0601_PCA.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "MYTkAs_9uJg8"
   },
   "source": [
    "# Dimension Reduction"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "PDgGqcpE3Zau"
   },
   "source": [
    "# Principal components analysis (PCA) \n",
    "\n",
    "* It is a useful method for dimension reduction, especially when the number of variables is large.\n",
    "* PCA is intended for use with numerical variables. \n",
    "   * For categorical variables, other methods such as correspondence analysis are more suitable. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "id": "eZIIbxdR27FX"
   },
   "outputs": [],
   "source": [
    "Cereals = 'https://raw.githubusercontent.com/tec03/Datasets/main/datasets/Cereals.csv'"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 411
    },
    "id": "x6I7njK8YHQO",
    "outputId": "873b1363-740d-49d0-c814-5bf353e7af68"
   },
   "outputs": [
    {
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>name</th>\n",
       "      <th>mfr</th>\n",
       "      <th>type</th>\n",
       "      <th>calories</th>\n",
       "      <th>protein</th>\n",
       "      <th>fat</th>\n",
       "      <th>sodium</th>\n",
       "      <th>fiber</th>\n",
       "      <th>carbo</th>\n",
       "      <th>sugars</th>\n",
       "      <th>potass</th>\n",
       "      <th>vitamins</th>\n",
       "      <th>shelf</th>\n",
       "      <th>weight</th>\n",
       "      <th>cups</th>\n",
       "      <th>rating</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>100%_Bran</td>\n",
       "      <td>N</td>\n",
       "      <td>C</td>\n",
       "      <td>70</td>\n",
       "      <td>4</td>\n",
       "      <td>1</td>\n",
       "      <td>130</td>\n",
       "      <td>10.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>6.0</td>\n",
       "      <td>280.0</td>\n",
       "      <td>25</td>\n",
       "      <td>3</td>\n",
       "      <td>1.0</td>\n",
       "      <td>0.33</td>\n",
       "      <td>68.402973</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>100%_Natural_Bran</td>\n",
       "      <td>Q</td>\n",
       "      <td>C</td>\n",
       "      <td>120</td>\n",
       "      <td>3</td>\n",
       "      <td>5</td>\n",
       "      <td>15</td>\n",
       "      <td>2.0</td>\n",
       "      <td>8.0</td>\n",
       "      <td>8.0</td>\n",
       "      <td>135.0</td>\n",
       "      <td>0</td>\n",
       "      <td>3</td>\n",
       "      <td>1.0</td>\n",
       "      <td>1.00</td>\n",
       "      <td>33.983679</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>All-Bran</td>\n",
       "      <td>K</td>\n",
       "      <td>C</td>\n",
       "      <td>70</td>\n",
       "      <td>4</td>\n",
       "      <td>1</td>\n",
       "      <td>260</td>\n",
       "      <td>9.0</td>\n",
       "      <td>7.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>320.0</td>\n",
       "      <td>25</td>\n",
       "      <td>3</td>\n",
       "      <td>1.0</td>\n",
       "      <td>0.33</td>\n",
       "      <td>59.425505</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>All-Bran_with_Extra_Fiber</td>\n",
       "      <td>K</td>\n",
       "      <td>C</td>\n",
       "      <td>50</td>\n",
       "      <td>4</td>\n",
       "      <td>0</td>\n",
       "      <td>140</td>\n",
       "      <td>14.0</td>\n",
       "      <td>8.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>330.0</td>\n",
       "      <td>25</td>\n",
       "      <td>3</td>\n",
       "      <td>1.0</td>\n",
       "      <td>0.50</td>\n",
       "      <td>93.704912</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>Almond_Delight</td>\n",
       "      <td>R</td>\n",
       "      <td>C</td>\n",
       "      <td>110</td>\n",
       "      <td>2</td>\n",
       "      <td>2</td>\n",
       "      <td>200</td>\n",
       "      <td>1.0</td>\n",
       "      <td>14.0</td>\n",
       "      <td>8.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>25</td>\n",
       "      <td>3</td>\n",
       "      <td>1.0</td>\n",
       "      <td>0.75</td>\n",
       "      <td>34.384843</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
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       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>72</th>\n",
       "      <td>Triples</td>\n",
       "      <td>G</td>\n",
       "      <td>C</td>\n",
       "      <td>110</td>\n",
       "      <td>2</td>\n",
       "      <td>1</td>\n",
       "      <td>250</td>\n",
       "      <td>0.0</td>\n",
       "      <td>21.0</td>\n",
       "      <td>3.0</td>\n",
       "      <td>60.0</td>\n",
       "      <td>25</td>\n",
       "      <td>3</td>\n",
       "      <td>1.0</td>\n",
       "      <td>0.75</td>\n",
       "      <td>39.106174</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>73</th>\n",
       "      <td>Trix</td>\n",
       "      <td>G</td>\n",
       "      <td>C</td>\n",
       "      <td>110</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>140</td>\n",
       "      <td>0.0</td>\n",
       "      <td>13.0</td>\n",
       "      <td>12.0</td>\n",
       "      <td>25.0</td>\n",
       "      <td>25</td>\n",
       "      <td>2</td>\n",
       "      <td>1.0</td>\n",
       "      <td>1.00</td>\n",
       "      <td>27.753301</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>74</th>\n",
       "      <td>Wheat_Chex</td>\n",
       "      <td>R</td>\n",
       "      <td>C</td>\n",
       "      <td>100</td>\n",
       "      <td>3</td>\n",
       "      <td>1</td>\n",
       "      <td>230</td>\n",
       "      <td>3.0</td>\n",
       "      <td>17.0</td>\n",
       "      <td>3.0</td>\n",
       "      <td>115.0</td>\n",
       "      <td>25</td>\n",
       "      <td>1</td>\n",
       "      <td>1.0</td>\n",
       "      <td>0.67</td>\n",
       "      <td>49.787445</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>75</th>\n",
       "      <td>Wheaties</td>\n",
       "      <td>G</td>\n",
       "      <td>C</td>\n",
       "      <td>100</td>\n",
       "      <td>3</td>\n",
       "      <td>1</td>\n",
       "      <td>200</td>\n",
       "      <td>3.0</td>\n",
       "      <td>17.0</td>\n",
       "      <td>3.0</td>\n",
       "      <td>110.0</td>\n",
       "      <td>25</td>\n",
       "      <td>1</td>\n",
       "      <td>1.0</td>\n",
       "      <td>1.00</td>\n",
       "      <td>51.592193</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>76</th>\n",
       "      <td>Wheaties_Honey_Gold</td>\n",
       "      <td>G</td>\n",
       "      <td>C</td>\n",
       "      <td>110</td>\n",
       "      <td>2</td>\n",
       "      <td>1</td>\n",
       "      <td>200</td>\n",
       "      <td>1.0</td>\n",
       "      <td>16.0</td>\n",
       "      <td>8.0</td>\n",
       "      <td>60.0</td>\n",
       "      <td>25</td>\n",
       "      <td>1</td>\n",
       "      <td>1.0</td>\n",
       "      <td>0.75</td>\n",
       "      <td>36.187559</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>77 rows × 16 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "                         name mfr type  calories  protein  fat  sodium  fiber  \\\n",
       "0                   100%_Bran   N    C        70        4    1     130   10.0   \n",
       "1           100%_Natural_Bran   Q    C       120        3    5      15    2.0   \n",
       "2                    All-Bran   K    C        70        4    1     260    9.0   \n",
       "3   All-Bran_with_Extra_Fiber   K    C        50        4    0     140   14.0   \n",
       "4              Almond_Delight   R    C       110        2    2     200    1.0   \n",
       "..                        ...  ..  ...       ...      ...  ...     ...    ...   \n",
       "72                    Triples   G    C       110        2    1     250    0.0   \n",
       "73                       Trix   G    C       110        1    1     140    0.0   \n",
       "74                 Wheat_Chex   R    C       100        3    1     230    3.0   \n",
       "75                   Wheaties   G    C       100        3    1     200    3.0   \n",
       "76        Wheaties_Honey_Gold   G    C       110        2    1     200    1.0   \n",
       "\n",
       "    carbo  sugars  potass  vitamins  shelf  weight  cups     rating  \n",
       "0     5.0     6.0   280.0        25      3     1.0  0.33  68.402973  \n",
       "1     8.0     8.0   135.0         0      3     1.0  1.00  33.983679  \n",
       "2     7.0     5.0   320.0        25      3     1.0  0.33  59.425505  \n",
       "3     8.0     0.0   330.0        25      3     1.0  0.50  93.704912  \n",
       "4    14.0     8.0     NaN        25      3     1.0  0.75  34.384843  \n",
       "..    ...     ...     ...       ...    ...     ...   ...        ...  \n",
       "72   21.0     3.0    60.0        25      3     1.0  0.75  39.106174  \n",
       "73   13.0    12.0    25.0        25      2     1.0  1.00  27.753301  \n",
       "74   17.0     3.0   115.0        25      1     1.0  0.67  49.787445  \n",
       "75   17.0     3.0   110.0        25      1     1.0  1.00  51.592193  \n",
       "76   16.0     8.0    60.0        25      1     1.0  0.75  36.187559  \n",
       "\n",
       "[77 rows x 16 columns]"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cereals_df = pd.read_csv(Cereals)\n",
    "df = cereals_df.copy()\n",
    "df"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
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   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(77, 16)"
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     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>calories</th>\n",
       "      <th>protein</th>\n",
       "      <th>fat</th>\n",
       "      <th>sodium</th>\n",
       "      <th>fiber</th>\n",
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       "      <th>calories</th>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.019066</td>\n",
       "      <td>0.498610</td>\n",
       "      <td>0.300649</td>\n",
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       "    <tr>\n",
       "      <th>protein</th>\n",
       "      <td>0.019066</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.208431</td>\n",
       "      <td>-0.054674</td>\n",
       "      <td>0.500330</td>\n",
       "      <td>-0.025012</td>\n",
       "      <td>-0.291853</td>\n",
       "      <td>0.563706</td>\n",
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       "      <td>0.133865</td>\n",
       "      <td>0.216158</td>\n",
       "      <td>-0.244469</td>\n",
       "      <td>0.470618</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>fat</th>\n",
       "      <td>0.498610</td>\n",
       "      <td>0.208431</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>-0.005407</td>\n",
       "      <td>0.016719</td>\n",
       "      <td>-0.300003</td>\n",
       "      <td>0.302497</td>\n",
       "      <td>0.200445</td>\n",
       "      <td>-0.031156</td>\n",
       "      <td>0.263691</td>\n",
       "      <td>0.214625</td>\n",
       "      <td>-0.175892</td>\n",
       "      <td>-0.409284</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>sodium</th>\n",
       "      <td>0.300649</td>\n",
       "      <td>-0.054674</td>\n",
       "      <td>-0.005407</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>-0.070675</td>\n",
       "      <td>0.297687</td>\n",
       "      <td>0.058866</td>\n",
       "      <td>-0.042632</td>\n",
       "      <td>0.361477</td>\n",
       "      <td>-0.069719</td>\n",
       "      <td>0.308576</td>\n",
       "      <td>0.119665</td>\n",
       "      <td>-0.401295</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>fiber</th>\n",
       "      <td>-0.293413</td>\n",
       "      <td>0.500330</td>\n",
       "      <td>0.016719</td>\n",
       "      <td>-0.070675</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>-0.380357</td>\n",
       "      <td>-0.138760</td>\n",
       "      <td>0.911528</td>\n",
       "      <td>-0.032243</td>\n",
       "      <td>0.297539</td>\n",
       "      <td>0.247226</td>\n",
       "      <td>-0.513061</td>\n",
       "      <td>0.584160</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>carbo</th>\n",
       "      <td>0.257638</td>\n",
       "      <td>-0.025012</td>\n",
       "      <td>-0.300003</td>\n",
       "      <td>0.297687</td>\n",
       "      <td>-0.380357</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>-0.471184</td>\n",
       "      <td>-0.365003</td>\n",
       "      <td>0.219202</td>\n",
       "      <td>-0.192650</td>\n",
       "      <td>0.138467</td>\n",
       "      <td>0.367460</td>\n",
       "      <td>0.088712</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>sugars</th>\n",
       "      <td>0.566533</td>\n",
       "      <td>-0.291853</td>\n",
       "      <td>0.302497</td>\n",
       "      <td>0.058866</td>\n",
       "      <td>-0.138760</td>\n",
       "      <td>-0.471184</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.001414</td>\n",
       "      <td>0.098231</td>\n",
       "      <td>0.068377</td>\n",
       "      <td>0.455844</td>\n",
       "      <td>-0.048961</td>\n",
       "      <td>-0.763902</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>potass</th>\n",
       "      <td>-0.072063</td>\n",
       "      <td>0.563706</td>\n",
       "      <td>0.200445</td>\n",
       "      <td>-0.042632</td>\n",
       "      <td>0.911528</td>\n",
       "      <td>-0.365003</td>\n",
       "      <td>0.001414</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>-0.005427</td>\n",
       "      <td>0.385784</td>\n",
       "      <td>0.419933</td>\n",
       "      <td>-0.501607</td>\n",
       "      <td>0.416009</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>vitamins</th>\n",
       "      <td>0.265356</td>\n",
       "      <td>0.007335</td>\n",
       "      <td>-0.031156</td>\n",
       "      <td>0.361477</td>\n",
       "      <td>-0.032243</td>\n",
       "      <td>0.219202</td>\n",
       "      <td>0.098231</td>\n",
       "      <td>-0.005427</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.299262</td>\n",
       "      <td>0.320324</td>\n",
       "      <td>0.128405</td>\n",
       "      <td>-0.240544</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>shelf</th>\n",
       "      <td>0.097234</td>\n",
       "      <td>0.133865</td>\n",
       "      <td>0.263691</td>\n",
       "      <td>-0.069719</td>\n",
       "      <td>0.297539</td>\n",
       "      <td>-0.192650</td>\n",
       "      <td>0.068377</td>\n",
       "      <td>0.385784</td>\n",
       "      <td>0.299262</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.190762</td>\n",
       "      <td>-0.335269</td>\n",
       "      <td>0.025159</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>weight</th>\n",
       "      <td>0.696091</td>\n",
       "      <td>0.216158</td>\n",
       "      <td>0.214625</td>\n",
       "      <td>0.308576</td>\n",
       "      <td>0.247226</td>\n",
       "      <td>0.138467</td>\n",
       "      <td>0.455844</td>\n",
       "      <td>0.419933</td>\n",
       "      <td>0.320324</td>\n",
       "      <td>0.190762</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>-0.199583</td>\n",
       "      <td>-0.298124</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>cups</th>\n",
       "      <td>0.087200</td>\n",
       "      <td>-0.244469</td>\n",
       "      <td>-0.175892</td>\n",
       "      <td>0.119665</td>\n",
       "      <td>-0.513061</td>\n",
       "      <td>0.367460</td>\n",
       "      <td>-0.048961</td>\n",
       "      <td>-0.501607</td>\n",
       "      <td>0.128405</td>\n",
       "      <td>-0.335269</td>\n",
       "      <td>-0.199583</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>-0.203160</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>rating</th>\n",
       "      <td>-0.689376</td>\n",
       "      <td>0.470618</td>\n",
       "      <td>-0.409284</td>\n",
       "      <td>-0.401295</td>\n",
       "      <td>0.584160</td>\n",
       "      <td>0.088712</td>\n",
       "      <td>-0.763902</td>\n",
       "      <td>0.416009</td>\n",
       "      <td>-0.240544</td>\n",
       "      <td>0.025159</td>\n",
       "      <td>-0.298124</td>\n",
       "      <td>-0.203160</td>\n",
       "      <td>1.000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "          calories   protein       fat    sodium     fiber     carbo  \\\n",
       "calories  1.000000  0.019066  0.498610  0.300649 -0.293413  0.257638   \n",
       "protein   0.019066  1.000000  0.208431 -0.054674  0.500330 -0.025012   \n",
       "fat       0.498610  0.208431  1.000000 -0.005407  0.016719 -0.300003   \n",
       "sodium    0.300649 -0.054674 -0.005407  1.000000 -0.070675  0.297687   \n",
       "fiber    -0.293413  0.500330  0.016719 -0.070675  1.000000 -0.380357   \n",
       "carbo     0.257638 -0.025012 -0.300003  0.297687 -0.380357  1.000000   \n",
       "sugars    0.566533 -0.291853  0.302497  0.058866 -0.138760 -0.471184   \n",
       "potass   -0.072063  0.563706  0.200445 -0.042632  0.911528 -0.365003   \n",
       "vitamins  0.265356  0.007335 -0.031156  0.361477 -0.032243  0.219202   \n",
       "shelf     0.097234  0.133865  0.263691 -0.069719  0.297539 -0.192650   \n",
       "weight    0.696091  0.216158  0.214625  0.308576  0.247226  0.138467   \n",
       "cups      0.087200 -0.244469 -0.175892  0.119665 -0.513061  0.367460   \n",
       "rating   -0.689376  0.470618 -0.409284 -0.401295  0.584160  0.088712   \n",
       "\n",
       "            sugars    potass  vitamins     shelf    weight      cups    rating  \n",
       "calories  0.566533 -0.072063  0.265356  0.097234  0.696091  0.087200 -0.689376  \n",
       "protein  -0.291853  0.563706  0.007335  0.133865  0.216158 -0.244469  0.470618  \n",
       "fat       0.302497  0.200445 -0.031156  0.263691  0.214625 -0.175892 -0.409284  \n",
       "sodium    0.058866 -0.042632  0.361477 -0.069719  0.308576  0.119665 -0.401295  \n",
       "fiber    -0.138760  0.911528 -0.032243  0.297539  0.247226 -0.513061  0.584160  \n",
       "carbo    -0.471184 -0.365003  0.219202 -0.192650  0.138467  0.367460  0.088712  \n",
       "sugars    1.000000  0.001414  0.098231  0.068377  0.455844 -0.048961 -0.763902  \n",
       "potass    0.001414  1.000000 -0.005427  0.385784  0.419933 -0.501607  0.416009  \n",
       "vitamins  0.098231 -0.005427  1.000000  0.299262  0.320324  0.128405 -0.240544  \n",
       "shelf     0.068377  0.385784  0.299262  1.000000  0.190762 -0.335269  0.025159  \n",
       "weight    0.455844  0.419933  0.320324  0.190762  1.000000 -0.199583 -0.298124  \n",
       "cups     -0.048961 -0.501607  0.128405 -0.335269 -0.199583  1.000000 -0.203160  \n",
       "rating   -0.763902  0.416009 -0.240544  0.025159 -0.298124 -0.203160  1.000000  "
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.corr()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* `fiber` vs `potass` is higly correlated. \n",
    "* `fiber` vs `rating` is relatively highly correlated. \n",
    "\n",
    "\n",
    "In fact, we can put all those in a comon component. No need to study separately. \n",
    "\n",
    "Again; \n",
    "\n",
    "* `calories` vs `weight` is higly correlated. \n",
    "* `calories` vs `sugars` is relatively highly correlated. \n",
    "* `weight` vs `sugars` is relatively  correlated. \n",
    "\n",
    "In fact, we can put all those in a comon component. No need to study separately."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "xAYdWuxn6rHb"
   },
   "source": [
    "For each cereal, the data include 13 numerical variables, and we are interested in reducing this dimension.\n",
    "\n",
    "Principal Component Analysis can tell you what variables in your data are the most important. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We focus first on two variables: `calories` and consumer `rating`: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 109
    },
    "id": "pwvuGk-13Itd",
    "outputId": "3976f628-cc49-4f2b-f8e5-ccb8047bcd7f"
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>calories</th>\n",
       "      <th>rating</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>calories</th>\n",
       "      <td>1.000000</td>\n",
       "      <td>-0.689376</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>rating</th>\n",
       "      <td>-0.689376</td>\n",
       "      <td>1.000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "          calories    rating\n",
       "calories  1.000000 -0.689376\n",
       "rating   -0.689376  1.000000"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ndf = df[['calories', 'rating']]\n",
    "ndf.corr()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "106.88"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.calories.mean().round(2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "42.67"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.rating.mean().round(2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 379.63089542, -188.68156228],\n",
       "       [-188.68156228,  197.32632105]])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Varience = np.cov([df.calories, df.rating])\n",
    "Varience"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "576.9572164702394"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ndf = df[['calories','rating']]\n",
    "ndf.var().sum()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "t= ndf.var().sum()\n",
    "varC =Varience[0][0]\n",
    "varR = Varience[1][1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Total Varience \t\t\t: 576.96\n",
      "Varience prop. of Calory \t: 65.8 %\n",
      "Varience prop. of Rating \t: 34.2 %\n"
     ]
    }
   ],
   "source": [
    "print('Total Varience \\t\\t\\t:', t.round(2) )\n",
    "print('Varience prop. of Calory \\t:', ((varC/t)*100).round(2),'%' )\n",
    "print('Varience prop. of Rating \\t:', ((varR/t)*100).round(2),'%' )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we drop `Rating` for the sake of dimension reduction, we lose at least 34% of the total variability. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>calories</th>\n",
       "      <th>rating</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>calories</th>\n",
       "      <td>1.00</td>\n",
       "      <td>-0.69</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>rating</th>\n",
       "      <td>-0.69</td>\n",
       "      <td>1.00</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "          calories  rating\n",
       "calories      1.00   -0.69\n",
       "rating       -0.69    1.00"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ndf.corr().round(2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Roughly speaking, 69% of the total variation in both variables is actually “co-variation,” or variation in one variable that is duplicated by similar variation in the other variable. \n",
    "\n",
    "Can we use this fact to reduce the number of variables, while making maximum use of their unique contributions to the overall variation? \n",
    "\n",
    "Since there is redundancy in the information that the two variables contain, it might be possible to reduce the `two variables` to a `single variable` `without losing` `too much` information. \n",
    "\n",
    "The idea in PCA is to find a linear combination of the two variables that contains most, even if not all, of the information, so that this new variable can replace the two original variables.  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The line `z1 (first principal component = PC1)` is the direction in which the variability of the points is largest. \n",
    "\n",
    "The `z2 (second principal component = PC2)`-axis is chosen to be perpendicular to the z1-axis.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "id": "jEOhJk1Sg9fc"
   },
   "outputs": [],
   "source": [
    "import seaborn as sns"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 305
    },
    "id": "LA1DY6SFgsCN",
    "outputId": "69913999-3639-40b0-a363-c055a17de5e5"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<AxesSubplot:xlabel='calories', ylabel='rating'>"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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0BaNjPQiAiGFSV2qV6XA5bNy6pYZbt9TQH4zx7JFuWtu6ONkTpHMgzLdfOs3fv3Sad60pp7mpjvdsrp7wXvORq0BR5LKymNPzBU1tHc+l9Ezqcq+T7pEIX3iqjS+BBAkhknJW0U1r3QWcU0ql5hd2Ae3AU8C9yWP3Aj/LVRvmym5TFLsd1JS4WVPlY3WFj6oiN16XPafF9+6+vp54MjhprJ9xU3P39fWXPLeyyMWHrlvNtz62ncd+6zr+67WrKPc60cDrZwf58i+OcNc3XuHh/zjCr84NZm1FVziW4MJgmItDYUKx+OVfcBn33bweU1vDSqY2kz+t47n06AsdOO3WsKNS1k+nXfHoCx05Pa8Qi0muVzFdg7XM1QV0AJ/ACko/AtYAZ7CWufZP9z7bt2/XBw8ezFk7Z0Pr8d5FLia7J+xJXeqdVQXbeMLktdMDtLR38crJPozE+GdbV+qx9q5o8rMqi4UDXQ4b5T4XRfMInoVYxSTVXMVysGBXMWXTQgoQk6VPdodiCyf3Yjhs8NzRblraAhy5pHBgKc2Nddy6tYbiLBUOdNhslHodlHici2LzItkPQiwHEiAWmIUw2T3Z2b4QLe1dPN0eoDetcKArWThwTxYLByplDc+V+5w4F/CeFFLNVSwHEiAWsPTJ7ohR+NyLhKl54+wAre0BXjzeSzStcGBVsYs7tvlpbvKzLguFA5VSFLmtCe3LBYpCrSaSaq5iqZMAsYgYCZOIkSBimAUPGMFonBeO9dDSHuCtzqEJj231l9Dc5GdnQy1l8ywcmOpRVPgy73In3+SFyB0JEIvYQknWuzAY5ul2q3DgxaHI2HGHTbFjQxV7mvy8e/38Cgeq5Hao5ZO2Qy3kXIDkQYilTgLEEhIxrInuUCw+Yd+IfNFa8/b5IVrbAuw71kMoLVmtzOtkV0MtzU1+NtcWz3nF0uQ5ikKtJpKei1gO8lJqQyn1NlY11nRDwEHgL7TWfXNtgBiXStarLHKRMDWhWNyqTBsz85Ksp5TiqtXlXLW6nN/fuYmXT/TS0hbg0JkBhsIGT75xniffOM/66iKaG/3s3lZLVbL0+kxprRmJGIxGrX2zV5V56Q1G8743dHoeBFj7Y4RicR59oUMChBBJM13j+AsgAXw/+fvdWJVZu4B/BP5T1lu2zNltihKPk5JkKZBoWqHBSB5WR3mcdnZts6rH9oxEeeZwgNa2AGf6Q5zqDfLoCx1868UOtq+rZE+jn5s2Vc+qcKDWmtFInP/8rlX8VcsROvtDJLTGYbNR4nHwp7/RmMOrg3MDIconza94nXY6B0I5PS9I9VqxeMw0QOzWWl+b9vvbSqnXtdbXKqU+mouGiYncDjtuh51yn1WZNmyMB4tcT3bXlLi554Y13H19PUcDI7S0BXjuSDfDkTgHTvVz4FSycODWGpqb/DSuKJ3VEJQGtLL+otF5ySWpr/BdMveRj55LevVah8065yN7TwBIkBALzkwDhF0pdYPW+gCAUup6IFUsZ/71FsSs2GyKIreDomSSm5HcWS/Xk91KKRrqSmmoK+V3b93I/o4+WtoCvHqqzyoc+NZFfv6WVTiwudHPHY1+/JcpHPjEa+essiZpQ1VhI8HX9p7gxk1VOds7+4FbNvCFp9qsXfXS5iDmWi14pqR6rVhMZhog7gP+XilVjLX55zBwn1KqCPh/c9U4MTNOuw1nhsq0udwoyeWwccuWGm7ZUsNAKMazh7tpbQtwomeUzoEwf//yaf7h5dNcs6acPY1+3rO5JuMmRJkq2HqcNs4Phjg/YBUHLPU4x4JhttzWUMtdnYOXDPXkev6hkNVrhZitGf1fp7V+DbgyuQkQWuv0hfM/ykXDxNylV6ZNbZQUSlamzcVwVIXPxV3Xreau61ZzsmeU1rYAzxwOMBAyeOPsIG+cHeRvnz3OrVtqaG70c3V9ObbkENTlKtimKuo67TZKvU5KPY6sFEzcd6Sb7+4/QyxhYlMQS5h8d/8ZrlpdntMgUajqtULMxYyWuSql3MB/xdpHeiyoaK2/lLOWpVkuy1zzITUclbrx5mq8P2FqXjvdT0tbgF+e7J1QONBf6uaORj/NjX4uDETGdtHzOG1EDJO4qXlo5+aMRQrtNkWZ10mpx4ltHqVB7vyb5znRE8SuFEqB1pDQmk01RfzHH9w65/e9HNlBT+RTvnaU+xnWstZDQHSuJxOFN9VwVLZrR9mTSXY7NlQxEjF47mgPrW1dtF8cITAc5Z/3n+Wf95+laWUpO9ZXcbx7hN7R6GUr2CZMTX8wxlDYmFegONUXAjSGqdEaVPKGbR3PnQd3b+FU7yhPvdWFkdDYbYr3X1UnwUEsSDMNEKu11nfmtCWiICZvlBSKxceW02ard1HicfL+q1fy/qtXcrY/xNPtAZ5uD9A9EqXtwjBtF4ZxOWzctLGK5iY/162tuOx7pgLFYMig2OOgxOOY1YS2aWripjWhBlYPIq5BXZLuk137jnRz6OwQ66p8Y5Pjh84Ose9It+RfiAVnpgHil0qpK7XWb+e0NaKg0nMvtNZEDJNgMmBka+5iTaWPT928nk/ctI43zw7S0h7gxWM9ROImzx3t4bmjPVQWudi9rZY9TXWsr56+cKCpNcNhg+Gwgcdpp8w7swltp8OGEUtcEg6cs8jlmAtJ0BOLyUwDxM3Ax5VSp7CGmBTWltJX5axloqCUUnhd9rGVR9lO1LMpxbVrK7h2bQUP7drEC8d6aW3v4s1zQ/QHY/zoYCc/OtjJ5tpi9jTVsauhljLf9IUDrUKIibEhtBKPY8rhJ1vmrdCxT3E8WwqZoCfEbM00QPx6TlshFrz0RL3EWKKe1buYb97FO53DtLYFuDgcYVtdCasrvLRdHObCYITj3aMc7z7BN58/ybs3VNLcWMeODZXTlhA3EiZ9wSgDodjYyqfJhQZNnTlwJKY4ni2FStATYi6mDRBKqVKt9TAwMt3zxPKS2rM7tRtdxEgQjMbntAXrgY7+sVVMpR4Ho9E471wY5sHbN1HkcdDSFmDf0W6CsQQvn+jj5RN9lHoc7GywhqC2+KcuHGhqzWDImtAuclv5FJ7kclojWQwx/ZU67XiuFCpBT4i5uFwP4vvA+7BWL2ku/f9J/qsWYxPdVVjf3kPRBCEjTsS4fJLeE6+dw2FTY3kQqZvmDw928pUPX80Vq8r4/ds38vLJPlraujh0ZoDhSJyfvnmBn755gXVVPpqb6rhjmsKBqbpPo5H4WD6FSpa6MDVjq5jsinktnZ2J2xpq+RLIRkViUZg2QGit35f8uT4/zRGLndNuo8xnowznjGpGTZVJ3TUcHvvd7bSzs6GWnQ219I5GeaY9QEt7gDN9IU73hXjshQ4ef7GD7WsraG6q46aNVbidmVc0GQmTvtEoq8u9nOkP4VAKm12N5UGsr8r9UM9tDbUSEMSiMNNy389qrXdd7pgQ6TLVjEpldIeNBFrry2ZST1Zd7ObuG9bw4evrORYYpbU9wLOHA1bhwNMDHDg9QJHbzm1batnT5KdpZebCgfffspGHW44QjMWJm6aVgOdx8Llf35abfwwhFqHLzUF4sMp6VyulKhgfYioFVuW4bWKJcdptlHltlHmtZbRhI8EnblrHw/9xhLCRmJBJfff19dO+l1KKrXUlbK0r4Xdu3cD+jn5a27rYf6qfYDTBv719kX97+yKryscLB9aVjRcOvGFDJR+8eiU/OtRJOJHAbbPxgatXcXV9OfGEOa/d8y5HdrITi8W0pTaUUg8BnwFWAucZDxDDwLe01l/PdQNBSm0sdfuOdPON509yrj9EXZmH37xu6kzqyxkMxdh7pJuWtgDHu0cnPHZNfRnNjXXcsqWadzqHpyzx8e6NVZdMameL7GQn8ikvW44qpT6ttf7aXE8yXxIglpd4wiSY3Hp1JhPdUznVG6SlrYtnDnfTH4yNHfc4bHhddkzTJBq3AoPTbqPYbWdVeRFf+fDVY891OWyUuJ0UexzYszCBXcg9uEF6L8tNXmoxaa2/ppS6AmgEPGnHvzvXEwsxFUfaUFSq/EdoDsUF11cX8Tu3buS337OB107309oW4OWTvUTiJpFJy1njpknUMImbwQnHY3GTvniU/lCMIpedEo8zY9nymSpkolx676Xc66R7JMIXnmrjSyBBQmQ000nqPwNuwwoQ/46VOPcSIAFC5NTk8h+pVVGhaGLG+3RPLhy472gPf/vM8UvKbJjASCTOSMQY2+o1RWvNaDTOaHR8qWyJe+pM7akUMlFOynyI2ZrpTNxdwC6gS2v9CeBqoCxnrRIiA6Wsm1t1sZs1VT5WVXipLHLhddlnvEdEicfJf7p6JVPd1+Om5q5vvsKXft7O/o6+jFniqaWyZ/tD9I5Gic0iue6BWzZgJKxekdbWz3wlyp0bCE1YLQZS5kNMb6alNiJaa1MpFVdKlQLdwPTLTITIsbHyHzBWXDBsWEtoY5fZSW+6cGIkNPuO9bDvmFU4cFeDtWR2Q03xhOelFwqc6c53hUyUkzIfYrYuGyCU9dXsLaVUOfAtrKzqUeCV3DZNiJmbXFwwlaQXjMUJRS+du3A4bMSNS7/5ux2Kz+zeQktbgDfPDdIfjPHjQ538+FAnm2qL2dPkZ1dDLeU+14TXpTZgcthslHis3A/XFJVhC5UoJ2U+xGzNdBXT21rrK5N/XweUaq3fynHbxsgqJjEfqbmL0eh4sPgvf/cyw5E46SNINgVlHgf/8ns3AdA1HOHp9gCtbQHOD45ndtttinevr6S5yc+O9VVTBgK30z5WsyobK6CyIbWKScp8LA/5Wub6HeDryb2p804ChMgWa9w/wce+fYAz/UGC0ThGwsRpt1HksrO6YuIy19Rr2i4M09oe4Lmj3QSj4+XOSz0Obk8OQW31l2ScC7HmTqxg4ZvFfIkQ85WvAHEE2AScAYLkeT8ICRAi21JLPu02cNtthJLDLVPthZ0SNRL88mQfLe0BDp7un9ADWVvpo7nJz+5tfmpKMhcOTFXCLfE4p+x5CJEt+QoQazMd11qfmeuJZ0MChMiFycMtn7xpHdetq2Q0Ep/REtq+0SjPHO6mtT3Aqd7x/AmbgmvXVLCnyc9Nm6ovycY+0NHPE6+do2skTH25j9+9dSM7G/1Zvz4h8pUol5dAIEQhpL4iOe02KotcVBa5CMcSjESNjBPcKVXFbj58fT2/uX01x7tHaW0L8OyRbobCBgfPDHDwzAA+l53bttTQ3OTnylVlvHZqwCoSGI1b+2qPxvijn/yKP/2NRu5o8lPsdsgQlFgwZtSDKDTpQYhsm2lNpNScRWpDpMtlchsJkwOn+mlpC7C/o4942hjUijIPESPBUMjAblcoZe1FYZqatVVFPH7vduzJCrjFbkfW60ABfPWZYzz+0imCsQRFLjv33byeB3dvyfp5xMKQlx6EEEvNTLOKlRovWa61tmpEReMEY4mMeRZOu42bNlVz06ZqhkIGe49209oW4GhghItDkbHnmYmJrz2XTFZLmOO5FU67DZ/LTlGWgsVXnznGI3tPYEtulhQ2Ejyy9wTAkg4SUn9q7nI+S6aUsiul3lBK/Wvy9/VKqVeVUieUUj9USrku9x5CZNu5gRDxhElHzyhHuobp6BklnjCnzSpWyppgri31sLbSR22ph6JphoTKfE7+87tW8Y2PXsu3793Oh7evnvK9jYS+JGvbSJgMhQ0uDIY522dlbUeMxBTvcHmPv3QqGRxs2JQt+dM6vlSleordI5EJ9af2HekudNMWhXwso3gIOJz2+8PA32itNwEDwKfy0AYhJihxOzjXHx5bvRQyEpzrD4/ts305tuRqJP8Mg8X66iIeuHXjtO/53771Ko+/2MHZ/kuDVNw0GU4GizN9QXpGomPlOmYqGEtcUmLEpqzjS1V6TzFVqsVpVzz6Qkehm7Yo5HSISSm1GvgN4C+BP0xmZe8E/lvyKd8Bvgh8I5ftEGKykXAME8ZmqLW2/joSjk3zqsxSwaLY7cA0NcFYnGB0fNe8meoZjfL9A+f4/oFzbFtRQnNjHbdvraF0UvXXhKkZiRiMRAxsyRyLohnkWBS57IxG40BibB9uYMZBcTEqZPXcpSDXPYi/Bf4HVqFMgCpgUGsdT/7eyRQ70yml7ldKHVRKHezp6clxM8Vy0xM0cNisb9AKxsble4LGvN7Xlqw+W1fmYU2lj6pi95T7Y0/2sR1r8Zda+ROHL47wyLPH+dCjr/DFn7fxysk+4hn29DaTVWYDwxHO9IXoHokQnqJHsKuhBlODmQyGqb/vaqiZ8/UudPUVPsKThuWk/tTM5eyrg1LqfUC31vqQUuq22b5ea/0Y8BhYq5iy2zohwKYUzrStRRMzLB8+U3aboszrpMzrJBY3GY3GsTH+bWlCW4CP37SOj/3aWt7qHKKlrYvnj/UQMUxeONbLC8d6qfA52bWtlj1NdWycVDgQksEiEmc0Eh9bDVXkcozVp+oajlHpczIYNjC1FRTLvU66hmffa1ospP7U/OSyb3kT8H6l1HuxNhkqBR4BypVSjmQvYjXWVqZC5NWG6iKOd4+itB5fbqphc01RTs7nctiodLgyBgcYDxo2pbimvpxr6st5cNdmXjzeS2tbF2+cHWQgZPCTQ+f5yaHzbKwpormpjt3baqnwXbrOI301lN1mjb2f6Q9S4nEQjZvEEiYuu1VYcCkPtxSyeu5SkLMAobX+PPB5gGQP4r9rrT+ilPox1v4STwD3Aj/LVRuEmMpn72zgj3/yK0YiceIJE4fNRoXPyWfvbMjpeVPBKPVzOl6nneZGP82NfgKpwoHtAToHwpzsCfKNfSd59PmT3LC+kj1Nddy4IXPhwNSchcdh50xfELtNYVPW3hfnByNsrr20N7KUFKp67lJQiNmpzwJPKKX+AngD+HYB2iCWudsaavnru67O+zfLVWUeOgcjlwSHlWVuXA7blJsP+Us9fHTHWj7y7jW0X0wWDjzSw2g0zv6OfvZ39FPicXD7VqtwYENdhsKByZOOnVtrtNYkTGvvjFxmcC/HXISlcM2SSS1EHu070s3vfe8QobS9KHxOG3/3keu4raHWyrQOGwSj8WnexRKLm/zyZB+t7V0cODWxcOCaSh/NjX7uaBwvHHjPt/ZjVzAQMsYq2Fb4nJganrj/RnzJ/TR8ruyWJ59p1nquFOJGXehrTpFMaiHmqFDf8FJDPKmJ4vSbscdpx+O0jyXJjUSmznVwOWzctrWG27bW0B+M8ezhAC1tATp6g5ztD/H4S6f49kunuHZNOc1NddQWexgMx6hPW8ETNhLUFrnHVkNZy2CjuJ12ipIBw+2YXxZ3IffCTr9RpyfKfQlyeu6lsv+3BAixLBXqxvHlXxwmFEuMDfNoDaFYgi//4vCE8zrtNqqL3VT4XNZkc8TIuD92SmWRiw9tr+eu61ZzonuUlvYAew93Mxg2OHR2kENnB3E5bNiVothtp8zrIBrXxE3N3ddfuntw1EgQNRIQZN4lPwqZi1CoG/VSyb+QACGWpUdf6CAWT9A3Gp+woifXN46TvUES2sq9UArQkNDW8UzsNkVFkYtyn5PRaJyhsDHlPAVY5UA2+0vY7C/hd27ZwKun+mltD/DKyb6x14WNBP3BGNXFbj5247pp97+AVMkPq0fjsNnwuuwUue14nTPb/KiQe2EX6ka9VPb/lh1LxLJ0LDBMXzBGPKGxK0U8oekLxjgeGM7peeOpIn2p+6qadHwKSlkJeKsrfKwo847lNkzHkSwc+Ofvb+LHv3MjD+7cxNa6EsAKSoGRKH/depSHnniTf3/74ozmPeKmyUjEoGsowum+EF1DEYbCRsYkvpQHbtlglTNJlgYJxeJ5y0UoVKJcIa85m6QHIZYlI3lDtiXH/5Wyym7HLnOjni+7TRE39SWrmByzmBT2uux4XV4iRoLBkEEodvkbe5nXyQfftYoPvmsVp/uCtLYFePpwgL7RGG+fH+Lt80N8be8Jbt5UTXOTn2vXVFx2ojp14wvF4vRhzYn4XFbJj/ShqELmIhQqUW6p5F9IgBDLksthI5zc32EsJ0GT821A60rddA5GLjmeKrExGx6nnboyO9G4FShm0gNI7WZ3cTjM6jIvH7hqJWf6Q7x0opdo3OTZI908e6Sb6mIXu7f52dPkZ23VzJIHY3GTWDzGYMgKhF7n+KqoQuUiLIQb9cJfJzo1CRBiWdpcW8LpvlGGw+NzEKVFTtZV5TZprNjtwIZ109Ak5yKYX8E8t8OOv9QKFEMhI7kS6VIHOvp5ZO9xHDZFqcdBfyjGL9q6eGjnZj6zezPPH+uhpS3A2+eH6B2N8cRr53jitXNsrSthT6Of2xtqKZs0nj+VhHnpqihfMmDkYiOk6RQiOBVqEUS2SYAQy1Jq6KGuzJHXoYfRWIKqYid9QWMso7qqyJmVkttuh53aUjsVUyyRfeK1cziS3+yBset+4rVzfGXD1bz3yhW898oVXBgM09oe4On2ABeHIhztGuFo1wh/t+8kN26sornRz7vXV+Kwz7y3lVoVNZDqXSR7Fl6nPas5FwuFLHMVYhEr1NBDsctO11AEp802NrQ1EIqzqWb2Q0xTmWqJ7MXhMKWeif/Le5w2uobDE46tLPfy8V9bx8duXMvbnUO0tgd4/lgPoViCF4/38uLxXsq9TnZuq+XOpjo2zbJUR8IcLyoIFLR3kSuyzFWIRa4QQw9KKUxTk1B6rAehNDkpc5G+RHY4EmdlmZfe0ehYDwIgYpjUlXozvt6mFFfXl3N1fTmf3rmJl0700tIW4PUzAwyGDZ58/TxPvn6eDTVFNDf62b3NT2XR7DeITO9dpJbR+lzWMlrbIu1dyDJXIcSsdY9EUKlJCKyfygY9I5dOXGeLUlbZ8Yd2bUYDkXgCjSZsJKZMlJvM47Sze5ufv77rKn7w2+/mvpvXU19hBZaOniDffL6D33z0FT7/5NvsO9o9ba7GdFLLaAPDEc70h7gwGGYgGJvXVquFsFSWuUotJiHy6KovtjAcuXQSudTj4K0v7sn5+VPlRc72B6kr9fKh61ZfNlFuKlprjnSN0NIW4Lmj3YykXVex28HtW2vY01THthUZCgfOwWKbu0j9Wxdymet8azFJgBAijxr+178TiV/6/5zHoTjyF+/Ne3uMhLXX9UgkjjmPe0EsbvJKRx+tbQFePdU3oXDg6gove5r83LHNT22pJwuttnic9rECg/OtF7VUSbE+IRaRqRLxcp2gNxWn3UZVsZvyGdZ8morLYePWLTXcuiVZOPBIN61tXZzsCdI5EObbL53m2y+dxmVX1JV6ue/m9dy8pXpebY8YCWvoKbh05i6yJdV7cdasu3I+7yMBQog8mureO4d78pxMVcF28oT2UMggPsctWCuLXHzoutV86LrVnOwe5Tu/PM0vO6xeRSyhOTsQ4s9+3sa1ayr4yI41XLW6DNs8h6CsuQtr/kIphcdpw+d04HHZll3vIj0HA21ePntyGhIghMijqXaSy+FePWNmkryVmtAu9TgYjcYZTO4dMVcba4sZjSZYWebB1DAcMQhGE2jg0NkBDp0doK7Uwx2NtTQ31rGqIvOKqtnQWhOOJQjHxnsXHpfNyux22meVv7EYTc7BmA8JEELkUZXPSW/QyHg812aTvJUqDljicSYDRWzOK5NS+RcKRbHbYe2XHTGs4GNquoYj/NP+s/zT/rNcsbKU5qY6bttaM6/s8nRx02Q0Yo7lXaTKl/tcDjxOW0530iuETDkYcyUBQog8qi52ZwwQ1cXZS5SbylyTt4rdDordVjAZDBmzXnK6otRLX3A8/8JuU3icdppW+vjM7s20tnfxdHs3PaNR3rkwzDsXhvn6cye4aWMVe5rquG7t5QsHTie9/tSKUi93X1/PDRsqGQpbw1Fepx2f20rWWwq9i0w5GHO1+P81hFhEekaj2G0Tq33bbdA7Gs35uedb+trncrCy3MvKci9Fs/h2f/f19cRNK+9icv7Fmiof971nA9//7Xfz13ddxe5ttbiTe3M/d7SHzz35Nnc/tp9Hnz/JqSn2zJhOqv5UXzBKqcdBXzDKI3uPc6CjHxivSNs7EuVsf4jOgRD9wRjhWGLKnfwWuvQcjPmSHoQQeWQkNKY5IU8O08zPKqZslb5ObYsai5sMhmPWnMI0N9MbNlTyEJt54rVzdA2HqUv7Fp9itymuW1vBdWsrCMXiPH+sl9a2Ln7VOURfMMYPD3byw4OdbPEX09xYx66GWspmMCw3Xf2pTPkfYxVpYax34XVZ268ult5FehkZlG1e93jJgxAij7b96S8IG5eO5XudNg7/37+e8/PnInkrnlYccD65FJlcHArzdLu11/bFofFsc4dN8e4NldzZVMcN6ytxTnHzvudb+8fmP1I0mpFInO//9o5ZtWWq/S4WMsmDEGIRKXQeRC7qTznScimGwgbDYSNrgWJFmZeP3biO39qxlnfOD9PS1sW+ZOHAl0/08fKJPsq8TnY11NLc5GdzbfGESefJ8x8wff2p6Vyy38UiyuqeKwkQQuSROUXCw1THFxO7TVFZ5KLc62Q4YjAUnlvSXSZKKa5cXcaVq8v4/Z2beDlZOPDQmQGGwgZPvnGeJ984z/rqVOHAWqqK3dx9fT2P7D1O2EjgcdqIGOaM609NZ3JF2lRWd2r4bamQISYh8mjjn/x7xpum3aY4+f/kv9RGLmmt5510dzk9I1GeORygtS3Amf7x1Vg2BdvXVtDcVIfbbuNfXj8/5fxHtqXvplfovAsZYhJiEfE5bYxEL10m6nMujgnQ2UhPuhuJWoFiPkl3mdSUuLnnhjXcfX09RwPJwoFHuhmOxDlweoADpwcoctu5fWstn7x5HU0rS3Oe9zBxNz1r7iI9YCymvAsJEELk0RWryjnSNcRQOI6prW+6ZV4HDXVlhW5aziilKPU4Kc1C0t1052ioK6WhrpTfvXUj+0/18XRbgP2n+glGE/zrWxf517cusrrCyx2Nfu5o9FOXxcKB07HmLsyxvItUGRCvy57zPdDnS4aYhMij9HIX6UtNv/T+pkW1FeV8hWMJBsNWvkEuDYRi7D3STcs7AU70jE547Jr6cvY0+bllcw1eV2HmDVJFBlO9i2xPdku5byEWmYWwT8BCETES9OdpQ6CTPaO0tgV45nCAgdB4NrvHaeOWzTU0N/m5pr583oUD5yO1lNbrtGelDIjMQQixSC38r2a553HaWVnuJRxLMBDKbaDYWFPM795WzP23bOC10/20tgV4+WQvEcOktT1Aa3uA2hI3dzT62dPkL8j2oJMT9TzO8fmLQlSllR6EEHkkQ0zTy9fQU8pIxOC5oz20tgVovzg84bHGFaXsafJz+9Zaij2F/y6dWh3lcdnxOGY2fyFDTEIsIvc8tv+SQmqhWJzaEg8/uH92mb1LWcSwehT5ChQA5/pDtLYHeLo9QPfIeG0sp11x08Zqmpv8XL+ucsEkxTlsNjxOGx7X1IUGZYhJiEXk3ECImJHgVG9wbBVTdZEr66t6FjuP086KsvwMPaXUV/r41M3r+cRN63jz3CCtbQFeONZDJG6y71gP+471UFnkYve2WvY01bG+uijnbZpO3DQZjZpjy2lTZcyzuZxWAoQQeaSA7tEYKvl3ra3f67OwUc5SZK3w8RIxEgyGjKxUKL0cm1Jcu6aCa9dU8OCuTbxwrJfW9i7ePDdEfzDGjw528qODnWyuLWZPk5+dDbWU+1w5b9flGAmTofD4ctps7AkhAUKIPEqV9Z48sJuPct8w9ZajC53HaaeuzE40nmAoZIx9a841n8vBnVfUcecVdXQNRXg6OZl9fjDM8e5RjneP8o3nO9ixvpLmpjp2bJi6cGA+aa0xspC9LgFCiDyKJTQOm7UHtdbWVqM2lZ9ifTPZcnShczvs1JbaqUiYDCYDRb7mUevKPPzWjWv56I41tF0YpqUtwL6j3QRjCV4+2cfLJ/so9TjY2WANQW3xFy+qrOlMJEAIkUdFLmvlkjvtW2bcNPHlocDbbLYcXeicdhs1JW4qi1wMhmI5KTU+FaUUV6wq44pVZfz+7Rt5+WQfrW1dHDwzwHAkzk/fvMBP37zA2iofexr97Nrmp6Yk9zsG5oIECCHy6L6b1/PI3hPETRObsnoSpraO59pctxxdyOw2lbNS4zPhdtrZ2VDLzoZaekejPHO4m9a2Lk73hTjTF+KxF0/x+EunuG5tBc2Nddy8qQr3Iqr2mrMAoZSqB74L+LGGXB/TWj+ilKoEfgisA04Dv6m1HshVO4RYSB7cvQWAx186RTCWoMhl576b148dz6X6Ch+n+0YZDseJJUxcdhulXgfrqopzfu5cS5UaL/M6GQ4bDEeyV2p8pqqT5cU/vH01x7tHaWkL8OzhAMOROK+dHuC10wMUuezcurWGPY11XLEq94UD5ytneRBKqRXACq3160qpEuAQ8EHg40C/1vrLSqnPARVa689O916SByHE/H31mWM8svcEtuS8R6r38tDOTXkJUPlkmjrre1LMhZEwebWjn5b2LvZ39E9oy8pyD82Nfpob66gry37hwGKPA3+pd3EkyimlfgZ8PfnnNq31xWQQ2ae13jrdayVACDF/9zy2n1O9o4xExnsQJR4H66uLl2ySnmla24sOhmMFDRQAQyGDZ49009rexbHAxMKBV68uY09THbdsqZ6QRDkfiyZAKKXWAS8AVwBntdblyeMKGEj9Puk19wP3A6xZs+a6M2fO5LydQixlNz+8l3Kvc8KwhtaaobDBi5/dWcCW5Z7WmuHwwggUAKd6g7S2dfHM4W76grGx4x6HjfdsqaG50SocOJ+s7UURIJRSxcDzwF9qrZ9USg2mBwSl1IDWumK695AehBDzd89j+6ecg1iqPYjJFsrQU0rC1Bw8YxUOfOlEL0bacudU4cDmRj/1lbMvHJiNAJHTVUxKKSfwL8D3tNZPJg8HlFIr0oaYunPZBiGE5cYNlRw43T82BxFLmHSPxLjn+txtv7nQ2GyKcp+LUo+TobAVKPK56mkyu03x7vVVvHt9FaOROPuOddPSFqDtwjDdI1G+9+pZvvfqWRpXlNDcVMftW2so8cw/Q3qmcrmKSQHfBg5rrb+S9tBTwL3Al5M/f5arNgghxr3S0U9NseuSOYhXOvp5sNCNyzObTVFR5KLU6yzI8thMij0O3nfVSt531Uo6B6zCga1tVuHA9osjtF8c4f88d4Jf21jNnjwVDszlKqabgReBt4FUzvefAK8CPwLWAGewlrn2T/deMsQkxPwt5zmIy0mYesEEinSm1vzq3CCt7QGeP9ZDxBgvn1Hhc7J7m5/mJj8bay5dqrwo5iCyQQKEEPMnpcYvL2FqBkMxhiP5K+ExU+FYgheP99DSHuDNs4MT6nltqimmucnPrm21VCQLB0qAEELMmGxWNHPxhFUVdSEGCoDA8HjhwM6B8Nhxa06jkuYmP7u3+amvLJIAIYSYGdkPe3biCZPBsMHIAg0UWmvaLw7T2hZg79FugtHxfTNKvQ7e/uKdEiCEECKX4gmTgTxXj52tqJHglyf7aGkPcPB0P6aGMw+/b+EucxVCiKXAkaweW+5z5r3M+Ey5nXZub6jl9oZa+kajvHiilz96eH7vWfidLYQQYpFIlRlfXeGl2ONYsMX2qordfHTH2nm/j/QghBBilpx2G7UlHgxf/jcuyifpQQghxBylehT1FV5KJ+WYLAUSIIQQYp4cdhvVxVagKFtCgUKGmIQQIkscdltBd7jLNgkQQgiRZQthh7tskAAhhBA5Yk8WBSzzOhmJxBkKG8RN8/IvXCAkQAghRI7ZbIoyn5NSr4PRqBUoYvGFHygkQAghRJ4opSjxOCnxOAnHEgyFDUKxeKGbNSUJEEIIUQBelx2vy04sbhUGXIi5FBIghBCigFwOK5eissi14Ca0JUAIIcQCkJrQLvc5GYnGGQoZGInCzlNIgBBCiAVEKUWpx0mpx8loNM5gKFawCW0JEEIIsUAVux0Uux0Eo3EGChAoJEAIIcQCV+R2UFSAQCEBQgghFol8BwoJEEIIscjkK1BIgBBCiEUqFShGo3EGgrGsr3qSACGEEItcajI724FCAoQQQiwR6YEinoUgIQFCCCGWmGJ3dm7tsqOcEEKIjCRACCGEyEgChBBCiIwkQAghhMhIAoQQQoiMJEAIIYTISAKEEEKIjCRACCGEyEgChBBCiIzUQtskOxOlVA9wpgCnrgZ6C3DeQpJrXh7kmpeHrVrrkrm+eFGU2tBa1xTivEqpg1rr7YU4d6HINS8Pcs3Lg1Lq4HxeL0NMQgghMpIAIYQQIiMJENN7rNANKAC55uVBrnl5mNc1L4pJaiGEEPknPQghhBAZSYAQQgiRkQSINEqpcqXUT5RSR5RSh5VSNyqlKpVSTyuljid/VhS6ndmklPoDpVSbUuodpdQPlFIepdR6pdSrSqkTSqkfKqVchW7nfCil/l4p1a2UeiftWMbPVVm+mrz2t5RS1xau5XM3xTX/dfK/7beUUv+fUqo87bHPJ6/5qFJqT0EaPQ+ZrjftsT9SSmmlVHXy9yX7GSePfzr5Obcppf4q7fisP2MJEBM9AvyH1roBuBo4DHwOeFZrvRl4Nvn7kqCUWgU8CGzXWl8B2IG7gYeBv9FabwIGgE8VrpVZ8Y/AnZOOTfW5/jqwOfnnfuAbeWpjtv0jl17z08AVWuurgGPA5wGUUo1Yn3tT8jV/p5Sy56+pWfGPXHq9KKXqgWbgbNrhJfsZK6VuBz4AXK21bgL+d/L4nD5jCRBJSqky4Bbg2wBa65jWehDrH/s7yad9B/hgIdqXQw7Aq5RyAD7gIrAT+Eny8UV/zVrrF4D+SYen+lw/AHxXW/YD5UqpFXlpaBZlumatdavWOp78dT+wOvn3DwBPaK2jWutTwAnghrw1Ngum+IwB/gb4H0D6apwl+xkDvwt8WWsdTT6nO3l8Tp+xBIhx64Ee4B+UUm8opR5XShUBfq31xeRzugB/wVqYZVrr81jfMM5iBYYh4BAwmHYj6QRWFaaFOTXV57oKOJf2vKV6/Z8EfpH8+5K8ZqXUB4DzWutfTXpoSV5v0hbgPckh4ueVUtcnj8/pmiVAjHMA1wLf0Fq/CwgyaThJW2uCl8y64OS4+wewguNKoIgM3fSlbql9rpejlPqfQBz4XqHbkitKKR/wJ8AXCt2WPHMAlcAO4I+BHyml1FzfTALEuE6gU2v9avL3n2AFjECq+5n82T3F6xej3cAprXWP1toAngRuwupyp+p0rQbOF6qBOTTV53oeqE973pK6fqXUx4H3AR/R40lQS/GaN2J98fmVUuo01jW9rpSqY2leb0on8GRy+OwAYGIVKZzTNUuASNJadwHnlFJbk4d2Ae3AU8C9yWP3Aj8rQPNy5SywQynlS37LSF3zc8BdyecstWtOmepzfQr4WHKlyw5gKG0oalFTSt2JNR7/fq11KO2hp4C7lVJupdR6rMnbA4VoY7Zord/WWtdqrddprddh3TivTf5/vmQ/Y+CnwO0ASqktgAurgu3cPmOttfxJ/gGuAQ4CbyX/oSuAKqxVLseBZ4DKQrczy9f858AR4B3gnwA3sCH5H88J4MeAu9DtnOc1/gBrjsXAulF8aqrPFVDA/wFOAm9jrfAq+DVk6ZpPYI1Dv5n888205//P5DUfBX690O3PxvVOevw0UL0MPmMX8M/J/59fB3bO5zOWUhtCCCEykiEmIYQQGUmAEEIIkZEECCGEEBlJgBBCCJGRBAghhBAZSYAQYhpKqY8rpb4+y9e8Xym1ZIo6iuXLcfmnCCFmSinl0Fo/hZWYJMSiJgFCLEtKqY8B/x2rBtNbwI+A/4WVaNSHVYoiMOk164C/xypd0AN8Qmt9Vin1j0AEeBfwslLqLazkq99XStUA3wTWJN/mM1rrl5VSt2KVlyfZhlu01iO5ul4h5kIChFh2lFJNWMHg17TWvUqpSqyb9A6ttVZK3YdVkuKPJr30a8B3tNbfUUp9Evgq42XCVyffL5Gsd5TyCNbeGi8ppdYALcA2rOD0fyWDRTFWgBFiQZEAIZajncCPtda9AFrrfqXUlcAPk4X7XMCpDK+7Efgvyb//E/BXaY/9WGudyPCa3UBjWkHN0mRAeBn4ilLqe1jF1Trne1FCZJtMUgth+Rrwda31lcADgGeWrw9OcdyG1TO5JvlnldZ6VGv9ZeA+wIs1LNUw55YLkSMSIMRytBf4kFKqCqz9qYEyxssf3zvF636JtW0jwEeAF2dwrlbg06lflFLXJH9u1FbF0YeB1wAJEGLBkQAhlh2tdRvwl8DzSqlfAV8Bvgj8WCl1CKs8ciafBj6RnIT+LeChGZzuQWC7UuotpVQ78DvJ459RSr2TfC+D8d3dhFgwpJqrEEKIjKQHIYQQIiMJEEIIITKSACGEECIjCRBCCCEykgAhhBAiIwkQQgghMpIAIYQQIqP/H6+arHjqVEg4AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "p = sns.regplot(data = df,  \n",
    "                x = df.calories, \n",
    "                y = df.rating\n",
    "               ) \n",
    "p"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "RtHrbeHxiVPX"
   },
   "source": [
    "The weights for z1 are given by (−0.847, 0.532): Means, when x axis move -0.847, y axis move 0.532 along regression (line fit) line. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "id": "Nn3-e5zE3ouV"
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from sklearn.decomposition import PCA"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 109
    },
    "id": "MjLgwAOV3EIU",
    "outputId": "b480fb9f-e823-4088-9262-d94579dcdd08"
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Standard deviation</th>\n",
       "      <th>Proportion of variance</th>\n",
       "      <th>Cumulative proportion</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>22.316462</td>\n",
       "      <td>0.863191</td>\n",
       "      <td>0.863191</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>8.884410</td>\n",
       "      <td>0.136809</td>\n",
       "      <td>1.000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   Standard deviation  Proportion of variance  Cumulative proportion\n",
       "0           22.316462                0.863191               0.863191\n",
       "1            8.884410                0.136809               1.000000"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pcs = PCA(n_components=2)\n",
    "\n",
    "pcs.fit(df[['calories', 'rating']])\n",
    "\n",
    "pcsSummary = pd.DataFrame({'Standard deviation': np.sqrt(pcs.explained_variance_),\n",
    "                          'Proportion of variance': pcs.explained_variance_ratio_,\n",
    "                          'Cumulative proportion': np.cumsum(pcs.explained_variance_ratio_)#column \n",
    "                          }\n",
    "                          )\n",
    "pcsSummary"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 140
    },
    "id": "2RXvhDOn3yRc",
    "outputId": "0eadf7a8-3d6e-4f99-a3b9-261715844b25"
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>0</th>\n",
       "      <th>1</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>Standard deviation</th>\n",
       "      <td>22.32</td>\n",
       "      <td>8.88</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Proportion of variance</th>\n",
       "      <td>0.86</td>\n",
       "      <td>0.14</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Cumulative proportion</th>\n",
       "      <td>0.86</td>\n",
       "      <td>1.00</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                            0     1\n",
       "Standard deviation      22.32  8.88\n",
       "Proportion of variance   0.86  0.14\n",
       "Cumulative proportion    0.86  1.00"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pcsSummary = pcsSummary.transpose()\n",
    "pcsSummary.round(2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 140
    },
    "id": "O9Y616c04NBl",
    "outputId": "0bb37e3b-7336-4b86-e0d0-202ebd5d735d"
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>PC1</th>\n",
       "      <th>PC2</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>Standard deviation</th>\n",
       "      <td>22.3165</td>\n",
       "      <td>8.8844</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Proportion of variance</th>\n",
       "      <td>0.8632</td>\n",
       "      <td>0.1368</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Cumulative proportion</th>\n",
       "      <td>0.8632</td>\n",
       "      <td>1.0000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                            PC1     PC2\n",
       "Standard deviation      22.3165  8.8844\n",
       "Proportion of variance   0.8632  0.1368\n",
       "Cumulative proportion    0.8632  1.0000"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pcsSummary.columns = ['PC1', 'PC2']\n",
    "pcsSummary = pcsSummary.round(4)\n",
    "pcsSummary"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The summary gives the reallocated variance: \n",
    "* `z1` accounts for 86.32% of the total variability and \n",
    "* `z2` for the remaining 13.68%. \n",
    "\n",
    "Therefore, if we drop `z2 = PC2`, we still maintain 86.32% of the total variability. \n",
    "\n",
    "https://www.youtube.com/watch?v=FgakZw6K1QQ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Scree Plot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib as mpl\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 469
    },
    "id": "7HvNPyJi_d3D",
    "outputId": "6d575c83-261d-470d-cbc0-4b6f43106862"
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x504 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "PC_components = np.arange(pcs.n_components_) + 1\n",
    "#PC_components\n",
    "\n",
    "_ = sns.set(style = 'whitegrid', \n",
    "            font_scale = 1.2\n",
    "            )\n",
    "\n",
    "fig, ax = plt.subplots(figsize=(10, 7))\n",
    "\n",
    "_ = sns.barplot(x = PC_components, \n",
    "                y = pcs.explained_variance_ratio_, \n",
    "                color = 'b'\n",
    "                )\n",
    "\n",
    "_ = sns.lineplot(x = PC_components-1, \n",
    "                 y = np.cumsum(pcs.explained_variance_ratio_), \n",
    "                 color = 'black', \n",
    "                 linestyle = '-', \n",
    "                 linewidth = 2, \n",
    "                 marker = 'o', \n",
    "                 markersize = 8\n",
    "                 )\n",
    "\n",
    "plt.title('Scree Plot')\n",
    "_.set(xlabel='N-th Principal Component', ylabel = 'Variance Explained')\n",
    "plt.ylim(0, 1)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* The first principal component has the largest possible variance (that is, accounts for as much of the variability in the data as possible)\n",
    "* Each succeeding component, in turn, has the highest variance possible under the constraint that it is orthogonal the preceding components."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "rk1ke8Lj4i2B",
    "outputId": "ced343db-02a6-4635-bbfb-4c8a668e2463"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-0.8471,  0.5315],\n",
       "       [ 0.5315,  0.8471]])"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pcs.components_.round(4) #is the set of all eigenvectors for your projection space (one eigenvector for each principal component)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 109
    },
    "id": "dTo86uS14RbJ",
    "outputId": "6d74c1cd-7e1e-4c39-924e-0384bc48ff93"
   },
   "outputs": [
    {
     "data": {
      "text/html": [
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       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>PC1</th>\n",
       "      <th>PC2</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>calories</th>\n",
       "      <td>-0.847053</td>\n",
       "      <td>0.531508</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>rating</th>\n",
       "      <td>0.531508</td>\n",
       "      <td>0.847053</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "               PC1       PC2\n",
       "calories -0.847053  0.531508\n",
       "rating    0.531508  0.847053"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pcsComponents_df = pd.DataFrame(pcs.components_.transpose(), \n",
    "                                columns=['PC1', 'PC2'],\n",
    "                                index=['calories', 'rating']\n",
    "                                )\n",
    "pcsComponents_df"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "LlWava2Cu-Ei"
   },
   "source": [
    "> The first column here shows coefficients of linear combination that defines PC1 (z1)\n",
    "\n",
    "> The second column shows coefficients for PC2 (z2)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "hrcrhoCookhL"
   },
   "source": [
    "The weights are used to compute principal component scores, which are the projected values of calories and rating onto the new axes (after subtracting the means). \n",
    "\n",
    "* The average calories across the 77 cereals is `106.88` and the average consumer rating is `42.67`.\n",
    "\n",
    "* The first column is the projection onto z1 using the weights (-0.847, 0.532). \n",
    "\n",
    "    * The second column is the projection onto z2 using the weights (0.532, 0.847). \n",
    "    \n",
    "For example:  \n",
    "\n",
    "First, we need to center the original variables by subtracting column means\n",
    "(70 − `106.88`)   & (68.4 − `42.67`)\n",
    "\n",
    "Second multiply them with corresponding Principal components. \n",
    "\n",
    "The first score for the 100% Bran cereal (with 70 calories and a rating of 68.4) is \n",
    "$$( − 0.847)(70 − 106.88) + (0.532)(68.4 − 42.67) = 44.92$$\n",
    "$$ Z_i = a_{i1} (X_1 - \\bar{X_1}) + a_{i2} (X_2 - \\bar{X_2}) + \\cdots + a_{in} (X_n - \\bar{X_n}) ; i = 1, 2, 3, \\cdots n$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 201
    },
    "id": "7nkdeQp159hZ",
    "outputId": "a2917b4d-0579-4039-c46e-4b4b716d1959"
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>x</th>\n",
       "      <th>y</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>44.921528</td>\n",
       "      <td>2.197183</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>-15.725265</td>\n",
       "      <td>-0.382416</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>40.149935</td>\n",
       "      <td>-5.407212</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>75.310772</td>\n",
       "      <td>12.999126</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>-7.041508</td>\n",
       "      <td>-5.357686</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "           x          y\n",
       "0  44.921528   2.197183\n",
       "1 -15.725265  -0.382416\n",
       "2  40.149935  -5.407212\n",
       "3  75.310772  12.999126\n",
       "4  -7.041508  -5.357686"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "scores = pd.DataFrame(pcs.transform(df[['calories', 'rating']]),\n",
    "                      columns=['x', 'y']\n",
    "                      )\n",
    "scores.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "id": "oPKWgp0C6G9-"
   },
   "outputs": [],
   "source": [
    "x = (-0.847)*(70-106.88)+(0.532)*(68.4-42.67)\n",
    "\n",
    "y = (0.531508)*(70-106.88)+(0.847053)*(68.4-42.67)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "du4dDNj70dc_",
    "outputId": "2da1bfb5-a18d-489b-d464-0e526b9f15ac"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "For the PC1\n",
      " \t x =  44.92572 \n",
      "\t y =  2.1926586500000056\n"
     ]
    }
   ],
   "source": [
    "print('For the PC1\\n \\t x = ', x, '\\n\\t y = ', y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "w1cfTB1t095i",
    "outputId": "dcb3b0ce-3b1b-42ff-cb92-cfdfa66aee0b"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.1880828211573103e-15"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "scores.x.mean()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "0EyKIGk11YeQ",
    "outputId": "7f978f0c-594a-40c9-85fa-142d49511d4e"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1.3322676295501878e-14"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "scores.y.sum()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The sum of the variances `var(z1) + var(z2)` is equal to the sum of the variances of the original variables, `var(calories) + var(rating)`.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "q7QLcJLN1eaG",
    "outputId": "cf31e975-e9dd-4d37-fb02-5e6aa82aea9b"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "576.9572164702394"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "scores.x.var() + scores.y.var()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "p0gsYAF01vKA",
    "outputId": "73b3ae28-6d7b-46a8-989e-fb643f7f5b4f"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "576.9572164702394"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.calories.var() + df.rating.var()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "Hykb7IsW10q-",
    "outputId": "42dd852e-b189-4c92-894a-deb2f17aa655"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "86.0"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "((scores.x.var()/\n",
    "  (scores.x.var()+scores.y.var())\n",
    " )*100).round(0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 140
    },
    "id": "0q4Y1djY2PYj",
    "outputId": "bc85fca6-cae9-46cb-d948-5bf8615f1a61"
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>0</th>\n",
       "      <th>1</th>\n",
       "      <th>2</th>\n",
       "      <th>3</th>\n",
       "      <th>4</th>\n",
       "      <th>5</th>\n",
       "      <th>6</th>\n",
       "      <th>7</th>\n",
       "      <th>8</th>\n",
       "      <th>9</th>\n",
       "      <th>10</th>\n",
       "      <th>11</th>\n",
       "      <th>12</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>Standard deviation</th>\n",
       "      <td>83.764075</td>\n",
       "      <td>70.914255</td>\n",
       "      <td>22.643745</td>\n",
       "      <td>19.181482</td>\n",
       "      <td>8.423228</td>\n",
       "      <td>2.091670</td>\n",
       "      <td>1.699423</td>\n",
       "      <td>0.779635</td>\n",
       "      <td>0.657833</td>\n",
       "      <td>0.370428</td>\n",
       "      <td>0.186441</td>\n",
       "      <td>6.302264e-02</td>\n",
       "      <td>5.333701e-08</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Proportion of variance</th>\n",
       "      <td>0.539503</td>\n",
       "      <td>0.386674</td>\n",
       "      <td>0.039425</td>\n",
       "      <td>0.028291</td>\n",
       "      <td>0.005456</td>\n",
       "      <td>0.000336</td>\n",
       "      <td>0.000222</td>\n",
       "      <td>0.000047</td>\n",
       "      <td>0.000033</td>\n",
       "      <td>0.000011</td>\n",
       "      <td>0.000003</td>\n",
       "      <td>3.054014e-07</td>\n",
       "      <td>2.187435e-19</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Cumulative proportion</th>\n",
       "      <td>0.539503</td>\n",
       "      <td>0.926177</td>\n",
       "      <td>0.965602</td>\n",
       "      <td>0.993892</td>\n",
       "      <td>0.999348</td>\n",
       "      <td>0.999684</td>\n",
       "      <td>0.999906</td>\n",
       "      <td>0.999953</td>\n",
       "      <td>0.999986</td>\n",
       "      <td>0.999997</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>1.000000e+00</td>\n",
       "      <td>1.000000e+00</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                               0          1          2          3         4   \\\n",
       "Standard deviation      83.764075  70.914255  22.643745  19.181482  8.423228   \n",
       "Proportion of variance   0.539503   0.386674   0.039425   0.028291  0.005456   \n",
       "Cumulative proportion    0.539503   0.926177   0.965602   0.993892  0.999348   \n",
       "\n",
       "                              5         6         7         8         9   \\\n",
       "Standard deviation      2.091670  1.699423  0.779635  0.657833  0.370428   \n",
       "Proportion of variance  0.000336  0.000222  0.000047  0.000033  0.000011   \n",
       "Cumulative proportion   0.999684  0.999906  0.999953  0.999986  0.999997   \n",
       "\n",
       "                              10            11            12  \n",
       "Standard deviation      0.186441  6.302264e-02  5.333701e-08  \n",
       "Proportion of variance  0.000003  3.054014e-07  2.187435e-19  \n",
       "Cumulative proportion   1.000000  1.000000e+00  1.000000e+00  "
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pcs = PCA()\n",
    "pcs.fit(cereals_df.iloc[:, 3:].dropna(axis=0))\n",
    "pcsSummary_df = pd.DataFrame({'Standard deviation': np.sqrt(pcs.explained_variance_),\n",
    "                             'Proportion of variance': pcs.explained_variance_ratio_,\n",
    "                             'Cumulative proportion': np.cumsum(pcs.explained_variance_ratio_)\n",
    "                             })\n",
    "pcsSummary_df = pcsSummary_df.transpose()\n",
    "pcsSummary_df"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Scree Plot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 469
    },
    "id": "dp5szqxU_O50",
    "outputId": "3955463c-f429-4386-c8dc-44b40a296c02"
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x504 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "PC_components = np.arange(pcs.n_components_) + 1\n",
    "#PC_components\n",
    "\n",
    "_ = sns.set(style = 'whitegrid', \n",
    "            font_scale = 1.2\n",
    "            )\n",
    "\n",
    "fig, ax = plt.subplots(figsize=(10, 7))\n",
    "\n",
    "_ = sns.barplot(x = PC_components, \n",
    "                y = pcs.explained_variance_ratio_, \n",
    "                color = 'b'\n",
    "                )\n",
    "\n",
    "_ = sns.lineplot(x = PC_components-1, \n",
    "                 y = np.cumsum(pcs.explained_variance_ratio_), \n",
    "                 color = 'black', \n",
    "                 linestyle = '-', \n",
    "                 linewidth = 2, \n",
    "                 marker = 'o', \n",
    "                 markersize = 8\n",
    "                 )\n",
    "\n",
    "plt.title('Scree Plot')\n",
    "plt.xlabel('N-th Principal Component')\n",
    "plt.ylabel('Variance Explained')\n",
    "plt.ylim(0, 1)\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "NZqgwnZSSTzD"
   },
   "source": [
    "Note that the first three components account for more than 96% of the total variation associated with all 13 of the original variables."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "Pr9NgpKpRslo",
    "outputId": "f60673a1-19a2-4957-9b0f-88114f189f30"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "97.0"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#pcsSummary_df.iloc[1][0]+pcsSummary_df.iloc[1][1]+pcsSummary_df.iloc[1][2]\n",
    "#OR\n",
    "((pcsSummary_df.iloc[2][2])*100).round(0)\n",
    "#type(pcsSummary_df)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "FjO0RDicSunS"
   },
   "source": [
    "In fact, the first two principal components alone capture 92.6% of the total variation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "STkHp3faSx9D",
    "outputId": "023e9a98-9935-4b7b-95e2-d0a12cdaad29"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "92.62"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#pcsSummary_df.iloc[1][0]+pcsSummary_df.iloc[1][1]\n",
    "#OR\n",
    "((pcsSummary_df.iloc[2][1])*100).round(2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 140
    },
    "id": "CXrtccVROj9n",
    "outputId": "de319130-7ae6-4d1b-a63d-0783e3008eef"
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
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       "\n",
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       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>PC1</th>\n",
       "      <th>PC2</th>\n",
       "      <th>PC3</th>\n",
       "      <th>PC4</th>\n",
       "      <th>PC5</th>\n",
       "      <th>PC6</th>\n",
       "      <th>PC7</th>\n",
       "      <th>PC8</th>\n",
       "      <th>PC9</th>\n",
       "      <th>PC10</th>\n",
       "      <th>PC11</th>\n",
       "      <th>PC12</th>\n",
       "      <th>PC13</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>Standard deviation</th>\n",
       "      <td>83.7641</td>\n",
       "      <td>70.9143</td>\n",
       "      <td>22.6437</td>\n",
       "      <td>19.1815</td>\n",
       "      <td>8.4232</td>\n",
       "      <td>2.0917</td>\n",
       "      <td>1.6994</td>\n",
       "      <td>0.7796</td>\n",
       "      <td>0.6578</td>\n",
       "      <td>0.3704</td>\n",
       "      <td>0.1864</td>\n",
       "      <td>0.063</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Proportion of variance</th>\n",
       "      <td>0.5395</td>\n",
       "      <td>0.3867</td>\n",
       "      <td>0.0394</td>\n",
       "      <td>0.0283</td>\n",
       "      <td>0.0055</td>\n",
       "      <td>0.0003</td>\n",
       "      <td>0.0002</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.0000</td>\n",
       "      <td>0.000</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Cumulative proportion</th>\n",
       "      <td>0.5395</td>\n",
       "      <td>0.9262</td>\n",
       "      <td>0.9656</td>\n",
       "      <td>0.9939</td>\n",
       "      <td>0.9993</td>\n",
       "      <td>0.9997</td>\n",
       "      <td>0.9999</td>\n",
       "      <td>1.0000</td>\n",
       "      <td>1.0000</td>\n",
       "      <td>1.0000</td>\n",
       "      <td>1.0000</td>\n",
       "      <td>1.000</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                            PC1      PC2      PC3      PC4     PC5     PC6  \\\n",
       "Standard deviation      83.7641  70.9143  22.6437  19.1815  8.4232  2.0917   \n",
       "Proportion of variance   0.5395   0.3867   0.0394   0.0283  0.0055  0.0003   \n",
       "Cumulative proportion    0.5395   0.9262   0.9656   0.9939  0.9993  0.9997   \n",
       "\n",
       "                           PC7     PC8     PC9    PC10    PC11   PC12  PC13  \n",
       "Standard deviation      1.6994  0.7796  0.6578  0.3704  0.1864  0.063   0.0  \n",
       "Proportion of variance  0.0002  0.0000  0.0000  0.0000  0.0000  0.000   0.0  \n",
       "Cumulative proportion   0.9999  1.0000  1.0000  1.0000  1.0000  1.000   1.0  "
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pcsSummary_df.columns = ['PC{}'.format(i) for i in range(1, len(pcsSummary_df.columns) + 1)]\n",
    "pcsSummary_df.round(4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "KcwhtymLVVgC",
    "outputId": "fb0faf05-1e06-4a6c-ae55-aaff9ad27c0e"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-7.79841812e-02,  7.56780568e-04,  1.01783356e-04,\n",
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       "         2.32799160e-04,  8.10661351e-04,  1.41471041e-02,\n",
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       "        -1.03280533e-02, -9.95312585e-01, -9.24845014e-02,\n",
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       "       [ 4.19359171e-02, -6.16293914e-01,  3.18469006e-01,\n",
       "         1.02602312e-02, -6.48360192e-01, -2.05693589e-01,\n",
       "         1.36487845e-01,  6.40048350e-03,  9.64251396e-03,\n",
       "         5.14519362e-09,  9.52224067e-08, -2.12711164e-08,\n",
       "         1.88286335e-01]])"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pcs.components_"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 444
    },
    "id": "6l5Ao-1gOsVD",
    "outputId": "80615287-f3fd-47fa-b05f-baec4e72bb21"
   },
   "outputs": [
    {
     "data": {
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>PC1</th>\n",
       "      <th>PC2</th>\n",
       "      <th>PC3</th>\n",
       "      <th>PC4</th>\n",
       "      <th>PC5</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>calories</th>\n",
       "      <td>-0.077984</td>\n",
       "      <td>-0.009312</td>\n",
       "      <td>0.629206</td>\n",
       "      <td>-0.601021</td>\n",
       "      <td>0.454959</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>protein</th>\n",
       "      <td>0.000757</td>\n",
       "      <td>0.008801</td>\n",
       "      <td>0.001026</td>\n",
       "      <td>0.003200</td>\n",
       "      <td>0.056176</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>fat</th>\n",
       "      <td>0.000102</td>\n",
       "      <td>0.002699</td>\n",
       "      <td>0.016196</td>\n",
       "      <td>-0.025262</td>\n",
       "      <td>-0.016098</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>sodium</th>\n",
       "      <td>-0.980215</td>\n",
       "      <td>0.140896</td>\n",
       "      <td>-0.135902</td>\n",
       "      <td>-0.000968</td>\n",
       "      <td>0.013948</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>fiber</th>\n",
       "      <td>0.005413</td>\n",
       "      <td>0.030681</td>\n",
       "      <td>-0.018191</td>\n",
       "      <td>0.020472</td>\n",
       "      <td>0.013605</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>carbo</th>\n",
       "      <td>-0.017246</td>\n",
       "      <td>-0.016783</td>\n",
       "      <td>0.017370</td>\n",
       "      <td>0.025948</td>\n",
       "      <td>0.349267</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>sugars</th>\n",
       "      <td>-0.002989</td>\n",
       "      <td>-0.000253</td>\n",
       "      <td>0.097705</td>\n",
       "      <td>-0.115481</td>\n",
       "      <td>-0.299066</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>potass</th>\n",
       "      <td>0.134900</td>\n",
       "      <td>0.986562</td>\n",
       "      <td>0.036782</td>\n",
       "      <td>-0.042176</td>\n",
       "      <td>-0.047151</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>vitamins</th>\n",
       "      <td>-0.094293</td>\n",
       "      <td>0.016729</td>\n",
       "      <td>0.691978</td>\n",
       "      <td>0.714118</td>\n",
       "      <td>-0.037009</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>shelf</th>\n",
       "      <td>0.001541</td>\n",
       "      <td>0.004360</td>\n",
       "      <td>0.012489</td>\n",
       "      <td>0.005647</td>\n",
       "      <td>-0.007876</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>weight</th>\n",
       "      <td>-0.000512</td>\n",
       "      <td>0.000999</td>\n",
       "      <td>0.003806</td>\n",
       "      <td>-0.002546</td>\n",
       "      <td>0.003022</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>cups</th>\n",
       "      <td>-0.000510</td>\n",
       "      <td>-0.001591</td>\n",
       "      <td>0.000694</td>\n",
       "      <td>0.000985</td>\n",
       "      <td>0.002148</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>rating</th>\n",
       "      <td>0.075296</td>\n",
       "      <td>0.071742</td>\n",
       "      <td>-0.307947</td>\n",
       "      <td>0.334534</td>\n",
       "      <td>0.757708</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "               PC1       PC2       PC3       PC4       PC5\n",
       "calories -0.077984 -0.009312  0.629206 -0.601021  0.454959\n",
       "protein   0.000757  0.008801  0.001026  0.003200  0.056176\n",
       "fat       0.000102  0.002699  0.016196 -0.025262 -0.016098\n",
       "sodium   -0.980215  0.140896 -0.135902 -0.000968  0.013948\n",
       "fiber     0.005413  0.030681 -0.018191  0.020472  0.013605\n",
       "carbo    -0.017246 -0.016783  0.017370  0.025948  0.349267\n",
       "sugars   -0.002989 -0.000253  0.097705 -0.115481 -0.299066\n",
       "potass    0.134900  0.986562  0.036782 -0.042176 -0.047151\n",
       "vitamins -0.094293  0.016729  0.691978  0.714118 -0.037009\n",
       "shelf     0.001541  0.004360  0.012489  0.005647 -0.007876\n",
       "weight   -0.000512  0.000999  0.003806 -0.002546  0.003022\n",
       "cups     -0.000510 -0.001591  0.000694  0.000985  0.002148\n",
       "rating    0.075296  0.071742 -0.307947  0.334534  0.757708"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pcsComponents_df = pd.DataFrame(pcs.components_.transpose(), \n",
    "                                columns = pcsSummary_df.columns,\n",
    "                                index = cereals_df.iloc[:, 3:].columns\n",
    "                                )\n",
    "pcsComponents_df.iloc[:,:5]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 444
    },
    "id": "DFeaIOovPu6b",
    "outputId": "56a26372-352e-418f-9164-dd4565a57f4d"
   },
   "outputs": [
    {
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       "      <th>PC13</th>\n",
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       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>calories</th>\n",
       "      <td>-0.077984</td>\n",
       "      <td>-0.009312</td>\n",
       "      <td>0.629206</td>\n",
       "      <td>-0.601021</td>\n",
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       "      <td>0.064517</td>\n",
       "      <td>0.008975</td>\n",
       "      <td>0.003853</td>\n",
       "      <td>4.193592e-02</td>\n",
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       "    <tr>\n",
       "      <th>protein</th>\n",
       "      <td>0.000757</td>\n",
       "      <td>0.008801</td>\n",
       "      <td>0.001026</td>\n",
       "      <td>0.003200</td>\n",
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       "      <td>0.112745</td>\n",
       "      <td>0.258103</td>\n",
       "      <td>0.655037</td>\n",
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       "      <td>-0.255922</td>\n",
       "      <td>-0.045273</td>\n",
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       "      <td>-6.162939e-01</td>\n",
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       "    <tr>\n",
       "      <th>fat</th>\n",
       "      <td>0.000102</td>\n",
       "      <td>0.002699</td>\n",
       "      <td>0.016196</td>\n",
       "      <td>-0.025262</td>\n",
       "      <td>-0.016098</td>\n",
       "      <td>-0.131816</td>\n",
       "      <td>0.372584</td>\n",
       "      <td>-0.117932</td>\n",
       "      <td>0.123749</td>\n",
       "      <td>-0.841440</td>\n",
       "      <td>-0.062438</td>\n",
       "      <td>-0.008603</td>\n",
       "      <td>3.184690e-01</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>sodium</th>\n",
       "      <td>-0.980215</td>\n",
       "      <td>0.140896</td>\n",
       "      <td>-0.135902</td>\n",
       "      <td>-0.000968</td>\n",
       "      <td>0.013948</td>\n",
       "      <td>0.022793</td>\n",
       "      <td>0.004508</td>\n",
       "      <td>0.000617</td>\n",
       "      <td>-0.004071</td>\n",
       "      <td>-0.000917</td>\n",
       "      <td>0.000243</td>\n",
       "      <td>0.000233</td>\n",
       "      <td>1.026023e-02</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>fiber</th>\n",
       "      <td>0.005413</td>\n",
       "      <td>0.030681</td>\n",
       "      <td>-0.018191</td>\n",
       "      <td>0.020472</td>\n",
       "      <td>0.013605</td>\n",
       "      <td>0.262841</td>\n",
       "      <td>0.043114</td>\n",
       "      <td>-0.658776</td>\n",
       "      <td>0.226880</td>\n",
       "      <td>-0.144477</td>\n",
       "      <td>0.021072</td>\n",
       "      <td>0.000811</td>\n",
       "      <td>-6.483602e-01</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>carbo</th>\n",
       "      <td>-0.017246</td>\n",
       "      <td>-0.016783</td>\n",
       "      <td>0.017370</td>\n",
       "      <td>0.025948</td>\n",
       "      <td>0.349267</td>\n",
       "      <td>-0.537837</td>\n",
       "      <td>-0.672432</td>\n",
       "      <td>0.006300</td>\n",
       "      <td>-0.025163</td>\n",
       "      <td>-0.300149</td>\n",
       "      <td>-0.041873</td>\n",
       "      <td>0.014147</td>\n",
       "      <td>-2.056936e-01</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>sugars</th>\n",
       "      <td>-0.002989</td>\n",
       "      <td>-0.000253</td>\n",
       "      <td>0.097705</td>\n",
       "      <td>-0.115481</td>\n",
       "      <td>-0.299066</td>\n",
       "      <td>0.647923</td>\n",
       "      <td>-0.566975</td>\n",
       "      <td>0.103466</td>\n",
       "      <td>-0.117133</td>\n",
       "      <td>-0.319664</td>\n",
       "      <td>-0.028131</td>\n",
       "      <td>0.019273</td>\n",
       "      <td>1.364878e-01</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>potass</th>\n",
       "      <td>0.134900</td>\n",
       "      <td>0.986562</td>\n",
       "      <td>0.036782</td>\n",
       "      <td>-0.042176</td>\n",
       "      <td>-0.047151</td>\n",
       "      <td>-0.049999</td>\n",
       "      <td>-0.017959</td>\n",
       "      <td>0.015376</td>\n",
       "      <td>0.001214</td>\n",
       "      <td>0.006339</td>\n",
       "      <td>0.000470</td>\n",
       "      <td>0.000951</td>\n",
       "      <td>6.400483e-03</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>vitamins</th>\n",
       "      <td>-0.094293</td>\n",
       "      <td>0.016729</td>\n",
       "      <td>0.691978</td>\n",
       "      <td>0.714118</td>\n",
       "      <td>-0.037009</td>\n",
       "      <td>0.015757</td>\n",
       "      <td>0.012102</td>\n",
       "      <td>0.004025</td>\n",
       "      <td>0.012422</td>\n",
       "      <td>0.000539</td>\n",
       "      <td>-0.001544</td>\n",
       "      <td>0.000790</td>\n",
       "      <td>9.642514e-03</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>shelf</th>\n",
       "      <td>0.001541</td>\n",
       "      <td>0.004360</td>\n",
       "      <td>0.012489</td>\n",
       "      <td>0.005647</td>\n",
       "      <td>-0.007876</td>\n",
       "      <td>-0.059901</td>\n",
       "      <td>0.092215</td>\n",
       "      <td>-0.327611</td>\n",
       "      <td>-0.934608</td>\n",
       "      <td>-0.046006</td>\n",
       "      <td>0.067769</td>\n",
       "      <td>-0.010328</td>\n",
       "      <td>5.145194e-09</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>weight</th>\n",
       "      <td>-0.000512</td>\n",
       "      <td>0.000999</td>\n",
       "      <td>0.003806</td>\n",
       "      <td>-0.002546</td>\n",
       "      <td>0.003022</td>\n",
       "      <td>0.009052</td>\n",
       "      <td>-0.023613</td>\n",
       "      <td>-0.002816</td>\n",
       "      <td>0.002057</td>\n",
       "      <td>0.005990</td>\n",
       "      <td>-0.092919</td>\n",
       "      <td>-0.995313</td>\n",
       "      <td>9.522241e-08</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>cups</th>\n",
       "      <td>-0.000510</td>\n",
       "      <td>-0.001591</td>\n",
       "      <td>0.000694</td>\n",
       "      <td>0.000985</td>\n",
       "      <td>0.002148</td>\n",
       "      <td>-0.010305</td>\n",
       "      <td>-0.019594</td>\n",
       "      <td>0.062186</td>\n",
       "      <td>0.053596</td>\n",
       "      <td>-0.080385</td>\n",
       "      <td>0.988810</td>\n",
       "      <td>-0.092485</td>\n",
       "      <td>-2.127112e-08</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>rating</th>\n",
       "      <td>0.075296</td>\n",
       "      <td>0.071742</td>\n",
       "      <td>-0.307947</td>\n",
       "      <td>0.334534</td>\n",
       "      <td>0.757708</td>\n",
       "      <td>0.413021</td>\n",
       "      <td>0.018324</td>\n",
       "      <td>0.011991</td>\n",
       "      <td>-0.035610</td>\n",
       "      <td>-0.022702</td>\n",
       "      <td>0.002681</td>\n",
       "      <td>0.003127</td>\n",
       "      <td>1.882863e-01</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "               PC1       PC2       PC3       PC4       PC5       PC6  \\\n",
       "calories -0.077984 -0.009312  0.629206 -0.601021  0.454959  0.118848   \n",
       "protein   0.000757  0.008801  0.001026  0.003200  0.056176  0.112745   \n",
       "fat       0.000102  0.002699  0.016196 -0.025262 -0.016098 -0.131816   \n",
       "sodium   -0.980215  0.140896 -0.135902 -0.000968  0.013948  0.022793   \n",
       "fiber     0.005413  0.030681 -0.018191  0.020472  0.013605  0.262841   \n",
       "carbo    -0.017246 -0.016783  0.017370  0.025948  0.349267 -0.537837   \n",
       "sugars   -0.002989 -0.000253  0.097705 -0.115481 -0.299066  0.647923   \n",
       "potass    0.134900  0.986562  0.036782 -0.042176 -0.047151 -0.049999   \n",
       "vitamins -0.094293  0.016729  0.691978  0.714118 -0.037009  0.015757   \n",
       "shelf     0.001541  0.004360  0.012489  0.005647 -0.007876 -0.059901   \n",
       "weight   -0.000512  0.000999  0.003806 -0.002546  0.003022  0.009052   \n",
       "cups     -0.000510 -0.001591  0.000694  0.000985  0.002148 -0.010305   \n",
       "rating    0.075296  0.071742 -0.307947  0.334534  0.757708  0.413021   \n",
       "\n",
       "               PC7       PC8       PC9      PC10      PC11      PC12  \\\n",
       "calories  0.093857 -0.026191  0.009027  0.064517  0.008975  0.003853   \n",
       "protein   0.258103  0.655037 -0.202406 -0.255922 -0.045273 -0.004508   \n",
       "fat       0.372584 -0.117932  0.123749 -0.841440 -0.062438 -0.008603   \n",
       "sodium    0.004508  0.000617 -0.004071 -0.000917  0.000243  0.000233   \n",
       "fiber     0.043114 -0.658776  0.226880 -0.144477  0.021072  0.000811   \n",
       "carbo    -0.672432  0.006300 -0.025163 -0.300149 -0.041873  0.014147   \n",
       "sugars   -0.566975  0.103466 -0.117133 -0.319664 -0.028131  0.019273   \n",
       "potass   -0.017959  0.015376  0.001214  0.006339  0.000470  0.000951   \n",
       "vitamins  0.012102  0.004025  0.012422  0.000539 -0.001544  0.000790   \n",
       "shelf     0.092215 -0.327611 -0.934608 -0.046006  0.067769 -0.010328   \n",
       "weight   -0.023613 -0.002816  0.002057  0.005990 -0.092919 -0.995313   \n",
       "cups     -0.019594  0.062186  0.053596 -0.080385  0.988810 -0.092485   \n",
       "rating    0.018324  0.011991 -0.035610 -0.022702  0.002681  0.003127   \n",
       "\n",
       "                  PC13  \n",
       "calories  4.193592e-02  \n",
       "protein  -6.162939e-01  \n",
       "fat       3.184690e-01  \n",
       "sodium    1.026023e-02  \n",
       "fiber    -6.483602e-01  \n",
       "carbo    -2.056936e-01  \n",
       "sugars    1.364878e-01  \n",
       "potass    6.400483e-03  \n",
       "vitamins  9.642514e-03  \n",
       "shelf     5.145194e-09  \n",
       "weight    9.522241e-08  \n",
       "cups     -2.127112e-08  \n",
       "rating    1.882863e-01  "
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pcsComponents_dfX = pd.DataFrame(pcs.components_.transpose(), \n",
    "                                columns = pcsSummary_df.columns,\n",
    "                                index = cereals_df.iloc[:, 3:].columns\n",
    "                                )\n",
    "pcsComponents_dfX"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sodium      0.980215\n",
       "potass      0.134900\n",
       "vitamins    0.094293\n",
       "Name: PC1, dtype: float64"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pcsComponents_dfX.PC1.abs().nlargest(3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here, it is clear that: \n",
    "* the first principal component is dominated by the sodium content of the cereal: \n",
    "  * it has the highest (in this case, positive) weight (-0.980215). \n",
    "  * This means that the first principal component is in fact measuring how much sodium is in the cereal. \n",
    "  \n",
    "* The second principal component seems to be measuring the amount of potassium. \n",
    "\n",
    "\n",
    "**Since both these variables are measured in `milligrams`, whereas the other nutrients are measured in `grams`, the scale is obviously leading to this result.** \n",
    "\n",
    "The variances of potassium and sodium are much larger than the variances of the other variables, and thus the total variance is dominated by these two variances. \n",
    "\n",
    "**Normalize before applying PCA!!!**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* Normalize and then performing PCA on the correlation matrix instead of the covariance matrix. \n",
    "* Using the correlation matrix means that you are operating on the normalized data. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "id": "Gd_UIdk2budY"
   },
   "outputs": [],
   "source": [
    "from sklearn import preprocessing"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "L8DAaQrQfmhC"
   },
   "source": [
    "#### Normalize the 13 variables due to the different scales of the variables and then perform PCA (or equivalently, we use PCA applied to the correlation matrix)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>calories</th>\n",
       "      <th>protein</th>\n",
       "      <th>fat</th>\n",
       "      <th>sodium</th>\n",
       "      <th>fiber</th>\n",
       "      <th>carbo</th>\n",
       "      <th>sugars</th>\n",
       "      <th>potass</th>\n",
       "      <th>vitamins</th>\n",
       "      <th>shelf</th>\n",
       "      <th>weight</th>\n",
       "      <th>cups</th>\n",
       "      <th>rating</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>70</td>\n",
       "      <td>4</td>\n",
       "      <td>1</td>\n",
       "      <td>130</td>\n",
       "      <td>10.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>6.0</td>\n",
       "      <td>280.0</td>\n",
       "      <td>25</td>\n",
       "      <td>3</td>\n",
       "      <td>1.0</td>\n",
       "      <td>0.33</td>\n",
       "      <td>68.402973</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>120</td>\n",
       "      <td>3</td>\n",
       "      <td>5</td>\n",
       "      <td>15</td>\n",
       "      <td>2.0</td>\n",
       "      <td>8.0</td>\n",
       "      <td>8.0</td>\n",
       "      <td>135.0</td>\n",
       "      <td>0</td>\n",
       "      <td>3</td>\n",
       "      <td>1.0</td>\n",
       "      <td>1.00</td>\n",
       "      <td>33.983679</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>70</td>\n",
       "      <td>4</td>\n",
       "      <td>1</td>\n",
       "      <td>260</td>\n",
       "      <td>9.0</td>\n",
       "      <td>7.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>320.0</td>\n",
       "      <td>25</td>\n",
       "      <td>3</td>\n",
       "      <td>1.0</td>\n",
       "      <td>0.33</td>\n",
       "      <td>59.425505</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>50</td>\n",
       "      <td>4</td>\n",
       "      <td>0</td>\n",
       "      <td>140</td>\n",
       "      <td>14.0</td>\n",
       "      <td>8.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>330.0</td>\n",
       "      <td>25</td>\n",
       "      <td>3</td>\n",
       "      <td>1.0</td>\n",
       "      <td>0.50</td>\n",
       "      <td>93.704912</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>110</td>\n",
       "      <td>2</td>\n",
       "      <td>2</td>\n",
       "      <td>200</td>\n",
       "      <td>1.0</td>\n",
       "      <td>14.0</td>\n",
       "      <td>8.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>25</td>\n",
       "      <td>3</td>\n",
       "      <td>1.0</td>\n",
       "      <td>0.75</td>\n",
       "      <td>34.384843</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   calories  protein  fat  sodium  fiber  carbo  sugars  potass  vitamins  \\\n",
       "0        70        4    1     130   10.0    5.0     6.0   280.0        25   \n",
       "1       120        3    5      15    2.0    8.0     8.0   135.0         0   \n",
       "2        70        4    1     260    9.0    7.0     5.0   320.0        25   \n",
       "3        50        4    0     140   14.0    8.0     0.0   330.0        25   \n",
       "4       110        2    2     200    1.0   14.0     8.0     NaN        25   \n",
       "\n",
       "   shelf  weight  cups     rating  \n",
       "0      3     1.0  0.33  68.402973  \n",
       "1      3     1.0  1.00  33.983679  \n",
       "2      3     1.0  0.33  59.425505  \n",
       "3      3     1.0  0.50  93.704912  \n",
       "4      3     1.0  0.75  34.384843  "
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cereals_df.iloc[:, 3:].head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 140
    },
    "id": "HY8QEOSqQWBm",
    "outputId": "1ab0efd4-0821-4edc-84b6-3b9bbb502cde"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "PCA()"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pcs = PCA()\n",
    "pcs.fit(preprocessing.scale(cereals_df.iloc[:, 3:].dropna(axis=0))) \n",
    "                # Standardize a dataset along any axis.\n",
    "                # Center to the mean and component wise scale to unit variance."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>PC1</th>\n",
       "      <th>PC2</th>\n",
       "      <th>PC3</th>\n",
       "      <th>PC4</th>\n",
       "      <th>PC5</th>\n",
       "      <th>PC6</th>\n",
       "      <th>PC7</th>\n",
       "      <th>PC8</th>\n",
       "      <th>PC9</th>\n",
       "      <th>PC10</th>\n",
       "      <th>PC11</th>\n",
       "      <th>PC12</th>\n",
       "      <th>PC13</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>Standard deviation</th>\n",
       "      <td>1.9192</td>\n",
       "      <td>1.7864</td>\n",
       "      <td>1.3912</td>\n",
       "      <td>1.0166</td>\n",
       "      <td>1.0015</td>\n",
       "      <td>0.8555</td>\n",
       "      <td>0.8251</td>\n",
       "      <td>0.6496</td>\n",
       "      <td>0.5658</td>\n",
       "      <td>0.3051</td>\n",
       "      <td>0.2537</td>\n",
       "      <td>0.1399</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Proportion of variance</th>\n",
       "      <td>0.2795</td>\n",
       "      <td>0.2422</td>\n",
       "      <td>0.1469</td>\n",
       "      <td>0.0784</td>\n",
       "      <td>0.0761</td>\n",
       "      <td>0.0555</td>\n",
       "      <td>0.0517</td>\n",
       "      <td>0.0320</td>\n",
       "      <td>0.0243</td>\n",
       "      <td>0.0071</td>\n",
       "      <td>0.0049</td>\n",
       "      <td>0.0015</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Cumulative proportion</th>\n",
       "      <td>0.2795</td>\n",
       "      <td>0.5217</td>\n",
       "      <td>0.6685</td>\n",
       "      <td>0.7470</td>\n",
       "      <td>0.8231</td>\n",
       "      <td>0.8786</td>\n",
       "      <td>0.9303</td>\n",
       "      <td>0.9623</td>\n",
       "      <td>0.9866</td>\n",
       "      <td>0.9936</td>\n",
       "      <td>0.9985</td>\n",
       "      <td>1.0000</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                           PC1     PC2     PC3     PC4     PC5     PC6  \\\n",
       "Standard deviation      1.9192  1.7864  1.3912  1.0166  1.0015  0.8555   \n",
       "Proportion of variance  0.2795  0.2422  0.1469  0.0784  0.0761  0.0555   \n",
       "Cumulative proportion   0.2795  0.5217  0.6685  0.7470  0.8231  0.8786   \n",
       "\n",
       "                           PC7     PC8     PC9    PC10    PC11    PC12  PC13  \n",
       "Standard deviation      0.8251  0.6496  0.5658  0.3051  0.2537  0.1399   0.0  \n",
       "Proportion of variance  0.0517  0.0320  0.0243  0.0071  0.0049  0.0015   0.0  \n",
       "Cumulative proportion   0.9303  0.9623  0.9866  0.9936  0.9985  1.0000   1.0  "
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pcsSummary_df = pd.DataFrame({'Standard deviation': np.sqrt(pcs.explained_variance_),\n",
    "                             'Proportion of variance': pcs.explained_variance_ratio_,\n",
    "                             'Cumulative proportion': np.cumsum(pcs.explained_variance_ratio_)\n",
    "                             })\n",
    "\n",
    "pcsSummary_df = pcsSummary_df.transpose()\n",
    "pcsSummary_df.columns = ['PC{}'.format(i) for i in range(1, len(pcsSummary_df.columns) + 1)]\n",
    "pcsSummary_df.round(4) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.9302633875160602"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pcsSummary_df.iloc[2,6]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* Now we find that we need `seven principal components (PC1 - PC7)` to account for more than `90%` of the total variability. \n",
    "* The first two principal components account for only 52% of the total variability, and thus reducing the number of variables to two would mean losing a lot of information. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 444
    },
    "id": "iZD7YxMOQnFA",
    "outputId": "9f352f51-922f-4582-d2c3-56081b892d7b"
   },
   "outputs": [
    {
     "data": {
      "text/html": [
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       "<style scoped>\n",
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       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>PC1</th>\n",
       "      <th>PC2</th>\n",
       "      <th>PC3</th>\n",
       "      <th>PC4</th>\n",
       "      <th>PC5</th>\n",
       "      <th>PC6</th>\n",
       "      <th>PC7</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>calories</th>\n",
       "      <td>0.299542</td>\n",
       "      <td>0.393148</td>\n",
       "      <td>0.114857</td>\n",
       "      <td>0.204359</td>\n",
       "      <td>0.203899</td>\n",
       "      <td>0.255906</td>\n",
       "      <td>0.025595</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>protein</th>\n",
       "      <td>0.307356</td>\n",
       "      <td>0.165323</td>\n",
       "      <td>0.277282</td>\n",
       "      <td>0.300743</td>\n",
       "      <td>0.319749</td>\n",
       "      <td>0.120752</td>\n",
       "      <td>0.282705</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>fat</th>\n",
       "      <td>0.039915</td>\n",
       "      <td>0.345724</td>\n",
       "      <td>0.204890</td>\n",
       "      <td>0.186833</td>\n",
       "      <td>0.586893</td>\n",
       "      <td>0.347967</td>\n",
       "      <td>0.051155</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>sodium</th>\n",
       "      <td>0.183397</td>\n",
       "      <td>0.137221</td>\n",
       "      <td>0.389431</td>\n",
       "      <td>0.120337</td>\n",
       "      <td>0.338364</td>\n",
       "      <td>0.664372</td>\n",
       "      <td>0.283703</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>fiber</th>\n",
       "      <td>0.453490</td>\n",
       "      <td>0.179812</td>\n",
       "      <td>0.069766</td>\n",
       "      <td>0.039174</td>\n",
       "      <td>0.255119</td>\n",
       "      <td>0.064244</td>\n",
       "      <td>0.112325</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>carbo</th>\n",
       "      <td>0.192449</td>\n",
       "      <td>0.149448</td>\n",
       "      <td>0.562452</td>\n",
       "      <td>0.087835</td>\n",
       "      <td>0.182743</td>\n",
       "      <td>0.326393</td>\n",
       "      <td>0.260468</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>sugars</th>\n",
       "      <td>0.228068</td>\n",
       "      <td>0.351434</td>\n",
       "      <td>0.355405</td>\n",
       "      <td>0.022707</td>\n",
       "      <td>0.314872</td>\n",
       "      <td>0.152082</td>\n",
       "      <td>0.227985</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>potass</th>\n",
       "      <td>0.401964</td>\n",
       "      <td>0.300544</td>\n",
       "      <td>0.067620</td>\n",
       "      <td>0.090878</td>\n",
       "      <td>0.148360</td>\n",
       "      <td>0.025154</td>\n",
       "      <td>0.148808</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>vitamins</th>\n",
       "      <td>0.115980</td>\n",
       "      <td>0.172909</td>\n",
       "      <td>0.387859</td>\n",
       "      <td>0.604111</td>\n",
       "      <td>0.049287</td>\n",
       "      <td>0.129486</td>\n",
       "      <td>0.294276</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>shelf</th>\n",
       "      <td>0.171263</td>\n",
       "      <td>0.265050</td>\n",
       "      <td>0.001531</td>\n",
       "      <td>0.638879</td>\n",
       "      <td>0.329101</td>\n",
       "      <td>0.052044</td>\n",
       "      <td>0.174834</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>weight</th>\n",
       "      <td>0.050299</td>\n",
       "      <td>0.450309</td>\n",
       "      <td>0.247138</td>\n",
       "      <td>0.153429</td>\n",
       "      <td>0.221283</td>\n",
       "      <td>0.398774</td>\n",
       "      <td>0.013921</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>cups</th>\n",
       "      <td>0.294636</td>\n",
       "      <td>0.212248</td>\n",
       "      <td>0.140000</td>\n",
       "      <td>0.047489</td>\n",
       "      <td>0.120816</td>\n",
       "      <td>0.099461</td>\n",
       "      <td>0.748567</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>rating</th>\n",
       "      <td>0.438378</td>\n",
       "      <td>0.251539</td>\n",
       "      <td>0.181842</td>\n",
       "      <td>0.038316</td>\n",
       "      <td>0.057584</td>\n",
       "      <td>0.186145</td>\n",
       "      <td>0.063445</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "               PC1       PC2       PC3       PC4       PC5       PC6       PC7\n",
       "calories  0.299542  0.393148  0.114857  0.204359  0.203899  0.255906  0.025595\n",
       "protein   0.307356  0.165323  0.277282  0.300743  0.319749  0.120752  0.282705\n",
       "fat       0.039915  0.345724  0.204890  0.186833  0.586893  0.347967  0.051155\n",
       "sodium    0.183397  0.137221  0.389431  0.120337  0.338364  0.664372  0.283703\n",
       "fiber     0.453490  0.179812  0.069766  0.039174  0.255119  0.064244  0.112325\n",
       "carbo     0.192449  0.149448  0.562452  0.087835  0.182743  0.326393  0.260468\n",
       "sugars    0.228068  0.351434  0.355405  0.022707  0.314872  0.152082  0.227985\n",
       "potass    0.401964  0.300544  0.067620  0.090878  0.148360  0.025154  0.148808\n",
       "vitamins  0.115980  0.172909  0.387859  0.604111  0.049287  0.129486  0.294276\n",
       "shelf     0.171263  0.265050  0.001531  0.638879  0.329101  0.052044  0.174834\n",
       "weight    0.050299  0.450309  0.247138  0.153429  0.221283  0.398774  0.013921\n",
       "cups      0.294636  0.212248  0.140000  0.047489  0.120816  0.099461  0.748567\n",
       "rating    0.438378  0.251539  0.181842  0.038316  0.057584  0.186145  0.063445"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pcsComponents_df = pd.DataFrame(pcs.components_.transpose(), \n",
    "                                columns=pcsSummary_df.columns,\n",
    "                                index=cereals_df.iloc[:, 3:].columns\n",
    "                                ).abs()\n",
    "pcsComponents_df.iloc[:,:7]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [],
   "source": [
    "import seaborn as sns"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "['calories',\n",
       " 'protein',\n",
       " 'fat',\n",
       " 'sodium',\n",
       " 'fiber',\n",
       " 'carbo',\n",
       " 'sugars',\n",
       " 'potass',\n",
       " 'vitamins',\n",
       " 'shelf',\n",
       " 'weight',\n",
       " 'cups',\n",
       " 'rating']"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = pcsComponents_df.index\n",
    "x = x.to_list()\n",
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7fb76d580c70>"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sns.scatterplot(data=pcsComponents_df, x=\"PC1\", y=\"PC2\", hue=pcsComponents_df.index)\n",
    "plt.legend(bbox_to_anchor=(1.02, 1), loc='upper left', borderaxespad=0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'fiber'"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pcsComponents_df.PC1.idxmax()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### In these results: \n",
    "\n",
    "* First principal component (PC1) has large positive associations with `fiber`, `rating`, & `potass`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "fiber     0.453490\n",
       "rating    0.438378\n",
       "potass    0.401964\n",
       "Name: PC1, dtype: float64"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pcsComponents_df.PC1.abs().nlargest(3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "weight      0.450309\n",
       "calories    0.393148\n",
       "sugars      0.351434\n",
       "Name: PC2, dtype: float64"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pcsComponents_df.PC2.abs().nlargest(3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* Decond component (PC2) has large associations with `weight`, `calories`  & `sugars`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "carbo       0.562452\n",
       "sodium      0.389431\n",
       "vitamins    0.387859\n",
       "Name: PC3, dtype: float64"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pcsComponents_df.PC3.abs().nlargest(3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* Third component (PC3) has large associations with `carbo`, `sodium` & `vitamins`."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> * The cereal is low in calories and the amount per bowl (cups)\n",
    "> * The cereal high in protein, and potassium. \n",
    "> * This type of cereal is associated with a high consumer rating. \n",
    "\n",
    "> The second principal component is most affected by the weight of a serving (cups)\n",
    "\n",
    "> The third principal component by the carbohydrate (carbo) content"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<!--NAVIGATION-->\n",
    "< [previous](prev) | [Contents](toc.ipynb) | [next](next.ipynb) >"
   ]
  }
 ],
 "metadata": {
  "colab": {
   "collapsed_sections": [],
   "provenance": []
  },
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.9.5"
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 },
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