SVD_ACP.ipynb

{
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  {
   "cell_type": "markdown",
   "id": "32c3dfdf-f9a5-4410-90e7-b908bdfb1cc7",
   "metadata": {},
   "source": [
    "# Décomposition en valeurs singulières d’une matrice"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a8577f78-15f4-4b7c-be91-8aad450acc57",
   "metadata": {},
   "source": [
    "## I- Installation des bib"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "3181c60d-63fd-418b-ae84-faed0c6d9083",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Requirement already satisfied: numpy in c:\\users\\bydevmar\\appdata\\local\\programs\\python\\python312\\lib\\site-packages (1.26.4)\n",
      "Requirement already satisfied: pandas in c:\\users\\bydevmar\\appdata\\local\\programs\\python\\python312\\lib\\site-packages (2.2.2)\n",
      "Requirement already satisfied: numpy>=1.26.0 in c:\\users\\bydevmar\\appdata\\local\\programs\\python\\python312\\lib\\site-packages (from pandas) (1.26.4)\n",
      "Requirement already satisfied: python-dateutil>=2.8.2 in c:\\users\\bydevmar\\appdata\\local\\programs\\python\\python312\\lib\\site-packages (from pandas) (2.9.0.post0)\n",
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      "Requirement already satisfied: python-dateutil>=2.7 in c:\\users\\bydevmar\\appdata\\local\\programs\\python\\python312\\lib\\site-packages (from matplotlib) (2.9.0.post0)\n",
      "Requirement already satisfied: six>=1.5 in c:\\users\\bydevmar\\appdata\\local\\programs\\python\\python312\\lib\\site-packages (from python-dateutil>=2.7->matplotlib) (1.16.0)\n",
      "Requirement already satisfied: scikit-learn in c:\\users\\bydevmar\\appdata\\local\\programs\\python\\python312\\lib\\site-packages (1.4.2)\n",
      "Requirement already satisfied: numpy>=1.19.5 in c:\\users\\bydevmar\\appdata\\local\\programs\\python\\python312\\lib\\site-packages (from scikit-learn) (1.26.4)\n",
      "Requirement already satisfied: scipy>=1.6.0 in c:\\users\\bydevmar\\appdata\\local\\programs\\python\\python312\\lib\\site-packages (from scikit-learn) (1.13.0)\n",
      "Requirement already satisfied: joblib>=1.2.0 in c:\\users\\bydevmar\\appdata\\local\\programs\\python\\python312\\lib\\site-packages (from scikit-learn) (1.4.0)\n",
      "Requirement already satisfied: threadpoolctl>=2.0.0 in c:\\users\\bydevmar\\appdata\\local\\programs\\python\\python312\\lib\\site-packages (from scikit-learn) (3.4.0)\n"
     ]
    }
   ],
   "source": [
    "!pip install numpy\n",
    "!pip install pandas\n",
    "!pip install scipy\n",
    "!pip install matplotlib\n",
    "!pip install -U scikit-learn"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b1c0eeba-8304-4395-a9b0-19a0977c878a",
   "metadata": {},
   "source": [
    "## II- Importation de bib"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "06f5f9bf-7411-4fd0-9d24-9402b206bbbd",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import scipy\n",
    "import matplotlib.pyplot as plt\n",
    "from sklearn.preprocessing import StandardScaler"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9a9aae67-e256-494c-912a-382dcdcb41e8",
   "metadata": {},
   "source": [
    "## III- Importation des donnees"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "ea0754ed-49d4-4334-a204-708f3cf5cf8e",
   "metadata": {},
   "outputs": [
    {
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Modele</th>\n",
       "      <th>CYL</th>\n",
       "      <th>PUISS</th>\n",
       "      <th>LONG</th>\n",
       "      <th>LARG</th>\n",
       "      <th>POIDS</th>\n",
       "      <th>V.MAX</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>Alfasud TI</td>\n",
       "      <td>1350</td>\n",
       "      <td>79</td>\n",
       "      <td>393</td>\n",
       "      <td>161</td>\n",
       "      <td>870</td>\n",
       "      <td>165</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>Audi 100</td>\n",
       "      <td>1588</td>\n",
       "      <td>85</td>\n",
       "      <td>468</td>\n",
       "      <td>177</td>\n",
       "      <td>1110</td>\n",
       "      <td>160</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>Simca 1300</td>\n",
       "      <td>1294</td>\n",
       "      <td>68</td>\n",
       "      <td>424</td>\n",
       "      <td>168</td>\n",
       "      <td>1050</td>\n",
       "      <td>152</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>Citroen GS Club</td>\n",
       "      <td>1222</td>\n",
       "      <td>59</td>\n",
       "      <td>412</td>\n",
       "      <td>161</td>\n",
       "      <td>930</td>\n",
       "      <td>151</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>Fiat 132</td>\n",
       "      <td>1585</td>\n",
       "      <td>98</td>\n",
       "      <td>439</td>\n",
       "      <td>164</td>\n",
       "      <td>1105</td>\n",
       "      <td>165</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>Lancia Beta</td>\n",
       "      <td>1297</td>\n",
       "      <td>82</td>\n",
       "      <td>429</td>\n",
       "      <td>169</td>\n",
       "      <td>1080</td>\n",
       "      <td>160</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>Peugeot 504</td>\n",
       "      <td>1796</td>\n",
       "      <td>79</td>\n",
       "      <td>449</td>\n",
       "      <td>169</td>\n",
       "      <td>1160</td>\n",
       "      <td>154</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>Renault 16 TL</td>\n",
       "      <td>1565</td>\n",
       "      <td>55</td>\n",
       "      <td>424</td>\n",
       "      <td>163</td>\n",
       "      <td>1010</td>\n",
       "      <td>140</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>Renault 30</td>\n",
       "      <td>2664</td>\n",
       "      <td>128</td>\n",
       "      <td>452</td>\n",
       "      <td>173</td>\n",
       "      <td>1320</td>\n",
       "      <td>180</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>Toyota Corolla</td>\n",
       "      <td>1166</td>\n",
       "      <td>55</td>\n",
       "      <td>399</td>\n",
       "      <td>157</td>\n",
       "      <td>815</td>\n",
       "      <td>140</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>Alfetta 1.66</td>\n",
       "      <td>1570</td>\n",
       "      <td>109</td>\n",
       "      <td>428</td>\n",
       "      <td>162</td>\n",
       "      <td>1060</td>\n",
       "      <td>175</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>Princess 1800</td>\n",
       "      <td>1798</td>\n",
       "      <td>82</td>\n",
       "      <td>445</td>\n",
       "      <td>172</td>\n",
       "      <td>1160</td>\n",
       "      <td>158</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>Datsun 200L</td>\n",
       "      <td>1998</td>\n",
       "      <td>115</td>\n",
       "      <td>469</td>\n",
       "      <td>169</td>\n",
       "      <td>1370</td>\n",
       "      <td>160</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>Taunus 2000</td>\n",
       "      <td>1993</td>\n",
       "      <td>98</td>\n",
       "      <td>438</td>\n",
       "      <td>170</td>\n",
       "      <td>1080</td>\n",
       "      <td>167</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>Rancho</td>\n",
       "      <td>1442</td>\n",
       "      <td>80</td>\n",
       "      <td>431</td>\n",
       "      <td>166</td>\n",
       "      <td>1129</td>\n",
       "      <td>144</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>Mazda 9295</td>\n",
       "      <td>1769</td>\n",
       "      <td>83</td>\n",
       "      <td>440</td>\n",
       "      <td>165</td>\n",
       "      <td>1095</td>\n",
       "      <td>165</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>Opel Rekord</td>\n",
       "      <td>1979</td>\n",
       "      <td>100</td>\n",
       "      <td>459</td>\n",
       "      <td>173</td>\n",
       "      <td>1120</td>\n",
       "      <td>173</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>Lada 1300</td>\n",
       "      <td>1294</td>\n",
       "      <td>68</td>\n",
       "      <td>404</td>\n",
       "      <td>161</td>\n",
       "      <td>955</td>\n",
       "      <td>140</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "             Modele   CYL  PUISS  LONG  LARG  POIDS  V.MAX\n",
       "0        Alfasud TI  1350     79   393   161    870    165\n",
       "1          Audi 100  1588     85   468   177   1110    160\n",
       "2        Simca 1300  1294     68   424   168   1050    152\n",
       "3   Citroen GS Club  1222     59   412   161    930    151\n",
       "4          Fiat 132  1585     98   439   164   1105    165\n",
       "5       Lancia Beta  1297     82   429   169   1080    160\n",
       "6       Peugeot 504  1796     79   449   169   1160    154\n",
       "7     Renault 16 TL  1565     55   424   163   1010    140\n",
       "8        Renault 30  2664    128   452   173   1320    180\n",
       "9    Toyota Corolla  1166     55   399   157    815    140\n",
       "10     Alfetta 1.66  1570    109   428   162   1060    175\n",
       "11    Princess 1800  1798     82   445   172   1160    158\n",
       "12      Datsun 200L  1998    115   469   169   1370    160\n",
       "13      Taunus 2000  1993     98   438   170   1080    167\n",
       "14           Rancho  1442     80   431   166   1129    144\n",
       "15       Mazda 9295  1769     83   440   165   1095    165\n",
       "16      Opel Rekord  1979    100   459   173   1120    173\n",
       "17        Lada 1300  1294     68   404   161    955    140"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df = pd.read_csv(\"autos-acp-diapos.csv\")\n",
    "df"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c6aca57a-3df8-48b9-9943-f8f840259a4a",
   "metadata": {},
   "source": [
    "## IV- Prétraiter les données:"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "aa3add26-2f50-4d34-9f4c-2b97d8d24f7d",
   "metadata": {},
   "source": [
    "### 1- Calculer la moyenne de chaque variable"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "2bb4b29c-26fa-4da3-a23b-632fd2da2b08",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "CYL      1631.666667\n",
       "PUISS      84.611111\n",
       "LONG      433.500000\n",
       "LARG      166.666667\n",
       "POIDS    1078.833333\n",
       "V.MAX     158.277778\n",
       "dtype: float64"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "means = df.drop(columns=[\"Modele\"]).mean()\n",
    "means"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6e01fe84-41ae-4333-9d70-803cd278355f",
   "metadata": {},
   "source": [
    "### 2- Soustraire la moyenne de chaque variable pour centrer les données"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "dc3f3332-b984-4f33-920c-4f0d27714b4f",
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>CYL</th>\n",
       "      <th>PUISS</th>\n",
       "      <th>LONG</th>\n",
       "      <th>LARG</th>\n",
       "      <th>POIDS</th>\n",
       "      <th>V.MAX</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>-281.666667</td>\n",
       "      <td>-5.611111</td>\n",
       "      <td>-40.5</td>\n",
       "      <td>-5.666667</td>\n",
       "      <td>-208.833333</td>\n",
       "      <td>6.722222</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>-43.666667</td>\n",
       "      <td>0.388889</td>\n",
       "      <td>34.5</td>\n",
       "      <td>10.333333</td>\n",
       "      <td>31.166667</td>\n",
       "      <td>1.722222</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>-337.666667</td>\n",
       "      <td>-16.611111</td>\n",
       "      <td>-9.5</td>\n",
       "      <td>1.333333</td>\n",
       "      <td>-28.833333</td>\n",
       "      <td>-6.277778</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>-409.666667</td>\n",
       "      <td>-25.611111</td>\n",
       "      <td>-21.5</td>\n",
       "      <td>-5.666667</td>\n",
       "      <td>-148.833333</td>\n",
       "      <td>-7.277778</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>-46.666667</td>\n",
       "      <td>13.388889</td>\n",
       "      <td>5.5</td>\n",
       "      <td>-2.666667</td>\n",
       "      <td>26.166667</td>\n",
       "      <td>6.722222</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>-334.666667</td>\n",
       "      <td>-2.611111</td>\n",
       "      <td>-4.5</td>\n",
       "      <td>2.333333</td>\n",
       "      <td>1.166667</td>\n",
       "      <td>1.722222</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>164.333333</td>\n",
       "      <td>-5.611111</td>\n",
       "      <td>15.5</td>\n",
       "      <td>2.333333</td>\n",
       "      <td>81.166667</td>\n",
       "      <td>-4.277778</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>-66.666667</td>\n",
       "      <td>-29.611111</td>\n",
       "      <td>-9.5</td>\n",
       "      <td>-3.666667</td>\n",
       "      <td>-68.833333</td>\n",
       "      <td>-18.277778</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>1032.333333</td>\n",
       "      <td>43.388889</td>\n",
       "      <td>18.5</td>\n",
       "      <td>6.333333</td>\n",
       "      <td>241.166667</td>\n",
       "      <td>21.722222</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>-465.666667</td>\n",
       "      <td>-29.611111</td>\n",
       "      <td>-34.5</td>\n",
       "      <td>-9.666667</td>\n",
       "      <td>-263.833333</td>\n",
       "      <td>-18.277778</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>-61.666667</td>\n",
       "      <td>24.388889</td>\n",
       "      <td>-5.5</td>\n",
       "      <td>-4.666667</td>\n",
       "      <td>-18.833333</td>\n",
       "      <td>16.722222</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>166.333333</td>\n",
       "      <td>-2.611111</td>\n",
       "      <td>11.5</td>\n",
       "      <td>5.333333</td>\n",
       "      <td>81.166667</td>\n",
       "      <td>-0.277778</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>366.333333</td>\n",
       "      <td>30.388889</td>\n",
       "      <td>35.5</td>\n",
       "      <td>2.333333</td>\n",
       "      <td>291.166667</td>\n",
       "      <td>1.722222</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>361.333333</td>\n",
       "      <td>13.388889</td>\n",
       "      <td>4.5</td>\n",
       "      <td>3.333333</td>\n",
       "      <td>1.166667</td>\n",
       "      <td>8.722222</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>-189.666667</td>\n",
       "      <td>-4.611111</td>\n",
       "      <td>-2.5</td>\n",
       "      <td>-0.666667</td>\n",
       "      <td>50.166667</td>\n",
       "      <td>-14.277778</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>137.333333</td>\n",
       "      <td>-1.611111</td>\n",
       "      <td>6.5</td>\n",
       "      <td>-1.666667</td>\n",
       "      <td>16.166667</td>\n",
       "      <td>6.722222</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>347.333333</td>\n",
       "      <td>15.388889</td>\n",
       "      <td>25.5</td>\n",
       "      <td>6.333333</td>\n",
       "      <td>41.166667</td>\n",
       "      <td>14.722222</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>-337.666667</td>\n",
       "      <td>-16.611111</td>\n",
       "      <td>-29.5</td>\n",
       "      <td>-5.666667</td>\n",
       "      <td>-123.833333</td>\n",
       "      <td>-18.277778</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "            CYL      PUISS  LONG       LARG       POIDS      V.MAX\n",
       "0   -281.666667  -5.611111 -40.5  -5.666667 -208.833333   6.722222\n",
       "1    -43.666667   0.388889  34.5  10.333333   31.166667   1.722222\n",
       "2   -337.666667 -16.611111  -9.5   1.333333  -28.833333  -6.277778\n",
       "3   -409.666667 -25.611111 -21.5  -5.666667 -148.833333  -7.277778\n",
       "4    -46.666667  13.388889   5.5  -2.666667   26.166667   6.722222\n",
       "5   -334.666667  -2.611111  -4.5   2.333333    1.166667   1.722222\n",
       "6    164.333333  -5.611111  15.5   2.333333   81.166667  -4.277778\n",
       "7    -66.666667 -29.611111  -9.5  -3.666667  -68.833333 -18.277778\n",
       "8   1032.333333  43.388889  18.5   6.333333  241.166667  21.722222\n",
       "9   -465.666667 -29.611111 -34.5  -9.666667 -263.833333 -18.277778\n",
       "10   -61.666667  24.388889  -5.5  -4.666667  -18.833333  16.722222\n",
       "11   166.333333  -2.611111  11.5   5.333333   81.166667  -0.277778\n",
       "12   366.333333  30.388889  35.5   2.333333  291.166667   1.722222\n",
       "13   361.333333  13.388889   4.5   3.333333    1.166667   8.722222\n",
       "14  -189.666667  -4.611111  -2.5  -0.666667   50.166667 -14.277778\n",
       "15   137.333333  -1.611111   6.5  -1.666667   16.166667   6.722222\n",
       "16   347.333333  15.388889  25.5   6.333333   41.166667  14.722222\n",
       "17  -337.666667 -16.611111 -29.5  -5.666667 -123.833333 -18.277778"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "centered_data = df.drop(columns=[\"Modele\"]) - means\n",
    "centered_data"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0976ed51-0272-4e31-9088-f7a77661c0bc",
   "metadata": {},
   "source": [
    "### 3- Calculer l'écart-type de chaque variable"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "b3f493ec-1e37-496c-9337-2a32345c436d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "CYL      373.929846\n",
       "PUISS     20.376281\n",
       "LONG      22.107358\n",
       "LARG       5.313689\n",
       "POIDS    136.957808\n",
       "V.MAX     12.140383\n",
       "dtype: float64"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "std_devs = df.drop(columns=[\"Modele\"]).std()\n",
    "std_devs"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3cdefb17-3cac-409e-97d8-48c7945fd66f",
   "metadata": {},
   "source": [
    "### 4- Diviser chaque variable par son écart-type pour réduire les données"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "fa340787-be5b-49f3-9d1b-03e701a7a0ba",
   "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>CYL</th>\n",
       "      <th>PUISS</th>\n",
       "      <th>LONG</th>\n",
       "      <th>LARG</th>\n",
       "      <th>POIDS</th>\n",
       "      <th>V.MAX</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>-0.753261</td>\n",
       "      <td>-0.275375</td>\n",
       "      <td>-1.831969</td>\n",
       "      <td>-1.066428</td>\n",
       "      <td>-1.524800</td>\n",
       "      <td>0.553708</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>-0.116778</td>\n",
       "      <td>0.019085</td>\n",
       "      <td>1.560566</td>\n",
       "      <td>1.944663</td>\n",
       "      <td>0.227564</td>\n",
       "      <td>0.141859</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>-0.903021</td>\n",
       "      <td>-0.815218</td>\n",
       "      <td>-0.429721</td>\n",
       "      <td>0.250924</td>\n",
       "      <td>-0.210527</td>\n",
       "      <td>-0.517099</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>-1.095571</td>\n",
       "      <td>-1.256908</td>\n",
       "      <td>-0.972527</td>\n",
       "      <td>-1.066428</td>\n",
       "      <td>-1.086709</td>\n",
       "      <td>-0.599469</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>-0.124801</td>\n",
       "      <td>0.657082</td>\n",
       "      <td>0.248786</td>\n",
       "      <td>-0.501848</td>\n",
       "      <td>0.191056</td>\n",
       "      <td>0.553708</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>-0.894999</td>\n",
       "      <td>-0.128145</td>\n",
       "      <td>-0.203552</td>\n",
       "      <td>0.439117</td>\n",
       "      <td>0.008518</td>\n",
       "      <td>0.141859</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>0.439476</td>\n",
       "      <td>-0.275375</td>\n",
       "      <td>0.701124</td>\n",
       "      <td>0.439117</td>\n",
       "      <td>0.592640</td>\n",
       "      <td>-0.352359</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>-0.178287</td>\n",
       "      <td>-1.453215</td>\n",
       "      <td>-0.429721</td>\n",
       "      <td>-0.690042</td>\n",
       "      <td>-0.502588</td>\n",
       "      <td>-1.505536</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>2.760767</td>\n",
       "      <td>2.129382</td>\n",
       "      <td>0.836825</td>\n",
       "      <td>1.191890</td>\n",
       "      <td>1.760883</td>\n",
       "      <td>1.789254</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>-1.245332</td>\n",
       "      <td>-1.453215</td>\n",
       "      <td>-1.560566</td>\n",
       "      <td>-1.819201</td>\n",
       "      <td>-1.926384</td>\n",
       "      <td>-1.505536</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>-0.164915</td>\n",
       "      <td>1.196925</td>\n",
       "      <td>-0.248786</td>\n",
       "      <td>-0.878235</td>\n",
       "      <td>-0.137512</td>\n",
       "      <td>1.377405</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>0.444825</td>\n",
       "      <td>-0.128145</td>\n",
       "      <td>0.520189</td>\n",
       "      <td>1.003697</td>\n",
       "      <td>0.592640</td>\n",
       "      <td>-0.022880</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>0.979685</td>\n",
       "      <td>1.491385</td>\n",
       "      <td>1.605800</td>\n",
       "      <td>0.439117</td>\n",
       "      <td>2.125959</td>\n",
       "      <td>0.141859</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>0.966313</td>\n",
       "      <td>0.657082</td>\n",
       "      <td>0.203552</td>\n",
       "      <td>0.627311</td>\n",
       "      <td>0.008518</td>\n",
       "      <td>0.718447</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>-0.507225</td>\n",
       "      <td>-0.226298</td>\n",
       "      <td>-0.113085</td>\n",
       "      <td>-0.125462</td>\n",
       "      <td>0.366293</td>\n",
       "      <td>-1.176057</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>0.367270</td>\n",
       "      <td>-0.079068</td>\n",
       "      <td>0.294020</td>\n",
       "      <td>-0.313655</td>\n",
       "      <td>0.118041</td>\n",
       "      <td>0.553708</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>0.928873</td>\n",
       "      <td>0.755235</td>\n",
       "      <td>1.153462</td>\n",
       "      <td>1.191890</td>\n",
       "      <td>0.300579</td>\n",
       "      <td>1.212665</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>-0.903021</td>\n",
       "      <td>-0.815218</td>\n",
       "      <td>-1.334397</td>\n",
       "      <td>-1.066428</td>\n",
       "      <td>-0.904171</td>\n",
       "      <td>-1.505536</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "         CYL     PUISS      LONG      LARG     POIDS     V.MAX\n",
       "0  -0.753261 -0.275375 -1.831969 -1.066428 -1.524800  0.553708\n",
       "1  -0.116778  0.019085  1.560566  1.944663  0.227564  0.141859\n",
       "2  -0.903021 -0.815218 -0.429721  0.250924 -0.210527 -0.517099\n",
       "3  -1.095571 -1.256908 -0.972527 -1.066428 -1.086709 -0.599469\n",
       "4  -0.124801  0.657082  0.248786 -0.501848  0.191056  0.553708\n",
       "5  -0.894999 -0.128145 -0.203552  0.439117  0.008518  0.141859\n",
       "6   0.439476 -0.275375  0.701124  0.439117  0.592640 -0.352359\n",
       "7  -0.178287 -1.453215 -0.429721 -0.690042 -0.502588 -1.505536\n",
       "8   2.760767  2.129382  0.836825  1.191890  1.760883  1.789254\n",
       "9  -1.245332 -1.453215 -1.560566 -1.819201 -1.926384 -1.505536\n",
       "10 -0.164915  1.196925 -0.248786 -0.878235 -0.137512  1.377405\n",
       "11  0.444825 -0.128145  0.520189  1.003697  0.592640 -0.022880\n",
       "12  0.979685  1.491385  1.605800  0.439117  2.125959  0.141859\n",
       "13  0.966313  0.657082  0.203552  0.627311  0.008518  0.718447\n",
       "14 -0.507225 -0.226298 -0.113085 -0.125462  0.366293 -1.176057\n",
       "15  0.367270 -0.079068  0.294020 -0.313655  0.118041  0.553708\n",
       "16  0.928873  0.755235  1.153462  1.191890  0.300579  1.212665\n",
       "17 -0.903021 -0.815218 -1.334397 -1.066428 -0.904171 -1.505536"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "scaled_data = centered_data / std_devs\n",
    "scaled_data"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5155d1e6-5358-4c8a-a90f-cf5826f3837e",
   "metadata": {},
   "source": [
    "### 5- Créer la matrice de données Z à partir du DataFrame contenant les données centrées et réduites"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "445604f9-8d04-4546-88ae-7ca64de2177e",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[-0.75326073 -0.27537465 -1.83196922 -1.06642792 -1.52480049  0.55370761]\n",
      " [-0.1167777   0.01908537  1.56056637  1.94466269  0.227564    0.14185897]\n",
      " [-0.90302144 -0.81521802 -0.42972117  0.25092422 -0.21052712 -0.51709884]\n",
      " [-1.09557092 -1.25690805 -0.97252687 -1.06642792 -1.08670937 -0.59946857]\n",
      " [-0.12480059  0.65708208  0.24878594 -0.50184844  0.19105641  0.55370761]\n",
      " [-0.89499854 -0.12814464 -0.20355214  0.43911738  0.00851844  0.14185897]\n",
      " [ 0.43947638 -0.27537465  0.70112402  0.43911738  0.59263994 -0.35235939]\n",
      " [-0.17828656 -1.45321473 -0.42972117 -0.6900416  -0.50258787 -1.50553557]\n",
      " [ 2.76076742  2.12938219  0.83682544  1.19189003  1.76088293  1.78925352]\n",
      " [-1.24533164 -1.45321473 -1.56056637 -1.81920058 -1.92638402 -1.50553557]\n",
      " [-0.16491507  1.19692545 -0.24878594 -0.87823476 -0.13751194  1.37740488]\n",
      " [ 0.44482497 -0.12814464  0.52018879  1.00369687  0.59263994 -0.02288048]\n",
      " [ 0.97968466  1.49138547  1.60580018  0.43911738  2.12595887  0.14185897]\n",
      " [ 0.96631317  0.65708208  0.20355214  0.62731054  0.00851844  0.71844706]\n",
      " [-0.50722527 -0.22629798 -0.11308452 -0.12546211  0.36629286 -1.17605666]\n",
      " [ 0.36727032 -0.07906797  0.29401975 -0.31365527  0.11804122  0.55370761]\n",
      " [ 0.92887299  0.75523542  1.1534621   1.19189003  0.30057919  1.21266543]\n",
      " [-0.90302144 -0.81521802 -1.33439733 -1.06642792 -0.9041714  -1.50553557]]\n"
     ]
    }
   ],
   "source": [
    "Z = scaled_data.values\n",
    "print(Z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "ba4607b3-429f-4c3b-8183-94718cf26c20",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Matrice des valeurs numériques :\n",
      "[[1350   79  393  161  870  165]\n",
      " [1588   85  468  177 1110  160]\n",
      " [1294   68  424  168 1050  152]\n",
      " [1222   59  412  161  930  151]\n",
      " [1585   98  439  164 1105  165]\n",
      " [1297   82  429  169 1080  160]\n",
      " [1796   79  449  169 1160  154]\n",
      " [1565   55  424  163 1010  140]\n",
      " [2664  128  452  173 1320  180]\n",
      " [1166   55  399  157  815  140]\n",
      " [1570  109  428  162 1060  175]\n",
      " [1798   82  445  172 1160  158]\n",
      " [1998  115  469  169 1370  160]\n",
      " [1993   98  438  170 1080  167]\n",
      " [1442   80  431  166 1129  144]\n",
      " [1769   83  440  165 1095  165]\n",
      " [1979  100  459  173 1120  173]\n",
      " [1294   68  404  161  955  140]]\n"
     ]
    }
   ],
   "source": [
    "# Exclure la colonne 'Modele' (car elle contient des chaînes de caractères)\n",
    "M = df.drop(columns=['Modele']).values\n",
    "\n",
    "# Afficher la matrice\n",
    "print(\"Matrice des valeurs numériques :\")\n",
    "print(M)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "1ea486ee-ac8e-4be5-80d9-c305bc20cf23",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[-0.77509889 -0.28335818 -1.88508077 -1.09734528 -1.56900676  0.56976043]\n",
      " [-0.12016326  0.01963869  1.60580955  2.0010414   0.23416142  0.14597168]\n",
      " [-0.92920139 -0.83885242 -0.44217944  0.25819889 -0.21663062 -0.53209032]\n",
      " [-1.12733318 -1.29334771 -1.00072189 -1.09734528 -1.11821472 -0.61684807]\n",
      " [-0.12841875  0.67613189  0.25599862 -0.51639778  0.19659542  0.56976043]\n",
      " [-0.9209459  -0.13185975 -0.20945342  0.45184806  0.0087654   0.14597168]\n",
      " [ 0.45221746 -0.28335818  0.72145067  0.45184806  0.60982146 -0.36257482]\n",
      " [-0.18345536 -1.49534562 -0.44217944 -0.71004695 -0.51715865 -1.54918332]\n",
      " [ 2.84080623  2.19111619  0.86108628  1.22644473  1.81193359  1.84112668]\n",
      " [-1.28143568 -1.49534562 -1.60580955 -1.87194195 -1.98223281 -1.54918332]\n",
      " [-0.16969621  1.23162613 -0.25599862 -0.90369611 -0.14149861  1.41733793]\n",
      " [ 0.45772112 -0.13185975  0.53526985  1.03279556  0.60982146 -0.02354382]\n",
      " [ 1.0080872   1.53462299  1.65235475  0.45184806  2.18759363  0.14597168]\n",
      " [ 0.99432805  0.67613189  0.20945342  0.64549722  0.0087654   0.73927593]\n",
      " [-0.5219305  -0.2328587  -0.11636301 -0.12909944  0.37691224 -1.21015232]\n",
      " [ 0.37791804 -0.08136027  0.30254383 -0.32274861  0.12146341  0.56976043]\n",
      " [ 0.95580242  0.77713084  1.18690271  1.22644473  0.30929343  1.24782243]\n",
      " [-0.92920139 -0.83885242 -1.37308353 -1.09734528 -0.9303847  -1.54918332]]\n"
     ]
    }
   ],
   "source": [
    "#instanciation\n",
    "sc = StandardScaler()\n",
    "#transformation – centrage-réduction\n",
    "Z = sc.fit_transform(M)\n",
    "print(Z)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6cc4e486-9f76-4b92-96e7-c00936cf39e9",
   "metadata": {},
   "source": [
    "## V- Effectuer la décomposition en valeurs singulières (SVD):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "342c4104-beb6-4929-90b6-c1324e9fe8ff",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Effectuer la SVD\n",
    "Us, delta, Vs = np.linalg.svd(Z)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "29bab4de-adfc-431b-bc5b-886ed6a1717a",
   "metadata": {},
   "source": [
    "## VI-Interpréter les résultats:"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7348d52e-0944-4777-8b43-0ab942965bf9",
   "metadata": {},
   "source": [
    "### 1- Les valeurs singulières (delta) représentent la variance restituée par chaque axe"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "b7409624-288d-472c-990b-9698eb0002d1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "18"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n = df.shape[0]  # Nombre de lignes dans le DataFrame\n",
    "n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "057f4551-30b0-418b-b673-ff52df769c83",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "les valeurs singulières : \n",
      " [8.920507   3.92544535 2.59136825 1.96229396 1.29244799 0.88273717]\n",
      "les variances restituées par les axes : \n",
      " [4.42085806 0.85606229 0.37306608 0.21392209 0.09280121 0.04329027]\n"
     ]
    }
   ],
   "source": [
    "print(\"les valeurs singulières : \\n\",delta)  # Afficher les valeurs singulières\n",
    "print(\"les variances restituées par les axes : \\n\" , (delta**2) / n)  # Calculer les variances restituées par les axes"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9ac49632-3afe-4893-833a-df7aa96f26d1",
   "metadata": {},
   "source": [
    "### 2-Les matrices Us et Vs contiennent les vecteurs singuliers gauche et droit, respectivement. Elles peuvent être utilisées pour calculer les coordonnées factorielles des individus et des variables."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "259462f5-0a0b-4e5e-b750-919a565b9c14",
   "metadata": {},
   "source": [
    "Les \"coordonnées des variables\" font référence aux relations entre les variables originales et les axes principaux dans l'espace factoriel de l'Analyse en Composantes Principales (ACP). La matrice V, de taille (p, p), contient ces relations sous forme de vecteurs orthonormés appelés vecteurs propres. En d'autres termes, chaque colonne de la matrice V représente un vecteur propre qui définit la direction et l'importance de chaque variable dans l'espace des composantes principales.\n",
    "\n",
    "Dans votre exemple, la matrice V est transposée (numpy.transpose(Vs)), ce qui signifie que chaque ligne représente un vecteur propre. Ces vecteurs propres ont été obtenus à partir de la décomposition en valeurs singulières (SVD) de la matrice des données centrées et réduites. Plus précisément, ils correspondent aux vecteurs propres de la matrice de covariance des variables, soit la matrice (𝑍𝑇𝑍). Ces vecteurs propres fournissent les coordonnées des variables dans l'espace des composantes principales, ce qui permet de visualiser la contribution de chaque variable à chaque composante principale.\n",
    "\n",
    "La vérification est effectuée en comparant ces vecteurs propres avec ceux obtenus directement à partir de la matrice de covariance des variables (Z'Z). La correspondance entre les deux jeux de vecteurs propres confirme la validité des résultats de l'ACP."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "87868e4f-4f8d-4697-a5cc-8a305c71521e",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "US : \n",
      " [[-2.39776014e-01 -4.54898964e-01 -2.20679667e-01 -1.02903716e-01\n",
      "   2.33167573e-01 -6.10836321e-02  3.09320468e-01  2.05229523e-01\n",
      "   1.68064146e-01 -2.34085031e-01 -2.04159172e-01  1.40463939e-01\n",
      "   3.95358988e-01 -3.49834904e-02 -3.93461452e-02  2.41500697e-01\n",
      "   1.73380329e-01 -2.97443451e-01]\n",
      " [ 1.75041462e-01  3.89010707e-01 -5.07558644e-01  1.07706359e-01\n",
      "  -1.14910727e-01  3.70708800e-01  9.00069412e-02  2.52945639e-01\n",
      "   4.15412390e-01 -5.82343935e-02 -1.13344927e-01  9.80924454e-02\n",
      "  -3.67023008e-02 -6.16059652e-02 -1.64282461e-02  1.86511819e-01\n",
      "  -2.91114820e-01  5.22288034e-02]\n",
      " [-1.25484491e-01  1.71828955e-01 -1.76195762e-01  8.54234098e-02\n",
      "   2.90428694e-01 -3.07909582e-01 -1.60090566e-01 -1.59533442e-01\n",
      "   4.31207959e-01  1.17699608e-01  4.68094776e-01 -2.49283009e-01\n",
      "   3.92400010e-04  2.38261304e-01 -2.89987077e-01  1.58675754e-01\n",
      "   1.74571429e-01 -9.26648578e-02]\n",
      " [-2.88519667e-01 -2.87570330e-02 -5.73325680e-02  8.83823928e-03\n",
      "  -1.75522123e-01 -2.98476422e-01 -8.76367739e-02 -1.66573379e-01\n",
      "   4.51240076e-01 -2.04602737e-01 -3.06720387e-01  2.05261738e-01\n",
      "  -6.36637300e-02  2.56655193e-01  2.21791413e-01 -4.45224372e-01\n",
      "  -1.87839261e-02  2.40536033e-01]\n",
      " [ 4.79631036e-02 -1.77194497e-01  7.45882699e-02  3.19908481e-01\n",
      "  -2.03880783e-01  4.20880135e-02  7.68573138e-02  1.94514477e-01\n",
      "   3.92234597e-02  1.46646311e-01 -2.73690374e-01  1.11175235e-01\n",
      "  -5.51273335e-01  2.71092199e-01 -3.41644122e-01  1.20516352e-01\n",
      "   3.59011882e-01 -1.53223457e-01]\n",
      " [-3.41054208e-02  4.99685428e-02 -2.60790093e-01  2.83311969e-01\n",
      "   3.44374354e-01 -2.26662021e-01  8.72311429e-02  3.34941770e-01\n",
      "  -2.25937549e-01  5.07047647e-01 -2.28713927e-01 -1.32422544e-01\n",
      "   1.44426627e-01  9.54567968e-02  1.74055075e-01 -1.45156167e-01\n",
      "   3.46615527e-02  3.12680914e-01]\n",
      " [ 7.66692405e-02  2.37694499e-01  9.91071351e-02 -1.03518787e-01\n",
      "  -1.61439107e-01 -1.74334736e-01  8.93076033e-01 -1.33745181e-01\n",
      "   2.54503084e-02  4.70851144e-02  1.34989147e-01 -8.00810448e-02\n",
      "  -2.24318776e-02  7.48221721e-02 -1.01039130e-02 -8.54570210e-02\n",
      "   4.30405731e-02  4.79398649e-02]\n",
      " [-2.18428456e-01  2.49767341e-01  2.39091792e-01 -3.21215460e-01\n",
      "  -2.26829341e-01 -1.23073773e-01 -1.08731611e-01  7.60768330e-01\n",
      "   2.88988001e-02 -7.64155433e-02  1.81581005e-01 -6.17772880e-02\n",
      "   5.42077660e-02  6.78034003e-02 -1.38207352e-02 -9.31084494e-02\n",
      "   1.04945381e-01 -3.87198618e-02]\n",
      " [ 4.94336765e-01 -2.70958598e-01  2.29040554e-01 -4.31763946e-01\n",
      "   2.90061705e-01 -4.98410765e-02  9.48786229e-04  1.18853453e-01\n",
      "   4.40151369e-01  2.48134523e-01 -1.00900592e-01 -3.57818208e-02\n",
      "  -1.94270080e-01 -1.36615029e-01  5.89393123e-02 -4.28959156e-02\n",
      "  -5.07889096e-02  1.07344159e-01]\n",
      " [-4.46811198e-01 -6.01818128e-02  1.16978110e-01 -1.35108093e-01\n",
      "  -2.15427203e-01  3.72582014e-01  1.55505657e-02 -1.51317357e-01\n",
      "   2.05191065e-01  6.79214434e-01  2.62295296e-02  1.02818160e-01\n",
      "   1.53546530e-01 -3.92248264e-02  3.57501454e-02 -1.72400230e-03\n",
      "  -2.47899988e-02 -1.21362816e-01]\n",
      " [ 4.90620304e-02 -4.87192660e-01 -9.62512358e-03  3.86746550e-01\n",
      "  -1.30065323e-01  6.14460873e-02  1.04019809e-01  2.16429914e-01\n",
      "   1.04730832e-01 -1.90664502e-02  6.13364161e-01  1.82592445e-01\n",
      "  -4.70692986e-02 -1.57221034e-02  1.77403084e-01 -1.16551636e-01\n",
      "  -1.79397101e-01  1.77837628e-01]\n",
      " [ 1.14138735e-01  2.14424612e-01 -8.35891291e-02 -1.54628420e-01\n",
      "   1.43043113e-01 -2.09454829e-01 -4.52672728e-02 -3.18554219e-02\n",
      "  -1.54083757e-01  1.44169883e-01  1.61682069e-01  8.69741639e-01\n",
      "  -5.25608553e-04  1.07297905e-02 -7.24690508e-02  4.55535568e-02\n",
      "   8.70360529e-02 -2.37666628e-02]\n",
      " [ 3.29698785e-01  1.42448701e-01  4.80050708e-01  3.93504277e-01\n",
      "  -4.21000532e-02  6.48694323e-02 -4.96887411e-02  1.34240986e-02\n",
      "   2.06870305e-01 -7.20924624e-03 -1.25295590e-01  1.00377349e-01\n",
      "   5.96248356e-01  1.64393616e-01 -1.48476717e-01 -1.82562062e-02\n",
      "   5.94116052e-02  3.94033481e-03]\n",
      " [ 1.47399737e-01 -1.23940703e-01 -1.09062616e-01 -2.96706939e-01\n",
      "   5.16343180e-02  2.86712478e-01 -6.11644786e-03 -1.91123650e-02\n",
      "  -1.76301721e-01 -2.13648674e-02  6.72192927e-02 -2.37952732e-02\n",
      "   8.39640503e-02  8.42641438e-01  7.88950166e-02  1.74911164e-02\n",
      "  -1.31172588e-01  1.59671118e-02]\n",
      " [-7.74744497e-02  2.28692878e-01  2.42500036e-01  1.82305349e-01\n",
      "   2.91761007e-01  1.37675821e-01  4.10710548e-02  7.59051243e-03\n",
      "   1.01353540e-01 -8.02915311e-02  3.47468876e-02  4.21860949e-02\n",
      "  -2.09394262e-01  8.16518924e-02  7.21490977e-01  2.14637859e-01\n",
      "   2.04620844e-01 -2.55195228e-01]\n",
      " [ 4.32384467e-02 -9.07373662e-02  2.91672128e-02 -5.24383448e-02\n",
      "  -4.07786626e-01 -3.83842391e-01 -8.56176425e-02 -5.24186948e-02\n",
      "   4.00661800e-03  1.05315406e-01 -7.41885585e-02 -2.73517289e-02\n",
      "   7.12196674e-02  6.47378223e-02  2.21335360e-01  7.10151375e-01\n",
      "  -9.23688306e-02  2.72276800e-01]\n",
      " [ 2.56685886e-01 -2.65816068e-02 -3.07321732e-01 -1.20444483e-01\n",
      "  -2.61876011e-01  1.77494594e-01 -8.51935012e-02 -7.55984979e-02\n",
      "   1.99109654e-02  4.49579971e-03  1.16460979e-01 -6.59357773e-02\n",
      "   1.72153522e-01 -1.02758184e-01  1.88694401e-01 -1.14562203e-01\n",
      "   7.55875905e-01  1.81090938e-01]\n",
      " [-3.03634494e-01  3.66070046e-02  2.21631517e-01 -4.90164443e-02\n",
      "   2.95366533e-01  3.21101223e-01  8.99623718e-02 -8.82736907e-03\n",
      "   2.12214955e-02 -1.93185134e-01  1.24162353e-02  8.41945044e-02\n",
      "  -7.37406440e-02 -2.21405322e-02 -2.06638915e-01  2.24211673e-01\n",
      "   1.61145003e-01  6.97527423e-01]]\n"
     ]
    }
   ],
   "source": [
    "print(\"US : \\n\" ,Us)  # Matrice des vecteurs singuliers gauche"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "97532f99-25e0-4022-885e-e6f08aedc309",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.42493602 -0.12419108  0.35361252 -0.80778648 -0.15158003  0.05889517]\n",
      " [ 0.42179441 -0.41577389  0.18492049  0.35779199  0.29373465  0.63303302]\n",
      " [ 0.42145993  0.41181773 -0.06763394  0.27975231 -0.73056903  0.19029153]\n",
      " [ 0.38692224  0.446087   -0.60486812 -0.21156941  0.47819008  0.10956624]\n",
      " [ 0.43051198  0.24267581  0.48439601  0.30171136  0.30455842 -0.5808122 ]\n",
      " [ 0.35894427 -0.6198626  -0.48547226  0.0735743  -0.18865511 -0.45852167]]\n"
     ]
    }
   ],
   "source": [
    "print(np.transpose(Vs))  # Matrice des vecteurs singuliers droit (transposée)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "68b714db-bb84-4dc1-8566-d1b6367da1be",
   "metadata": {},
   "source": [
    "## VII - Calculer les coordonnées factorielles des individus:"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "226a2f4f-9aca-4098-9c4c-1130003be9b1",
   "metadata": {},
   "source": [
    "### 1- Utilisez la formule 𝐹𝑖𝑘 = 𝛿𝑘 × 𝑢𝑖𝑘 pour calculer les coordonnées factorielles des individus pour chaque axe."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "b48ee547-f92a-4804-80b8-74df9db2d374",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                      F_1\n",
      "Modele                   \n",
      "Alfasud TI      -2.138924\n",
      "Audi 100         1.561459\n",
      "Simca 1300      -1.119385\n",
      "Citroen GS Club -2.573742\n",
      "Fiat 132         0.427855\n",
      "Lancia Beta     -0.304238\n",
      "Peugeot 504      0.683928\n",
      "Renault 16 TL   -1.948493\n",
      "Renault 30       4.409735\n",
      "Toyota Corolla  -3.985782\n",
      "Alfetta 1.66     0.437658\n",
      "Princess 1800    1.018175\n",
      "Datsun 200L      2.941080\n",
      "Taunus 2000      1.314880\n",
      "Rancho          -0.691111\n",
      "Mazda 9295       0.385709\n",
      "Opel Rekord      2.289768\n",
      "Lada 1300       -2.708574\n"
     ]
    }
   ],
   "source": [
    "# Coordonnées factorielles - 1er facteur\n",
    "coordinates = pd.DataFrame(Us[:,0] * delta[0], columns=['F_1'], index=df['Modele'])\n",
    "print(coordinates)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "fe9c2a70-26e6-4f27-bd61-1da6c1edae60",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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