Random-Forest_WIne-Quality.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "32410f24-88a5-483d-867e-da546257db5b",
   "metadata": {},
   "source": [
    "# Wine Quality\n",
    "\n",
    "Practice example from: https://www.simplilearn.com/tutorials/data-science-tutorial/random-forest-in-r "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f36330ef-5584-469f-88c6-c33942214ad3",
   "metadata": {},
   "source": [
    "# Import and Explore Data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "71b74b78-61f0-4e4d-9bba-ef71ada29e64",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>fixed.acidity</th><th scope=col>volatile.acidity</th><th scope=col>citric.acid</th><th scope=col>residual.sugar</th><th scope=col>chlorides</th><th scope=col>free.sulfur.dioxide</th><th scope=col>total.sulfur.dioxide</th><th scope=col>density</th><th scope=col>pH</th><th scope=col>sulphates</th><th scope=col>alcohol</th><th scope=col>quality</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td> 7.4  </td><td>0.70  </td><td>0.00  </td><td>1.9   </td><td>0.076 </td><td>11    </td><td>34    </td><td>0.9978</td><td>3.51  </td><td>0.56  </td><td>9.4   </td><td>5     </td></tr>\n",
       "\t<tr><td> 7.8  </td><td>0.88  </td><td>0.00  </td><td>2.6   </td><td>0.098 </td><td>25    </td><td>67    </td><td>0.9968</td><td>3.20  </td><td>0.68  </td><td>9.8   </td><td>5     </td></tr>\n",
       "\t<tr><td> 7.8  </td><td>0.76  </td><td>0.04  </td><td>2.3   </td><td>0.092 </td><td>15    </td><td>54    </td><td>0.9970</td><td>3.26  </td><td>0.65  </td><td>9.8   </td><td>5     </td></tr>\n",
       "\t<tr><td>11.2  </td><td>0.28  </td><td>0.56  </td><td>1.9   </td><td>0.075 </td><td>17    </td><td>60    </td><td>0.9980</td><td>3.16  </td><td>0.58  </td><td>9.8   </td><td>6     </td></tr>\n",
       "\t<tr><td> 7.4  </td><td>0.70  </td><td>0.00  </td><td>1.9   </td><td>0.076 </td><td>11    </td><td>34    </td><td>0.9978</td><td>3.51  </td><td>0.56  </td><td>9.4   </td><td>5     </td></tr>\n",
       "\t<tr><td> 7.4  </td><td>0.66  </td><td>0.00  </td><td>1.8   </td><td>0.075 </td><td>13    </td><td>40    </td><td>0.9978</td><td>3.51  </td><td>0.56  </td><td>9.4   </td><td>5     </td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|llllllllllll}\n",
       " fixed.acidity & volatile.acidity & citric.acid & residual.sugar & chlorides & free.sulfur.dioxide & total.sulfur.dioxide & density & pH & sulphates & alcohol & quality\\\\\n",
       "\\hline\n",
       "\t  7.4   & 0.70   & 0.00   & 1.9    & 0.076  & 11     & 34     & 0.9978 & 3.51   & 0.56   & 9.4    & 5     \\\\\n",
       "\t  7.8   & 0.88   & 0.00   & 2.6    & 0.098  & 25     & 67     & 0.9968 & 3.20   & 0.68   & 9.8    & 5     \\\\\n",
       "\t  7.8   & 0.76   & 0.04   & 2.3    & 0.092  & 15     & 54     & 0.9970 & 3.26   & 0.65   & 9.8    & 5     \\\\\n",
       "\t 11.2   & 0.28   & 0.56   & 1.9    & 0.075  & 17     & 60     & 0.9980 & 3.16   & 0.58   & 9.8    & 6     \\\\\n",
       "\t  7.4   & 0.70   & 0.00   & 1.9    & 0.076  & 11     & 34     & 0.9978 & 3.51   & 0.56   & 9.4    & 5     \\\\\n",
       "\t  7.4   & 0.66   & 0.00   & 1.8    & 0.075  & 13     & 40     & 0.9978 & 3.51   & 0.56   & 9.4    & 5     \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| fixed.acidity | volatile.acidity | citric.acid | residual.sugar | chlorides | free.sulfur.dioxide | total.sulfur.dioxide | density | pH | sulphates | alcohol | quality |\n",
       "|---|---|---|---|---|---|---|---|---|---|---|---|\n",
       "|  7.4   | 0.70   | 0.00   | 1.9    | 0.076  | 11     | 34     | 0.9978 | 3.51   | 0.56   | 9.4    | 5      |\n",
       "|  7.8   | 0.88   | 0.00   | 2.6    | 0.098  | 25     | 67     | 0.9968 | 3.20   | 0.68   | 9.8    | 5      |\n",
       "|  7.8   | 0.76   | 0.04   | 2.3    | 0.092  | 15     | 54     | 0.9970 | 3.26   | 0.65   | 9.8    | 5      |\n",
       "| 11.2   | 0.28   | 0.56   | 1.9    | 0.075  | 17     | 60     | 0.9980 | 3.16   | 0.58   | 9.8    | 6      |\n",
       "|  7.4   | 0.70   | 0.00   | 1.9    | 0.076  | 11     | 34     | 0.9978 | 3.51   | 0.56   | 9.4    | 5      |\n",
       "|  7.4   | 0.66   | 0.00   | 1.8    | 0.075  | 13     | 40     | 0.9978 | 3.51   | 0.56   | 9.4    | 5      |\n",
       "\n"
      ],
      "text/plain": [
       "  fixed.acidity volatile.acidity citric.acid residual.sugar chlorides\n",
       "1  7.4          0.70             0.00        1.9            0.076    \n",
       "2  7.8          0.88             0.00        2.6            0.098    \n",
       "3  7.8          0.76             0.04        2.3            0.092    \n",
       "4 11.2          0.28             0.56        1.9            0.075    \n",
       "5  7.4          0.70             0.00        1.9            0.076    \n",
       "6  7.4          0.66             0.00        1.8            0.075    \n",
       "  free.sulfur.dioxide total.sulfur.dioxide density pH   sulphates alcohol\n",
       "1 11                  34                   0.9978  3.51 0.56      9.4    \n",
       "2 25                  67                   0.9968  3.20 0.68      9.8    \n",
       "3 15                  54                   0.9970  3.26 0.65      9.8    \n",
       "4 17                  60                   0.9980  3.16 0.58      9.8    \n",
       "5 11                  34                   0.9978  3.51 0.56      9.4    \n",
       "6 13                  40                   0.9978  3.51 0.56      9.4    \n",
       "  quality\n",
       "1 5      \n",
       "2 5      \n",
       "3 5      \n",
       "4 6      \n",
       "5 5      \n",
       "6 5      "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "wine <- read.csv(url(\"https://archive.ics.uci.edu/ml/machine-learning-databases/wine-quality/winequality-red.csv\"), header = TRUE, sep = \";\") \n",
    "\n",
    "head(wine)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "364e4212-5605-494e-b985-fcc33aacb3c0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th></th><th scope=col>Min.</th><th scope=col>1st Qu.</th><th scope=col>Median</th><th scope=col>Mean</th><th scope=col>3rd Qu.</th><th scope=col>Max.</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>fixed.acidity</th><td>4.60000    </td><td> 7.1000    </td><td> 7.90000   </td><td> 8.31963727</td><td> 9.200000  </td><td> 15.90000  </td></tr>\n",
       "\t<tr><th scope=row>volatile.acidity</th><td>0.12000    </td><td> 0.3900    </td><td> 0.52000   </td><td> 0.52782051</td><td> 0.640000  </td><td>  1.58000  </td></tr>\n",
       "\t<tr><th scope=row>citric.acid</th><td>0.00000    </td><td> 0.0900    </td><td> 0.26000   </td><td> 0.27097561</td><td> 0.420000  </td><td>  1.00000  </td></tr>\n",
       "\t<tr><th scope=row>residual.sugar</th><td>0.90000    </td><td> 1.9000    </td><td> 2.20000   </td><td> 2.53880550</td><td> 2.600000  </td><td> 15.50000  </td></tr>\n",
       "\t<tr><th scope=row>chlorides</th><td>0.01200    </td><td> 0.0700    </td><td> 0.07900   </td><td> 0.08746654</td><td> 0.090000  </td><td>  0.61100  </td></tr>\n",
       "\t<tr><th scope=row>free.sulfur.dioxide</th><td>1.00000    </td><td> 7.0000    </td><td>14.00000   </td><td>15.87492183</td><td>21.000000  </td><td> 72.00000  </td></tr>\n",
       "\t<tr><th scope=row>total.sulfur.dioxide</th><td>6.00000    </td><td>22.0000    </td><td>38.00000   </td><td>46.46779237</td><td>62.000000  </td><td>289.00000  </td></tr>\n",
       "\t<tr><th scope=row>density</th><td>0.99007    </td><td> 0.9956    </td><td> 0.99675   </td><td> 0.99674668</td><td> 0.997835  </td><td>  1.00369  </td></tr>\n",
       "\t<tr><th scope=row>pH</th><td>2.74000    </td><td> 3.2100    </td><td> 3.31000   </td><td> 3.31111320</td><td> 3.400000  </td><td>  4.01000  </td></tr>\n",
       "\t<tr><th scope=row>sulphates</th><td>0.33000    </td><td> 0.5500    </td><td> 0.62000   </td><td> 0.65814884</td><td> 0.730000  </td><td>  2.00000  </td></tr>\n",
       "\t<tr><th scope=row>alcohol</th><td>8.40000    </td><td> 9.5000    </td><td>10.20000   </td><td>10.42298311</td><td>11.100000  </td><td> 14.90000  </td></tr>\n",
       "\t<tr><th scope=row>quality</th><td>3.00000    </td><td> 5.0000    </td><td> 6.00000   </td><td> 5.63602251</td><td> 6.000000  </td><td>  8.00000  </td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|llllll}\n",
       "  & Min. & 1st Qu. & Median & Mean & 3rd Qu. & Max.\\\\\n",
       "\\hline\n",
       "\tfixed.acidity & 4.60000     &  7.1000     &  7.90000    &  8.31963727 &  9.200000   &  15.90000  \\\\\n",
       "\tvolatile.acidity & 0.12000     &  0.3900     &  0.52000    &  0.52782051 &  0.640000   &   1.58000  \\\\\n",
       "\tcitric.acid & 0.00000     &  0.0900     &  0.26000    &  0.27097561 &  0.420000   &   1.00000  \\\\\n",
       "\tresidual.sugar & 0.90000     &  1.9000     &  2.20000    &  2.53880550 &  2.600000   &  15.50000  \\\\\n",
       "\tchlorides & 0.01200     &  0.0700     &  0.07900    &  0.08746654 &  0.090000   &   0.61100  \\\\\n",
       "\tfree.sulfur.dioxide & 1.00000     &  7.0000     & 14.00000    & 15.87492183 & 21.000000   &  72.00000  \\\\\n",
       "\ttotal.sulfur.dioxide & 6.00000     & 22.0000     & 38.00000    & 46.46779237 & 62.000000   & 289.00000  \\\\\n",
       "\tdensity & 0.99007     &  0.9956     &  0.99675    &  0.99674668 &  0.997835   &   1.00369  \\\\\n",
       "\tpH & 2.74000     &  3.2100     &  3.31000    &  3.31111320 &  3.400000   &   4.01000  \\\\\n",
       "\tsulphates & 0.33000     &  0.5500     &  0.62000    &  0.65814884 &  0.730000   &   2.00000  \\\\\n",
       "\talcohol & 8.40000     &  9.5000     & 10.20000    & 10.42298311 & 11.100000   &  14.90000  \\\\\n",
       "\tquality & 3.00000     &  5.0000     &  6.00000    &  5.63602251 &  6.000000   &   8.00000  \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| <!--/--> | Min. | 1st Qu. | Median | Mean | 3rd Qu. | Max. |\n",
       "|---|---|---|---|---|---|---|\n",
       "| fixed.acidity | 4.60000     |  7.1000     |  7.90000    |  8.31963727 |  9.200000   |  15.90000   |\n",
       "| volatile.acidity | 0.12000     |  0.3900     |  0.52000    |  0.52782051 |  0.640000   |   1.58000   |\n",
       "| citric.acid | 0.00000     |  0.0900     |  0.26000    |  0.27097561 |  0.420000   |   1.00000   |\n",
       "| residual.sugar | 0.90000     |  1.9000     |  2.20000    |  2.53880550 |  2.600000   |  15.50000   |\n",
       "| chlorides | 0.01200     |  0.0700     |  0.07900    |  0.08746654 |  0.090000   |   0.61100   |\n",
       "| free.sulfur.dioxide | 1.00000     |  7.0000     | 14.00000    | 15.87492183 | 21.000000   |  72.00000   |\n",
       "| total.sulfur.dioxide | 6.00000     | 22.0000     | 38.00000    | 46.46779237 | 62.000000   | 289.00000   |\n",
       "| density | 0.99007     |  0.9956     |  0.99675    |  0.99674668 |  0.997835   |   1.00369   |\n",
       "| pH | 2.74000     |  3.2100     |  3.31000    |  3.31111320 |  3.400000   |   4.01000   |\n",
       "| sulphates | 0.33000     |  0.5500     |  0.62000    |  0.65814884 |  0.730000   |   2.00000   |\n",
       "| alcohol | 8.40000     |  9.5000     | 10.20000    | 10.42298311 | 11.100000   |  14.90000   |\n",
       "| quality | 3.00000     |  5.0000     |  6.00000    |  5.63602251 |  6.000000   |   8.00000   |\n",
       "\n"
      ],
      "text/plain": [
       "                     Min.    1st Qu. Median   Mean        3rd Qu.   Max.     \n",
       "fixed.acidity        4.60000  7.1000  7.90000  8.31963727  9.200000  15.90000\n",
       "volatile.acidity     0.12000  0.3900  0.52000  0.52782051  0.640000   1.58000\n",
       "citric.acid          0.00000  0.0900  0.26000  0.27097561  0.420000   1.00000\n",
       "residual.sugar       0.90000  1.9000  2.20000  2.53880550  2.600000  15.50000\n",
       "chlorides            0.01200  0.0700  0.07900  0.08746654  0.090000   0.61100\n",
       "free.sulfur.dioxide  1.00000  7.0000 14.00000 15.87492183 21.000000  72.00000\n",
       "total.sulfur.dioxide 6.00000 22.0000 38.00000 46.46779237 62.000000 289.00000\n",
       "density              0.99007  0.9956  0.99675  0.99674668  0.997835   1.00369\n",
       "pH                   2.74000  3.2100  3.31000  3.31111320  3.400000   4.01000\n",
       "sulphates            0.33000  0.5500  0.62000  0.65814884  0.730000   2.00000\n",
       "alcohol              8.40000  9.5000 10.20000 10.42298311 11.100000  14.90000\n",
       "quality              3.00000  5.0000  6.00000  5.63602251  6.000000   8.00000"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t(do.call(cbind, lapply(wine, summary)))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "55a640f1-4626-4f58-9cbf-22dba30e793d",
   "metadata": {},
   "source": [
    "## Check Missing Values"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "bffc6d63-f006-470e-a68a-480fe20f6841",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>fixed.acidity</th><td>0</td></tr>\n",
       "\t<tr><th scope=row>volatile.acidity</th><td>0</td></tr>\n",
       "\t<tr><th scope=row>citric.acid</th><td>0</td></tr>\n",
       "\t<tr><th scope=row>residual.sugar</th><td>0</td></tr>\n",
       "\t<tr><th scope=row>chlorides</th><td>0</td></tr>\n",
       "\t<tr><th scope=row>free.sulfur.dioxide</th><td>0</td></tr>\n",
       "\t<tr><th scope=row>total.sulfur.dioxide</th><td>0</td></tr>\n",
       "\t<tr><th scope=row>density</th><td>0</td></tr>\n",
       "\t<tr><th scope=row>pH</th><td>0</td></tr>\n",
       "\t<tr><th scope=row>sulphates</th><td>0</td></tr>\n",
       "\t<tr><th scope=row>alcohol</th><td>0</td></tr>\n",
       "\t<tr><th scope=row>quality</th><td>0</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|l}\n",
       "\tfixed.acidity & 0\\\\\n",
       "\tvolatile.acidity & 0\\\\\n",
       "\tcitric.acid & 0\\\\\n",
       "\tresidual.sugar & 0\\\\\n",
       "\tchlorides & 0\\\\\n",
       "\tfree.sulfur.dioxide & 0\\\\\n",
       "\ttotal.sulfur.dioxide & 0\\\\\n",
       "\tdensity & 0\\\\\n",
       "\tpH & 0\\\\\n",
       "\tsulphates & 0\\\\\n",
       "\talcohol & 0\\\\\n",
       "\tquality & 0\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| fixed.acidity | 0 |\n",
       "| volatile.acidity | 0 |\n",
       "| citric.acid | 0 |\n",
       "| residual.sugar | 0 |\n",
       "| chlorides | 0 |\n",
       "| free.sulfur.dioxide | 0 |\n",
       "| total.sulfur.dioxide | 0 |\n",
       "| density | 0 |\n",
       "| pH | 0 |\n",
       "| sulphates | 0 |\n",
       "| alcohol | 0 |\n",
       "| quality | 0 |\n",
       "\n"
      ],
      "text/plain": [
       "                     [,1]\n",
       "fixed.acidity        0   \n",
       "volatile.acidity     0   \n",
       "citric.acid          0   \n",
       "residual.sugar       0   \n",
       "chlorides            0   \n",
       "free.sulfur.dioxide  0   \n",
       "total.sulfur.dioxide 0   \n",
       "density              0   \n",
       "pH                   0   \n",
       "sulphates            0   \n",
       "alcohol              0   \n",
       "quality              0   "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "cbind(colSums(is.na(wine)))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fbc678cd-29f5-48c8-af09-d2284398fb20",
   "metadata": {},
   "source": [
    "No missing values - good!"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "268dec08-ce36-4ff8-b4b7-169ae4999bdd",
   "metadata": {},
   "source": [
    "## Check Distributions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "c97f310d-8a22-4973-9386-1e075d312ff4",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Loading required package: lattice\n",
      "Loading required package: survival\n",
      "Loading required package: Formula\n",
      "Loading required package: ggplot2\n",
      "Registered S3 methods overwritten by 'ggplot2':\n",
      "  method         from \n",
      "  [.quosures     rlang\n",
      "  c.quosures     rlang\n",
      "  print.quosures rlang\n",
      "\n",
      "Attaching package: 'Hmisc'\n",
      "\n",
      "The following objects are masked from 'package:base':\n",
      "\n",
      "    format.pval, units\n",
      "\n"
     ]
    },
    {
     "data": {
      "image/png": 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Pep/X4CKvedmv5lRKR4rYO8m/dSVblehEWKRmlIpN8z802JtL1eJL8LBgU+UqQ1\nZxhTRHLp4Wo+a0TyQ1wgknP+S14kP2/uHMmISIkAfjUuiOQ28caJke0EaR1cZuW36pZFqpJk\n1Yj0Z4PbX/IiXfJGO1FapNoYVoqU16RCpEQh0uT249LrsyJ9/jEsktCoNcK1zKPfFEQq5fUS\ngtpXh7BQpFTzHONN+sKkGpEyn/C3/g8VyQ/8RSLVR7BOpOieLptpH5FK9XjsiHQKvCbEySJ9\na+y9pEUq5vUSLrVvCGAs1v2+1W8xmSE14Dh/0hodtEr7kCe1nz2G192QvQR+8w3Zb/Hb6eW6\ns5a8nl6n2if6bLxu7eEAwBVEAhAAkQAEQCQAARAJQABEAhAAkQAEQCQAARAJQABEAhAAkQAE\nQCQAARAJQABEAhAAkQAEQCQAARAJQABEAhAAkQAEQCR4MKnuLd/tEQkeTOmJL0v2BGCdx4kU\nfdxW5hlcGqh7EJNL5Yw9Fk1fxKnH81mOaed44PfnuVz+Q7m8Z7ELVfxOkdZWoZX655mln3E3\nVrQsrrR4SrQRUxa3bZ4s38X93U8iRJoMImXLaFh1Ez9nrg9a/S2l/sxL396mchpYTyPtPoMI\nH4apgePvHIRV857huU8Sfm/2LEfjnTNfI14az/dfb/Fof5Mx5TmGotMs7vfo308eKyIFA2uY\nEBw1dJCsWnCIO0d13vbX47z8t4R67fu/RbsxZYlGuG1+eMZE2i4tlH1RQ9jhvFi2S0iXAIJw\nT53unmNGaM+lm13rbSCmLIWe5o5MYnubSaVIzp9u6GCvnfc89X3NZWZT0+mOfy4RL43nVM3f\njOf77lpvAzHlcdc57Omd+4UgsjOpgvLll0RaV6FqzodYl1o5cPRey0Wk6CdvLaZxTIl0baHS\niw7CDhccFAY73dpQw2C8EcloTOOYESl2hcdLOA+/Gqd256pdUpwfld+VvLeR+L2IF8bj1eb7\nj4tFaSmmUQTrayxyAJ0gEoAAiAQgACIBCIBIAAIgEoAAwiId10mdf000sRNlN8LjEBERVddB\nrjT3K/O76KXEdq29lYiIiOoqIVuYhpAkIaLUGj3oiGi8SO+nRt5t8GhIn6yXH4Tog4iIqKNC\nwyU4r5R9eur2xf3ll9X54etsJCIiouYKiZTggjd/lXfei79WeyNtREREHRUaL+EUkr8Qvlhs\nJH+BiFSgLiJECiGijYh6KjReQmyQ/S76q602EhGpQ11EYiL51Uz9xsids2u92UdEGxH1VAgA\nBkEkAAEQCUAARLejrQwAAA+hSURBVAIQAJEABEAkAAEQCUAARAIQAJEABEAkAAEQCUAARAIQ\nAJEABEAkAAEQCUAARAIQAJEABEAkAAEQCUAARAIQAJEABEAkAAEQCUAARAIQAJEABEAkAAEQ\nCUAARAIQAJEABEAkAAEQCUAARAIQAJEABEAkAAEQCUAARAIQAJEABEAkAAEQCUAARAIQAJEA\nBEAkAAEQCUAARAIQAJEABEAkAAEQCUAARAIQAJEABEAkAAEQCUAARAIQAJEABEAkAAEQCUAA\nRAIQAJEABEAkAAEQCUAARAIQAJEABEAkAAEQCUAARAIQAJEABECkp+C+3F2Pl/LWz/1x3c4F\nC7CSl37sz+t2z4vIFi/92J/X7Z4XkS1e+rE/r9s9LyJbvPRjf2C3e9xZXxS1Ueqr0RrUNgjU\noK/d9NUI+njVoUFflKtrpKWtG7udklpnaJ2s6o8oR0vV1wS6XKQb9hmhudupqHWOaESZw4X+\niHI0ibQkUETyEi13u9yhwWZEORBJs0jpJAPd7nkR5UCkr0h3nyo9sNvlx9NYmvaIMiDSLtIN\nuz7X4+3d7nkRpfMi0i0Y7Xb7YeG1Y2yyGESaWY//2+dRI9Ih0WNEap1+p8tBpNnVeFy3i9fT\nckTbHJHET9ERKboukmaj28V7TTy/jYgmiSQdPSJF10XSjHS72JHWdkSIlNqfEpHOC5F15zTl\n3e5nUnrVJU19RBMvNiCSYD0SdTDa7TK8I6J0XkS6hXd0OwMRMSIV9odIi7EZEedIpf0h0mJs\nRhQVqeNXVw8W6fNBaG1Hm90uh82IGJFK+2NEWozNiN4iUt9PmxHpBoxG9K6LDYikn3dElM6L\nSLfwjm5nICLvfDq2qr4YRLoFo90ug82Ijj7y8HOkvgI0i2T6mQ05bEaESKXsWkX6YrPb5Qiq\n/jlcKI8IkUrZEWkxoUgWIjokQqREdmsimTh+57ApEpe/S9nNifT5z+6jSY2KlAGRDIukt8Yl\nEKkucQRECkAk/SASIt0AItUljoBIAYikH0RCpBtApLrEERAp4EEipS4gGw3nByIh0g08SCT9\nv5DtvUmiV6RsRC03+9SIJNZGOsLZ1vS65SNS3w1HvSLlInLBQmSdl6Cl5wm1kZZwthW97o6p\nXWrQFPqyxrZ+aleqtTGRtnQb5TaJJGgJZ5OJKJv3nhEp17Vqul15f6tHpNGIFImUjii7VSRB\nRTibWETZvPecI80TaR/UVp8jPUakXETZ7SIJCsLZBCPK5l0/tStlGhTp9/9lU7tkXSxebOhF\nr0i9IJIWkTIgkn40i5Tr2+MXG7xp3TKRpCLSI1Lnx6ZYJKmIsnnXniOd37YXUM62WKRcREbP\nkbx/27c7JygIZ5ve67551Yj0lzo4Im36RSp+oebunict0v2/VcxGZPMXsqVu5xKrHyRSKumx\nInUVJopYG2X3sfgcKX18iorU9q2nG0QqRXReiKzzEnSI1DuG6BVJrI1yu1gsUiGTvREpXx+T\nFxs6USxShieK5NzTRMqASDqwKlJ+opZ8IpVikXIR7WsMTu2edI5UbCOrFxsGCihnWyxSrvxD\nIlMiibWRGpEm97pvXkQaoihSwg1EWkd294xIiWzqREp9xmECIs2hoo2snSPlLmY7lz44KBYp\nF9GRJ73OT1Ai0oRzpHvvyda00an6bTddvpsvFak3l16RivXxXyOr/AQdIvWSH5G0RmV0ROrN\nNiDS/d9QiWJUpK47Y5qjsirScbOot4BitsiINLURpSL6iXS3+ZmImrqdHpFybWT2YkPXh6pX\nJLGI3Pm/+8hVwaZIk3vdNy8iDYFIySRLIkVXIhIiDTBfpNWT12xE49eKv3mXiiT2zeJ0tsUi\nyX9X+m6RshGJXGxYHmE+or9/o6tadrFYpLkF3CFSJ4pF6kSxSHmSf3IUkRBpFjZHpBKJ8Uqx\nSGJ3zdPZVp8jiX8P4O5u1vw9gFSSGpEq2siaSJF3HQVks90xIjV94olmVSPSXpts8smLUkR3\ni7TvdOYmt0ztnjQi/XY8uIUukRJVsDkiHXuduAUiiYBIwbTDuYhIp9TFPEqkzhMKxSKJRaRG\npGxE1RcbTrG4s0KrI5zc6755bxiRphVw24jUjl6ReimIZDBCREKkG0CkusQRLlO79oFWr0hi\nEanpZjIRaRJpcq/75l07InX2a70iiUWkRqRcRPXfTNMkUnMbdXin6JHF8S2bQrpjRPL+bd/u\nnKBDpGxEmZrpFUmsjbJ5dYt0LaCcDZEGyUeUrhoi1SWO4E3tpl2I3EtePrWbcfm7q0gpRCLS\nJNLk3xx8864VaV4BLvz/EpE6uXS74C6LxjrnUSxSJ4ikRaSu25eGetoJRKpLHOE0tespXLFI\nmYhcsBBZt7/T09NE2kiVSPN6nZd3qUi9n6JekXIR2RRJpo00iTSx13l5EWkIRPK2i5WCSILc\nLdLEq2DTRLrtwt08kc5XVNaR3WX1eWxpH0tFmnn5O/z/NWkK4t+Vvn1QetXl7/qDXWEXi0Wa\nV8BdInWiWKROakXaxyUdGBWpt1y9ImXLNTkiybSRphEptzNEimVRJlJ9IyHSTJ4oUuePFRWL\nlIko2kjRUUqVSCJtpEqkXERGLzbMK6Ak0g2XwWyOSL20idSr60oQKTEiIdJMWkekO1okgsER\naaBQpSKVCrR3sUGsjdSIVNhF/cGuVM5akToLViySSESqRMrtt+vQcLtImd1EReqYeFaLNDSf\nRaTopqd3JkSqP36bFimdlCmnVqTxUR+RLpue3iHSHBApLKCYRZlIzqXnCIikQ6TlFxskRMr0\nK399soB0+d/NlotUiiiz6fVd5L8bLg8XIrIoUqGNcps25V0nUm2m9kE2ZswKkYqk91g3It0x\nKGWRv9hw992kff/tve6SF5FuwahIMYrf1ciLdG+Uh0SItCHSMj7dTnZEUiBSohKIhEiTOD7F\nYhvZEinlQUs5akQaudgQ/z8iSSMukrv7x0n7vstXVkvl6BEpumXVFRdEWsTzRqSfSelVdcWo\nEenTQIxIqkXan2slK5LSr4EbFqmukZKblkRS1GI2Rdo+LkXrNTQiKQwUkbIjkpYWMytSEkSq\nSkQkWRAJkSbVqH7+HV2NSDeDSFWJS67aVc6/o6utiFT8HgAiacGuSF0FxL+wqlakL5Ui6bk6\nUgSRqhIVi2RrRPpSPSJpqXARr6LHfVajIk39hSwiifJskRiRkomIJEv6+P1ikZTNZBHJmEiM\nSL9nOOq9aV7Ki0i3gEjJFWpCRiREugFEqkpEJFkQCZHm1qi5gKeLpOecIQ8iVSUikiwtI5KW\nOudBpKpERJIFkTIiKRmEE5VIPKfwfSKpuMKKSNkRSUXIKZFqnXm2SL9v5t0KIiGSWI3GCvCe\noNk4IknUaBREQiSxGo0VUG+MeZEUTEXLIFJVoi6RLg/77hDp7s6577z4RbufS8p1mibSbYG/\nQaTz//tGJB0itf6nVidpkfyJ+9NFqm1URAroE+n25ylmOCIKxthOkS6p90UUJMuLVBlhZa76\nJ60aEakiojaRbuxV3+oWIorOVHtECm28L6IgWbdILli4rPPClRHJ51uu6MypMqLm/yIV3+Jn\nUdKnVuWI6n3Ji5Q5dqyO6BEi+d3FLaIqJCIiovURDY9INiEi/diKCJEeAhHdS+W4JT1s3w4R\n6cdURDZqCaAcUZHuP+WbwsOimhnO0ogmB9IWkaxI+ff5t425F5LfdaFiQxtPoe1zH80+kdiu\noruvTqzfujNXJYgkvPEUEElg685clSCS8MZTQCSBrTtzVYJIwhtPAZEEtu7MVQkiCW88BUQS\n2LozVyWIJLzxFBBJYOvOXJUgkvDGU0Akga07c1WCSMIbTwGRBLbuzAUAWRAJQABEAhAAkQAE\nQCQAARAJQABEAhAAkQAEQCQAARAJQAARkbzf48Z+mju2+lfHtT9kDioWVjySMbX6+sWaUsyz\nyO/NnV4S69NFHKsXBJTYS5gWb7owLf6z8qU/NXe/grxFodX/YgkyLsHfX7TNXCRjZHWscX/J\na6PK7+1b5WTX+f0huPiX0hbGkthVpOpJ3XLHvlLGik17KXQLF12MbR2u0ynS71m20Yp5j7rN\nHb81ieT21ckKZ0Ny97TQOaVBpJr0J4l0NN+NIiUMDzMmVyc3VyRS5dQuncMl18wh0u/jKlQl\nFqYOrXXpZW+D2JzynBabh+5romciCkSKhxVmjK7OTufXi5SZ9NeJlD1HWnSKFK9FTKT42W3+\nrPa8eV11qnJVl1MxImUOEbE/xuzKZU/iOiIlD3Y1I1Jy9eKo/vWL+SPSonjC6XTl5Cz+V79T\naq0ckQqdpqoJ4lurEGm7LJ4TCiJ1bTyTqSIV9iBKjUipjDpFOn+COZGys9PI1t5FyUeJVP7E\n5vFUkdLPRa0SKdlt14l0mbeFY0quiYozKAUjkvjULv+JzSM6pTmvzuUozN0WTu1Su6mc2iUT\nK3dTseMO9iOB+y7H1n92FT1m/FZn7mym186jVLHjU87ckI2vLn1i88jvrGpEqrgh21m5Fppu\nyFYlxseyW579DfBWEAlAAEQCEACRAARAJAABEAlAAEQCEACRAARAJAABEAlAAEQCEACRAARA\nJAABEAlAAEQCEACRAARAJAABEAlAAEQCEOA2kbynitjFRZZAN25OYyHSCCmRjIf1aCZ1PEQa\nAZHs8RiRnP/AWHs9zntE1/e/b4Jzx+pvHi/FDoWnVJmKZfs8n9l7ptp2tJmXLrMrqYLq9+i8\nR9zZapfNN2gX6fxyWr1Jt9YCniTSr10ubXZqIbF9LcYLy1i7/MN7hGJUne3aTNYifJhI0bbZ\nmyX39wR69rUW2yJFpnZndU5PUPVmePo5PVfWTzi/M0SqbRBJB9dpQvLF20A/yZCCRDOk2uYh\nInmR2GqXf4SNcu1n/mls6JRu/KZJHyLMcG6NU2OJd787RqQHXLXzpwn+XG+/ThRMkAzgz3z+\n3m/WRfq2xvHXENzPIvE/HH3P1O5BPCac89Rue4RIW6HCiKSHx4QTPR1CJJH9TMFcU+R5Tjje\n3cvrdTp3emeKTIVF59zmPhgAjSASgACIBCAAIgEIgEgAAiASgADCIh13/Z13Pzn51+Qt3PQn\nIiKqroNcad4XL76L6a9iSH9JYwpERER1lZAtTENIkhBRao0edEQ0XmTwJVTnlXwO6ZP194Nf\nkf3LQ0RE1FGh4RKcV8o+PXX74v7yy+r88HU2EhERUXOFREpwwZu/yjvvxV+rvZE2IiKijgqN\nl3AKyV8IXyw2kr9ARCpQFxEihRDRRkQ9FRovITbIfhf91VYbiYjUoS4iMZH8aqYekOjO2bXe\n7COijYh6KgQAgyASgACIBCAAIgEI8B/PUl9FUxJ+owAAAABJRU5ErkJggg==",
      "text/plain": [
       "Plot with title \"\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "library(Hmisc)\n",
    "\n",
    "hist.data.frame(wine)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d41167ee-db66-4a30-9b06-fc6f62efa001",
   "metadata": {},
   "source": [
    "Not all variables are normally distributed. This would cause problems with regression models."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7716d0a0-a4bd-4b9a-81e0-2a761a679a4f",
   "metadata": {},
   "source": [
    "## Check Correlations\n",
    "Use Pearson's Correlation Coefficient to check for correlations between the continuous-valued independent variables."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "66ddeb98-1e47-4ce8-ae9b-8868bc2d3e28",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "corrplot 0.84 loaded\n"
     ]
    },
    {
     "data": {
      "image/png": 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99VvxUeaCJtulxNn2zKWJ6aZPXbRhSp+HA5UXp+L05sThfJ16PSpMlF+uILu0mW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\nEYBIAEQAIgEQAYgEQAT+f4sj6IBefWljAAAAAElFTkSuQmCC",
      "text/plain": [
       "Plot with title \"\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "library(corrplot)\n",
    "\n",
    "corrplot(cor(wine[, !(colnames(wine) == \"quality\")]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "1d6f8fb5-842c-43f4-a470-a0b428304b14",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th></th><th scope=col>fixed.acidity</th><th scope=col>volatile.acidity</th><th scope=col>citric.acid</th><th scope=col>residual.sugar</th><th scope=col>chlorides</th><th scope=col>free.sulfur.dioxide</th><th scope=col>total.sulfur.dioxide</th><th scope=col>density</th><th scope=col>pH</th><th scope=col>sulphates</th><th scope=col>alcohol</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>fixed.acidity</th><td> 1.00000000 </td><td>-0.256130895</td><td> 0.67170343 </td><td> 0.114776724</td><td> 0.093705186</td><td>-0.153794193</td><td>-0.11318144 </td><td> 0.66804729 </td><td>-0.68297819 </td><td> 0.183005664</td><td>-0.06166827 </td></tr>\n",
       "\t<tr><th scope=row>volatile.acidity</th><td>-0.25613089 </td><td> 1.000000000</td><td>-0.55249568 </td><td> 0.001917882</td><td> 0.061297772</td><td>-0.010503827</td><td> 0.07647000 </td><td> 0.02202623 </td><td> 0.23493729 </td><td>-0.260986685</td><td>-0.20228803 </td></tr>\n",
       "\t<tr><th scope=row>citric.acid</th><td> 0.67170343 </td><td>-0.552495685</td><td> 1.00000000 </td><td> 0.143577162</td><td> 0.203822914</td><td>-0.060978129</td><td> 0.03553302 </td><td> 0.36494718 </td><td>-0.54190414 </td><td> 0.312770044</td><td> 0.10990325 </td></tr>\n",
       "\t<tr><th scope=row>residual.sugar</th><td> 0.11477672 </td><td> 0.001917882</td><td> 0.14357716 </td><td> 1.000000000</td><td> 0.055609535</td><td> 0.187048995</td><td> 0.20302788 </td><td> 0.35528337 </td><td>-0.08565242 </td><td> 0.005527121</td><td> 0.04207544 </td></tr>\n",
       "\t<tr><th scope=row>chlorides</th><td> 0.09370519 </td><td> 0.061297772</td><td> 0.20382291 </td><td> 0.055609535</td><td> 1.000000000</td><td> 0.005562147</td><td> 0.04740047 </td><td> 0.20063233 </td><td>-0.26502613 </td><td> 0.371260481</td><td>-0.22114054 </td></tr>\n",
       "\t<tr><th scope=row>free.sulfur.dioxide</th><td>-0.15379419 </td><td>-0.010503827</td><td>-0.06097813 </td><td> 0.187048995</td><td> 0.005562147</td><td> 1.000000000</td><td> 0.66766645 </td><td>-0.02194583 </td><td> 0.07037750 </td><td> 0.051657572</td><td>-0.06940835 </td></tr>\n",
       "\t<tr><th scope=row>total.sulfur.dioxide</th><td>-0.11318144 </td><td> 0.076470005</td><td> 0.03553302 </td><td> 0.203027882</td><td> 0.047400468</td><td> 0.667666450</td><td> 1.00000000 </td><td> 0.07126948 </td><td>-0.06649456 </td><td> 0.042946836</td><td>-0.20565394 </td></tr>\n",
       "\t<tr><th scope=row>density</th><td> 0.66804729 </td><td> 0.022026232</td><td> 0.36494718 </td><td> 0.355283371</td><td> 0.200632327</td><td>-0.021945831</td><td> 0.07126948 </td><td> 1.00000000 </td><td>-0.34169933 </td><td> 0.148506412</td><td>-0.49617977 </td></tr>\n",
       "\t<tr><th scope=row>pH</th><td>-0.68297819 </td><td> 0.234937294</td><td>-0.54190414 </td><td>-0.085652422</td><td>-0.265026131</td><td> 0.070377499</td><td>-0.06649456 </td><td>-0.34169933 </td><td> 1.00000000 </td><td>-0.196647602</td><td> 0.20563251 </td></tr>\n",
       "\t<tr><th scope=row>sulphates</th><td> 0.18300566 </td><td>-0.260986685</td><td> 0.31277004 </td><td> 0.005527121</td><td> 0.371260481</td><td> 0.051657572</td><td> 0.04294684 </td><td> 0.14850641 </td><td>-0.19664760 </td><td> 1.000000000</td><td> 0.09359475 </td></tr>\n",
       "\t<tr><th scope=row>alcohol</th><td>-0.06166827 </td><td>-0.202288027</td><td> 0.10990325 </td><td> 0.042075437</td><td>-0.221140545</td><td>-0.069408354</td><td>-0.20565394 </td><td>-0.49617977 </td><td> 0.20563251 </td><td> 0.093594750</td><td> 1.00000000 </td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|lllllllllll}\n",
       "  & fixed.acidity & volatile.acidity & citric.acid & residual.sugar & chlorides & free.sulfur.dioxide & total.sulfur.dioxide & density & pH & sulphates & alcohol\\\\\n",
       "\\hline\n",
       "\tfixed.acidity &  1.00000000  & -0.256130895 &  0.67170343  &  0.114776724 &  0.093705186 & -0.153794193 & -0.11318144  &  0.66804729  & -0.68297819  &  0.183005664 & -0.06166827 \\\\\n",
       "\tvolatile.acidity & -0.25613089  &  1.000000000 & -0.55249568  &  0.001917882 &  0.061297772 & -0.010503827 &  0.07647000  &  0.02202623  &  0.23493729  & -0.260986685 & -0.20228803 \\\\\n",
       "\tcitric.acid &  0.67170343  & -0.552495685 &  1.00000000  &  0.143577162 &  0.203822914 & -0.060978129 &  0.03553302  &  0.36494718  & -0.54190414  &  0.312770044 &  0.10990325 \\\\\n",
       "\tresidual.sugar &  0.11477672  &  0.001917882 &  0.14357716  &  1.000000000 &  0.055609535 &  0.187048995 &  0.20302788  &  0.35528337  & -0.08565242  &  0.005527121 &  0.04207544 \\\\\n",
       "\tchlorides &  0.09370519  &  0.061297772 &  0.20382291  &  0.055609535 &  1.000000000 &  0.005562147 &  0.04740047  &  0.20063233  & -0.26502613  &  0.371260481 & -0.22114054 \\\\\n",
       "\tfree.sulfur.dioxide & -0.15379419  & -0.010503827 & -0.06097813  &  0.187048995 &  0.005562147 &  1.000000000 &  0.66766645  & -0.02194583  &  0.07037750  &  0.051657572 & -0.06940835 \\\\\n",
       "\ttotal.sulfur.dioxide & -0.11318144  &  0.076470005 &  0.03553302  &  0.203027882 &  0.047400468 &  0.667666450 &  1.00000000  &  0.07126948  & -0.06649456  &  0.042946836 & -0.20565394 \\\\\n",
       "\tdensity &  0.66804729  &  0.022026232 &  0.36494718  &  0.355283371 &  0.200632327 & -0.021945831 &  0.07126948  &  1.00000000  & -0.34169933  &  0.148506412 & -0.49617977 \\\\\n",
       "\tpH & -0.68297819  &  0.234937294 & -0.54190414  & -0.085652422 & -0.265026131 &  0.070377499 & -0.06649456  & -0.34169933  &  1.00000000  & -0.196647602 &  0.20563251 \\\\\n",
       "\tsulphates &  0.18300566  & -0.260986685 &  0.31277004  &  0.005527121 &  0.371260481 &  0.051657572 &  0.04294684  &  0.14850641  & -0.19664760  &  1.000000000 &  0.09359475 \\\\\n",
       "\talcohol & -0.06166827  & -0.202288027 &  0.10990325  &  0.042075437 & -0.221140545 & -0.069408354 & -0.20565394  & -0.49617977  &  0.20563251  &  0.093594750 &  1.00000000 \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| <!--/--> | fixed.acidity | volatile.acidity | citric.acid | residual.sugar | chlorides | free.sulfur.dioxide | total.sulfur.dioxide | density | pH | sulphates | alcohol |\n",
       "|---|---|---|---|---|---|---|---|---|---|---|---|\n",
       "| fixed.acidity |  1.00000000  | -0.256130895 |  0.67170343  |  0.114776724 |  0.093705186 | -0.153794193 | -0.11318144  |  0.66804729  | -0.68297819  |  0.183005664 | -0.06166827  |\n",
       "| volatile.acidity | -0.25613089  |  1.000000000 | -0.55249568  |  0.001917882 |  0.061297772 | -0.010503827 |  0.07647000  |  0.02202623  |  0.23493729  | -0.260986685 | -0.20228803  |\n",
       "| citric.acid |  0.67170343  | -0.552495685 |  1.00000000  |  0.143577162 |  0.203822914 | -0.060978129 |  0.03553302  |  0.36494718  | -0.54190414  |  0.312770044 |  0.10990325  |\n",
       "| residual.sugar |  0.11477672  |  0.001917882 |  0.14357716  |  1.000000000 |  0.055609535 |  0.187048995 |  0.20302788  |  0.35528337  | -0.08565242  |  0.005527121 |  0.04207544  |\n",
       "| chlorides |  0.09370519  |  0.061297772 |  0.20382291  |  0.055609535 |  1.000000000 |  0.005562147 |  0.04740047  |  0.20063233  | -0.26502613  |  0.371260481 | -0.22114054  |\n",
       "| free.sulfur.dioxide | -0.15379419  | -0.010503827 | -0.06097813  |  0.187048995 |  0.005562147 |  1.000000000 |  0.66766645  | -0.02194583  |  0.07037750  |  0.051657572 | -0.06940835  |\n",
       "| total.sulfur.dioxide | -0.11318144  |  0.076470005 |  0.03553302  |  0.203027882 |  0.047400468 |  0.667666450 |  1.00000000  |  0.07126948  | -0.06649456  |  0.042946836 | -0.20565394  |\n",
       "| density |  0.66804729  |  0.022026232 |  0.36494718  |  0.355283371 |  0.200632327 | -0.021945831 |  0.07126948  |  1.00000000  | -0.34169933  |  0.148506412 | -0.49617977  |\n",
       "| pH | -0.68297819  |  0.234937294 | -0.54190414  | -0.085652422 | -0.265026131 |  0.070377499 | -0.06649456  | -0.34169933  |  1.00000000  | -0.196647602 |  0.20563251  |\n",
       "| sulphates |  0.18300566  | -0.260986685 |  0.31277004  |  0.005527121 |  0.371260481 |  0.051657572 |  0.04294684  |  0.14850641  | -0.19664760  |  1.000000000 |  0.09359475  |\n",
       "| alcohol | -0.06166827  | -0.202288027 |  0.10990325  |  0.042075437 | -0.221140545 | -0.069408354 | -0.20565394  | -0.49617977  |  0.20563251  |  0.093594750 |  1.00000000  |\n",
       "\n"
      ],
      "text/plain": [
       "                     fixed.acidity volatile.acidity citric.acid residual.sugar\n",
       "fixed.acidity         1.00000000   -0.256130895      0.67170343  0.114776724  \n",
       "volatile.acidity     -0.25613089    1.000000000     -0.55249568  0.001917882  \n",
       "citric.acid           0.67170343   -0.552495685      1.00000000  0.143577162  \n",
       "residual.sugar        0.11477672    0.001917882      0.14357716  1.000000000  \n",
       "chlorides             0.09370519    0.061297772      0.20382291  0.055609535  \n",
       "free.sulfur.dioxide  -0.15379419   -0.010503827     -0.06097813  0.187048995  \n",
       "total.sulfur.dioxide -0.11318144    0.076470005      0.03553302  0.203027882  \n",
       "density               0.66804729    0.022026232      0.36494718  0.355283371  \n",
       "pH                   -0.68297819    0.234937294     -0.54190414 -0.085652422  \n",
       "sulphates             0.18300566   -0.260986685      0.31277004  0.005527121  \n",
       "alcohol              -0.06166827   -0.202288027      0.10990325  0.042075437  \n",
       "                     chlorides    free.sulfur.dioxide total.sulfur.dioxide\n",
       "fixed.acidity         0.093705186 -0.153794193        -0.11318144         \n",
       "volatile.acidity      0.061297772 -0.010503827         0.07647000         \n",
       "citric.acid           0.203822914 -0.060978129         0.03553302         \n",
       "residual.sugar        0.055609535  0.187048995         0.20302788         \n",
       "chlorides             1.000000000  0.005562147         0.04740047         \n",
       "free.sulfur.dioxide   0.005562147  1.000000000         0.66766645         \n",
       "total.sulfur.dioxide  0.047400468  0.667666450         1.00000000         \n",
       "density               0.200632327 -0.021945831         0.07126948         \n",
       "pH                   -0.265026131  0.070377499        -0.06649456         \n",
       "sulphates             0.371260481  0.051657572         0.04294684         \n",
       "alcohol              -0.221140545 -0.069408354        -0.20565394         \n",
       "                     density     pH          sulphates    alcohol    \n",
       "fixed.acidity         0.66804729 -0.68297819  0.183005664 -0.06166827\n",
       "volatile.acidity      0.02202623  0.23493729 -0.260986685 -0.20228803\n",
       "citric.acid           0.36494718 -0.54190414  0.312770044  0.10990325\n",
       "residual.sugar        0.35528337 -0.08565242  0.005527121  0.04207544\n",
       "chlorides             0.20063233 -0.26502613  0.371260481 -0.22114054\n",
       "free.sulfur.dioxide  -0.02194583  0.07037750  0.051657572 -0.06940835\n",
       "total.sulfur.dioxide  0.07126948 -0.06649456  0.042946836 -0.20565394\n",
       "density               1.00000000 -0.34169933  0.148506412 -0.49617977\n",
       "pH                   -0.34169933  1.00000000 -0.196647602  0.20563251\n",
       "sulphates             0.14850641 -0.19664760  1.000000000  0.09359475\n",
       "alcohol              -0.49617977  0.20563251  0.093594750  1.00000000"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "cor(wine[, !(colnames(wine) == \"quality\")])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bc32c074-d65f-434f-b236-075df4cd18ec",
   "metadata": {},
   "source": [
    "Some independent variables are correlated with eachother. This may cause problems with model buidling.\n",
    "\n",
    "Typically Spearman's Rank Correlation would be used to measure correlation between all continuous-ordinal pairings (independent-dependent variables), but becaue this data has so many ties between rankings of wines, Kendall's Tau will be used instead. Kendall's Tau is designed to handle ties."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "459a742e-4ccd-4a09-b140-de486e903119",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Loading required package: stats4\n",
      "Loading required package: splines\n",
      "\n",
      "Attaching package: 'VGAM'\n",
      "\n",
      "The following object is masked _by_ '.GlobalEnv':\n",
      "\n",
      "    wine\n",
      "\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA0gAAANICAMAAADKOT/pAAAAMFBMVEUAAABNTU1oaGh8fHyM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\nQYCQIEBIEPAfJzHO65+CENwAAAAASUVORK5CYII=",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "library(VGAM)\n",
    "\n",
    "var <- names(wine)[1:11]\n",
    "kendal.tau <- rep(NA, 11)\n",
    "\n",
    "for (i in 1:11) {\n",
    "    kendal.tau[i] <- kendall.tau(wine[, var[i]], as.numeric(wine$quality), exact = TRUE)\n",
    "}\n",
    "\n",
    "kendal.tau <- data.frame(var, kendal.tau)\n",
    "\n",
    "plot(kendal.tau)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a84fe553-4ac5-4c23-8cb8-37638f302dee",
   "metadata": {},
   "source": [
    "None of the independent variables are strongly correlated with wine quality, but alcohol content is the most correlated."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "62917ca2-47c4-4dcd-acab-14f41add163e",
   "metadata": {},
   "source": [
    "# Prepare Data for Model\n",
    "## Convert the quality values into factors."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "1f31587b-8db4-4f7e-86c5-14c092162a19",
   "metadata": {},
   "outputs": [],
   "source": [
    "wine$quality <- as.ordered(wine$quality)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "eb055715-966b-47d3-9ccb-01daadd2d4db",
   "metadata": {},
   "source": [
    "## Split Data Into Training and Testing"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "b43c41cf-8420-420e-911f-e2bf174f3712",
   "metadata": {},
   "outputs": [],
   "source": [
    "set.seed(42)\n",
    "samp <- sample(nrow(wine), 0.8 * nrow(wine))\n",
    "train <- wine[samp, ]\n",
    "test <- wine[-samp, ]"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c2087f27-6e60-45c7-9d05-6346a69b3666",
   "metadata": {},
   "source": [
    "# Build Model"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a504e29d-4a9c-4ba4-a0d3-b1734b283ff3",
   "metadata": {},
   "source": [
    "## Create Class Weights\n",
    "Creating class weights that are inversely related to the prevelance of that class will help encourage the model to make class predictions on uncommon classes."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "ef8b6eaa-eb71-4485-9e01-3482647008ae",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<dl class=dl-horizontal>\n",
       "\t<dt>3</dt>\n",
       "\t\t<dd>159.875</dd>\n",
       "\t<dt>4</dt>\n",
       "\t\t<dd>29.0681818181818</dd>\n",
       "\t<dt>5</dt>\n",
       "\t\t<dd>2.35110294117647</dd>\n",
       "\t<dt>6</dt>\n",
       "\t\t<dd>2.47388781431335</dd>\n",
       "\t<dt>7</dt>\n",
       "\t\t<dd>8.35947712418301</dd>\n",
       "\t<dt>8</dt>\n",
       "\t\t<dd>98.3846153846154</dd>\n",
       "</dl>\n"
      ],
      "text/latex": [
       "\\begin{description*}\n",
       "\\item[3] 159.875\n",
       "\\item[4] 29.0681818181818\n",
       "\\item[5] 2.35110294117647\n",
       "\\item[6] 2.47388781431335\n",
       "\\item[7] 8.35947712418301\n",
       "\\item[8] 98.3846153846154\n",
       "\\end{description*}\n"
      ],
      "text/markdown": [
       "3\n",
       ":   159.8754\n",
       ":   29.06818181818185\n",
       ":   2.351102941176476\n",
       ":   2.473887814313357\n",
       ":   8.359477124183018\n",
       ":   98.3846153846154\n",
       "\n"
      ],
      "text/plain": [
       "         3          4          5          6          7          8 \n",
       "159.875000  29.068182   2.351103   2.473888   8.359477  98.384615 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "class.wt <- nrow(train) / summary(train$quality)\n",
    "class.wt"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f683cd96-e322-45b8-a942-037c620eee75",
   "metadata": {},
   "source": [
    "## Fit Model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "a6060a30-86e0-47c5-a9e5-55616ea1ff65",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "randomForest 4.6-14\n",
      "Type rfNews() to see new features/changes/bug fixes.\n",
      "\n",
      "Attaching package: 'randomForest'\n",
      "\n",
      "The following object is masked from 'package:ggplot2':\n",
      "\n",
      "    margin\n",
      "\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "\n",
       "Call:\n",
       " randomForest(formula = quality ~ ., data = train, ntree = 1000,      mtry = 5, classwt = class.wt, Importance = TRUE) \n",
       "               Type of random forest: classification\n",
       "                     Number of trees: 1000\n",
       "No. of variables tried at each split: 5\n",
       "\n",
       "        OOB estimate of  error rate: 30.26%\n",
       "Confusion matrix:\n",
       "  3 4   5   6  7 8 class.error\n",
       "3 0 0   6   2  0 0   1.0000000\n",
       "4 0 0  32  11  1 0   1.0000000\n",
       "5 1 0 441  99  3 0   0.1893382\n",
       "6 0 0 115 381 21 0   0.2630561\n",
       "7 0 0   9  76 68 0   0.5555556\n",
       "8 0 0   0   8  3 2   0.8461538"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "library(randomForest)\n",
    "\n",
    "model <- randomForest(quality ~ ., data = train, ntree = 1000, mtry = 5, classwt = class.wt, Importance = TRUE)\n",
    "\n",
    "model"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "52232b4c-7354-4026-9619-872c54eb4c48",
   "metadata": {},
   "source": [
    "## Variable Importance"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "c59d3123-aaef-4071-b80f-6280958bc0fb",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th></th><th scope=col>MeanDecreaseGini</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>alcohol</th><td>138.37505</td></tr>\n",
       "\t<tr><th scope=row>sulphates</th><td>133.06089</td></tr>\n",
       "\t<tr><th scope=row>volatile.acidity</th><td>115.20571</td></tr>\n",
       "\t<tr><th scope=row>chlorides</th><td>106.56542</td></tr>\n",
       "\t<tr><th scope=row>density</th><td> 49.47299</td></tr>\n",
       "\t<tr><th scope=row>pH</th><td> 45.09919</td></tr>\n",
       "\t<tr><th scope=row>citric.acid</th><td> 43.76360</td></tr>\n",
       "\t<tr><th scope=row>total.sulfur.dioxide</th><td> 40.59024</td></tr>\n",
       "\t<tr><th scope=row>fixed.acidity</th><td> 23.50309</td></tr>\n",
       "\t<tr><th scope=row>residual.sugar</th><td> 22.32345</td></tr>\n",
       "\t<tr><th scope=row>free.sulfur.dioxide</th><td> 19.73800</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|l}\n",
       "  & MeanDecreaseGini\\\\\n",
       "\\hline\n",
       "\talcohol & 138.37505\\\\\n",
       "\tsulphates & 133.06089\\\\\n",
       "\tvolatile.acidity & 115.20571\\\\\n",
       "\tchlorides & 106.56542\\\\\n",
       "\tdensity &  49.47299\\\\\n",
       "\tpH &  45.09919\\\\\n",
       "\tcitric.acid &  43.76360\\\\\n",
       "\ttotal.sulfur.dioxide &  40.59024\\\\\n",
       "\tfixed.acidity &  23.50309\\\\\n",
       "\tresidual.sugar &  22.32345\\\\\n",
       "\tfree.sulfur.dioxide &  19.73800\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| <!--/--> | MeanDecreaseGini |\n",
       "|---|---|\n",
       "| alcohol | 138.37505 |\n",
       "| sulphates | 133.06089 |\n",
       "| volatile.acidity | 115.20571 |\n",
       "| chlorides | 106.56542 |\n",
       "| density |  49.47299 |\n",
       "| pH |  45.09919 |\n",
       "| citric.acid |  43.76360 |\n",
       "| total.sulfur.dioxide |  40.59024 |\n",
       "| fixed.acidity |  23.50309 |\n",
       "| residual.sugar |  22.32345 |\n",
       "| free.sulfur.dioxide |  19.73800 |\n",
       "\n"
      ],
      "text/plain": [
       "                     MeanDecreaseGini\n",
       "alcohol              138.37505       \n",
       "sulphates            133.06089       \n",
       "volatile.acidity     115.20571       \n",
       "chlorides            106.56542       \n",
       "density               49.47299       \n",
       "pH                    45.09919       \n",
       "citric.acid           43.76360       \n",
       "total.sulfur.dioxide  40.59024       \n",
       "fixed.acidity         23.50309       \n",
       "residual.sugar        22.32345       \n",
       "free.sulfur.dioxide   19.73800       "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "variable.importance <- data.frame(importance(model), rep(NA))\n",
    "# Sort by MeanDecreaseGini\n",
    "variable.importance <- variable.importance[order(variable.importance$MeanDecreaseGini, decreasing = TRUE), ]\n",
    "variable.importance <- subset(variable.importance, select = c(MeanDecreaseGini))\n",
    "\n",
    "variable.importance"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "21329310-a64a-48f2-9e04-d631b45bd79e",
   "metadata": {},
   "source": [
    "MeanDecreaseGini measures the average amount (accross all decision trees) that grouping the remaining wine samples on a particular variable reduces the level of impurity (or degree to which the samples have different outcomes). This can be interpreted as a measure of the importance of each variable in determining (or helping to determine) the correct outcome of each sample."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "44a0ede8-2a03-4aaf-bc14-8e59d7dc0d85",
   "metadata": {},
   "source": [
    "## Model Performance"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "0e8b7123-e76d-4c33-b15e-665911e06bcb",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "\n",
      "Attaching package: 'caret'\n",
      "\n",
      "The following object is masked from 'package:VGAM':\n",
      "\n",
      "    predictors\n",
      "\n",
      "The following object is masked from 'package:survival':\n",
      "\n",
      "    cluster\n",
      "\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "Confusion Matrix and Statistics\n",
       "\n",
       "          Reference\n",
       "Prediction   3   4   5   6   7   8\n",
       "         3   0   0   0   0   0   0\n",
       "         4   0   0   0   0   0   0\n",
       "         5   1   2 111  21   1   0\n",
       "         6   1   7  26  95  22   2\n",
       "         7   0   0   0   5  23   3\n",
       "         8   0   0   0   0   0   0\n",
       "\n",
       "Overall Statistics\n",
       "                                          \n",
       "               Accuracy : 0.7156          \n",
       "                 95% CI : (0.6628, 0.7644)\n",
       "    No Information Rate : 0.4281          \n",
       "    P-Value [Acc > NIR] : < 2.2e-16       \n",
       "                                          \n",
       "                  Kappa : 0.5438          \n",
       "                                          \n",
       " Mcnemar's Test P-Value : NA              \n",
       "\n",
       "Statistics by Class:\n",
       "\n",
       "                     Class: 3 Class: 4 Class: 5 Class: 6 Class: 7 Class: 8\n",
       "Sensitivity           0.00000  0.00000   0.8102   0.7851  0.50000  0.00000\n",
       "Specificity           1.00000  1.00000   0.8634   0.7085  0.97080  1.00000\n",
       "Pos Pred Value            NaN      NaN   0.8162   0.6209  0.74194      NaN\n",
       "Neg Pred Value        0.99375  0.97188   0.8587   0.8443  0.92042  0.98438\n",
       "Prevalence            0.00625  0.02813   0.4281   0.3781  0.14375  0.01562\n",
       "Detection Rate        0.00000  0.00000   0.3469   0.2969  0.07187  0.00000\n",
       "Detection Prevalence  0.00000  0.00000   0.4250   0.4781  0.09688  0.00000\n",
       "Balanced Accuracy     0.50000  0.50000   0.8368   0.7468  0.73540  0.50000"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "library(caret)\n",
    "\n",
    "# Make predictions\n",
    "predictions <- predict(model, newdata = test)\n",
    "\n",
    "confusionMatrix(predictions, test$quality)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c888c219-b5c8-4fdb-8ba3-3ec8db78f3d1",
   "metadata": {},
   "source": [
    "### Compare Outcome Distributions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "f5a4ea55-84b0-4017-97ce-8af9b809f836",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th></th><th scope=col>train</th><th scope=col>test</th><th scope=col>prediction</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>3</th><td> 0.6254887</td><td> 0.6250   </td><td> 0.0000   </td></tr>\n",
       "\t<tr><th scope=row>4</th><td> 3.4401876</td><td> 2.8125   </td><td> 0.0000   </td></tr>\n",
       "\t<tr><th scope=row>5</th><td>42.5332291</td><td>42.8125   </td><td>42.5000   </td></tr>\n",
       "\t<tr><th scope=row>6</th><td>40.4222048</td><td>37.8125   </td><td>47.8125   </td></tr>\n",
       "\t<tr><th scope=row>7</th><td>11.9624707</td><td>14.3750   </td><td> 9.6875   </td></tr>\n",
       "\t<tr><th scope=row>8</th><td> 1.0164191</td><td> 1.5625   </td><td> 0.0000   </td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|lll}\n",
       "  & train & test & prediction\\\\\n",
       "\\hline\n",
       "\t3 &  0.6254887 &  0.6250    &  0.0000   \\\\\n",
       "\t4 &  3.4401876 &  2.8125    &  0.0000   \\\\\n",
       "\t5 & 42.5332291 & 42.8125    & 42.5000   \\\\\n",
       "\t6 & 40.4222048 & 37.8125    & 47.8125   \\\\\n",
       "\t7 & 11.9624707 & 14.3750    &  9.6875   \\\\\n",
       "\t8 &  1.0164191 &  1.5625    &  0.0000   \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| <!--/--> | train | test | prediction |\n",
       "|---|---|---|---|\n",
       "| 3 |  0.6254887 |  0.6250    |  0.0000    |\n",
       "| 4 |  3.4401876 |  2.8125    |  0.0000    |\n",
       "| 5 | 42.5332291 | 42.8125    | 42.5000    |\n",
       "| 6 | 40.4222048 | 37.8125    | 47.8125    |\n",
       "| 7 | 11.9624707 | 14.3750    |  9.6875    |\n",
       "| 8 |  1.0164191 |  1.5625    |  0.0000    |\n",
       "\n"
      ],
      "text/plain": [
       "  train      test    prediction\n",
       "3  0.6254887  0.6250  0.0000   \n",
       "4  3.4401876  2.8125  0.0000   \n",
       "5 42.5332291 42.8125 42.5000   \n",
       "6 40.4222048 37.8125 47.8125   \n",
       "7 11.9624707 14.3750  9.6875   \n",
       "8  1.0164191  1.5625  0.0000   "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "outcome.distributions <- data.frame(\n",
    "    train=summary(train$quality) * 100 / length(train$quality), \n",
    "    test=summary(test$quality) * 100 / length(test$quality),\n",
    "    prediction=summary(predictions) * 100 / length(predictions)\n",
    ")\n",
    "\n",
    "outcome.distributions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "328c4bae-ab43-4e53-95e9-becb5a903a61",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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QTvrt356IorXTsJjO4M2WPFJvkPNji3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\nRU5ErkJggg==",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "lwd <- 2\n",
    "\n",
    "plot(outcome.distributions$train, type = 'l', lty = 1, lwd = lwd, ylim = c(0,50), xlab = \"Wine Quality\", ylab = \"Percent of Data Subset\", xaxt = 'n')\n",
    "axis(1, at = c(1:6), labels = 3:8)\n",
    "lines(outcome.distributions$test, type = 'l', lty = 2, lwd = lwd)\n",
    "lines(outcome.distributions$prediction, type = 'l', lty = 3, lwd = lwd)\n",
    "\n",
    "legend(\"topright\",                                       # Add legend to plot\n",
    "       legend = c(\"Train\", \"Test\", \"Prediction\"),\n",
    "       lty = c(1, 2, 3),\n",
    "       lwd = lwd)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "49fca7ee-d091-413c-9804-c252717aa57b",
   "metadata": {},
   "source": [
    "Most of the data has wine qualities of 5 and 6."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "ba9692b4-4fb8-48ab-9490-446c6a149eb0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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qdRpBFFmtDr6vY6jSKNKNKEXle312kUaUSRJvS6ur1Oo0gjijSh19XtdRpFGlGk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\nTkSuQmCC",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "library(ggplot2)\n",
    "\n",
    "# Create this dataframe to plot both Predictions and Actuals in the same histogram\n",
    "group <- c(rep(\"Prediction\", length(predictions)), \n",
    "           rep(\"Actual\", length(test$quality)))\n",
    "\n",
    "outcomes <- data.frame(values=c(as.numeric(predictions), as.numeric(test$quality)), group)\n",
    "\n",
    "ggplot(outcomes, aes(x=values, fill=group)) +\n",
    "    geom_histogram(position = \"identity\", alpha = 0.4, bins = 6) +\n",
    "    scale_x_continuous(\"Wine Quality\", breaks = c(1:6), labels = c(3:8))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9bfa3263-41f6-45ba-ade4-c82de762383e",
   "metadata": {},
   "source": [
    "The model is overpredicting wine qualities 5, 6 and underpredicting all others."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1fea7a12-de25-4b63-92d4-b094b32951ee",
   "metadata": {},
   "source": [
    "# Conclusion\n",
    "\n",
    "## Important Variables\n",
    "This Random Forest model found the following importance ranking for the variables in determining the wine quality for each wine:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "8c2541ea-80ee-4bf8-a2bd-6d68c0002592",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th></th><th scope=col>MeanDecreaseGini</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>alcohol</th><td>138.37505</td></tr>\n",
       "\t<tr><th scope=row>sulphates</th><td>133.06089</td></tr>\n",
       "\t<tr><th scope=row>volatile.acidity</th><td>115.20571</td></tr>\n",
       "\t<tr><th scope=row>chlorides</th><td>106.56542</td></tr>\n",
       "\t<tr><th scope=row>density</th><td> 49.47299</td></tr>\n",
       "\t<tr><th scope=row>pH</th><td> 45.09919</td></tr>\n",
       "\t<tr><th scope=row>citric.acid</th><td> 43.76360</td></tr>\n",
       "\t<tr><th scope=row>total.sulfur.dioxide</th><td> 40.59024</td></tr>\n",
       "\t<tr><th scope=row>fixed.acidity</th><td> 23.50309</td></tr>\n",
       "\t<tr><th scope=row>residual.sugar</th><td> 22.32345</td></tr>\n",
       "\t<tr><th scope=row>free.sulfur.dioxide</th><td> 19.73800</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|l}\n",
       "  & MeanDecreaseGini\\\\\n",
       "\\hline\n",
       "\talcohol & 138.37505\\\\\n",
       "\tsulphates & 133.06089\\\\\n",
       "\tvolatile.acidity & 115.20571\\\\\n",
       "\tchlorides & 106.56542\\\\\n",
       "\tdensity &  49.47299\\\\\n",
       "\tpH &  45.09919\\\\\n",
       "\tcitric.acid &  43.76360\\\\\n",
       "\ttotal.sulfur.dioxide &  40.59024\\\\\n",
       "\tfixed.acidity &  23.50309\\\\\n",
       "\tresidual.sugar &  22.32345\\\\\n",
       "\tfree.sulfur.dioxide &  19.73800\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| <!--/--> | MeanDecreaseGini |\n",
       "|---|---|\n",
       "| alcohol | 138.37505 |\n",
       "| sulphates | 133.06089 |\n",
       "| volatile.acidity | 115.20571 |\n",
       "| chlorides | 106.56542 |\n",
       "| density |  49.47299 |\n",
       "| pH |  45.09919 |\n",
       "| citric.acid |  43.76360 |\n",
       "| total.sulfur.dioxide |  40.59024 |\n",
       "| fixed.acidity |  23.50309 |\n",
       "| residual.sugar |  22.32345 |\n",
       "| free.sulfur.dioxide |  19.73800 |\n",
       "\n"
      ],
      "text/plain": [
       "                     MeanDecreaseGini\n",
       "alcohol              138.37505       \n",
       "sulphates            133.06089       \n",
       "volatile.acidity     115.20571       \n",
       "chlorides            106.56542       \n",
       "density               49.47299       \n",
       "pH                    45.09919       \n",
       "citric.acid           43.76360       \n",
       "total.sulfur.dioxide  40.59024       \n",
       "fixed.acidity         23.50309       \n",
       "residual.sugar        22.32345       \n",
       "free.sulfur.dioxide   19.73800       "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "variable.importance"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fc6f0fb6-99bf-403c-a818-cdcaf53c0371",
   "metadata": {},
   "source": [
    "The first four variables, 'alcohol', 'sulphates', 'volatile.acidity', and 'chlorides' contribute much more to determining the outcome than the other variables. If cost of data collection were an issue - these would be the measurements to focus on.\n",
    "\n",
    "## Performance\n",
    "Overall model performance is relatively high given that there are 6 possible outcome classes - accuracy is 72%, but the model does a poor job of correctly classifying the extreme wine qualities. The model did not capture any of the 3, 4, or 8 wine qualities (sensitivity = 0% for those classes). Performance is high for the 5 and 6 wine qualities and good for the 7 wine quality. This behaviour is likely due to the relative distributions of the wine qualities in the data - if the model were to always guess that the wine quality is 5 or 6, it would be correct a large portion of the time. This issue was mitigated to some extent by using class weights, but clearly the model is still not predicting any of the rarer classes."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "4e74b2e2-eb9f-4c34-9938-be067fdd8487",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "R",
   "language": "R",
   "name": "ir"
  },
  "language_info": {
   "codemirror_mode": "r",
   "file_extension": ".r",
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