRax3cGAF0S1zYc1kUVtXV61BXM1hxH+dJ0Qt3QyPj43/fW5zOnBt2SphX8MUkflnSdmb0i6V5JniQ5535f0qck/QMzW5JUlfQZ59gYByBfwtqGWz9g/dqQg7vKevTUS6kn546KFeciW6+TjvJaAxfm1GA9JQY3zrm7Eh7/XUm/27U7AoAuSDPXJqqL6j+f+ymBTYi4I6O4o7yooXzMqcF6YUIxgKETlZGRtOrDNOoDeaFa06ZxT5cWa6GPD6uCmZYjEu9JR0ZRR3nlUjH1UD6gWwhuAAydpLk2fkZn1fjhFs7VP9BHqWtqJSKwKRU9Hb59W2yWJWxtBTU06BeCGwBDJyoj42dwmh/AMWdPb1RrevDQzmYgVPTGtDiAaxayaP12XLWxoH/+P4cXELeihgZ5QnADIPfS1M/MzlV03zPnY4+SCmapMzFhnT2j5u3aSqbghBoa5AXBDYC2pF1E2Y33SaqfmZ2raOrJc7H7krIeMY1vHNP7vvytyBqUUTDKv3cMNoIbAJmlLdhNe624ICmqfuaeJ841f33PE+diP4hLRU9myhTc/L8/eyvD72I4xS3EBPKM4AZAZnEFu1mCm7ggyX+fqGF6y87pHz9xVispkguXl1ZGqjDYF1Mvvcp4RD3RXbfe2PV7AnqB4AZAZlkXUUrhGZqoIOm+Z87r7VpyQJImsPGvOUr8uTKnX3xdj5x6Kfa5Ra+g/+OO7Tr94ut67LmXteycCma669Ybdf/k9h7dMdBdBDcAMouaaRI15C0qQxMVdIzafJluKhW9Zvv10TOV2OcGh+tNTpQJZjA0CG4AZJZ1pklUhiZuaBzas1Ct6ctPPa8rNowlZqwYrodhRXADILOsM02ijquWnRu5QXm9UK0tJ35P4zZzA4OO4AZAW7LMNLmm6GmhuvaoqVT09PEd1yfWhaD79m7d3O9bANbNWL9vAMDwi+ooNpNOXJjv7c1AEt93DDcyNwDWRbA7KqqqZmGxpgWKh/sirrMNGHQENwC6rrU7KopT/CZqrJ+ozjZgGHAsBaDrwrqjohDY9B7bujHsyNwAiBU2fE+K75TiyCM/il5BB3eVdeLCPNu6MTIIbgBEChu+N/XNc5KpuaQybK9U1JA/ZGeSSuOe3nx7SbXASGaT9Dd/8Sr96Gdvralp8tculAlkMKIIbgBECjteqoXsPGjdKxU25A/ZFcy04pzGN27QgQ9eH5p96dV2dmCQENwAiJTleCn43OCQPzI47fPrkSoLVR09U9EDd2xfE7hkmTcEjAoKigFEytJR0/rcyYmyTk7vYxJuwKZxTz+ZOaCfzBzQQ4d2Znqtnx0DkIzgBkCkqf1bVPQKic+L676Z2r9FXiFiit+IWVisaXauoomvfVtfevxs5tdTqA2kw7EUgEj+ccfhp8+Hrk+Q6nUhv3LTNTpy7KLufvysbigVtXfr5mZ9yPjGQrP4eNSNbyzo7sfPRg41TMJsGiAdMjcAYk1OlHXVFdF/D1p2Tif/8nVVGpOIKwtVPXLqpeav33qHomLfW+8stx3YeGPGbBogJTI3ABJxHNIbcdOar75yw5rCYTqlgHBkbgAk4jikN75+5w5FVSe17uDyZxAFM2Zffup5zc5V1v0+gbwjuAFGzOxcRXtmjuuW6We1Z+Z4qg/DtIXFaN+Y1Y8AowLJ1q+HzSCiowqo41gKGCFhE4eD04Wjjjn8o45/9p+ep4Zmnfzq37hWe2aOq7JQbU4Y9oV1o0UdFXKECBDcACMl6W/7cYGPRHFwEm/MVq2mSGvP+67V9196o/m9d0peoRC14oIjRIBjKWCkxP1tPyrw+Wf/6XntvK+9uSyjwlQPQo58eoeOfGqHNo17qV5X9Ap66NBO/eS16prvvVO9wNj/2bQeH4YdFbLtG6gjcwOMkKi/7Y+ZRa5JqGdryNjEeWHmwKojvTFLN7TQz5pFfe+D6xdas2jBFRd0SwGrmYtoO1xvu3fvdqdPn+7LewOjIKx+RhILLddJa53Mery2XCrq5PS+Nt8FGC5mdsY5tzvsMY6lgCEU1SYsSQ/csV3lUlGm+rEHuqOTvyb6NTZJKBYG0iG4AYZQXOGwv9DyhZkDWulT5nYUJQUvfvFwXNBJsTCQDsENMITStgnzYdkbJumzt92kn8wciNyS7h85vTBzQF+/cwfFwkAHCG6AIRQVtIyZrRrex4dlbzhJJy7MS5L2bt0c+pzg1ycnyquOD8uloh64YzvFwkBKdEsBOdbu7qCp/VtCC4dbu28euGO7SkUvcuM3usfPmvlBTqvWrweHJwLIhswNkFOd7A6anCjr4K5ybJ2HX4Pz8R3XpypmHWWtR0TtfL/8bBqThYH1R3AD5FSnu4NOXJhP7OCpLFR19Eylo06fYVcqemuOiB48tFMPHdpZn0icQrBeJu3uKADt41gKyKlO/4af5nljJmbexPDGTIdv3xZ7RBQ3hE9auz4h7MiQYmGguwhugJzqdHfQNSlqaVZI2YQyKVWNkx/0tC4kleoBS1gRMJOFgfXHhGIgp8I+ML0x09VXbtDCYi3xQ3Hia9/WpUUKhbMqFT2dvfcjsc+Jmv5MwAL0TtyEYoIbIMeCH6LXFD299c7Smo3TpaLXPDoJumX6WWpp2rBp3NPcV6ODmyxZGgDrJy644VgKyDH/w/Lw0+cjj5gWqjVNPXlOh58+rzeq72Z0oo61EG8hIduVNP0ZQP/RLQXk2OxcRVPfPJdYO1Nbdlqo1la1jO/dunlNCzOSJdU00coN5B/BDZBjR45dVK2Nqt9qbVmPnnpJxsFUZlEThH20cgP5R3AD9NnsXEV7Zo6vWovg6yQb4CQt1la6cIejJWqCsG9q/xb2PgE5R80N0Eetxan+kZJUr7ehbqb3kgJKWrmB/CO4Afooqjj18NPnNTlR1tT+LZr65rm2jqYQr2DW3LUVlOZ4ib1PQL5xLAX0UVSWYKFa0+xcRZMTZR359A6Ne/xRTZJl31PRK+iuW2/keAkYUmRugDa0u6279bVjEdkDSbrvmfPN0f4stkwWl9vyCqarNm5Y1So/OVHW7vdey/ESMIQIboCMkupksrw2KrCRpEuLteaEYQ6l2rdp3JNzWhPYSOmPlzoJZgH0HrluIKNOtnWHvRbr6+3aypoZQMGOtCR+QFpZqLZ9DQC9RXADZNTJEDcGvfVeu4Gor5NgFkB/ENwAGXUyxC3qOVdtLKhg9coa/99YP1mCTCYSA4OH4AbIqJMhbmGv9Qqmd5ZWmvU3cXU46I4s04SZSAwMHoIbIKPJibIeuGO7yqWiTFK5VEy9ETrstVdt3MAcm3XUmgfL2u7NRGJg8CR2S5nZNyR9XNLPnHO/HPK4SfqXkj4maVHS551z3+/2jQL9FNYtc3J6X1euc/fjZ7t/w5BUD0IO7irrxIX5tjudmEgMDB5zCSlwM/t1SW9K+qOI4OZjkn5H9eDmVkn/0jl3a9Ib7969250+fbqtmwZ6qbV9W6p/aKbN1iRd54oNY4lbvxHPVG+Xj2v7BjBczOyMc2532GOJmRvn3HfM7OaYp3xS9cDHSTplZiUzu94599P2bhfondZMyt6tm9f8LT+uWybLB2fUdS4v0RqexA9eyo2fiUQmBUC0bgzxK0t6OfDrVxpfWxPcmNkXJX1Rkm666aYuvDXQvrBhfI+ceqn5uD/PJGouTdZumagFmJTbrFUqerrqig2xwQvBDIAoPZ1Q7Jx7WNLDUv1YqpfvDbRKM1CvWlvuaMGib3au0sw+IJ43Zjp8+zaCFwBt60a3VEXSjYFfv6fxNSDX0mZelp3ruFvm8NPnCWxSuvrKDQQ2ADrSjczN05J+28z+WPWC4jeot8EguKFUjDwqCioHam/S1HiE1fFQMJzewiLfKwCdSczcmNljkv5U0hYze8XMvmBmf9/M/n7jKd+S9GNJP5L0f0n639btboEuCptf0iprhiZsD9GjgToeJGM4HoBOpemWuivhcSfpt7p2R0CPhM0vufkXijr140tadk4FMx3cVX9O2i3gYXU8o3AcVSp6OnvvRzTxtW83N5m3g+F4ALqBCcUYaZMTZZ2c3qcXZg5oav8Wff+lN1atQTh6pqL7njmfenHiKO4b8guAJeneT2yTV2hvN1bBLPPsIAAI09NuKSDPoubQZGkFT1vHMyzMpNqKawZ6kxNlnX7x9VUt9WmtOEdgA6AryNxgZMzOVbRn5rhumX5We2aOa3ZudVNf1qzLmNmaa6Sp4xkEY1b/J45J8jvk/aO62bmKTlyYb+s9qbUB0C2J6xfWC+sX0Ethqw+8gumqjRuao/rfurwU2tVUKnq6vLQSmsEJW8Pgd0sNcgbnc7fdpN3vvTZyiGHUzJ5yqahXG8XUWbSzziKNsF1eZIeA4RC3foHMDUZC2JFTbdlpoVprdjW99c6SvJZ0RdEr6PDt2/TAHdtVsLWpjLDaG7+OpzzAmYgTF+ZXbTCX1Pz9l0vFyODFDyLS2DTuZd6qnkVY55qfXQIw3Ki5wUhIk0WpLTtdsWFMBadV3VL+h27U9u7KQlV7Zo6vyg5I0uI7S127/16rLFT1ldnnm3u2yi1Zjz0zx0O/p/7v/0spNp2Pb9ygua9+pNu33tStnWAABg/BDYZeltUHl5dWmv/td0tJ9UxG3Ov9D/rKQjXVB/sgCNuzJdUzU1P7t4RuOPcDoDRFxevdWRZ1/VHsaANGDcdSGHpHjl1se9ZMtbasR0+9NND1M90SPIILHlmFHS3dP7ldDx3aGXs0t94FxFHXp3AZGH5kbjD0Ov2b+igM4Usr+L2cnCjHHu/4j4cVc/diWF9cdgnAcCO4wdAbtdkz66mdrEfYJOhedC31630B9B+t4Bh6YZkDZLde7doA0I64VnAyNxgKcfNMgn+DJ4OTnUlkPQAMFDI3GHhxA/oWqjUVzLTsnMocT62yadzT27VlVWsrkc8pl4o6Ob2vh3cFAOmQucFQC1ts6Q/ok9RchElg867P3XaT7p/cLil6Zo1J2rt185oZPmRvAOQdreAYaLNzFV1aXLsyAfGC+5/C9mGZpF9737U6eqbChF8AA4fgBgPtvmfO9/sWBlJrS3frzJoHD+3UT16rRk74BYA8I7jBQCNr057gRvOoYmwm/AIYVNTcAANszCTnsg8aXHZOX37qeZ1+8XUdPVNpZmiCaxai5gMx4RdA3pG5wUCYnatoz8xx3TL9rPbMHG9mHUpFr6PrmtXrSzaNewP5h2ElIbCxxj9hqrVlPfbcy5FHT2G1OEz4BTAIBvH/zzFi/FbvYGHrlx4/q4mvfVsf33G9vLGoj+8UnPTCzAHNffUjuma8/UCpgztYNxsLpg0Fiw1+liNGQby6UE3cHwUAecWxFHIvrNVbqtfbHD1T0aEP3agTF+b1aiP4ycKp3go9tX+LFjqo38nj/qnasku8L38GUCv/6Clpf5QUP0ARAPqBzA1yLanVu1pb1okL8zo5vU8PHtqpgmXPofh1JqUOMjd5lBTYFL2C7rr1xo6OnsKyarSLA+g3ghvkWpq241cXqs0P2ahjliTV2vJIdV5tGvf0wB3bdf/k9o6Ono4cu0i7OIDc4VgKufCV2ef12HMva9k5Fcx016036v7J7ammCt9QKoZ+yIaJOoYZFOPemN5eWtFKyt+CKTyD83Zg5UKao6cotIsDyCMyN+i7r8w+r0dOvdQMOpad0yOnXtJXZp9PPGbyj1DSfpgOamBTMNPnbrtJTpY6sCl6BX32tptCv4fdyq5EtYXTLg6gnwhu0HePPfdy5NfjgpHgEcqwf5iuOKcTF+Zjs1Obxr01x0v3T27XSkxHVKdoFweQRxxLoe+iApi4Td6t26qn9m9Zsxl8mNxQKsYGI0WvoHs/sa15vOR3MN39+FmNJXREdcJ/P7qlAOQJwQ36LqoOxkx66/LSmq+3Zgb8D/Jqbbl5rXKpqL1bN7fdIp4n/u/3yLGLoYFewWxVEbBfXO0HemHf225mVzqp2QGA9UBwg67oZNbJXbfeqEdOvbTm685JC9XVHUybxr01GYrWD3L/gzv4/jvv+/aaaw2Ccsv3sjU7VfQKa7qbooqrC2ZacY7sCoChR3CDjrUGGMH9RGk+QO+f3C5Jq7qlxswp0NDTtLBY0+kXX28GUmFHLn6xbDAAamP8Td/5gY1/vHRDqaiDu8rNbFRUkBJ1fLXinF6YOdCLWweAviK4QcfiZp2kCW5m5yo6cWFeK43jpKn9W/Slx8+GPtdJq7I8UfU6/vFNa+AVxRoLKPOi6BW0d+vmNUHj0TOVxDk0LLwEMOrolkLHOpl1EjXhtlN++3Oa+TeloqcH79zZ2Y6qLvI7ncK6o9K0cNPBBGDUEdygY53MOonK+nTKz+ikCbDeqNY0OVHW1VdmS2Ray787VS4V9ZOZAzo5vU+TE+W2g0YWXgIYdRxLoSOzc5VUHU1R1muS7abGnqioI5ogPwhLszjTb02P6sqKartOo/U+OzleooMJwCgjc4NEs3MV7Zk5rlumn9WemePNpYj+kVJYR1PaTMF6Lau8tFjTzvu+rb1bN685ogkKBmFpggY/2PADGL8OZmr/Fr0wc0Bfv3NH6PttGvdUKsb/XlsnCXO8BADtIbhBrLitz1H1LOMbNyQGNn7AtJ7LKheqNT3+Zy/r4K5y84jGDzLCjmv2bt3c1vsE62DCjoQeOrRTc1/9iA7fvi020GrN+HC8BADt4VgKq7TOq3nr8lJkUWu7NSFpO5i6obbi9NSZV7Tpqisk1QOvsPbp2bmKHg2ZtZNW8PccdSTkf+2eJ86FHl2VQzJHHC8BQHYEN2gKm1cTxQ9+2qkJSbvBu1sWaytabNxnZaGqqW+e033PnNfCYk03NGpmjp6pdDTF2EnaM3M8cThe3DA+jpsAoDs4lkJTlqDDHyDXesxiqgcQwdqcVkmZnaJX0Hp2ZddWnC4t1prHbI+eeqkrwVbwyC4Ox00AsL7I3KApbedS63oDf+eRSc3sR9yU4muKXuQqhKQhflJ9Lk03Vyl0c3Zftbase544J2n17ztsPUVw8ScAoHvI3KAp6jhp07gXmWWYnCjr5PQ+lUvFNUGC/0HfmsmIWoWwadxrzngJqz+R6h1Fh2/fpocO7Yx8Tr8tO7cqgxNXlA0A6D6CGzRFtR7f+4ltOjm9Ty8EBsy1isr6tH7QS4rskArOmQm7l+D1JOnk9D49dCh6snA/90kFO6ji1lNI0a32AID2ENygqZNakLgi4tYP8qiYw79GXJt56/UkKWS/psYs++TgglmzXbwb/IAvrquMrA4AdB81N1il3dbjqf1bYtu7/Q/4I8cuhta4WOMaadvEg9dbXll7xZAvJfK3Zndr/o4frMV1lXW6dBQAsBaZmxGynscfftandcquz/+gj2ovd41rpO3YGjPTLdPPJq5WyMK/x6TC6nKpmJjdCbZ2x00a7mTpKAAgHMHNiOjV8cf/ELJ80v8gjzuSKicEP62Wnetql1PaNQxFr6C9WzfrzbfX7tPyhRVdRx33dbJ0FAAQjmOpEbHexx9Rx0mbxj3d+4ltmpwoa8/M8dgjKUnNhZRhTEq9mHLM6teqBc6ngq3qQQWzVcFI1BGb/3s5cuziquv6SkVPZ+/9SOj9RB33hb0XA/0AoDNkbkZEO8cfWY6xoo6T/Dhkz8zxxCMpae1+paAXZg5oJeXG7b9960068ukdq7Iln73tptDjobtuvVFHjl1s/j5Pv/i6rvTe/aNRKnrN/VCTE+XI79kbbczeYaAfAHQfmZsRkXVVQtgqhqihfFJ0kLRQrWnqyXOqLUcHJcF5NeWI+5TqAVJp3EtV7Hviwrzun1wbJOx+77Wrhun5qxeCv89HWnZMXV5a3Y/V7tqJKOyPAoDuInMzIuKKWsNEHWPd98z50GxO3Ad7XGDTeg9R822keuDx5ttL8grJTd5hwVbYlOATF+YTC5hbW8+zfi8BAL1F5mZEBFclBD/cozIGUZmYS4u1ZuYkmM1JWpkQphxyD60rHVrVVpxKRU9XXbFBry5UI2twWoOt2bnKqgxSZaGamFEKat367d9jmu8lAKC3zKWsYei23bt3u9OnT/flvZEsrkamVblU1MnpfZr42rdTz4fxXxPnlulnIwuQX5g5ICm8kLnoFdbUrUTd25ilm4mT5n4BAL1jZmecc7vDHuNYCqHijoda+VmNez+xLdVrvDFLdYSTpk06bUFuVNC14pR4zxw5AcBg4VgKocKOXt66vBS6jdsPNlpfE3VkdPWVG0KPcFprYlqLfaXwQKPTgtwH7ti+5n1PXJjnyAkABhTHUkgt7RGQL+pYSaofLQUDh6g5OVdtLMgrjOmNaq2jQGPnfd8ODcziZtMAAPIr7liKzA1Sy1pIG9UyLWnVlGT/mmFdS2+9s6yiJz14aGdbQY2fDQoLbLwx0+Hbt2W+JgAg38jcoCmsVbqT45i0SzDLpaJebayFiHtO1oLesPf3pxSHdWpluS6dUgDQXx0XFJvZR83sopn9yMymQx7/vJnNm9nZxj9/r9ObRm+tx+6p1mLfKJVGfU6cdhZJhmWDnOrrFl5dqOrIsYuRv7+o6cy92tEFAGhfYubGzAqS/kLS35L0iqTvSbrLOfeDwHM+L2m3c+63074xmZt8iWr97mYLdJb28lYFM339zh2ZMiRxNT++sExOXG1R1PwdWsUBoLc6rbn5kKQfOed+3LjYH0v6pKQfxL4KmXVy3NHpUUlU0NFOxiTK1P4tiYPzopZbLjsXu/4hTFzNj89/r6T6H39KcTs7ugAAvZXmWKos6eXAr19pfK3VQTP7czN70sxuDLuQmX3RzE6b2en5+fk2bnd4dXLc0elRyexcJfLYKMu+pNajnK/MPr/q15J01cbkePqhQztVCDmmal2DkCTLrJ7g9eMCmDSzdwAA/dWtIX7PSLrZOfdBSf9V0r8Pe5Jz7mHn3G7n3O7Nmzd36a2HQ1y2YD1fK0mHnz4feXyz+M5S2wHWI6deWhNwhXUtBd1QKmpyohy5/TtLhqS15icsYAq7flwAw14pAMi/NMFNRVIwE/OexteanHOvOecuN375B5J2def2Rkcnxx2dvHZ2rhIbcFxarKXKAkW1cgdVa8uxAUYwSOhWhmRyoqyT0/v0wswBff3OHYmZnKQAJu1EZABA/6SpufmepPeb2S2qBzWfkfS3g08ws+udcz9t/PJ2ST/s6l2OgKj6kDQf5p28Nk12x88CxX2Ap82oLDunoldYEwiVip4O376t+R5T+7eEFvV2kiFpXcrZWt8TDGD854XVMHU6ERkAsL4Sgxvn3JKZ/bakY5IKkr7hnDtvZl+TdNo597Skf2hmt0takvS6pM+v4z0PpU4+zDt5bdqgJOl5aYp3pXe7kpKKn9dr83YwMIkrwiaAAYDBxRC/HOlHt1TaTd5Jrc5pBvbFrWoAACAL1i8MiCzZgrAlk2me1xr0pIlt02SBwjItLKAEAPQDmZsBlDZLcnBXOXSrdjB7kmW5ZZb7Yz0BAGA9kbkZMmk7kx577mUttwSvrcXBUbUy7U7cbQ28gsPxCHAAAL3QrTk36KEsnUlJr293bkvU7qVOZ+4AANApMjcDKG1nUsEsNMAJtoi305UUl51hPQEAoN8IbgZEsI7lmqInr2CxO5riam5aszJh7dF3P342MtCJy850MnMHAIBu4FhqnUUd32S9RnC1wUK1tiqwKRU9fe62m9ZMzb1/cnumabppd1TFZWdYTwAA6DcyN+uoW8W1SQXEl5dWtPu91+r+ye1rHsvSXh6XkQleIy47s17D9wAASIvgpk1p2p3TBgtJkupV2rlmlvdp/XrSRGSm+wIA+oljqTZ04/gmizT1Kt0o2E27rJLlkQCAPCNz04ZuHN9kEZYp6fSaad8nql6G7AwAIK/I3LQhy/FNN4prg5kSqT45uNNrJr0PGRkAwKAic9OGtBmZbhbXpt1m3SkyMgCAQcduqTaE7XZi4zUAAL3Dbqku60W7M8sn6/g+AACyIrhp03oe37B8so7vAwCgHRQU5xDLJ+v4PgAA2kFwk0Msn6zj+wAAaAfHUn0SV0uSZT7OMNeksIQTANAOMjd9kDThOO18nLSTkgcVSzgBAO0guOmDpFqStMP0hr0mhaGCAIB2cCzVB2lqSdJ0Y41CTQpDBQEAWZG56YO0Cyp7dR0AAIYJwU0ftFtLMjtX0Z6Z47pl+lntmTmuvVs3U5MCAEALgps+aKeWJKx4+OiZig7uKlOTAgBAADU3fZK1liSqePjEhXmdnN7X7dsDAGBgkbkZEKNQPAwAQDeQuWlTr4fnMdAOAIB0yNy0oR/D8xhoBwBAOgQ3bejH8DwG2gEAkA7HUm3oV/0LA+0AAEhG5qYNDM8DACC/CG7aQP0LAAD5xbFUG/yjoV52SwEAgHQIbtpE/QsAAPnEsRQAABgqI5+56fUwPgAAsL5GOrjxh/H5M2v8YXySCHAAABhQI30s1Y9hfAAAYH2NdHDDMkoAAIbPSAc3DOMDAGD4jHRwwzA+AACGz0gXFDOMDwCA4TN0wU3W1m6G8QEAMFyGKrgZldZuZvMAABBtqIKbuNbubn349zuwGJUADgCAdg1VQfF6t3b7gUVloSqndwOL2blKV66fBrN5AACIN1TBzXq3duchsGA2DwAA8YYquFnv1u48BBbM5gEAIN5QBTeTE2U9cMd2lUtFmaRyqagH7tjetVqUPAQWzOYBACDeUBUUS+vb2j21f8uqYl6p94EFs3kAAIg3dMHNespLYMFsHgAAohHcZERgAQBAvg1VzQ0AAADBDQAAGCoENwAAYKikCm7M7KNmdtHMfmRm0yGPX2Fmjzcef87Mbu76nQIAAKSQGNyYWUHS70n6DUkfkHSXmX2g5WlfkHTJOfc3JT0o6f/s9o0CAACkkSZz8yFJP3LO/dg5946kP5b0yZbnfFLSv2/895OS/iczs+7dJgAAQDppgpuypJcDv36l8bXQ5zjnliS9IekXunGDAAAAWfS0oNjMvmhmp83s9Pz8fC/fGgAAjIg0wU1F0o2BX7+n8bXQ55jZBknXSHqt9ULOuYedc7udc7s3b97c3h0DAADESBPcfE/S+83sFjPbKOkzkp5uec7Tkv7Xxn9/StJx55zr3m0CAACkY2liEDP7mKSHJBUkfcM598/N7GuSTjvnnjazKyX9B0kTkl6X9Bnn3I8Trjkv6cUO71+SrpP037twHawffkb5xs8n//gZ5R8/o957r3Mu9BgoVXCTZ2Z22jm3u9/3gWj8jPKNn0/+8TPKP35G+cKEYgAAMFQIbgAAwFAZhuDm4X7fABLxM8o3fj75x88o//gZ5cjA19wAAAAEDUPmBgAAoGlgghs2k+dbip/P581s3szONv75e/24z1FmZt8ws5+Z2X+LeNzM7F81foZ/bma/0ut7HGUpfj4fNrM3An+Gvtrrexx1ZnajmZ0wsx+Y2Xkz+0chz+HPUQ4MRHDDZvJ8S/nzkaTHnXM7G//8QU9vEpL0h5I+GvP4b0h6f+OfL0r6tz24J7zrDxX/85Gk7wb+DH2tB/eE1ZYk3eOc+4Ck2yT9Vsj/1/HnKAcGIrgRm8nzLs3PB33mnPuO6kM2o3xS0h+5ulOSSmZ2fW/uDil+Pugz59xPnXPfb/z3X0n6odYukubPUQ4MSnDDZvJ8S/PzkaSDjTTtk2Z2Y8jj6K+0P0f0z6+a2Tkz+xMz29bvmxlljdKHCUnPtTzEn6McGJTgBoPvGUk3O+c+KOm/6t0sG4B0vq/6uPkdkv61pNn+3s7oMrOrJR2V9CXn3M/7fT9Ya1CCm65tJse6SPz5OOdec85dbvzyDyTt6tG9Ib00f87QJ865nzvn3mz897ckeWZ2XZ9va+SYmad6YPOoc+6pkKfw5ygHBiW4YTN5viX+fFrOnG9X/awa+fK0pL/T6Pa4TdIbzrmf9vumUGdmf92vIzSzD6n+/9/8Ba6HGt//fyfph865fxHxNP4c5cCGft9AGs65JTP7bUnH9O5m8vPBzeSq/w/uP5jZj9TYTN6/Ox4tKX8+/9DMble92+B1SZ/v2w2PKDN7TNKHJV1nZq9IuleSJ0nOud+X9C1JH5P0I0mLkv5uf+50NKX4+XxK0j8wsyVJVUmf4S9wPbdH0v8i6XkzO9v42j+VdJPEn6M8YUIxAAAYKoNyLAUAAJAKwQ0AABgqBDcAAGCoENwAAIChQnADAAB6KmlRbMjz7wwsLP2Pic+nWwoAAPSSmf26pDdV38P1ywnPfb+kJyTtc85dMrNfdM79LO41ZG4AAEBPhS2KNbP3mdl/MbMzZvZdM9vaeOg3Jf2ec+5S47WxgY1EcAMAAPLhYUm/45zbJemfSPo3ja//kqRfMrOTZnbKzD6adKGBmFAMAACGV2MZ6a9J+mZjy4gkXdH49wZJ71d9gvd7JH3HzLY75xairkdwAwAA+m1M0oJzbmfIY69Ies45V5P0gpn9herBzvfiLgYAANA3zrmfqx64fFqqLyk1sx2Nh2dVz9rIzK5T/Zjqx3HXI7gBAAA91VgU+6eStpjZK2b2BUmflfQFMzsn6bykTzaefkzSa2b2A0knJE05516LvT6t4AAAYJiQuQEAAEOF4AYAAAwVghsAADBUCG4AAMBQIbgBAABDheAGAAAMFYIbAAAwVAhuAADAUPn/ARDP5KEk+P+RAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 684x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Compare our predictions to the actual values for the test set\n",
    "plt.scatter( y_test, predictions )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A perfectly straight line would have been a perfect fit, but seeing a tight positive correlation like this shows that our model is pretty good."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<AxesSubplot:xlabel='Price', ylabel='Count'>"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 684x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Look at our residuals\n",
    "sns.histplot( data=(y_test - predictions), bins=50, kde=True )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Normally distributed residuals are good. That means that our model was a good choice for the data."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "MAE: 82288.22251914928\n",
      "MSE: 10460958907.208244\n",
      "RMSE: 102278.82922290538\n"
     ]
    }
   ],
   "source": [
    "from sklearn import metrics\n",
    "\n",
    "print('MAE:', metrics.mean_absolute_error(y_test, predictions))\n",
    "print('MSE:', metrics.mean_squared_error(y_test, predictions))\n",
    "print('RMSE:', np.sqrt(metrics.mean_squared_error(y_test, predictions)))"
   ]
  }
 ],
 "metadata": {
  "colab": {
   "name": "Linear Regression.ipynb",
   "provenance": [],
   "toc_visible": true
  },
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}