R/clases/3_estadistica/4_regresion_lineal.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Notas para contenedor de docker:**\n",
    "\n",
    "Comando de docker para ejecución de la nota de forma local:\n",
    "\n",
    "nota: cambiar `dir_montar` por la ruta de directorio que se desea mapear a `/datos` dentro del contenedor de docker.\n",
    "\n",
    "```\n",
    "dir_montar=<ruta completa de mi máquina a mi directorio>#aquí colocar la ruta al directorio a montar, por ejemplo: \n",
    "#dir_montar=/Users/erick/midirectorio.\n",
    "```\n",
    "\n",
    "Ejecutar:\n",
    "\n",
    "```\n",
    "VERSION=<versión imagen de docker>\n",
    "docker run --rm -v $dir_montar:/datos --name jupyterlab_prope_r_kernel_tidyverse -p 8888:8888 -d palmoreck/jupyterlab_prope_r_kernel_tidyverse:$VERSION\n",
    "\n",
    "```\n",
    "\n",
    "Ir a `localhost:8888` y escribir el password para jupyterlab: `qwerty`\n",
    "\n",
    "\n",
    "Detener el contenedor de docker:\n",
    "\n",
    "```\n",
    "docker stop jupyterlab_prope_r_kernel_tidyverse\n",
    "```\n",
    "\n",
    "Documentación de la imagen de docker `palmoreck/jupyterlab_prope_r_kernel_tidyverse:<versión imagen de docker>` en [liga](https://github.com/palmoreck/dockerfiles/tree/master/jupyterlab/prope_r_kernel_tidyverse)\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Para ejecución de la nota usar:\n",
    "\n",
    "[docker](https://www.docker.com/) (instalación de forma **local** con [Get docker](https://docs.docker.com/install/)) y ejecutar comandos que están al inicio de la nota de forma **local**. \n",
    "\n",
    "O bien dar click en alguno de los botones siguientes:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "[![Binder](https://mybinder.org/badge_logo.svg)](https://mybinder.org/v2/gh/palmoreck/dockerfiles-for-binder/jupyterlab_prope_r_kernel_tidyverse?urlpath=lab/tree/Propedeutico/R/clases/3_estadistica/4_regresion.ipynb) esta opción crea una máquina individual en un servidor de Google, clona el repositorio y permite la ejecución de los notebooks de jupyter."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "[![Run on Repl.it](https://repl.it/badge/github/palmoreck/dummy)](https://repl.it/languages/Rlang) esta opción no clona el repositorio, no ejecuta los notebooks de jupyter pero permite ejecución de instrucciones de Python de forma colaborativa con [repl.it](https://repl.it/). Al dar click se crearán nuevos ***repl*** debajo de sus users de ***repl.it***.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Lo siguiente está basado en los libros:\n",
    "\n",
    "### * B. Rumbos, Pensando Antes de Actuar: Fundamentos de Elección Racional, 2009\n",
    "\n",
    "### * G. J. Kerns, Introduction to Probability and Statistics Using R, 2014.\n",
    "\n",
    "### * M.D. Ugarte, A. F. Militino, A. T. Arnholt, Probability and Statistics with R\", 2ed.\n",
    "\n",
    "\n",
    "El libro de G. J. Kerns tiene github: [jkerns/IPSUR](https://github.com/gjkerns/IPSUR)\n",
    "\n",
    "El libro de M. D. Ugarte, A. F. Militino, A. T. Arnholttiene github de scripts [alanarnholt/PASWR2E-Rscripts](https://github.com/alanarnholt/PASWR2E-Rscripts)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Nota:**\n",
    "\n",
    "* En algunas líneas no es necesario colocar `print` y sólo se ha realizado para mostrar los resultados de las funciones en un formato similar al de R pues la nota se escribió con *jupyterlab* y *R*."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "library(ggplot2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "options(repr.plot.width=6, repr.plot.height=6) #esta línea sólo se ejecuta para jupyterlab con R"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Introducción"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Con frecuencia es necesario determinar si dos variables aleatorias están relacionadas de alguna manera. Por ejemplo ¿tendrán los años de educación efecto sobre el salario que percibe una persona? \n",
    "\n",
    "La relación entre dos variables cuantitativas puede visualizarse en un **diagrama de dispersión** en el plano, representando los valores de las variables en los ejes horizontal y vertical. También el coeficiente de correlación de Pearson ayuda a determinar si exsite alguna relación **lineal** entre las variables de modo que se escribe si están **positivamente, negativamente o no correlacionadas**. Ver [1_elementos_de_estadistica_descriptiva](https://github.com/ITAM-DS/Propedeutico/blob/master/R/clases/3_estadistica/1_elementos_de_estadistica_descriptiva.ipynb)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Un ejemplo de una relación lineal es considerar la altura y peso de una persona. Por ejemplo para los datos:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "heights <- c(1.94, 1.82, 1.75, 1.80, 1.62, 1.64, 1.68, 1.46, \n",
    "             1.50, 1.55, 1.72, 1.67, 1.57, 1.60, 1.70)\n",
    "weights <- c(98, 80, 72, 83, 65, 70, 67, 47, 45, 50, 70,\n",
    "             61, 50, 52, 200)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Se tiene la siguiente gráfica:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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O0BWgzQLgFaDNCAtgdoMUC7BGgxQAPa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\ngKa6CtBUVwGa6qr/BypbwcY0GOVLAAAAAElFTkSuQmCC",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 360,
       "width": 360
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "qplot(heights, weights, main = \"Relación entre altura y peso\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Se observa una clara relación **lineal** entre el peso y altura de una persona si no consideramos el punto que sobresale en todos los demás.  La altura y peso tienden a aumentar o disminuir juntos. Escribimos en este caso que ambas variables están **positivamente correlacionadas**. El dato atípico o *outlier* debe determinarse si ¿es un error de medición? ¿la persona es atípicamente obesa pues tiene una altura de $1.70$ y $200 kg$?."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "También puede darse la situación de que no exista relación alguna entre variables por ejemplo si a los datos de altura anteriores les graficamos con el ingreso de las personas se tiene:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "wages <- c(9500, 11000, 16000, 5000, 10000, 10500, 65000, 5000, 5000,\n",
    "           33000, 25000, 48000, 8000, 5000, 23000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "15"
      ],
      "text/latex": [
       "15"
      ],
      "text/markdown": [
       "15"
      ],
      "text/plain": [
       "[1] 15"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "length(wages)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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C2cAtoyQAsDNKANAVo4BbRlgBYGaEAb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\nAAAASUVORK5CYII=",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 360,
       "width": 360
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "qplot(heights, wages, main = \"Gráfica entre altura e ingreso\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "En esta gráfica no observamos relación alguna entre ingreso y altura de las personas. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Y la asociación más simple entre variables es cuando éstas se relacionan en forma lineal, sin embargo, no siempre es posible establecer este tipo de relación entre ellas. Por ejemplo, si tomamos el número promedio de infecciones respiratorias al año que sufre un niño de los 0 a los 12 años con los siguientes datos:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "ages <- c(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12)\n",
    "\n",
    "number_of_respiratory_infections <- c(2, 3, 5, 5, 8, 10, 15,\n",
    "                                      12, 8, 5, 3, 3, 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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j5wURDM+wzQgFa2ixC0eSj5z7Nfgk28XQXoFm8DWsgP6O7T0kFnCtC2bUAL+QE9\nrs3QkWbIyFSA1rYBLeQH9KrTv9aKLx8FtLZdhKCjM89mhgxo1zaghfyBPvcDQANa2S5S0MnW\nLlmyFtDyNqCFPIJ+d2iZMWXHvAdocRvQQv5AL+1sDh0//jDTeSmgtW1AC/kDPbrdgvBkQbsx\ngNa2AS3kD3TXC+PTC7oBWtsGtJA/0G3ujE9nWl7rG9DZbANayB/onmPj09N6AVrbBrSQP9AX\nmmlVQVA1xVwEaG0b0EL+QK/uayr2H1Bh+mXxarKAtm0DWsjj49Abru3fsaL/5A2ZPQPaug1o\nIX5ykjNAF2B7OwDNd6wo24AWArQzQBdgG9CuAK0EaEDbtgGtbAPaFaCVAA1o2zaglW1AuwK0\nEqABbdsGtLINaFeAVgJ0E9BpL43MT05StgEtxEsjOwN0AbaLEzQvjQxocbtIQfPSyIDWtosU\nNC+NDGhtu0hB87PtAK1tFyloXhoZ0Np2kYJuSYC2bQNayCfo6teeyuL7CQHd3DaghTyCnr+r\nMYuDj7vOA7S2DWghf6D/2OrAm5Kgg6OGA1rbBrSQP9BHDKyuCkFf3RvQ2jaghfyB7nhzEIGe\nVQ5obRvQQv5Al98Rg76+E6C1bUAL+QPdf1QEOjHoUEBr24AW8gf65rI5SdDrJ5i7Aa1tA1rI\nH+jq40w306etqawFtLYNaCGPj0PXzDx4p4oDbqvJ7BnQ1m1AC/HUtzNAF2Ab0K4ArQRoG+gP\nbpp4zk1ZvAYWoO3bgBbyBzrx8+jlvssuB7S4DWghjw/bmSP+sHTp4981twJa2wa0kMdvko2/\np3DroXsBWtsGtJA/0G1nxKe387Ju4jaghfyB3mNafDplT0Br24AW8gf6xp4rw5N/9bwR0No2\noIX8gA5/eMEjA7tc+tvfXvqVgY8AWtsGtJAf0CatzKCrXCUSzoszVF2dy9E5bm/dXreDHXW7\nNvq1CeiH0soMev0aR7W1rkszVVWVy9GJ3LY37Zjbm3Pbrslpe2MuRwfh9rqmoFsadzls29zl\nEOJrOZwBugDbRQt65eInnwgDtLYNaCF/oL8YXcbPtstpG9BC/kCPNCdNnxcFaG0b0EL+QFeM\nzQwZ0K5tQAv5A73zbYAGtLJdpKCHnQ5oQCvbRQp66a53ZfH93oBufhvQQh4ftnu4VcW3BoQB\nWtsGtJDH11gpM133iQK0tg1oIX+g9+3518ySAe3YBrSQP9DtpmXtGdDWbUAL+QPdZzKgAa1s\nFyno2/dcD+ictgEt5A/0Y4f1uuEhXngzh21AC/HCm84AXYDtIgXNC28CWtsuUtAtCdC2bUAL\nAdoZoAuwDWhXgFYCdFPQHesDtLYNaCGPXz4advw+pt8wQGvbgBbyfpfjkS5ZfEkHoG3bgBby\nfx/6x8cCWtsGtJB/0LdVAFrbBrRQHv4LzUsji9uAFvIH+tWoBRe0OgnQ2jaghfx/LcchHwFa\n2wa0kD/Qt4bd9l8vZ+YMaPs2oIV4ptAZoAuwDWhXgFYCtBV07ZrVYYDWtgEt5A907Z37tuUL\n/HPZBrSQP9CTTZ9RE6IArW0DWsgf6N3PzPongQHaug1oIX+g27yUtWdAW7cBLeTxtb4XABrQ\nynaRgp5yfALQOW0DWsjjd33vMWj6o9vFz+V4d2y39oc83pLth79d3v2M9zNv7/Cg/3F+j7b9\n5rRwu0hBbzc/l2PFftGtfDL77YejAwZ+knF7Rwe9cmj0N3VHy7aLFPR283M5psX/2vXPfnvv\n+IhbMm7v6KB/H/9Fdc74r37adpGCbkkFBT0q/ltv/Vm22x+l/s/zo4zbOzroq1J/Uy+1aBvQ\nrjKD/nH8l95hZbbbn7WJj/hJxu0dHfT1KdBvtWgb0K4yg34o/ksfkf328fERT2Tc3tFBv9Qu\n+os6oGXbgHaVxaMc54R/6XtaH7Swb/+tR3jEBZm3d3TQ8ecnXV5s2TagXWXzOPSj54z51YqW\nbP9z2uifZPzvM6CTPXPeyEn/v4XbgHbFEytKPLECaNs2oJVtQLsCtBKgAW3bBrSyDWhXgFYC\nNKBt24BWtgHtCtBKgAa0bRvQyjagXQFaCdCAtm0DWtkGtCtAKwEa0LZtQCvbgHYFaCVAA9q2\nDWhlG9CuAK0EaEDbtgGtbAPaFaCVAA1o2zaglW1AuwK0EqABbdsGtLINaFeAVgI0oG3bgFa2\nAe0K0EqABrRtG9DKNqBdAVoJ0IC2bQNa2Qa0K0ArARrQtm1AK9uAdgVoJUAD2rYNaGUb0K4A\nrQRoQNu2Aa1sA9oVoCr/RHgAAAnRSURBVJUADWjbNqCVbUC7ArQSoFsM+t3hwwDt3ga0UKFA\nr/3xZEBn2Aa0UIFAJ66Z/yigM2wDWqhAoOdfmUiBrl6b7It/O6qtdV2aqY0bczk6x+0NuRyd\nqMnl6E076HbV+lyODqqTv6xuOeg3T/8iSIF+5sBkL2f9bwKR72rr38oW9BfjXg/qQL85Mdnb\nWx0lEq5LM1VTk8vRAdvb2XZt7ttbWgz69cphw4adWDnsvrrf4D60bZv70EIFuQ9d9WGye4Z9\nuAbQrm1ACxXuiRUe5ci0DWghQDsDdAG2t1PQDQO0bRvQQoB2BugCbAPaFaCVAA1o2zaglW1A\nuwK0EqABbdsGtLINaFeAVgI0oG3bgFa2Ae0K0EqABrRtG9DKNqBdAVoJ0IC2bQNa2Qa0K0Ar\nARrQtm1AK9uAdgVoJUAD2rYNaGUb0K4ArQRoQNu2Aa1sA9oVoJUADWjbNqCVbUC7ArQSoAFt\n2wa0sg1oV4BWAjSgbduAVrYB7QrQSoAGtG0b0Mo2oF0BWgnQgLZtA1rZBrQrQCsBGtC2bUAr\n24B2BWglQAPatg1oZRvQrgCtBGhA27YBrWwD2hWglQBdPKBXTt+7TY/LV9SdBbRS4UAvGlzR\nedjrLTjgs6l7tOl93ad1Z7MA/cbJnTsc/rT1omIEPdmE/aDuLKCVCgb6xQ7hR+/r72d/xAXR\nx/ucurOZQS/rER5Q/pztsiIE/fd20Tto/pg6D2ilgoE+Nv7onZv1AUtax0e8kjqfGfTF8QFH\n2i4rQtB/jG+uuTF1HtBKBQO9W/zROzzrA+anPt73pM5nBj00PuArtsuKEPTzqXdwZuo8oJUK\nBrpP/NE7JusDHkt9vB9Inc8Menh8wO62y4oQ9L/2jG5uxV9T5wGtVDDQ58fabs/6gI92jQ7o\nsjx1PjPoWfHEeNtlRQh61dM7J29t27vrzgJaqWCgVxwUYjtpZfZHPBx+Gtm+7j/Q2TzKMSqc\nGPAP20XFCHrVe1ePuuil+nOAVircw3afzZzw03tbdMSSy0dd+kb9uWweh35gwukzPrVeUpSg\n0wO0Ek+sANq2DWhlG9CuAK0EaEDbtgGtbAPaFaCVAA1o2zaglW1AuwK0EqABbdsGtLINaFeA\nVgI0oG3bgFa2Ae0K0EqABrRtG9DKNqBdAVoJ0IC2bQNa2Qa0K0ArARrQtm1AK9uAdgVoJUAD\n2rYNaGUb0K4ArQRoQNu2Aa1sA9oVoJUADWjbNqCVbUC7ArQSoAFt2wa0sg1oV4BWAjSgbduA\nVrYB7QrQSoAGtG0b0Mo2oF0BWgnQgLZtA1rZBrQrQCsBGtC2bUAr24B2BWglQAPatg1oZRvQ\nrgCtBGhA27YBrWyXNug/H73TTke/oN9EQCsB2hfoNzuHr8G1yxvNXiFTgFYCtC/QY+JXSRwl\n30RAKwE6N9BVzTYgBt2v+WtkqLpaPjRZIpHL0dVbt9ftYEfdro1+zRn0+jXNdUgMelCzV8hU\nVZV8aLJEbS5HV23aMbc357Zdk9P2xlyODsLtdbmDbv4uxy9j0NfK/xPhLocSdzl8gf7se6Hn\nIZ/JNxHQSoD2BXrVyrnn/mROC17LvHGAVgK0N9A8saIFaCVAu7cBrWwD2hWglQANaNs2oJVt\nQLsCtBKgAW3bBrSyDWhXgFYCNKBt24BWtgHtCtBKgAa0bRvQyjagXQFaCdCAtm0DWtkGtCtA\nKwEa0LZtQCvbgHYFaCVAA9q2DWhlG9CuAK0EaEDbtgGtbAPaFaCVAA1o2zaglW1AuwK0EqAB\nbdsGtLINaFeAVgI0oG3bgFa2SwC0sxnTvfyxWTX9jsJt33Jn4bZ/fXfhtn81q3Db0+5JO+sH\n9PHHevljs+qYEwq3fdTwwm0PPqVw298ZVbjtg8emnQX0tgzQ+Q/QHgN0/gO0xwCd//ICmqhA\nAZpKKkBTSQVoKqm2LehXzzv5jPsS9nN+W3j12FPPf7ru3JOVYW/mZ7rxWj7f7eDCaPvETdZb\n4rH3p55ZeXv0Vv4/5vXbtg/6NgX93rDffLhoxDzrOc9dce+rf5td+VTq3JNjliWrch6x7Upf\ny+u7HawIp8dPtt4Sny2Z+6fxEaoCfMzrt20f9G0KeupPkr/ce+pm27l8dOVVqTeeHJe/0cZr\n+X+3l1a+ar0lnjsvQlWYj3m8HdXog75NQY/7z+Qv71S+YzuXjy65MfXGk8PHjf75C/maTV/L\n/7s9/ay6/8fn9f2OURXmY94AdKMP+rYEnah8JPnrp5UvWM7lo4XDP0i9tWTBe2/NqHw8T7tp\na/l/t9ePeNh6S3wXoSrQx/xL0I0/6CUE+vkRz6Wdv+H0fC03XMs/6MdOXmO9Jb4rCtBNPuil\nc5fjqRGL03/j8crqPE2nreX7Lkdi/M3N3BLPFcNdjqYf9JL5pPD+Uxs/WnVDXj8zrF/L9yeF\nrzfmk6/3uwg+KbR80Lf5w3bPhA/avHDpxgbn8tGs4U8tW7bsn6npOxa98+btlY/mZ/rLtfy/\n28mu+1l0kuf3e8uyZedMXfb3gnzM67dtH/Rt+8TKK+ed9KN7E+H/9dY2OJePxkSPqp+dmp41\nfsToS57Pz3KDtfy/20Gwclj8OGye3+9l0d/3sKAQH/P6bdsHnae+qaQCNJVUgKaSCtBUUgGa\nSipAU0kFaCqpAE0lFaCppAI0lVSAppIK0PlrzVWDurTtffH68O2PT9ul4+AXR7QL366+eUB5\nxeAF4Vu/6ltRsdcP1xX0Zm7fATp/vd114q0zR7Y6PBEEa/csmzjrp532C0HXHFc2csZNA1rN\nD4JLzJhZ//mLgSsKfUu34wCdvzZvDX+dYhYGwS9M+BOV/8uEoGeaOclftx6wa3XQe0ghb19J\nBOj8trXqHXNdEPT7ak3yTGL3EPTB3arCbjKvBQO7v1LoG7i9B+g8NvfQDibZhUHQ8ZDoN44K\nQe9kUj0VLOpivjFmzsbC3srtO0Dnr5tN5fw/LX7SnJ8EfWj0OxHoij6L41YHwboHJ+5nenAf\nWg/Q+Wu/3uG3cjwfgk7d5egRgj6g7fq0q91vLi3ErSuRAJ2/+vaqDoKaY0PQV5vw26PvjT4p\nvM1MiL5n6eMg+Dw8XW7OKuSt3M4DdP661nx/1s3fPigEvaZ363Nn/6zTfuXJ364+wQyaMusX\n3+8aBO1GTP3dTfu0fi7jH0XNBej8VX39nm17XLg8BB2sGL1zh+++MLRz+Pu1vzmkorzX8HlB\ncOVhX22z2/AXC3w7t+sAXch671/oW1ByAbowRT/y9gFzWaFvR8kF6MI05Md3zj679W4rC307\nSi5AF6YbB+z8H7v9+KNC34zSC9BUUgGaSipAU0kFaCqpAE0lFaCppAI0lVSAppLqfwEd5rQm\nxjYDFgAAAABJRU5ErkJggg==",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 360,
       "width": 360
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "qplot(ages, number_of_respiratory_infections, main=\"Enfermedades respiratorias\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Observamos en la gráfica anterior que existe relación y podría determinarse una curva con un valor máximo alrededor de los 6 años como una posible relación apropiada en este caso y no una recta."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Análisis de Regresión Lineal Simple (ARLS)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Objetivos en el ARLS"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "El **análisis de regresión** es la herramienta en Estadística que estudia las relaciones entre variables para:\n",
    "\n",
    "* Explicar/Estimar/Predecir/Controlar a una variable dependiente en términos de la variables independientes.\n",
    "\n",
    "* Identificar a las variables independientes estadísticamente significativas.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Modelo y suposiciones en el ARLS\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "En lo siguiente consideramos $x$ variable **no aleatoria**, puede ser cuantitativa (discreta o continua) o cualitativa (nominal u ordinal) que nombramos variable explicativa, independiente, regresora o predictora, $Y$ variable **aleatoria continua** que depende de $x$ y por tanto nombramos dependiente o respuesta.\n",
    "\n",
    "Como se describió en la sección anterior encontrar relaciones entre variables puede ser complicado y en el ARLS se considera a la cantidad $E[Y]$ asumiendo el modelo:\n",
    "\n",
    "$$E[Y] = \\mu(x | \\beta) = \\beta_0 + \\beta_1 x$$\n",
    "\n",
    "donde: $\\beta_0 , \\beta_1$ son parámetros desconocidos. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "\n",
    "**Observación** \n",
    "\n",
    "Si existe relación entre $Y$ y $x$ esta relación típicamente se escribe como $E[Y|x] = \\beta_0 + \\beta_1 x$ para denotar que la media de la variable $Y$ es función de x (la variable explicativa únicamente tiene efecto sobre la media de la distribución de $Y$). Si $Y$ no fuera función de $x$ entonces $\\bar{Y}$ sería una buena opción para los objetivos del ARLS $E[Y]$.\n",
    "\n",
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "En el ARLS asumimos que el valor aleatorio de $Y$ se distribuye como una Normal con media igual a la altura de la recta de regresión en el valor x, esto es, por cada valor de $x$ observamos un valor aleatorio de $Y$ que en promedio se encuentra localizado sobre la recta de regresión:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"https://dl.dropboxusercontent.com/s/v9qu9msa02dn0hn/regression_normal_curves.png?dl=0\" heigth=\"500\" width=\"500\">"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Gráfica creada con el código en [PASWR2E-chapter12](https://github.com/alanarnholt/PASWR2E-Rscripts/blob/master/ChapterScripts/chapter12.R). "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Además como se observa en la gráfica anterior asumimos que la varianza es idéntica para todos los valores de $x$ (obsérvese que las distribuciones normales para cualquier valor $x$ tienen misma varianza). Si se denota a cada punto donde se evalúan $E(Y|x_1), E(Y|x_2) \\dots, E(Y|x_m)$ como: $Y_1 = Y|x_1, Y_2 = Y|x_2 \\dots, Y_m = Y|x_m$ entonces tendremos un conjunto de puntos $(x_i, Y_i)$ en los que las $Y_i$'s tienen un ruido aleatorio y además asumimos $Y_i | x_i$ es independiente de $Y_j | x_j$ si $i \\neq j$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "El **modelo** considerando la nube de puntos $(x_i, Y_i)$ es:\n",
    "\n",
    "$$Y_i = \\beta_0 + \\beta_1 x_i + \\epsilon_i$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\forall i=1,\\dots, m$ donde: $Y_i$ variable aleatoria dependiente cuantitativa, $x_i$ variable no aleatoria explicativa de cualquier tipo (cuantitativa o cualitativa), $\\epsilon_i$ término de error aleatorio, $\\beta_0$ y $\\beta_1$ parámetros del modelo en ARLS."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Supuestos del modelo:**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "0)Los puntos $(x_i,Y_i)$  $\\forall i=1, \\dots,m$ cumplen que $Y_i$ es dependiente de $x_i$ y la nube de puntos son mediciones independientes."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1)$E[\\epsilon_i | x_i] = 0$ $\\forall i=1, \\dots, m$ la media de los errores es igual a cero. Este supuesto ayuda a tener correctamente determinado al coeficiente $\\beta_0$.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2)$\\text{Var}(\\epsilon_i|x_i) = \\sigma^2$ la varianza del error es homocedástica (misma dispersión) $\\forall i=1, \\dots, m$.\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "3)$\\text{Cov}(\\epsilon_i, \\epsilon_j | x_i, x_j) = 0$ para $i \\neq j$, la covarianza\\* entre los errores es cero.\n",
    "\n",
    "\\*La covarianza entre dos variables aleatorias $z_1, z_2$ se define como $\\text{Cov}(z_1, z_2) = E[(z_1 - E[z_1])(z_2 - E[z_2])]$ y mide la variabilidad conjunta entre $z_1, z_2$. Puede ser positiva (comportamiento similar: valores grandes en $z_1$ muestran valores grandes en $z_2$), negativa o cero y su magnitud no tiene directamente una interpretación, en este caso se utiliza la correlación que es una versión de la covarianza normalizada para medir la relación lineal entre $z_1, z_2$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "4)$\\epsilon_i | x_i \\sim N(0, \\sigma^2)$: los errores tienen una distribución normal, ver [2_distribuciones_comunes](https://github.com/ITAM-DS/Propedeutico/blob/main/R/clases/3_estadistica/2_distribuciones_comunes.ipynb). Por lo tanto los errores $\\epsilon_i$ son independientes e idénticamente distribuidas (i.i.d) $N(0,\\sigma^2)$ $\\forall i=1, \\dots, m$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Con los supuestos anteriores se tiene:\n",
    "\n",
    "* $Y_i|x_i \\sim N(\\beta_0 + \\beta_1 x_i, \\sigma^2).$ Obsérvese que las $Y_i$'s son independientes pero **no** son idénticamente distribuidas.\n",
    "\n",
    "* Modelar $E[Y|x]$ es equivalente a modelar $Y = E[Y|x] + \\epsilon$ con $\\epsilon$ error aleatorio.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "\n",
    "**Comentario** \n",
    "\n",
    "Aunque la relación entre las variables no sea inicialmente lineal, puede volverse lineal mediante alguna transformación. Muchas transformaciones útiles caen en la familia del nombre de [Power transform](https://en.wikipedia.org/wiki/Power_transform) entre las que se encuentran la de Box-Cox.\n",
    "\n",
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Mínimos cuadrados lineales y el ARLS: proceso de ajuste para $\\beta_0, \\beta_1$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "El ARLS no sólo se ocupa de la correlación lineal entre las variables, sino determina la línea recta que mejor aproxima a los puntos. El método más común para calcular tal recta minimiza las sumas de los cuadrados de las distancias verticales y se conoce como mínimos cuadrados lineales u ordinarios, ver [3_minimos_cuadrados](https://github.com/ITAM-DS/Propedeutico/blob/main/Python/clases/3_algebra_lineal/3_minimos_cuadrados.ipynb). En este método se calculan los cuadrados de las distancias de los datos a la recta para evitar que una distancia positiva y una negativa se cancelen por ejemplo:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "var1 <- c(1,2,3,4)\n",
    "var2 <- c(3,2,5,4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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RqzdGRQoUNpcwQOAEBesW1pw0eup1Ya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\neZMnEo13KB7N/vF4//WjAw61lVUBIH8QOACBardrVTVG3fHmHcr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      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 360,
       "width": 360
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "qplot(var1, var2) + \n",
    "geom_smooth(formula = \"y~x\", aes(x=var1, y=var2),method=\"lm\", se = FALSE)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Considerando un conjunto de observaciones de las $x_i$'s y de las $Y_i$'s**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Supóngase que se han realizado mediciones de un fenómeno de interés en diferentes puntos $x_i$'s resultando en cantidades $y_i$'s $\\forall i=1,\\dots, m$, esto es, se tienen $m$ puntos $(x_i,y_i)$ en los que  el punto $(x_i, y_i)$ es una medición independiente del punto $(x_j, y_j)$ con $i \\neq j$ y además las $y_i$'s contienen un ruido aleatorio causado por errores de medición:\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"https://dl.dropboxusercontent.com/s/z0ksltumd4ibyjp/mcuadrados_1.jpg?dl=0\" heigth=\"400\" width=\"400\">\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Entonces de acuerdo a [3_minimos_cuadrados](https://github.com/ITAM-DS/Propedeutico/blob/main/Python/clases/3_algebra_lineal/3_minimos_cuadrados.ipynb) el proceso de ajuste en ARLS consiste en resolver las **ecuaciones normales**:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$A^TA \\beta = A^Ty$$\n",
    "\n",
    "donde:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$A = \\left[\\begin{array}{cc}\n",
    "1 & x_1\\\\\n",
    "1 &x_2\\\\\n",
    "\\vdots &\\vdots\\\\\n",
    "1 &x_n\\\\\n",
    "\\vdots &\\vdots\\\\\n",
    "1 & x_{m-1}\\\\\n",
    "1 &x_m\n",
    "\\end{array}\n",
    "\\right] \\in \\mathbb{R}^{mx2},\n",
    "\\beta=\n",
    "\\left[\\begin{array}{c}\n",
    "\\beta_0\\\\\n",
    "\\beta_1\\\\\n",
    "\\end{array}\n",
    "\\right] \\in \\mathbb{R}^2,\n",
    "y=\n",
    "\\left[\\begin{array}{c}\n",
    "y_1\\\\\n",
    "y_2\\\\\n",
    "\\vdots \\\\\n",
    "y_m\n",
    "\\end{array}\n",
    "\\right] \\in \\mathbb{R}^m\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Otra forma es considerar la función de verosimilitud de los parámetros $\\beta_0, \\beta_1$ y maximizarla. Ver [Likelihood function](https://en.wikipedia.org/wiki/Likelihood_function). Cualquiera que sea el método (mínimos cuadrados o máxima verosimilitud) las expresiones para los coeficientes ajustados $\\hat{\\beta_0}, \\hat{\\beta_1}$ son los siguientes: "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\hat{\\beta}_1 = \\frac{\\displaystyle \\sum_{i=1}^m x_iy_i - m \\bar{x} \\bar{y}}{\\displaystyle \\sum_{i=1}^m x_i^2 - m \\bar{x}^2} = \\frac{\\displaystyle \\sum_{i=1}^m (x_i - \\bar{x}) y_i}{\\displaystyle \\sum_{i=1}^m(x_i - \\bar{x})^2} =\\frac{\\displaystyle \\sum_{i=1}^m(x_i - \\bar{x})(y_i - \\bar{y})}{\\displaystyle \\sum_{i=1}^m(x_i -\\bar{x})^2} = \\frac{SC_{xy}}{SC_{xx}}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\hat{\\beta}_0 = \\bar{y} - \\hat{\\beta}_1 \\bar{x} = \\displaystyle \\sum_{i=1}^m \\left( \\frac{1}{m} - \\frac{\\bar{x}(x_i - \\bar{x})}{\\displaystyle \\sum_{i=1}^m(x_i - \\bar{x})^2} \\right ) y_i$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "donde: $SC_{xy}$ y $SC_{xx}$ son la suma de cuadrados respecto a $x,y$ y la suma de cuadrados respecto a $x,x$ respectivamente."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Entonces la recta por mínimos cuadrados está dada por la forma funcional:\n",
    "\n",
    "$$\\mu(x | \\hat{\\beta}_0, \\hat{\\beta}_1) = \\hat{\\beta}_0 + \\hat{\\beta}_1 x .$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "\n",
    "**Observaciones** \n",
    "\n",
    "* Obsérvese que el ajuste para los parámetros $\\beta$'s, $\\hat{\\beta}_0 , \\hat{\\beta}_1$, son combinaciones lineales de las $y_i$'s. \n",
    "\n",
    "* Como $\\hat{\\beta}_0 = \\bar{y} - \\hat{\\beta}_1 \\bar{x}$ entonces $\\bar{y} = \\hat{\\beta}_0 + \\hat{\\beta}_1 \\bar{x}$ por lo que el punto $(\\bar{x}, \\bar{y})$ está en la recta de regresión.\n",
    "\n",
    "* Si $\\hat{\\beta_1} = 0$ (y por tanto $x$ no explica la variabilidad de $Y$) entonces $\\hat{\\beta_0} = \\bar{y}$ (la mejor estimación es con la media de los valores $y_i$'s).\n",
    "\n",
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Propiedades de $\\hat{\\beta}_0, \\hat{\\beta}_1$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Por la observación anterior: $\\hat{\\beta}_0 , \\hat{\\beta}_1$ son combinaciones lineales de las $y_i$'s. Entonces considerando que $Y_i$'s son aleatorias se tiene que $\\hat{\\beta}_0 , \\hat{\\beta}_1$, son aleatorias. De hecho cumplen:\n",
    "\n",
    "* $E[\\hat{\\beta}_0 | x] = \\beta_0$\n",
    "\n",
    "* $E[\\hat{\\beta}_1 | x] = \\beta_1$\n",
    "\n",
    "por lo que son estimadores insesgados (en promedio no sobreestiman ni subestiman los parámetros $\\beta_0, \\beta_1$). Aquí se utilizó la notación $x = (x_1, \\dots, x_m)^T$.\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* $\\text{Var}(\\hat{\\beta}_0 | x) = \\sigma^2 \\left(\\frac{1}{n} + \\frac{\\bar{x}^2}{SC_{xx}} \\right)$.\n",
    "\n",
    "* $\\text{Var}(\\hat{\\beta}_1 | x) = \\frac{\\sigma^2}{SC_{xx}}$.\n",
    "\n",
    "* $\\text{Cov}(\\hat{\\beta}_0, \\hat{\\beta}_1 | x) = -\\bar{x} \\text{Var}(\\hat{\\beta}_1)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Valores ajustados"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Denotamos como $\\hat{y}_i$ a los valores ajustados y se definen como:\n",
    "\n",
    "$$\\hat{y}_i = \\hat{\\beta}_0 + \\hat{\\beta}_1x_i \\forall i=1, \\dots, m.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "\n",
    "**Observación** \n",
    "\n",
    "Obsérvese que si $\\hat{y}$ denota al vector de valores ajustados matricialmente la definición anterior es $\\hat{y} = A \\hat{\\beta}$ con $\\hat{\\beta}=\n",
    "\\left[\\begin{array}{c}\n",
    "\\hat{\\beta}_0\\\\\n",
    "\\hat{\\beta}_1\\\\\n",
    "\\end{array}\n",
    "\\right]$ vector de parámetros ajustados.\n",
    "\n",
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Residuales"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Denotamos como $\\hat{e}_i$ a los residuales y se definen como:\n",
    "\n",
    "$$\\hat{e}_i = y_i - \\hat{y}_i \\forall i=1,\\dots, m.$$\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "\n",
    "**Observación** \n",
    "\n",
    "\n",
    "Se espera que $\\hat{e}_i$ refleje a $\\epsilon_i$. Para lo anterior se realiza un **análisis de residuales** que incluye verificación de:\n",
    "\n",
    "* $N(0, \\sigma^2)$ normalidad con media $0$ y homocedasticidad.\n",
    "\n",
    "* Correlación igual a $0$ entre los residuales.\n",
    "\n",
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## *Dataset cars*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "options(digits=4) #for number of digits in print lines"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Para describir al ARLS consideramos el *dataset* de [cars](https://stat.ethz.ch/R-manual/R-devel/library/datasets/html/cars.html) de R"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<caption>A data.frame: 6 × 2</caption>\n",
       "<thead>\n",
       "\t<tr><th></th><th scope=col>speed</th><th scope=col>dist</th></tr>\n",
       "\t<tr><th></th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th></tr>\n",
       "</thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>1</th><td>4</td><td> 2</td></tr>\n",
       "\t<tr><th scope=row>2</th><td>4</td><td>10</td></tr>\n",
       "\t<tr><th scope=row>3</th><td>7</td><td> 4</td></tr>\n",
       "\t<tr><th scope=row>4</th><td>7</td><td>22</td></tr>\n",
       "\t<tr><th scope=row>5</th><td>8</td><td>16</td></tr>\n",
       "\t<tr><th scope=row>6</th><td>9</td><td>10</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A data.frame: 6 × 2\n",
       "\\begin{tabular}{r|ll}\n",
       "  & speed & dist\\\\\n",
       "  & <dbl> & <dbl>\\\\\n",
       "\\hline\n",
       "\t1 & 4 &  2\\\\\n",
       "\t2 & 4 & 10\\\\\n",
       "\t3 & 7 &  4\\\\\n",
       "\t4 & 7 & 22\\\\\n",
       "\t5 & 8 & 16\\\\\n",
       "\t6 & 9 & 10\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A data.frame: 6 × 2\n",
       "\n",
       "| <!--/--> | speed &lt;dbl&gt; | dist &lt;dbl&gt; |\n",
       "|---|---|---|\n",
       "| 1 | 4 |  2 |\n",
       "| 2 | 4 | 10 |\n",
       "| 3 | 7 |  4 |\n",
       "| 4 | 7 | 22 |\n",
       "| 5 | 8 | 16 |\n",
       "| 6 | 9 | 10 |\n",
       "\n"
      ],
      "text/plain": [
       "  speed dist\n",
       "1 4      2  \n",
       "2 4     10  \n",
       "3 7      4  \n",
       "4 7     22  \n",
       "5 8     16  \n",
       "6 9     10  "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "head(cars)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "La columna `speed` representa qué tan rápido iban carros en millas por hora y `dist` mide qué tanta distancia les tomó a carros detenerse y está medida en pies. Ver `?cars`.\n",
    "\n",
    "Se considerará a la variable `speed` como la variable regresora ($x$) y `dist` como la variable dependiente ($Y$).\n",
    "\n",
    "Podemos hacer un *scatterplot*:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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ctrNVzQjxy+S+/vveWxCGjdDrsp6Pf2d167PM9jOVTQTzqvoz5ojbwKaN0Ouyno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\nFaCpqAI0FVWApqIK0FRUAZqKKkBTUfX/PnW8Mw/7z2oAAAAASUVORK5CYII=",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 360,
       "width": 360
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "qplot(speed, dist, data = cars)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Se observa una relación aproximadamente lineal entre `dist` y `speed`, no se observan datos atípicos o *outliers* y además una tendencia creciente."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Realizamos el ARLS con la función [lm](https://www.rdocumentation.org/packages/stats/versions/3.6.2/topics/lm) de R:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "cars_lm <- lm(dist ~ speed, data = cars)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "'lm'"
      ],
      "text/latex": [
       "'lm'"
      ],
      "text/markdown": [
       "'lm'"
      ],
      "text/plain": [
       "[1] \"lm\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "class(cars_lm)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "\n",
    "**Observación** \n",
    "\n",
    "`dist ~ speed` es una [formula](https://www.rdocumentation.org/packages/stats/versions/3.6.2/topics/formula) que se lee como \"la variable `dist` está descrita o modelada por la variable `speed`\". \n",
    "\n",
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Los coeficientes de la regresión pueden obtenerse a partir del objeto tipo `lm` anterior con la función [coef](https://www.rdocumentation.org/packages/stats/versions/3.6.2/topics/coef) y también usando el símbolo \\$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(Intercept)       speed \n",
      "    -17.579       3.932 \n"
     ]
    }
   ],
   "source": [
    "print(coef(cars_lm))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "gf <- ggplot(cars, aes(x = speed, y = dist))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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LHHWrdu3axZs6FDh+bl5cndERDvjpxW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\nh8ABAAAAtEPgAAAAANr9P1Ot1B2GLfNUAAAAAElFTkSuQmCC",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 360,
       "width": 360
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "gf + \n",
    "geom_point() + \n",
    "geom_smooth(formula = \"y~x\", method = lm, se = FALSE)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Pregunta: el dato de $8$ mph es un valor presente en los datos de la variable de `speed` ¿cuál es la media de la distancia (en pies) que le toma a un carro detenerse que va a una velocidad de 8 mph?**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "beta_0 <- as.double(cars_lm$coef[\"(Intercept)\"])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "beta_1 <- as.double(cars_lm$coef[\"speed\"])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] 13.88\n"
     ]
    }
   ],
   "source": [
    "speed_value <- 8\n",
    "dist_value <- beta_0 + beta_1 * 8\n",
    "print(dist_value)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Como se escribió en la pregunta el dato de $8$ mph es un valor medido con la variable `speed`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "ind <- cars$speed == 8 "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " [1] FALSE FALSE FALSE FALSE  TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE\n",
      "[13] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE\n",
      "[25] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE\n",
      "[37] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE\n",
      "[49] FALSE FALSE\n"
     ]
    }
   ],
   "source": [
    "print(ind)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "    5 \n",
      "13.88 \n"
     ]
    }
   ],
   "source": [
    "print(fitted(cars_lm)[ind])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ver [fitted](https://www.rdocumentation.org/packages/stats/versions/3.6.2/topics/fitted)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Pregunta: ¿qué interpretación se le puede dar a $\\hat{\\beta}_0$?**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] -17.58\n"
     ]
    }
   ],
   "source": [
    "print(beta_0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\hat{\\beta}_0$ se considera un parámetro de ruido y su interpretación para este ejemplo puede enunciarse como \"la distancia promedio que le tomó a carros detenerse que iban a una velocidad de $0$ mph fue de  $-17.58$ pies\". En este ejemplo no podemos escribir en un reporte tal enunciado pues las distancias son positivas. En este caso $\\beta_0$ solamente ayuda a realizar el ajuste a los datos de `cars` y si revisamos el valor mínimo de la variable `speed` es:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "4"
      ],
      "text/latex": [
       "4"
      ],
      "text/markdown": [
       "4"
      ],
      "text/plain": [
       "[1] 4"
      ]
     },
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   ],
   "source": [
    "min(cars$speed)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "por lo que realizar enunciados para valores de la variable `speed` que son lejanos (conocido con el nombre de extrapolación, ver [2_interpolacion](https://github.com/ITAM-DS/Propedeutico/blob/main/Python/clases/3_algebra_lineal/2_interpolacion.ipynb)) del intervalo de ajuste debe realizarse con cuidado."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Pregunta: ¿qué interpretación se le puede dar a $\\hat{\\beta}_1$?**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] 3.932\n"
     ]
    }
   ],
   "source": [
    "print(beta_1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "La interpretación se enuncia en términos de incrementos de unidades para la variable explicativa respecto a un incremento (pues el signo de $\\beta_1$ en este caso es positivo) en promedio para la variable dependiente. En el ejemplo se puede enunciar que \"un incremento de una milla por hora en la variable `speed` en promedio le toma una distancia de $3.93$ pies adicionales para detenerse al carro\"."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Pregunta: ¿cuáles son los valores ajustados $\\hat{y}_i$ para $i=1,\\dots,6$?**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "     1      2      3      4      5      6 \n",
      "-1.849 -1.849  9.948  9.948 13.880 17.813 \n"
     ]
    }
   ],
   "source": [
    "print(head(fitted(cars_lm)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Pregunta: ¿cuáles son los residuales $\\hat{e}_i$ para $i=1,\\dots, 6$?**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "     1      2      3      4      5      6 \n",
      " 3.849 11.849 -5.948 12.052  2.120 -7.813 \n"
     ]
    }
   ],
   "source": [
    "print(head(residuals(cars_lm)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ver [residuals](https://www.rdocumentation.org/packages/stats/versions/3.6.2/topics/residuals)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Pregunta: ¿cuáles son valores predichos por el modelo para datos de `speed` con valores de 6, 8 y 21 mph?**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "     1      2      3 \n",
      " 6.015 13.880 65.001 \n"
     ]
    }
   ],
   "source": [
    "df_new <- data.frame(speed = c(6, 8, 21))\n",
    "print(predict(cars_lm, newdata = df_new))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ver [predict](https://www.rdocumentation.org/packages/gstat/versions/2.0-6/topics/predict)"
   ]
  }
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