6.5_Chemometrics_data_manipulation.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Chemometrics\n",
    "<br>\n",
    "**Julien Wist / 2017 / Universidad del Valle**\n",
    "<br>\n",
    "**Andrés Bernal / 2017 / ???**\n",
    "\n",
    "An up-to-date version of this notebook can be found here: https://github.com/jwist/chemometrics/"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "options(repr.plot.width=4, repr.plot.height=4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# manipulating data with R"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "R is a language tought from the statistical standpoint. It is at first a little counter intuitive for people used to matlab like programming. Matlab and the likes, scilab, octave, python, are build from the linear algebra standpoint.\n",
    "\n",
    "Thus R is optimal for statistical data manipulation since it provide a large number of built-in functions. However most of these functions are ignored because they have no equivalent in matlab. \n",
    "\n",
    "In this section, I will list some very important functions that are usually discovered too late... and that can greatly simplify the code and more importantly make it more readeable."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First load some data to play with."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "load(url('https://github.com/jwist/chemometrics/raw/master/datasets/coffeeMulti.rda'))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "'coffeeMulti'"
      ],
      "text/latex": [
       "'coffeeMulti'"
      ],
      "text/markdown": [
       "'coffeeMulti'"
      ],
      "text/plain": [
       "[1] \"coffeeMulti\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# use ls to list the variables and discover that whas saved into this file\n",
    "ls()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<style>\n",
       ".list-inline {list-style: none; margin:0; padding: 0}\n",
       ".list-inline>li {display: inline-block}\n",
       ".list-inline>li:not(:last-child)::after {content: \"\\00b7\"; padding: 0 .5ex}\n",
       "</style>\n",
       "<ol class=list-inline><li>'nmrBin'</li><li>'nmrParam'</li><li>'irms'</li><li>'gc'</li></ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 'nmrBin'\n",
       "\\item 'nmrParam'\n",
       "\\item 'irms'\n",
       "\\item 'gc'\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 'nmrBin'\n",
       "2. 'nmrParam'\n",
       "3. 'irms'\n",
       "4. 'gc'\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] \"nmrBin\"   \"nmrParam\" \"irms\"     \"gc\"      "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# use the following command to explore the variable.\n",
    "names(coffeeMulti)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"dataframe\">\n",
       "<caption>A AsIs: 6 × 6</caption>\n",
       "<thead>\n",
       "\t<tr><th></th><th scope=col>code</th><th scope=col>country</th><th scope=col>department</th><th scope=col>caffeine1</th><th scope=col>caffeine2</th><th scope=col>mean</th></tr>\n",
       "\t<tr><th></th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;fct&gt;</th><th scope=col>&lt;fct&gt;</th><th scope=col>&lt;dbl&gt;</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>5</th><td>1172</td><td>Colombia</td><td>Tolima</td><td>-28.285</td><td>-27.936</td><td>-28.111</td></tr>\n",
       "\t<tr><th scope=row>6</th><td>1173</td><td>Colombia</td><td>Tolima</td><td>-28.931</td><td>-28.800</td><td>-28.866</td></tr>\n",
       "\t<tr><th scope=row>7</th><td>1198</td><td>Brasil  </td><td>Otro  </td><td>-28.056</td><td>-28.274</td><td>-28.165</td></tr>\n",
       "\t<tr><th scope=row>8</th><td>1199</td><td>Colombia</td><td>Huila </td><td>-28.154</td><td>-28.022</td><td>-28.088</td></tr>\n",
       "\t<tr><th scope=row>10</th><td>1201</td><td>Colombia</td><td>Huila </td><td>-28.198</td><td>-28.372</td><td>-28.285</td></tr>\n",
       "\t<tr><th scope=row>11</th><td>1204</td><td>Brasil  </td><td>Otro  </td><td>-27.949</td><td>-27.978</td><td>-27.964</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A AsIs: 6 × 6\n",
       "\\begin{tabular}{r|llllll}\n",
       "  & code & country & department & caffeine1 & caffeine2 & mean\\\\\n",
       "  & <int> & <fct> & <fct> & <dbl> & <dbl> & <dbl>\\\\\n",
       "\\hline\n",
       "\t5 & 1172 & Colombia & Tolima & -28.285 & -27.936 & -28.111\\\\\n",
       "\t6 & 1173 & Colombia & Tolima & -28.931 & -28.800 & -28.866\\\\\n",
       "\t7 & 1198 & Brasil   & Otro   & -28.056 & -28.274 & -28.165\\\\\n",
       "\t8 & 1199 & Colombia & Huila  & -28.154 & -28.022 & -28.088\\\\\n",
       "\t10 & 1201 & Colombia & Huila  & -28.198 & -28.372 & -28.285\\\\\n",
       "\t11 & 1204 & Brasil   & Otro   & -27.949 & -27.978 & -27.964\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A AsIs: 6 × 6\n",
       "\n",
       "| <!--/--> | code &lt;int&gt; | country &lt;fct&gt; | department &lt;fct&gt; | caffeine1 &lt;dbl&gt; | caffeine2 &lt;dbl&gt; | mean &lt;dbl&gt; |\n",
       "|---|---|---|---|---|---|---|\n",
       "| 5 | 1172 | Colombia | Tolima | -28.285 | -27.936 | -28.111 |\n",
       "| 6 | 1173 | Colombia | Tolima | -28.931 | -28.800 | -28.866 |\n",
       "| 7 | 1198 | Brasil   | Otro   | -28.056 | -28.274 | -28.165 |\n",
       "| 8 | 1199 | Colombia | Huila  | -28.154 | -28.022 | -28.088 |\n",
       "| 10 | 1201 | Colombia | Huila  | -28.198 | -28.372 | -28.285 |\n",
       "| 11 | 1204 | Brasil   | Otro   | -27.949 | -27.978 | -27.964 |\n",
       "\n"
      ],
      "text/plain": [
       "   code country  department caffeine1 caffeine2 mean   \n",
       "5  1172 Colombia Tolima     -28.285   -27.936   -28.111\n",
       "6  1173 Colombia Tolima     -28.931   -28.800   -28.866\n",
       "7  1198 Brasil   Otro       -28.056   -28.274   -28.165\n",
       "8  1199 Colombia Huila      -28.154   -28.022   -28.088\n",
       "10 1201 Colombia Huila      -28.198   -28.372   -28.285\n",
       "11 1204 Brasil   Otro       -27.949   -27.978   -27.964"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# use head to visualize the data\n",
    "head( coffeeMulti$irms )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's play with the Isotope Ratio Mass Spectrometry (IRMS) data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<style>\n",
       ".list-inline {list-style: none; margin:0; padding: 0}\n",
       ".list-inline>li {display: inline-block}\n",
       ".list-inline>li:not(:last-child)::after {content: \"\\00b7\"; padding: 0 .5ex}\n",
       "</style>\n",
       "<ol class=list-inline><li>'AsIs'</li><li>'oldClass'</li></ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 'AsIs'\n",
       "\\item 'oldClass'\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 'AsIs'\n",
       "2. 'oldClass'\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] \"AsIs\"     \"oldClass\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "d <- coffeeMulti$irms\n",
    "is(d)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's compute the mean colomn from the two columns \"caffeine1\" and \"caffeine2\". This is done easily using the apply function to manipulate arrays. The first argument is the array, in this case the two columns, the second argument is the \"MARGIN\", 1 for rows and 2 for columns, and the last arguments is the function to be applied. The MARGIN tells how to apply the mean function. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"dataframe\">\n",
       "<caption>A data.frame: 6 × 1</caption>\n",
       "<thead>\n",
       "\t<tr><th></th><th scope=col>mean</th></tr>\n",
       "\t<tr><th></th><th scope=col>&lt;dbl&gt;</th></tr>\n",
       "</thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>5</th><td>-28.1105</td></tr>\n",
       "\t<tr><th scope=row>6</th><td>-28.8655</td></tr>\n",
       "\t<tr><th scope=row>7</th><td>-28.1650</td></tr>\n",
       "\t<tr><th scope=row>8</th><td>-28.0880</td></tr>\n",
       "\t<tr><th scope=row>10</th><td>-28.2850</td></tr>\n",
       "\t<tr><th scope=row>11</th><td>-27.9635</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A data.frame: 6 × 1\n",
       "\\begin{tabular}{r|l}\n",
       "  & mean\\\\\n",
       "  & <dbl>\\\\\n",
       "\\hline\n",
       "\t5 & -28.1105\\\\\n",
       "\t6 & -28.8655\\\\\n",
       "\t7 & -28.1650\\\\\n",
       "\t8 & -28.0880\\\\\n",
       "\t10 & -28.2850\\\\\n",
       "\t11 & -27.9635\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A data.frame: 6 × 1\n",
       "\n",
       "| <!--/--> | mean &lt;dbl&gt; |\n",
       "|---|---|\n",
       "| 5 | -28.1105 |\n",
       "| 6 | -28.8655 |\n",
       "| 7 | -28.1650 |\n",
       "| 8 | -28.0880 |\n",
       "| 10 | -28.2850 |\n",
       "| 11 | -27.9635 |\n",
       "\n"
      ],
      "text/plain": [
       "   mean    \n",
       "5  -28.1105\n",
       "6  -28.8655\n",
       "7  -28.1650\n",
       "8  -28.0880\n",
       "10 -28.2850\n",
       "11 -27.9635"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "m <- apply(d[,c('caffeine1','caffeine2')], 1, mean)\n",
    "head( data.frame(mean = m) )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "to make it clear we can use ```MARGIN = 2```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"dataframe\">\n",
       "<caption>A data.frame: 2 × 1</caption>\n",
       "<thead>\n",
       "\t<tr><th></th><th scope=col>mean.by.columns</th></tr>\n",
       "\t<tr><th></th><th scope=col>&lt;dbl&gt;</th></tr>\n",
       "</thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>caffeine1</th><td>-29.00174</td></tr>\n",
       "\t<tr><th scope=row>caffeine2</th><td>-28.98335</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A data.frame: 2 × 1\n",
       "\\begin{tabular}{r|l}\n",
       "  & mean.by.columns\\\\\n",
       "  & <dbl>\\\\\n",
       "\\hline\n",
       "\tcaffeine1 & -29.00174\\\\\n",
       "\tcaffeine2 & -28.98335\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A data.frame: 2 × 1\n",
       "\n",
       "| <!--/--> | mean.by.columns &lt;dbl&gt; |\n",
       "|---|---|\n",
       "| caffeine1 | -29.00174 |\n",
       "| caffeine2 | -28.98335 |\n",
       "\n"
      ],
      "text/plain": [
       "          mean.by.columns\n",
       "caffeine1 -29.00174      \n",
       "caffeine2 -28.98335      "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "m <- apply(d[,c('caffeine1','caffeine2')], 2, mean)\n",
    "head( data.frame('mean by columns' = m) )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Instead of mean any function can be used, like max, min, quantile, etc."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"dataframe\">\n",
       "<caption>A data.frame: 5 × 2</caption>\n",
       "<thead>\n",
       "\t<tr><th></th><th scope=col>quantile.caffeine1</th><th scope=col>quantile.caffeine2</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>0%</th><td>-30.87700</td><td>-30.90900</td></tr>\n",
       "\t<tr><th scope=row>25%</th><td>-29.52950</td><td>-29.73650</td></tr>\n",
       "\t<tr><th scope=row>50%</th><td>-28.88300</td><td>-28.80000</td></tr>\n",
       "\t<tr><th scope=row>75%</th><td>-28.39175</td><td>-28.37375</td></tr>\n",
       "\t<tr><th scope=row>100%</th><td>-27.72200</td><td>-27.73500</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A data.frame: 5 × 2\n",
       "\\begin{tabular}{r|ll}\n",
       "  & quantile.caffeine1 & quantile.caffeine2\\\\\n",
       "  & <dbl> & <dbl>\\\\\n",
       "\\hline\n",
       "\t0\\% & -30.87700 & -30.90900\\\\\n",
       "\t25\\% & -29.52950 & -29.73650\\\\\n",
       "\t50\\% & -28.88300 & -28.80000\\\\\n",
       "\t75\\% & -28.39175 & -28.37375\\\\\n",
       "\t100\\% & -27.72200 & -27.73500\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A data.frame: 5 × 2\n",
       "\n",
       "| <!--/--> | quantile.caffeine1 &lt;dbl&gt; | quantile.caffeine2 &lt;dbl&gt; |\n",
       "|---|---|---|\n",
       "| 0% | -30.87700 | -30.90900 |\n",
       "| 25% | -29.52950 | -29.73650 |\n",
       "| 50% | -28.88300 | -28.80000 |\n",
       "| 75% | -28.39175 | -28.37375 |\n",
       "| 100% | -27.72200 | -27.73500 |\n",
       "\n"
      ],
      "text/plain": [
       "     quantile.caffeine1 quantile.caffeine2\n",
       "0%   -30.87700          -30.90900         \n",
       "25%  -29.52950          -29.73650         \n",
       "50%  -28.88300          -28.80000         \n",
       "75%  -28.39175          -28.37375         \n",
       "100% -27.72200          -27.73500         "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "m <- apply(d[,c('caffeine1','caffeine2')], 2, quantile)\n",
    "head( data.frame(quantile = m) )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The message is that the loop should be avoided as much as possible in R and that there are many build-in function for that purpose."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Another useful example is centering the data. Say you want to find the mean of a column and then substract this value to all the elements of that column. This way you will have centered your data. At first glance, this look fairly complex and implies several operations. R provide a simple framework for this, called ``` sweep()```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"dataframe\">\n",
       "<caption>A data.frame: 6 × 2</caption>\n",
       "<thead>\n",
       "\t<tr><th></th><th scope=col>caffeine1</th><th scope=col>caffeine2</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>5</th><td>0.71673529</td><td>1.0473529</td></tr>\n",
       "\t<tr><th scope=row>6</th><td>0.07073529</td><td>0.1833529</td></tr>\n",
       "\t<tr><th scope=row>7</th><td>0.94573529</td><td>0.7093529</td></tr>\n",
       "\t<tr><th scope=row>8</th><td>0.84773529</td><td>0.9613529</td></tr>\n",
       "\t<tr><th scope=row>10</th><td>0.80373529</td><td>0.6113529</td></tr>\n",
       "\t<tr><th scope=row>11</th><td>1.05273529</td><td>1.0053529</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A data.frame: 6 × 2\n",
       "\\begin{tabular}{r|ll}\n",
       "  & caffeine1 & caffeine2\\\\\n",
       "  & <dbl> & <dbl>\\\\\n",
       "\\hline\n",
       "\t5 & 0.71673529 & 1.0473529\\\\\n",
       "\t6 & 0.07073529 & 0.1833529\\\\\n",
       "\t7 & 0.94573529 & 0.7093529\\\\\n",
       "\t8 & 0.84773529 & 0.9613529\\\\\n",
       "\t10 & 0.80373529 & 0.6113529\\\\\n",
       "\t11 & 1.05273529 & 1.0053529\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A data.frame: 6 × 2\n",
       "\n",
       "| <!--/--> | caffeine1 &lt;dbl&gt; | caffeine2 &lt;dbl&gt; |\n",
       "|---|---|---|\n",
       "| 5 | 0.71673529 | 1.0473529 |\n",
       "| 6 | 0.07073529 | 0.1833529 |\n",
       "| 7 | 0.94573529 | 0.7093529 |\n",
       "| 8 | 0.84773529 | 0.9613529 |\n",
       "| 10 | 0.80373529 | 0.6113529 |\n",
       "| 11 | 1.05273529 | 1.0053529 |\n",
       "\n"
      ],
      "text/plain": [
       "   caffeine1  caffeine2\n",
       "5  0.71673529 1.0473529\n",
       "6  0.07073529 0.1833529\n",
       "7  0.94573529 0.7093529\n",
       "8  0.84773529 0.9613529\n",
       "10 0.80373529 0.6113529\n",
       "11 1.05273529 1.0053529"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "head( sweep( d[c('caffeine1', 'caffeine2')], 2, apply(d[c('caffeine1', 'caffeine2')], 2, mean), \"-\" ) )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"dataframe\">\n",
       "<caption>A matrix: 6 × 2 of type dbl</caption>\n",
       "<thead>\n",
       "\t<tr><th></th><th scope=col>caffeine1</th><th scope=col>caffeine2</th></tr>\n",
       "</thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>5</th><td>0.71673529</td><td>1.0473529</td></tr>\n",
       "\t<tr><th scope=row>6</th><td>0.07073529</td><td>0.1833529</td></tr>\n",
       "\t<tr><th scope=row>7</th><td>0.94573529</td><td>0.7093529</td></tr>\n",
       "\t<tr><th scope=row>8</th><td>0.84773529</td><td>0.9613529</td></tr>\n",
       "\t<tr><th scope=row>10</th><td>0.80373529</td><td>0.6113529</td></tr>\n",
       "\t<tr><th scope=row>11</th><td>1.05273529</td><td>1.0053529</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A matrix: 6 × 2 of type dbl\n",
       "\\begin{tabular}{r|ll}\n",
       "  & caffeine1 & caffeine2\\\\\n",
       "\\hline\n",
       "\t5 & 0.71673529 & 1.0473529\\\\\n",
       "\t6 & 0.07073529 & 0.1833529\\\\\n",
       "\t7 & 0.94573529 & 0.7093529\\\\\n",
       "\t8 & 0.84773529 & 0.9613529\\\\\n",
       "\t10 & 0.80373529 & 0.6113529\\\\\n",
       "\t11 & 1.05273529 & 1.0053529\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A matrix: 6 × 2 of type dbl\n",
       "\n",
       "| <!--/--> | caffeine1 | caffeine2 |\n",
       "|---|---|---|\n",
       "| 5 | 0.71673529 | 1.0473529 |\n",
       "| 6 | 0.07073529 | 0.1833529 |\n",
       "| 7 | 0.94573529 | 0.7093529 |\n",
       "| 8 | 0.84773529 | 0.9613529 |\n",
       "| 10 | 0.80373529 | 0.6113529 |\n",
       "| 11 | 1.05273529 | 1.0053529 |\n",
       "\n"
      ],
      "text/plain": [
       "   caffeine1  caffeine2\n",
       "5  0.71673529 1.0473529\n",
       "6  0.07073529 0.1833529\n",
       "7  0.94573529 0.7093529\n",
       "8  0.84773529 0.9613529\n",
       "10 0.80373529 0.6113529\n",
       "11 1.05273529 1.0053529"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "head( apply(d[c('caffeine1', 'caffeine2')], 2, function(x) x - mean(x)) )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This second example is my favourite and shows how apply works. Instead of using a predifined function such as ```mean()``` we use a user defined function. In this case it is clear that x selects the column and the function performes: the column minus its means."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Althought ```sweep()``` achieves the same results, the second example is much more general."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Another very useful example is the data aggregation. For example you want to find the mean value of your variable by country. In this case the isotope ratio mean by country. Again, if looked at from the matlab standpoint, this is not straighforward. However R provide a very handy solution. The first aggregation function you should know is ```table()```. Look how this works:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "  Brasil Colombia     Peru \n",
       "      11       15        8 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "table( d$country )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So, we know how many samples we have from each country. But now we want to compute the mean."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"dataframe\">\n",
       "<caption>A data.frame: 3 × 3</caption>\n",
       "<thead>\n",
       "\t<tr><th scope=col>Group.1</th><th scope=col>caffeine1</th><th scope=col>caffeine2</th></tr>\n",
       "\t<tr><th scope=col>&lt;fct&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th></tr>\n",
       "</thead>\n",
       "<tbody>\n",
       "\t<tr><td>Brasil  </td><td>-29.13264</td><td>-29.12582</td></tr>\n",
       "\t<tr><td>Colombia</td><td>-28.51793</td><td>-28.48793</td></tr>\n",
       "\t<tr><td>Peru    </td><td>-29.72888</td><td>-29.71637</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A data.frame: 3 × 3\n",
       "\\begin{tabular}{lll}\n",
       " Group.1 & caffeine1 & caffeine2\\\\\n",
       " <fct> & <dbl> & <dbl>\\\\\n",
       "\\hline\n",
       "\t Brasil   & -29.13264 & -29.12582\\\\\n",
       "\t Colombia & -28.51793 & -28.48793\\\\\n",
       "\t Peru     & -29.72888 & -29.71637\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A data.frame: 3 × 3\n",
       "\n",
       "| Group.1 &lt;fct&gt; | caffeine1 &lt;dbl&gt; | caffeine2 &lt;dbl&gt; |\n",
       "|---|---|---|\n",
       "| Brasil   | -29.13264 | -29.12582 |\n",
       "| Colombia | -28.51793 | -28.48793 |\n",
       "| Peru     | -29.72888 | -29.71637 |\n",
       "\n"
      ],
      "text/plain": [
       "  Group.1  caffeine1 caffeine2\n",
       "1 Brasil   -29.13264 -29.12582\n",
       "2 Colombia -28.51793 -28.48793\n",
       "3 Peru     -29.72888 -29.71637"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "aggregate(d[c('caffeine1', 'caffeine2')], by = list(unlist(d['country'], use.names = FALSE)), mean)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The ```by=``` argument must be a list object. Because our data are not perfectly stored, we have to first unlist our country column and create a new clean list. The ```unlist()``` function is very usefull to unformat any vector of data before reassigning it with a new type."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 248,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "country: Brasil\n",
       "[1] -27.964\n",
       "------------------------------------------------------------ \n",
       "country: Colombia\n",
       "[1] -27.729\n",
       "------------------------------------------------------------ \n",
       "country: Peru\n",
       "[1] -28.567"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "by(d['mean'], d['country'], function(x) max(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Another way to obtain the same results."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 249,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "d$country: Brasil\n",
       "[1] -27.964\n",
       "------------------------------------------------------------ \n",
       "d$country: Colombia\n",
       "[1] -27.729\n",
       "------------------------------------------------------------ \n",
       "d$country: Peru\n",
       "[1] -28.567"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "by(d, d$country, function(x) max(x$mean))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For more control over the data, aggregation can be performed without applying any function. Functions can be applied later."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 250,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>Group.1</th><th scope=col>caffeine1</th><th scope=col>caffeine2</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td>Brasil                                                                                           </td><td>-28.056, -27.949, -29.317, -29.089, -28.933, -28.154, -30.877, -29.816, -28.411, -29.589, -30.268</td><td>-28.274, -27.978, -29.504, -28.934, -28.800, -27.906, -30.909, -29.877, -28.440, -29.814, -29.948</td></tr>\n",
       "\t<tr><td>Colombia                                                                                                                             </td><td>-28.285, -28.931, -28.154, -28.198, -28.406, -27.722, -28.605, -28.503, -28.835, -28.387, -28.226, -28.731, -29.029, -28.577, -29.180</td><td>-27.936, -28.800, -28.022, -28.372, -28.505, -27.735, -28.716, -28.163, -29.207, -28.379, -28.057, -28.893, -28.771, -28.658, -29.105</td></tr>\n",
       "\t<tr><td>Peru                                                                  </td><td>-29.300, -30.027, -29.351, -30.261, -29.643, -30.517, -30.199, -28.533</td><td>-29.148, -30.038, -29.273, -30.089, -30.076, -30.247, -30.259, -28.601</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|lll}\n",
       " Group.1 & caffeine1 & caffeine2\\\\\n",
       "\\hline\n",
       "\t Brasil                                                                                            & -28.056, -27.949, -29.317, -29.089, -28.933, -28.154, -30.877, -29.816, -28.411, -29.589, -30.268 & -28.274, -27.978, -29.504, -28.934, -28.800, -27.906, -30.909, -29.877, -28.440, -29.814, -29.948\\\\\n",
       "\t Colombia                                                                                                                              & -28.285, -28.931, -28.154, -28.198, -28.406, -27.722, -28.605, -28.503, -28.835, -28.387, -28.226, -28.731, -29.029, -28.577, -29.180 & -27.936, -28.800, -28.022, -28.372, -28.505, -27.735, -28.716, -28.163, -29.207, -28.379, -28.057, -28.893, -28.771, -28.658, -29.105\\\\\n",
       "\t Peru                                                                   & -29.300, -30.027, -29.351, -30.261, -29.643, -30.517, -30.199, -28.533 & -29.148, -30.038, -29.273, -30.089, -30.076, -30.247, -30.259, -28.601\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "Group.1 | caffeine1 | caffeine2 | \n",
       "|---|---|---|\n",
       "| Brasil                                                                                            | -28.056, -27.949, -29.317, -29.089, -28.933, -28.154, -30.877, -29.816, -28.411, -29.589, -30.268 | -28.274, -27.978, -29.504, -28.934, -28.800, -27.906, -30.909, -29.877, -28.440, -29.814, -29.948 | \n",
       "| Colombia                                                                                                                              | -28.285, -28.931, -28.154, -28.198, -28.406, -27.722, -28.605, -28.503, -28.835, -28.387, -28.226, -28.731, -29.029, -28.577, -29.180 | -27.936, -28.800, -28.022, -28.372, -28.505, -27.735, -28.716, -28.163, -29.207, -28.379, -28.057, -28.893, -28.771, -28.658, -29.105 | \n",
       "| Peru                                                                   | -29.300, -30.027, -29.351, -30.261, -29.643, -30.517, -30.199, -28.533 | -29.148, -30.038, -29.273, -30.089, -30.076, -30.247, -30.259, -28.601 | \n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "  Group.1 \n",
       "1 Brasil  \n",
       "2 Colombia\n",
       "3 Peru    \n",
       "  caffeine1                                                                                                                            \n",
       "1 -28.056, -27.949, -29.317, -29.089, -28.933, -28.154, -30.877, -29.816, -28.411, -29.589, -30.268                                    \n",
       "2 -28.285, -28.931, -28.154, -28.198, -28.406, -27.722, -28.605, -28.503, -28.835, -28.387, -28.226, -28.731, -29.029, -28.577, -29.180\n",
       "3 -29.300, -30.027, -29.351, -30.261, -29.643, -30.517, -30.199, -28.533                                                               \n",
       "  caffeine2                                                                                                                            \n",
       "1 -28.274, -27.978, -29.504, -28.934, -28.800, -27.906, -30.909, -29.877, -28.440, -29.814, -29.948                                    \n",
       "2 -27.936, -28.800, -28.022, -28.372, -28.505, -27.735, -28.716, -28.163, -29.207, -28.379, -28.057, -28.893, -28.771, -28.658, -29.105\n",
       "3 -29.148, -30.038, -29.273, -30.089, -30.076, -30.247, -30.259, -28.601                                                               "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "a <- aggregate(d[c('caffeine1', 'caffeine2')], list(d$country), function(x) x)\n",
    "a"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For example to obtain boxplots"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 241,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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AQrAcEmAMEKQLAaEKwABCsBwSbAWHD5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\ncyCYORDMHAhmDgQzB4KZA8HM+X+MUKHucMXbDAAAAABJRU5ErkJggg==",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<dl>\n",
       "\t<dt>$caffeine1</dt>\n",
       "\t\t<dd><dl>\n",
       "\t<dt>$stats</dt>\n",
       "\t\t<dd><table>\n",
       "<tbody>\n",
       "\t<tr><td>-30.8770</td><td>-29.1800</td><td>-30.5170</td></tr>\n",
       "\t<tr><td>-29.7025</td><td>-28.7830</td><td>-30.2300</td></tr>\n",
       "\t<tr><td>-29.0890</td><td>-28.5030</td><td>-29.8350</td></tr>\n",
       "\t<tr><td>-28.2825</td><td>-28.2555</td><td>-29.3255</td></tr>\n",
       "\t<tr><td>-27.9490</td><td>-27.7220</td><td>-28.5330</td></tr>\n",
       "</tbody>\n",
       "</table>\n",
       "</dd>\n",
       "\t<dt>$n</dt>\n",
       "\t\t<dd><ol class=list-inline>\n",
       "\t<li>11</li>\n",
       "\t<li>15</li>\n",
       "\t<li>8</li>\n",
       "</ol>\n",
       "</dd>\n",
       "\t<dt>$conf</dt>\n",
       "\t\t<dd><table>\n",
       "<tbody>\n",
       "\t<tr><td>-29.76547</td><td>-28.7182 </td><td>-30.34027</td></tr>\n",
       "\t<tr><td>-28.41253</td><td>-28.2878 </td><td>-29.32973</td></tr>\n",
       "</tbody>\n",
       "</table>\n",
       "</dd>\n",
       "\t<dt>$out</dt>\n",
       "\t\t<dd></dd>\n",
       "\t<dt>$group</dt>\n",
       "\t\t<dd></dd>\n",
       "\t<dt>$names</dt>\n",
       "\t\t<dd><ol class=list-inline>\n",
       "\t<li>Brasil</li>\n",
       "\t<li>Colombia</li>\n",
       "\t<li>Peru</li>\n",
       "</ol>\n",
       "\n",
       "<details>\n",
       "\t<summary style=display:list-item;cursor:pointer>\n",
       "\t\t<strong>Levels</strong>:\n",
       "\t</summary>\n",
       "\t<ol class=list-inline>\n",
       "\t\t<li>'Brasil'</li>\n",
       "\t\t<li>'Colombia'</li>\n",
       "\t\t<li>'Peru'</li>\n",
       "\t</ol>\n",
       "</details></dd>\n",
       "</dl>\n",
       "</dd>\n",
       "\t<dt>$caffeine2</dt>\n",
       "\t\t<dd><dl>\n",
       "\t<dt>$stats</dt>\n",
       "\t\t<dd><table>\n",
       "<tbody>\n",
       "\t<tr><td>-30.9090</td><td>-29.2070</td><td>-30.2590</td></tr>\n",
       "\t<tr><td>-29.8455</td><td>-28.7855</td><td>-30.1680</td></tr>\n",
       "\t<tr><td>-28.9340</td><td>-28.5050</td><td>-30.0570</td></tr>\n",
       "\t<tr><td>-28.3570</td><td>-28.1100</td><td>-29.2105</td></tr>\n",
       "\t<tr><td>-27.9060</td><td>-27.7350</td><td>-28.6010</td></tr>\n",
       "</tbody>\n",
       "</table>\n",
       "</dd>\n",
       "\t<dt>$n</dt>\n",
       "\t\t<dd><ol class=list-inline>\n",
       "\t<li>11</li>\n",
       "\t<li>15</li>\n",
       "\t<li>8</li>\n",
       "</ol>\n",
       "</dd>\n",
       "\t<dt>$conf</dt>\n",
       "\t\t<dd><table>\n",
       "<tbody>\n",
       "\t<tr><td>-29.6431 </td><td>-28.78057</td><td>-30.59187</td></tr>\n",
       "\t<tr><td>-28.2249 </td><td>-28.22943</td><td>-29.52213</td></tr>\n",
       "</tbody>\n",
       "</table>\n",
       "</dd>\n",
       "\t<dt>$out</dt>\n",
       "\t\t<dd></dd>\n",
       "\t<dt>$group</dt>\n",
       "\t\t<dd></dd>\n",
       "\t<dt>$names</dt>\n",
       "\t\t<dd><ol class=list-inline>\n",
       "\t<li>Brasil</li>\n",
       "\t<li>Colombia</li>\n",
       "\t<li>Peru</li>\n",
       "</ol>\n",
       "\n",
       "<details>\n",
       "\t<summary style=display:list-item;cursor:pointer>\n",
       "\t\t<strong>Levels</strong>:\n",
       "\t</summary>\n",
       "\t<ol class=list-inline>\n",
       "\t\t<li>'Brasil'</li>\n",
       "\t\t<li>'Colombia'</li>\n",
       "\t\t<li>'Peru'</li>\n",
       "\t</ol>\n",
       "</details></dd>\n",
       "</dl>\n",
       "</dd>\n",
       "</dl>\n"
      ],
      "text/latex": [
       "\\begin{description}\n",
       "\\item[\\$caffeine1] \\begin{description}\n",
       "\\item[\\$stats] \\begin{tabular}{lll}\n",
       "\t -30.8770 & -29.1800 & -30.5170\\\\\n",
       "\t -29.7025 & -28.7830 & -30.2300\\\\\n",
       "\t -29.0890 & -28.5030 & -29.8350\\\\\n",
       "\t -28.2825 & -28.2555 & -29.3255\\\\\n",
       "\t -27.9490 & -27.7220 & -28.5330\\\\\n",
       "\\end{tabular}\n",
       "\n",
       "\\item[\\$n] \\begin{enumerate*}\n",
       "\\item 11\n",
       "\\item 15\n",
       "\\item 8\n",
       "\\end{enumerate*}\n",
       "\n",
       "\\item[\\$conf] \\begin{tabular}{lll}\n",
       "\t -29.76547 & -28.7182  & -30.34027\\\\\n",
       "\t -28.41253 & -28.2878  & -29.32973\\\\\n",
       "\\end{tabular}\n",
       "\n",
       "\\item[\\$out] \n",
       "\\item[\\$group] \n",
       "\\item[\\$names] \\begin{enumerate*}\n",
       "\\item Brasil\n",
       "\\item Colombia\n",
       "\\item Peru\n",
       "\\end{enumerate*}\n",
       "\n",
       "\\emph{Levels}: \\begin{enumerate*}\n",
       "\\item 'Brasil'\n",
       "\\item 'Colombia'\n",
       "\\item 'Peru'\n",
       "\\end{enumerate*}\n",
       "\n",
       "\\end{description}\n",
       "\n",
       "\\item[\\$caffeine2] \\begin{description}\n",
       "\\item[\\$stats] \\begin{tabular}{lll}\n",
       "\t -30.9090 & -29.2070 & -30.2590\\\\\n",
       "\t -29.8455 & -28.7855 & -30.1680\\\\\n",
       "\t -28.9340 & -28.5050 & -30.0570\\\\\n",
       "\t -28.3570 & -28.1100 & -29.2105\\\\\n",
       "\t -27.9060 & -27.7350 & -28.6010\\\\\n",
       "\\end{tabular}\n",
       "\n",
       "\\item[\\$n] \\begin{enumerate*}\n",
       "\\item 11\n",
       "\\item 15\n",
       "\\item 8\n",
       "\\end{enumerate*}\n",
       "\n",
       "\\item[\\$conf] \\begin{tabular}{lll}\n",
       "\t -29.6431  & -28.78057 & -30.59187\\\\\n",
       "\t -28.2249  & -28.22943 & -29.52213\\\\\n",
       "\\end{tabular}\n",
       "\n",
       "\\item[\\$out] \n",
       "\\item[\\$group] \n",
       "\\item[\\$names] \\begin{enumerate*}\n",
       "\\item Brasil\n",
       "\\item Colombia\n",
       "\\item Peru\n",
       "\\end{enumerate*}\n",
       "\n",
       "\\emph{Levels}: \\begin{enumerate*}\n",
       "\\item 'Brasil'\n",
       "\\item 'Colombia'\n",
       "\\item 'Peru'\n",
       "\\end{enumerate*}\n",
       "\n",
       "\\end{description}\n",
       "\n",
       "\\end{description}\n"
      ],
      "text/markdown": [
       "$caffeine1\n",
       ":   $stats\n",
       ":   \n",
       "| -30.8770 | -29.1800 | -30.5170 | \n",
       "| -29.7025 | -28.7830 | -30.2300 | \n",
       "| -29.0890 | -28.5030 | -29.8350 | \n",
       "| -28.2825 | -28.2555 | -29.3255 | \n",
       "| -27.9490 | -27.7220 | -28.5330 | \n",
       "\n",
       "\n",
       "\n",
       "$n\n",
       ":   1. 11\n",
       "2. 15\n",
       "3. 8\n",
       "\n",
       "\n",
       "\n",
       "$conf\n",
       ":   \n",
       "| -29.76547 | -28.7182  | -30.34027 | \n",
       "| -28.41253 | -28.2878  | -29.32973 | \n",
       "\n",
       "\n",
       "\n",
       "$out\n",
       ":   \n",
       "$group\n",
       ":   \n",
       "$names\n",
       ":   1. Brasil\n",
       "2. Colombia\n",
       "3. Peru\n",
       "\n",
       "\n",
       "\n",
       "**Levels**: 1. 'Brasil'\n",
       "2. 'Colombia'\n",
       "3. 'Peru'\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "$caffeine2\n",
       ":   $stats\n",
       ":   \n",
       "| -30.9090 | -29.2070 | -30.2590 | \n",
       "| -29.8455 | -28.7855 | -30.1680 | \n",
       "| -28.9340 | -28.5050 | -30.0570 | \n",
       "| -28.3570 | -28.1100 | -29.2105 | \n",
       "| -27.9060 | -27.7350 | -28.6010 | \n",
       "\n",
       "\n",
       "\n",
       "$n\n",
       ":   1. 11\n",
       "2. 15\n",
       "3. 8\n",
       "\n",
       "\n",
       "\n",
       "$conf\n",
       ":   \n",
       "| -29.6431  | -28.78057 | -30.59187 | \n",
       "| -28.2249  | -28.22943 | -29.52213 | \n",
       "\n",
       "\n",
       "\n",
       "$out\n",
       ":   \n",
       "$group\n",
       ":   \n",
       "$names\n",
       ":   1. Brasil\n",
       "2. Colombia\n",
       "3. Peru\n",
       "\n",
       "\n",
       "\n",
       "**Levels**: 1. 'Brasil'\n",
       "2. 'Colombia'\n",
       "3. 'Peru'\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "$caffeine1\n",
       "$caffeine1$stats\n",
       "         [,1]     [,2]     [,3]\n",
       "[1,] -30.8770 -29.1800 -30.5170\n",
       "[2,] -29.7025 -28.7830 -30.2300\n",
       "[3,] -29.0890 -28.5030 -29.8350\n",
       "[4,] -28.2825 -28.2555 -29.3255\n",
       "[5,] -27.9490 -27.7220 -28.5330\n",
       "\n",
       "$caffeine1$n\n",
       "[1] 11 15  8\n",
       "\n",
       "$caffeine1$conf\n",
       "          [,1]     [,2]      [,3]\n",
       "[1,] -29.76547 -28.7182 -30.34027\n",
       "[2,] -28.41253 -28.2878 -29.32973\n",
       "\n",
       "$caffeine1$out\n",
       "numeric(0)\n",
       "\n",
       "$caffeine1$group\n",
       "numeric(0)\n",
       "\n",
       "$caffeine1$names\n",
       "[1] Brasil   Colombia Peru    \n",
       "Levels: Brasil Colombia Peru\n",
       "\n",
       "\n",
       "$caffeine2\n",
       "$caffeine2$stats\n",
       "         [,1]     [,2]     [,3]\n",
       "[1,] -30.9090 -29.2070 -30.2590\n",
       "[2,] -29.8455 -28.7855 -30.1680\n",
       "[3,] -28.9340 -28.5050 -30.0570\n",
       "[4,] -28.3570 -28.1100 -29.2105\n",
       "[5,] -27.9060 -27.7350 -28.6010\n",
       "\n",
       "$caffeine2$n\n",
       "[1] 11 15  8\n",
       "\n",
       "$caffeine2$conf\n",
       "         [,1]      [,2]      [,3]\n",
       "[1,] -29.6431 -28.78057 -30.59187\n",
       "[2,] -28.2249 -28.22943 -29.52213\n",
       "\n",
       "$caffeine2$out\n",
       "numeric(0)\n",
       "\n",
       "$caffeine2$group\n",
       "numeric(0)\n",
       "\n",
       "$caffeine2$names\n",
       "[1] Brasil   Colombia Peru    \n",
       "Levels: Brasil Colombia Peru\n",
       "\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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D5AsA7wNEkBCFYCgk0AghWAYDXMF5xW\nGyOF8wGCdYBatAIQrARO0SYAwQpAsBoQrAAEKwHBJgDBCkCwGmEjuPwYBohmK3j3lNbxRLGp\nhZtVwyBYB+EguDSKUnrm5fVKJRqj1t0QBOsgDASvoCFfuVN7JtJSlUAI1kEYCO6TebUmWZWd\nrRIIwToIA8FJU73phQ1VAiFYB2EguE87b3eKuX1VAiFYB2EgeAUN2+VO7Sukx1UCIVgHYSBY\nKCFqnj1yVP90opFXVOIgWAfhIFjYUZBWlygmpUC9Rw8I1kFYCBapOnlcd0vWPI09J5dr7WK5\nHgQrYFtbdJ65/StLrA/5d7kOEKzAKa1dJ//7OxoDd5j6b83BwF3w2S5dVNZeRzAnuAsu0zNm\nAye4C67YoGPMBk5wF6yOUcHP/cukgoQOxoL1P/DXTOZKQ5tbAVPBxh74awaCPTjqgb9mINiD\nox74awaCPTjqgb9mINhD+Dzwv7rGOwDgPahFB08YCFZ94H+kbSsPyXRebx5OgaVgow/8OcFS\nsNEH/pxYm7DeZAoHKuXjqAf+mnHCNTjW9Eef/ZTycdQDf804oRbNVrDEmQCKIdihgn96vqh0\nt/BeM0ocdVQtLhIExz9pMgNzlfKxVPDZLPHvLHFDXFJuB2pyRiUwEgRzrEXPp7m71mcktBB/\nvW/SPJVACNZBGAjOkoagXUNPSOmcUL6y44RaNEfBCTPEyS5aLaVn1FMJjIT7YI6C0weJkwsl\nO6X0uGSVQAjWQRgInlDng5rkwYQ8lcBIEBw73mTa2y/4UD1q9b6U2D27YdTfVAIj4Bp8Ymax\n2byllI+198EHxjZZLs1XUpPVanERUIvWzJdzDW1ueUuWqw3r4Ba1F3YgWM7KTEOb83xtFoI9\n2CF4bX6gCAj24kDBywLuAIK9QLACDqhFKzKnmz8t4xUW9j6gdZc8BTuV9xUeEi2cqrDwqXNa\ndwnBzLFD8KUTgSIg2DR43iY59RocAngK5lSLNggEMweCmQPBzIFg5vAUjFq0B56CgQcIZg4E\nM4enYFyDPfAUjFq0BwhmDgQzJ3wEn1+0wMMQCDaL8BF8cpL3Jf1uEGwW4SNYDmrRpsFTMPAA\nwcyBYObwFIxrsAeeglGL9gDBzIFg5kAwcyCYOTwFoxbtgadg4AGCmQPBzOE5tB2uwR4wtB1z\nMLQdczC0HXMwtB1zwmdoOzkQbBrhM7SdHNSiTQND2zEHQ9sxh+fQdsADz6HtcA32wLMtGrVo\nDxDMHLsEn+0SyoGxINiDXYLLSG0vEGwadgmu2LBBZW0Qgo+WKAww06ifwsLnTCi283D8NVhR\ncJ8JEFyN4wUDdSCYORDMHAhmjpWCn29UC5VICDYNKwUfmB1HDTp48Fl7fNhgD+3pvM48gA/W\nnqLX0Yjrrru42NuN0iRSe5gIgsDia3Db6wuWswWCzcJiwYVjNIVtI2Aa24K2FPpatLBzuyFS\n737VAvr3tyKXu1ONHYudwR99CwQbxJqHDUVFVuRicPRRPRgXfCbgOx3GgGBD6Bf80/NFpbuF\n95pR4qijJhbIDwg2hG7BZ7PEa37ihrik3A7U5IyZRfIBgg2hW/B8mrtrfUZCC/HX+ybNM7NI\nPkCwIXQLzuotTtbQE1I6R+2VHaNAsCF0C06YIU520WopPaOeaeXxB4INoVtw+iBxcqHEdWc2\nLtm08vgDwYbQLXhCnQ9qkgcT8swpjCIQbAjdgg/Vo1bvS4ndsxtG/c208vgDwYbQfx98YGyT\n5dJ8JTVZbVpxFOj4x1DuvYbiYity+WNHK3KphaGWLFcb1sEtaj10GOd7Sx5GnQnlrbyHK99b\nkUstwr8tGhjCqOC1+aYUA4QKo4KX4RQQ3kAwcyCYORDMHKN+Lp0wpRggVOAHyBwIZg4EMweC\nmQPBzIFg5kAwcyCYOWEkeLnr36ti20wL9B79aLHQk+lnzTteN7Vr/bQ7PvBZGmgPyYNrZaiD\nne5/GGs2WK2XqVATVoK7TZ48eVQ6NTmpHhic4MrpFJ01umssTa693ArBLcQvNPFWIkteSlEm\nrAQvk2aV99D96oFlPwQj+GHq+q0429+bXqi1PAjBUoY62Enuh+Vv0S9+0rUDMwg/wcJ+6u9e\nUHn2+sGaBR+KaXnJlTga2z7QHi7LP8gE66RGsJBNX/usUvtq5hKGgg/QMEEoalo1O3GlIBy5\nq31883zX29d/6tnohgHrxER+MKfo+fRidWreiFOC8ON9nRK7L5B+UK49eD9Ob7S1Ld045sTF\nezMa5Eo+kgcfmXBz6thvqjOUl0QjHsET6SPh2tLeiWn3HRc8X21EorTqZ98rh9mEn+CqGbRK\nOgqLbvzlFuGfiXHjSnrENBbrXb+hlF9OaRi9KUjBnalc9ulEa+p7T1fqcMG9B9nH6XGNey0c\nSF26dn5wGKVfFQW3a95qaj9K/B93hrKSaKVGcEUr2n9lAHUvzqEWRzxfLRIF9ygqKspvE/uo\n+KEopmOZOLuPPhSkAUD+JB7wTFHpl3R3kILjk+SfZrr+iBbQEvceZB+n04RKQbiNBlwRhGEk\n/m6TaYSYxRt0mztDWUm04hZ87Zvx1OnaMikPYRWN83y1SBTsJmq8eCotorelZZtek17N/Uj0\nUBGTLr2eu+dwcIJ/Jvm75hV1O1RJC5s2c+1B/nE6SQO5zaG14vRJ2iwKjjkkbZInXj+lDL0l\n0czOmp41mu4SWmS43jHuXfdSzVeLRMGug3fiCfEPXjwKB6oXH//k0dbSmtHUYuFnrvpSUL/g\nuvJBuw5QqWs+li5Je5B/nE6nxdQDJI0QscwlOMO17jl6130N9pZEK+7bpMn3PvejcJF6vyaR\nS7tqvlrEChaEcbROPAqu7tNOFNxIiQMmSGsuP9aKqF7h0SAFZ7rESbyT/5eN1SM8zaJ90h7k\nH6eTdN58gPYKNYLd4/WtoRWuDGUl0YqnkiVewT395HxR89UiWPBT4jEtootSslfUrM8rha3V\na/a9PIA6VgUneCYtr04NpM376T5XMl88xuIe5B/9BbdxrVshXnulDGuXRBMywWU0y7O4+qu5\nBZdFouD59Ofqo/CdVCkRhE/FNQce+lRK3k5HghO8KyrF3ZXi0Tr1r1TEdpaSV25u6r4Gyz76\nC445LK0cSYekDGUl0YxMsHBDd9fsuUe8guOk6/9nESj4UOP4Y9VHoZyk/0Iu70/PCIeos1jJ\nqugedyU4wcIMypKkfd9T2n2xeHIQhEX0mHsPso/+gmmUWKF+m4a7rgmykmhGLngR/UacfhJV\n4BE8hT4XrzvZESVYuk2aOqyeNBxx9VEYQsOXFCcPok5rxZ9SRvGMNvRwkNdg4coYim4/PrcO\nTRB/McfSKHdmT+p00b0H2Ud/wTdRm2kDo276pztDWUm0Ihd8vgN1mzkq5uYfPF/tPWo459eZ\nCQ0iSbCLpJ5vCp6jUF6SmtRvlTCz4XTh/KPt6yf3fb0qWMFC1dv5WQkZw91izs3qWL/rA56W\nLO9Hf8Gl64YmpxVK/w8oZSgriVbkgoXLC7rWa11yzPvVhP/qEEeN12ZEjuDIo/JIaP/zVgKC\nmQPBzIFg5kAwcyCYORDMHAhmDgQzB4KZA8HMgWDmQDBzIJg5EMwcCGYOBDMHgpkDwcyBYOZA\nMHMgmDkQzBwIZg4EMweCmQPBzIFg5kAwcyCYORDMHAhmDgQzB4KZA8H+fJr2nt1FMA8I9ufP\n9JrdRTAPCJZRPZi8V7B13TqHjMgRXF6S1Wigq299byfRsu4Epzf6LieqboeXBWGw1NdPWXW3\nzk/Tf0shy6Uujh1JxAg+khYzpDiDflWrz2i54Pod0+aUJtG7wiezqfiVn6u7dT5Ed0khA+J+\ntLX0+okYwXfRGkGo6BO1X95JtFwwdRTPx5tpUs0purpb586NrwrC8eixthbeAJEi+HS01BWh\n8GH2enkn0bUEr5bSiYM9gt3dOi+mv0pn6NV2FdwokSJ4c3XPwbX7jK4l+KCUTpYJdnXr/LV0\nVh+QeFlhn44gUgS/Ti9Vp+SdRNcSLJ2Qawl291LbuqV4hg5xf4MhJFIEb6DfVqfknUTL+mxW\nEOzuUnI+7VxO2nsgDTciRfB3NFKafRy7Ut5JtKzP5usK/oIW928c+j4lQ0WkCBbyoj4WhKsD\no/bKO4mW9dlcS7B0u1wjuColPfoeGwtukIgR/M1NMSNmZUmj5sk6iZb12SwT/Al1X3LJI1go\nIaki7VQiRrBwbEqbxK5/kE7J3k6iZX02ywT/NCK+8Rmv4E8ppdK2UhsmcgRfh8B9Nv+vdKPk\nWCJecGDm0la7i2AACA7Aj18ltrW7DEaA4AAkU9S7dpfBCBAcgN/NcHAVWoBg9kAwcyCYORDM\nHAhmDgQzB4KZA8HMgWDmQDBzIJg5EMwcCGYOBDMHgpkDwcyBYOZAMHMgmDkQzBwIZg4EMweC\nmQPBzIFg5kAwc/4fdlll7AspPNUAAAAASUVORK5CYII=",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "apply(a[c(2,3)], 2, function(x) { \n",
    "    boxplot(x, main = names(x) , names = a[[1]], xlab = \"country\", ylab = \"IRMS\")\n",
    "    }\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The above example is, however, not a very good one since boxplot itself is a very powerful function to aggregate data. The same result is thus optained by the simple call:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 224,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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AcAXx1IRGl7kzoUDqKMWsFAABvIAcBr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",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "boxplot(mean ~ country, d)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "R",
   "language": "R",
   "name": "ir"
  },
  "language_info": {
   "codemirror_mode": "r",
   "file_extension": ".r",
   "mimetype": "text/x-r-source",
   "name": "R",
   "pygments_lexer": "r",
   "version": "4.0.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}