notebooks/22. Data Mining II/22.e. Data Mining II- Choosing Initial values .ipynb

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  10    "cells": [
  11     {
  12      "cell_type": "heading",
  13      "level": 1,
  14      "metadata": {},
  15      "source": [
  16       "Section 22.e. Data mining II: Choosing Initial Values"
  17      ]
  18     },
  19     {
  20      "cell_type": "heading",
  21      "level": 3,
  22      "metadata": {},
  23      "source": [
  24       "Exercise 22.7"
  25      ]
  26     },
  27     {
  28      "cell_type": "markdown",
  29      "metadata": {},
  30      "source": [
  31       "Try running the k-means algorithm with $k = 2$ and $k = 3$ on the English Premier League data using the `random` method for initialisation with the number of runs of only one time. Execute your Python code a few times to observe that different choices of initial cluster means can actually lead to different results.  \n"
  32      ]
  33     },
  34     {
  35      "cell_type": "code",
  36      "collapsed": false,
  37      "input": [
  38       "import numpy as np\n",
  39       "import matplotlib.pyplot as plt\n",
  40       "\n",
  41       "import pandas as pd\n",
  42       "\n",
  43       "%pylab inline\n",
  44       "\n",
  45       "\n",
  46       "np.set_printoptions(precision=2, linewidth=100)"
  47      ],
  48      "language": "python",
  49      "metadata": {},
  50      "outputs": [
  51       {
  52        "output_type": "stream",
  53        "stream": "stdout",
  54        "text": [
  55         "Populating the interactive namespace from numpy and matplotlib\n"
  56        ]
  57       }
  58      ],
  59      "prompt_number": 26
  60     },
  61     {
  62      "cell_type": "code",
  63      "collapsed": false,
  64      "input": [
  65       "premierLeague=pd.DataFrame([(68, 41), (39, 61), (32, 74), (71, 27),\n",
  66       "                            (33, 48), (61, 39), (40, 85), (38, 53), \n",
  67       "                            (101, 50), (102, 37), (64, 43), (43, 59), \n",
  68       "                            (28, 62), (54, 46), (45, 52), (41, 60), \n",
  69       "                            (54, 54), (55, 51), (43, 59), (40, 51)],\n",
  70       "                           index=['Arsenal', 'Aston Villa', 'Cardiff City', 'Chelsea', \n",
  71       "                                  'Crystal Palace', 'Everton', 'Fulham', 'Hull City', \n",
  72       "                                  'Liverpool', 'Manchester City', 'Manchester United',\n",
  73       "                                  'Newcastle United', 'Norwich City', 'Southampton',\n",
  74       "                                  'Stoke City', 'Sunderland', 'Swansea City',\n",
  75       "                                  'Tottenham Hotspur', 'West Bromwich Albion',\n",
  76       "                                  'West Ham United'],\n",
  77       "                           columns=['Goals for', 'Goals against'])\n",
  78       "\n",
  79       "premierLeague"
  80      ],
  81      "language": "python",
  82      "metadata": {},
  83      "outputs": [
  84       {
  85        "html": [
  86         "<div style=\"max-height:1000px;max-width:1500px;overflow:auto;\">\n",
  87         "<table border=\"1\" class=\"dataframe\">\n",
  88         "  <thead>\n",
  89         "    <tr style=\"text-align: right;\">\n",
  90         "      <th></th>\n",
  91         "      <th>Goals for</th>\n",
  92         "      <th>Goals against</th>\n",
  93         "    </tr>\n",
  94         "  </thead>\n",
  95         "  <tbody>\n",
  96         "    <tr>\n",
  97         "      <th>Arsenal</th>\n",
  98         "      <td>  68</td>\n",
  99         "      <td> 41</td>\n",
 100         "    </tr>\n",
 101         "    <tr>\n",
 102         "      <th>Aston Villa</th>\n",
 103         "      <td>  39</td>\n",
 104         "      <td> 61</td>\n",
 105         "    </tr>\n",
 106         "    <tr>\n",
 107         "      <th>Cardiff City</th>\n",
 108         "      <td>  32</td>\n",
 109         "      <td> 74</td>\n",
 110         "    </tr>\n",
 111         "    <tr>\n",
 112         "      <th>Chelsea</th>\n",
 113         "      <td>  71</td>\n",
 114         "      <td> 27</td>\n",
 115         "    </tr>\n",
 116         "    <tr>\n",
 117         "      <th>Crystal Palace</th>\n",
 118         "      <td>  33</td>\n",
 119         "      <td> 48</td>\n",
 120         "    </tr>\n",
 121         "    <tr>\n",
 122         "      <th>Everton</th>\n",
 123         "      <td>  61</td>\n",
 124         "      <td> 39</td>\n",
 125         "    </tr>\n",
 126         "    <tr>\n",
 127         "      <th>Fulham</th>\n",
 128         "      <td>  40</td>\n",
 129         "      <td> 85</td>\n",
 130         "    </tr>\n",
 131         "    <tr>\n",
 132         "      <th>Hull City</th>\n",
 133         "      <td>  38</td>\n",
 134         "      <td> 53</td>\n",
 135         "    </tr>\n",
 136         "    <tr>\n",
 137         "      <th>Liverpool</th>\n",
 138         "      <td> 101</td>\n",
 139         "      <td> 50</td>\n",
 140         "    </tr>\n",
 141         "    <tr>\n",
 142         "      <th>Manchester City</th>\n",
 143         "      <td> 102</td>\n",
 144         "      <td> 37</td>\n",
 145         "    </tr>\n",
 146         "    <tr>\n",
 147         "      <th>Manchester United</th>\n",
 148         "      <td>  64</td>\n",
 149         "      <td> 43</td>\n",
 150         "    </tr>\n",
 151         "    <tr>\n",
 152         "      <th>Newcastle United</th>\n",
 153         "      <td>  43</td>\n",
 154         "      <td> 59</td>\n",
 155         "    </tr>\n",
 156         "    <tr>\n",
 157         "      <th>Norwich City</th>\n",
 158         "      <td>  28</td>\n",
 159         "      <td> 62</td>\n",
 160         "    </tr>\n",
 161         "    <tr>\n",
 162         "      <th>Southampton</th>\n",
 163         "      <td>  54</td>\n",
 164         "      <td> 46</td>\n",
 165         "    </tr>\n",
 166         "    <tr>\n",
 167         "      <th>Stoke City</th>\n",
 168         "      <td>  45</td>\n",
 169         "      <td> 52</td>\n",
 170         "    </tr>\n",
 171         "    <tr>\n",
 172         "      <th>Sunderland</th>\n",
 173         "      <td>  41</td>\n",
 174         "      <td> 60</td>\n",
 175         "    </tr>\n",
 176         "    <tr>\n",
 177         "      <th>Swansea City</th>\n",
 178         "      <td>  54</td>\n",
 179         "      <td> 54</td>\n",
 180         "    </tr>\n",
 181         "    <tr>\n",
 182         "      <th>Tottenham Hotspur</th>\n",
 183         "      <td>  55</td>\n",
 184         "      <td> 51</td>\n",
 185         "    </tr>\n",
 186         "    <tr>\n",
 187         "      <th>West Bromwich Albion</th>\n",
 188         "      <td>  43</td>\n",
 189         "      <td> 59</td>\n",
 190         "    </tr>\n",
 191         "    <tr>\n",
 192         "      <th>West Ham United</th>\n",
 193         "      <td>  40</td>\n",
 194         "      <td> 51</td>\n",
 195         "    </tr>\n",
 196         "  </tbody>\n",
 197         "</table>\n",
 198         "</div>"
 199        ],
 200        "metadata": {},
 201        "output_type": "pyout",
 202        "prompt_number": 27,
 203        "text": [
 204         "                      Goals for  Goals against\n",
 205         "Arsenal                      68             41\n",
 206         "Aston Villa                  39             61\n",
 207         "Cardiff City                 32             74\n",
 208         "Chelsea                      71             27\n",
 209         "Crystal Palace               33             48\n",
 210         "Everton                      61             39\n",
 211         "Fulham                       40             85\n",
 212         "Hull City                    38             53\n",
 213         "Liverpool                   101             50\n",
 214         "Manchester City             102             37\n",
 215         "Manchester United            64             43\n",
 216         "Newcastle United             43             59\n",
 217         "Norwich City                 28             62\n",
 218         "Southampton                  54             46\n",
 219         "Stoke City                   45             52\n",
 220         "Sunderland                   41             60\n",
 221         "Swansea City                 54             54\n",
 222         "Tottenham Hotspur            55             51\n",
 223         "West Bromwich Albion         43             59\n",
 224         "West Ham United              40             51"
 225        ]
 226       }
 227      ],
 228      "prompt_number": 27
 229     },
 230     {
 231      "cell_type": "heading",
 232      "level": 3,
 233      "metadata": {},
 234      "source": [
 235       "Result for $k=2$"
 236      ]
 237     },
 238     {
 239      "cell_type": "markdown",
 240      "metadata": {},
 241      "source": [
 242       "For $k = 2$, we can observe that there are at least two different clustering results by running the following Python code a few times."
 243      ]
 244     },
 245     {
 246      "cell_type": "code",
 247      "collapsed": false,
 248      "input": [
 249       "import sklearn.cluster as sc\n",
 250       "\n",
 251       "# Run k-means algorithm on the data (k=2)\n",
 252       "kmeans = sc.KMeans(n_clusters=2,init='random',n_init=1)\n",
 253       "assigned_clusters = kmeans.fit(premierLeague).labels_\n",
 254       "\n",
 255       "# Print the assigned clusters\n",
 256       "cluster1 = find(equal(assigned_clusters,0))\n",
 257       "cluster2 = find(equal(assigned_clusters,1))\n",
 258       "print(\"Cluster 1:\" + str(premierLeague.index[cluster1]))\n",
 259       "print(\"Cluster 2:\" + str(premierLeague.index[cluster2]))\n",
 260       "\n",
 261       "# Plot the clustering result\n",
 262       "plot(premierLeague['Goals for'][cluster1], premierLeague['Goals against'][cluster1], 'ro')\n",
 263       "plot(premierLeague['Goals for'][cluster2], premierLeague['Goals against'][cluster2], 'bo')\n",
 264       "xlabel('Goals for')\n",
 265       "ylabel('Goals against')"
 266      ],
 267      "language": "python",
 268      "metadata": {},
 269      "outputs": [
 270       {
 271        "output_type": "stream",
 272        "stream": "stdout",
 273        "text": [
 274         "Cluster 1:Index(['Arsenal', 'Chelsea', 'Everton', 'Liverpool', 'Manchester City', 'Manchester United'], dtype='object')\n",
 275         "Cluster 2:Index(['Aston Villa', 'Cardiff City', 'Crystal Palace', 'Fulham', 'Hull City', 'Newcastle United', 'Norwich City', 'Southampton', 'Stoke City', 'Sunderland', 'Swansea City', 'Tottenham Hotspur', 'West Bromwich Albion', 'West Ham United'], dtype='object')\n"
 276        ]
 277       },
 278       {
 279        "metadata": {},
 280        "output_type": "pyout",
 281        "prompt_number": 28,
 282        "text": [
 283         "<matplotlib.text.Text at 0x10f098d50>"
 284        ]
 285       },
 286       {
 287        "metadata": {},
 288        "output_type": "display_data",
 289        "png": 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wYcMGAMBzzz2HqVOn4vLlywgLC5M8GHWfeXU3N1uT4+7utQ5OQtS9tVgIv+Xr6ytVDvqd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\nkxcAAAAASUVORK5CYII=\n",
 290        "text": [
 291         "<matplotlib.figure.Figure at 0x10edeb090>"
 292        ]
 293       }
 294      ],
 295      "prompt_number": 28
 296     },
 297     {
 298      "cell_type": "heading",
 299      "level": 3,
 300      "metadata": {},
 301      "source": [
 302       "Result for $k=3$"
 303      ]
 304     },
 305     {
 306      "cell_type": "markdown",
 307      "metadata": {},
 308      "source": [
 309       "Likewise, for $k = 3$, we can observe different resultant clusterings from the following Python code that sets \u2018random\u2019 for initialisation and the number of run to just one."
 310      ]
 311     },
 312     {
 313      "cell_type": "code",
 314      "collapsed": false,
 315      "input": [
 316       "import sklearn.cluster as sc\n",
 317       "\n",
 318       "# Run k-means algorithm on the data (k=3)\n",
 319       "kmeans = sc.KMeans(n_clusters=3,init='random',n_init=1)\n",
 320       "assigned_clusters = kmeans.fit(premierLeague).labels_\n",
 321       "\n",
 322       "# Print the assigned clusters\n",
 323       "cluster1 = find(equal(assigned_clusters,0))\n",
 324       "cluster2 = find(equal(assigned_clusters,1))\n",
 325       "cluster3 = find(equal(assigned_clusters,2))\n",
 326       "\n",
 327       "print(\"Cluster 1:\" + str(premierLeague.index[cluster1]))\n",
 328       "print(\"Cluster 2:\" + str(premierLeague.index[cluster2]))\n",
 329       "print(\"Cluster 2:\" + str(premierLeague.index[cluster3]))\n",
 330       "\n",
 331       "# Plot the clustering result\n",
 332       "plot(premierLeague['Goals for'][cluster1], premierLeague['Goals against'][cluster1], 'ro')\n",
 333       "plot(premierLeague['Goals for'][cluster2], premierLeague['Goals against'][cluster2], 'bo')\n",
 334       "plot(premierLeague['Goals for'][cluster3], premierLeague['Goals against'][cluster3], 'go')\n",
 335       "xlabel('Goals for')\n",
 336       "ylabel('Goals against')\n"
 337      ],
 338      "language": "python",
 339      "metadata": {},
 340      "outputs": [
 341       {
 342        "output_type": "stream",
 343        "stream": "stdout",
 344        "text": [
 345         "Cluster 1:Index(['Cardiff City', 'Fulham', 'Norwich City'], dtype='object')\n",
 346         "Cluster 2:Index(['Aston Villa', 'Crystal Palace', 'Hull City', 'Newcastle United', 'Southampton', 'Stoke City', 'Sunderland', 'Swansea City', 'Tottenham Hotspur', 'West Bromwich Albion', 'West Ham United'], dtype='object')\n",
 347         "Cluster 2:Index(['Arsenal', 'Chelsea', 'Everton', 'Liverpool', 'Manchester City', 'Manchester United'], dtype='object')\n"
 348        ]
 349       },
 350       {
 351        "metadata": {},
 352        "output_type": "pyout",
 353        "prompt_number": 29,
 354        "text": [
 355         "<matplotlib.text.Text at 0x10f87dd10>"
 356        ]
 357       },
 358       {
 359        "metadata": {},
 360        "output_type": "display_data",
 361        "png": 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UGzZsEEJc+zu0WCyyZ7qurq5OeHp6irKyMlkzlZaWCl9fX2G1WoUQQsTHx4t3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\nIRARUQMWAhERAWAhEBFRAxYCEREBAP4/VW8o3CfsGTIAAAAASUVORK5CYII=\n",
 362        "text": [
 363         "<matplotlib.figure.Figure at 0x10ed86810>"
 364        ]
 365       }
 366      ],
 367      "prompt_number": 29
 368     },
 369     {
 370      "cell_type": "markdown",
 371      "metadata": {},
 372      "source": [
 373       "If you run these two blocks of code several times, you may see different clusterings, which depend on the initial values chosen in the algorithm."
 374      ]
 375     },
 376     {
 377      "cell_type": "heading",
 378      "level": 3,
 379      "metadata": {},
 380      "source": [
 381       "Choosing k"
 382      ]
 383     },
 384     {
 385      "cell_type": "heading",
 386      "level": 4,
 387      "metadata": {},
 388      "source": [
 389       "Example:"
 390      ]
 391     },
 392     {
 393      "cell_type": "markdown",
 394      "metadata": {},
 395      "source": [
 396       "Let's start by creating a data set of four clusters in Python:\n"
 397      ]
 398     },
 399     {
 400      "cell_type": "code",
 401      "collapsed": false,
 402      "input": [
 403       "# data\n",
 404       "a1 = concatenate((random.rand(20)*3+1, random.rand(20)*3+3, random.rand(20)*3+6, random.rand(20)*3+8))\n",
 405       "a2 = concatenate((random.rand(20)*3+1, random.rand(20)*3+6, random.rand(20)*3+3, random.rand(20)*3+8))\n",
 406       "\n",
 407       "plt.plot(a1,a2,'.')"
 408      ],
 409      "language": "python",
 410      "metadata": {},
 411      "outputs": [
 412       {
 413        "metadata": {},
 414        "output_type": "pyout",
 415        "prompt_number": 30,
 416        "text": [
 417         "[<matplotlib.lines.Line2D at 0x10edc20d0>]"
 418        ]
 419       },
 420       {
 421        "metadata": {},
 422        "output_type": "display_data",
 423        "png": 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 424        "text": [
 425         "<matplotlib.figure.Figure at 0x10f54e250>"
 426        ]
 427       }
 428      ],
 429      "prompt_number": 30
 430     },
 431     {
 432      "cell_type": "markdown",
 433      "metadata": {},
 434      "source": [
 435       "To demonstrate the procedure of the Silhouette method for finding an optimal value of $k$, we will use the Silhouette method, as described in the notebook \"\" for the values $k = 2, 3, 4, 5, 6, 7$ and $8$.\n",
 436       "\n",
 437       "The definition of silhouette is repeated here in case this notebook is being run from scratch."
 438      ]
 439     },
 440     {
 441      "cell_type": "code",
 442      "collapsed": false,
 443      "input": [
 444       "from scipy.spatial.distance import pdist, squareform\n",
 445       "from matplotlib import cm\n",
 446       "    \n",
 447       "def silhouette(X, cIDX):\n",
 448       "    \"\"\"\n",
 449       "    Computes the silhouette score for each instance of a clustered dataset,\n",
 450       "    which is defined as:\n",
 451       "        s(i) = (b(i)-a(i)) / max{a(i),b(i)}\n",
 452       "    with:\n",
 453       "        -1 <= s(i) <= 1\n",
 454       "\n",
 455       "    Args:\n",
 456       "        X    : A M-by-N array of M observations in N dimensions\n",
 457       "        cIDX : array of len M containing cluster indices (starting from zero)\n",
 458       "\n",
 459       "    Returns:\n",
 460       "        s    : silhouette value of each observation\n",
 461       "    \"\"\"\n",
 462       "\n",
 463       "    N = X.shape[0]              # number of instances\n",
 464       "    K = len(np.unique(cIDX))    # number of clusters\n",
 465       "\n",
 466       "    # compute pairwise distance matrix\n",
 467       "    D = squareform(pdist(X))\n",
 468       "\n",
 469       "    # indices belonging to each cluster\n",
 470       "    kIndices = [np.flatnonzero(cIDX==k) for k in range(K)]\n",
 471       "\n",
 472       "    # compute a,b,s for each instance\n",
 473       "    a = np.zeros(N)\n",
 474       "    b = np.zeros(N)\n",
 475       "    for i in range(N):\n",
 476       "        # instances in same cluster other than instance itself\n",
 477       "        a[i] = np.mean( [D[i][ind] for ind in kIndices[cIDX[i]] if ind!=i] )\n",
 478       "        # instances in other clusters, one cluster at a time\n",
 479       "        b[i] = np.min( [np.mean(D[i][ind]) for k,ind in enumerate(kIndices) if cIDX[i]!=k] )\n",
 480       "    s = (b-a)/np.maximum(a,b)\n",
 481       "\n",
 482       "    # plot\n",
 483       "    order = np.lexsort((-s,cIDX))\n",
 484       "    indices = [np.flatnonzero(cIDX[order]==k) for k in range(K)]\n",
 485       "    ytick = [(np.max(ind)+np.min(ind))/2 for ind in indices]\n",
 486       "    ytickLabels = [\"%d\" % x for x in range(K)]\n",
 487       "    cmap = cm.jet( np.linspace(0,1,K) ).tolist()\n",
 488       "    clr = [cmap[i] for i in cIDX[order]]\n",
 489       "\n",
 490       "    fig = plt.figure()\n",
 491       "    ax = fig.add_subplot(111)\n",
 492       "    ax.barh(range(X.shape[0]), s[order], height=1.0, edgecolor='none', color=clr)\n",
 493       "    ax.set_ylim(ax.get_ylim()[::-1])\n",
 494       "    plt.yticks(ytick, ytickLabels)\n",
 495       "    plt.xlabel('Silhouette Coefficient')\n",
 496       "    plt.ylabel('Cluster')\n",
 497       "    \n",
 498       "    return s"
 499      ],
 500      "language": "python",
 501      "metadata": {},
 502      "outputs": [],
 503      "prompt_number": 31
 504     },
 505     {
 506      "cell_type": "markdown",
 507      "metadata": {},
 508      "source": [
 509       "The following code runs the silhouette function for each of these values of $k$, and stores the results in the variable $s$:"
 510      ]
 511     },
 512     {
 513      "cell_type": "code",
 514      "collapsed": false,
 515      "input": [
 516       "import sklearn.cluster as sc\n",
 517       "\n",
 518       "# rearrange data into [nxd] format (n data instances, d attribute dimensions)\n",
 519       "x = c_[a1,a2]\n",
 520       "\n",
 521       "# run k-means for k = 2, 3, 4, 5, 6, 7, 8\n",
 522       "s=[]\n",
 523       "for i in range(2,9):\n",
 524       "    kmeans = sc.KMeans(n_clusters=i)\n",
 525       "    assigned_clusters = kmeans.fit(x).labels_\n",
 526       "    s.append(average(silhouette(x, assigned_clusters)))\n",
 527       "\n",
 528       "data=pd.DataFrame(s, index=range(2, 9))"
 529      ],
 530      "language": "python",
 531      "metadata": {},
 532      "outputs": [
 533       {
 534        "metadata": {},
 535        "output_type": "display_data",
 536        "png": 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 537        "text": [
 538         "<matplotlib.figure.Figure at 0x10ef15a50>"
 539        ]
 540       },
 541       {
 542        "metadata": {},
 543        "output_type": "display_data",
 544        "png": 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 545        "text": [
 546         "<matplotlib.figure.Figure at 0x10ef15650>"
 547        ]
 548       },
 549       {
 550        "metadata": {},
 551        "output_type": "display_data",
 552        "png": 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 553        "text": [
 554         "<matplotlib.figure.Figure at 0x10ef1f9d0>"
 555        ]
 556       },
 557       {
 558        "metadata": {},
 559        "output_type": "display_data",
 560        "png": 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 561        "text": [
 562         "<matplotlib.figure.Figure at 0x10f8e8f10>"
 563        ]
 564       },
 565       {
 566        "metadata": {},
 567        "output_type": "display_data",
 568        "png": 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 569        "text": [
 570         "<matplotlib.figure.Figure at 0x10eff6650>"
 571        ]
 572       },
 573       {
 574        "metadata": {},
 575        "output_type": "display_data",
 576        "png": 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 577        "text": [
 578         "<matplotlib.figure.Figure at 0x10f5d8d90>"
 579        ]
 580       },
 581       {
 582        "metadata": {},
 583        "output_type": "display_data",
 584        "png": 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 585        "text": [
 586         "<matplotlib.figure.Figure at 0x10f37cb50>"
 587        ]
 588       }
 589      ],
 590      "prompt_number": 32
 591     },
 592     {
 593      "cell_type": "markdown",
 594      "metadata": {},
 595      "source": [
 596       "We can then plot the values of $s$ as a bar chart:"
 597      ]
 598     },
 599     {
 600      "cell_type": "code",
 601      "collapsed": false,
 602      "input": [
 603       "ax=data.plot(kind='bar', title='Silhouette coefficients for a range of values k')\n",
 604       "\n",
 605       "ax.set_xlabel('k')\n",
 606       "ax.set_ylabel('Average silhouette coefficient')"
 607      ],
 608      "language": "python",
 609      "metadata": {},
 610      "outputs": [
 611       {
 612        "metadata": {},
 613        "output_type": "pyout",
 614        "prompt_number": 33,
 615        "text": [
 616         "<matplotlib.text.Text at 0x10eb821d0>"
 617        ]
 618       },
 619       {
 620        "metadata": {},
 621        "output_type": "display_data",
 622        "png": 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q3Fpn/Pv27cO5c+cwceJELFy4sMabu/WVVL/CkBBCKuk8V8/9+/fh7e2NkpIS\nODo6WiqXGmsa9Uj1pTDl1lidclevLNHcRDPB5+PPyMhAhw4dEBAQAAC4cOEC/vGPf5g+ISGEEIvQ\n2fjnz5+PlJQU1ResBwcH49ixY2YPVjcwsQMIxMQOIBATO4ARmNgBBGJiBxCEZvx68PPzU7+RrV43\nI4QQizDne29A3Xv/TWcH9/Pzw8mTJwEAZWVlWLlyZY1TKxNtFGIHEEghdgCBFGIHMIJC7AACKcQO\nAEDIJ44NrW8dB7SY6tPGer25+8Ybb+Dw4cPgnGPAgAFYuXIlPD09TRJAH/TmrgkqU25N1Sl39cqU\nW9sWrKYHGcLgD3BZE+k2fgbDnhFZyy8GA+U2juGNiEH/7JS7RmWz5gaspfEbeq4ewUf1ZGdnY8SI\nEfD29oa3tzdGjRqFW7duGRSWEEKI9dD5jP+ll17ChAkTMHHiRADApk2bsGnTJhw6dMgiAQEpP+M3\nuLoVPSMyqDrlrl6ZcmuqLtHcgLU84zeU4FFPUFAQfv31V52XmRM1fhNUptyaqlPu6pUpt7YtWE0P\nMoTgUY+npyc2bNgApVKJ8vJybNy4UXVMP9GFiR1AICZ2AIGY2AGMwMQOIBATO4BATOwAgpjqOH6d\njX/NmjXYtm0bmjVrhubNm2P79u1Yu3atSTZOCCHE8uioHgPRS2GN1Sl39cqUW1N1ieYG6t2oZ/Lk\nyXjw4IFqOT8/H9OnTzdtOkIIIRajs/FfvHgRbm5uqmV3d3ecO3fOrKHqDiZ2AIGY2AEEYmIHMAIT\nO4BATOwAAjGxAwhisRk/5xx5eX+foyIvLw9KpdIkGyeEEGJ5Omf869evx8cff4wxY8aAc47t27fj\n3XffxeTJky2VkWb8pqhMuTVVp9zVK1NubVuwmh5kCKNO2fDbb7/hyJEjkMlkeOGFF9ChQwezhNSG\nGr8JKlNuTdUpd/XKlFvbFqymBxmCztVjIvXj3DEMlNs4dM4bjdUlmhuwlsZvsXP1GCMlJQUBAQFo\n27YtYmNjta73yy+/wNbWFjt37jRnHEIIITDjM36lUon27dvj8OHD8PHxQdeuXZGUlFTjXP5KpRJh\nYWFwcHDAtGnTMGrUqJohJfuM3+DqVvSMyKDqlLt6ZcqtqbpEcwPW8ozfUEY948/MzMThw4cBAKWl\npSgsLNR5m7S0NPj7+0Mul8POzg6RkZFITk6usd5//vMfvPLKK/D29tYnCiGEECPpbPwJCQkYPXo0\noqKiAAAmebutAAAcvElEQVS3bt3CiBEjdBbOyclBy5YtVcu+vr7IycmpsU5ycjJeffVVAJV/tesS\nJnYAgZjYAQRiYgcwAhM7gEBM7AACMbEDqLi4eEAmk5nlRxudX724atUqpKWloUePHgCAdu3a4d69\nezrvjD5NfP78+fj0009VL0dqeyk1depUyOVyAICbmxuCg4NVb3JUfqjBUst/P2h0LUPH9ZrXN1d+\n/fNc0DNv5bL6m060v4XkuaBH3spla9nfhi7/nd2Uec2/v82b/+nXRh7VM0/VLNquX/e/ZTmAD6CJ\nzhl/t27dkJaWhi5duuD8+fMoLy/Hc889h4sXL9Z2M6SmpiI6OhopKSkAgJiYGNjY2GDRokWqdVq3\nbq1q9rm5uXBwcEBiYiKGDh2qHpJm/MZXptyaqlPu6pUpt7YtSDS75tw6n/H369cPH3/8MUpLS3Ho\n0CF8+eWXGDJkiM7NhYaGIiMjA5mZmWjRogW2bt2KpKQktXWuX7+u+ve0adMwZMiQGk2fEEKIaemc\n8cfGxsLb2xudOnVCfHw8Bg8ejI8++khnYVtbW8TFxSE8PBwdOnTA2LFjERgYiPj4eMTHx5skvPVj\nYgcQiIkdQCAmdgAjMLEDCMTEDiAQEzuAQMw0ZbgO//73v/W6zJz0iGkxADjA9fw5asC65r2flNua\ncxuanXJbNrc1ZTdNbp0z/srZflXBwcG4cOGClluYHs34TVCZcmuqTrmrV6bc2rYg0ewGzviTkpKw\nefNm3LhxQ22mX1RUBE9PT/NkJIQQYnZaG3+vXr3QvHlz5ObmYuHChaq/Gi4uLujcubPFAkobg2Hn\nA7EWDJTb0hikmZ2BclsSgylya238rVq1QqtWrfD888+jX79+atctWrSo1nPvEEIIsV6CZvydOnXC\nf//7X7MGq4pm/CaoTLk1Vafc1StTbm1bkGh2A2f8X331Fb788ktcu3YNnTp1Ul1eVFSE3r17mycj\nIYQQs9P6jL+goAD5+fl4++23ERsbq/qr4ezsbPE3d6X7jJ/BWs75Tbk1VreS3ED9OK+9VHMD1pOd\nwRS5tX6Ay9XVFXK5HFu2bMHNmzdx9OhRyOVyVFRU4MaNGwZsmBBCiDXROeOPjo7G2bNnceXKFVy9\nehU5OTkYPXo0Tp06ZamMEn7Gb3B1K3lWYXB1yl29MuXWVF2iuQHpZhd4Pv7vv/8eycnJcHR0BAD4\n+PiguLjY9PkIIYRYhM7G37BhQ9jY/L1aSUmJWQPVLUzsAAIxsQMIxMQOYAQmdgCBmNgBBGJiBxCI\nmaSKzsZf+SUsDx48QEJCAl588UXMnDnTJBsnhBBieXp95+7Bgwdx8OBBAEB4eDjCwsLMHqwqmvGb\noDLl1lSdclevTLm1bUGi2TXnNtuXrZsSNX4TVKbcmqpT7uqVKbe2LUg0u8A3d52cnODs7AxnZ2fV\nvN/FxcUsEeseJnYAgZjYAQRiYgcwAhM7gEBM7AACMbEDCMRMUkXnN3BVPYKnoqICu3btQmpqqkk2\nTgghxPIEjXrofPz0UrhadcpdvTLl1lRdorkB6WYX+J273333nerfFRUVOHv2LBo3bmzabIQQQixG\nZ+PfvXv3//4iPf0eXblcjuTkZLMHqxsY6vM5vy2PQZq5AelmZ6DclsRg1vPxV1q3bp3RGyGEEGI9\ndB7Vk52djREjRsDb2xve3t4YNWoUbt26ZYlsdYBC7AACKcQOIJBC7ABGUIgdQCCF2AEEUogdQCCF\nSarobPzTpk3D0KFDcfv2bdy+fRtDhgzBtGnTTLJxQgghlqez8d+/fx/Tpk2DnZ0d7OzsMHXqVNy7\nd88S2eoAJnYAgZjYAQRiYgcwAhM7gEBM7AACMbEDCMRMUkVn4/f09MSGDRugVCpRXl6OjRs3wsvL\nS6/iKSkpCAgIQNu2bTV+R29ycjKCgoLQpUsXhISE4MiRI4bfA0IIIYbhOty4cYNHRERwLy8v7uXl\nxYcOHcqzsrJ03YyXl5fzNm3a8Bs3bvCysjIeFBTE09PT1dYpLi5W/fvixYu8TZs2GmvpEdNiAHCA\nm+nHfPeTclNuyl0fs2vOrfOoHrlcjt27dxv8ByUtLQ3+/v6Qy+UAgMjISCQnJyMwMFC1TuU5/oGn\nnxDW95UEIYQQ4XQ2/nv37iExMRGZmZkoLy8H8PSTZmvWrKn1djk5OWjZsqVq2dfXF6dPn66x3g8/\n/IDFixfjzp07qjOA1h0M0jx6gIFyWxqDNLMzUG5LYrDIcfzDhg1D3759ERYWpvpClsoPdNVGn3UA\nYPjw4Rg+fDiOHz+OSZMm4cqVKxrXmzp1qurVg5ubG4KDg6FQKAAAjDEAsNjy32+w6FqGjus1r2+u\n/PrnuaDj+urLT7dB+1t92bA8F/TIW7lsLfvb0OW/s5syr/n3t3nz/70NQ/NoWmYA1v1vWQ5tdJ6r\nR+h5eVJTUxEdHY2UlBQAQExMDGxsbLBo0SKtt2nTpg3S0tLg6empHpLO1WN8ZcqtqTrlrl6Zcmvb\ngkSzCzwtc0REBPbu3Wvw5kJDQ5GRkYHMzEyUlZVh69atGDp0qNo6165dU4U6d+4cANRo+oQQQkxL\n66jHyclJNa755JNPYG9vDzs7OwBP/0IVFhbWXtjWFnFxcQgPD4dSqcSMGTMQGBiI+Ph4AEBUVBS+\n++47rF+/HnZ2dnBycsKWLVtMdb+sBEN9niNaHoM0cwPSzc5AuS2JwRS56Ru4DGTYyzIGw/4nWcvL\nSQbKbRzDX74z6J+dcteobNbcgPVkZzBFbq2Nv3L0os1zzz1nwMaNI93Gb3B1K3lwGVydclevTLk1\nVZdobkC62Q1s/AqFotYjc44ePWq6bDpQ4zdBZcqtqTrlrl6ZcmvbgkSz05etm0T9GD0wUG7j0MhE\nY3WJ5gasJzuDKXJrfXP3yJEjeOGFF/Ddd99pfOY/cuRIAzZOCCHEWmht/MeOHcMLL7yg9g1cVVHj\n14dC7AACKcQOIJBC7ABGUIgdQCCF2AEEUogdQCCFSarQqMdANAPVWJ1yV69MuTVVl2huQLrZBX6A\n64svvkBhYSE455gxYwaee+45HDhwwCwR6x4mdgCBmNgBBGJiBzACEzuAQEzsAAIxsQMIxExSRWfj\nX716NVxcXHDw4EHk5eVh/fr1ePvtt02ycUIIIZans/FXvkzYu3cvJk2ahI4dO5o9VN2hEDuAQAqx\nAwikEDuAERRiBxBIIXYAgRRiBxBIYZIqOht/SEgIBgwYgH379mHgwIEoLCxUnaWTEEKI9Oh8c1ep\nVOLChQto06YN3Nzc8NdffyEnJwedO3e2VEYJv7nLUJ+PFTaF+pEbqB/Hw0s1N2A92RnMehx/pQYN\nGiAkJES17OnpSWfQJIQQCaPDOQ1Eh7tprE65q1em3JqqSzQ3IN3sAg/nJIQQUrfo1fiPHz+OtWvX\nAgDu37+PGzdumDVU3cHEDiAQEzuAQEzsAEZgYgcQiIkdQCAmdgCBmEmq6Gz80dHR+OyzzxATEwMA\nKCsrw8SJE02ycUIIIZanc8YfFBSE8+fPIyQkBOfPnwcAdO7cGRcvXrRIQIBm/CapTLk1Vafc1StT\nbm1bkGh2gTP+hg0bqh23X1JSYtpchBBCLEpn4x89ejSioqLw4MEDJCQk4MUXX8TMmTMtka0OYGIH\nEIiJHUAgJnYAIzCxAwjExA4gEBM7gEDMJFV0Hsf/1ltv4eDBg3B2dsbVq1exdOlShIWFmWTjhBBC\nLI+O4zcQzUA1Vqfc1StTbk3VJZobkG52gTN+Z2fnGj++vr4YMWIErl+/bpaohBBCzEdn43/jjTew\nbNky5OTkICcnB8uXL8eECRMwduxYTJ8+3RIZJYyJHUAgJnYAgZjYAYzAxA4gEBM7gEBM7AACMZNU\n0dn4d+3ahaioKLi4uMDFxQWzZ8/GgQMHEBkZifz8fJ0bSElJQUBAANq2bYvY2Nga12/atAlBQUHo\n3LkzevfubdHDRAkhpD7S2fgdHBywdetWVFRUoKKiAtu2bUOjRo0AVM6mtFMqlZg3bx5SUlKQnp6O\npKQkXL58WW2d1q1b46effsLFixexZMkSzJ4924i7Y20UYgcQSCF2AIEUYgcwgkLsAAIpxA4gkELs\nAAIpTFJFZ+PftGkTNmzYgCZNmqBJkyZYv349Nm7ciIcPHyIuLq7W26alpcHf3x9yuRx2dnaIjIxE\ncnKy2jo9e/aEq6srAKB79+64deuWEXeHEEKILjobf5s2bbBnzx7k5uYiNzcXe/bsgb+/Pxo3bow+\nffrUetucnBy0bNlStezr64ucnByt669evRqDBw82IL61Y2IHEIiJHUAgJnYAIzCxAwjExA4gEBM7\ngEDMJFV0Hsf/8OFDrF69Gunp6Xj06JHq8jVr1ugsrmsUVNXRo0exZs0anDx5UuP1U6dOhVwuBwC4\nubkhODgYCoUCAMAYAwCLLf+983UtQ8f1mtc3V37981zQM2/l8tNt0P5WXzYszwU98lYuW8v+NnT5\n7+ymzGv+/W3e/H9vw9A8mpYZgHX/W5ZDG53H8b/yyisIDAzEpk2b8P7772Pjxo0IDAzEypUra7sZ\nACA1NRXR0dFISUkBAMTExMDGxgaLFi1SW+/ixYsYOXIkUlJS4O/vXzMkHcdvfGXKrak65a5emXJr\n24JEs2vJzXUICgrinHPeqVMnzjnnZWVlvFu3brpuxjnn/MmTJ7x169b8xo0b/PHjxzwoKIinp6er\nrZOVlcXbtGnDf/75Z6119IhpMU//D3Ez/ZjvflJuyk2562N2zbl1zvjt7e0BAK6urvjvf/+LBw8e\n4P79+3r9rbG1tUVcXBzCw8PRoUMHjB07FoGBgYiPj0d8fDwA4MMPP0R+fj5effVVdOnSBd26ddOr\ntjQwsQMIxMQOIBATO4ARmNgBBGJiBxCIiR1AIGaaMrr+GiUmJvK//vqLM8a4XC7nXl5e/KuvvjL5\nX73a6BHTYmDQX+ejEn1WQbktm9vQ7JTbsrmtKbtpctc646+oqMD27dsxduxY0/yVEYhm/CaoTLk1\nVafc1StTbm1bkGh2AefqsbGxwWeffWamQIQQQsSgc8YfFhaGZcuWITs7G3l5eaofog8mdgCBmNgB\nBGJiBzACEzuAQEzsAAIxsQMIxExSRedx/Fu2bIFMJsOqVavULqcvXCeEEGmi8/EbiGagGqtT7uqV\nKbem6hLNDUg3u8Dz8ZeUlGDp0qWYNWsWACAjIwN79uwxfT5CCCEWobPxT5s2Dfb29jh16hQAoEWL\nFnj33XfNHqxuYGIHEIiJHUAgJnYAIzCxAwjExA4gEBM7gEDMJFV0Nv5r165h0aJFqg9yOTo6mmTD\nhBBCxKGz8Tds2BAPHz5ULV+7dg0NGzY0a6i6QyF2AIEUYgcQSCF2ACMoxA4gkELsAAIpxA4gkMIk\nVXQe1RMdHY2BAwfi1q1bGD9+PE6ePIl169aZZOOEEEIsT6+jenJzc5Gamgrg6ZeleHt7mz1YVdI9\nqofBsL/Q1nLkAAPlNo7hR2ow6J+dcteobNbcgPVkZzBFbp3P+IcMGYJx48Zh2LBhNN8nhJA6QOcz\nfsYYtm7din379qFr166IjIxERESE6nt3LUG6z/gNrm4lzyoMrk65q1em3JqqSzQ3IN3smnPr/QGu\n8vJyHD16FImJiUhJSUFhYaHJI2pDjd8ElSm3puqUu3plyq1tCxLNLvADXMDTr1/87rvv8PXXX+OX\nX37BlClTTB6vbmJiBxCIiR1AICZ2ACMwsQMIxMQOIBATO4BAzCRVdM74x4wZg9OnT2PgwIGYN28e\n+vXrBxsbvf5eEEIIsUI6Rz0pKSkICwtDgwYNAADHjx/Hli1bapy0zZxo1GOCypRbU3XKXb0y5da2\nBYlmF3hUz8CBA3Hu3DkkJSVh27ZteOaZZzBq1CizRCSEEGJ+Wmc2V65cQXR0NAIDAzF//nz4+fmB\ncw7GGF577TVLZpQwJnYAgZjYAQRiYgcwAhM7gEBM7AACMbEDCMRMUkXrM/7AwEBERETgwIED8PPz\nAwCsWLHCJBslhBAiHq3P+Hfu3InGjRujb9++mDNnDn788UermbNLh0LsAAIpxA4gkELsAEZQiB1A\nIIXYAQRSiB1AIIVJquh8c7e4uBjJyclISkrC0aNHMXnyZIwYMQIDBgwwSQB90Ju7JqhMuTVVp9zV\nK1NubVuQaHaBx/E7OTlhwoQJ2LNnD7Kzs9GlSxd8+umnZolY9zCxAwjExA4gEBM7gBGY2AEEYmIH\nEIiJHUAgZpIqBh2Q7+HhgdmzZ+PIkSN63yYlJQUBAQFo27YtYmNja1z/+++/o2fPnmjUqBGWL19u\nSBxCCCECmPU7d5VKJdq3b4/Dhw/Dx8cHXbt2RVJSEgIDA1Xr3L9/H1lZWfjhhx/g7u6OBQsW1AxJ\nox7jK1NuTdUpd/XKlFvbFiSa3YhTNgiVlpYGf39/yOVy2NnZITIyEsnJyWrreHt7IzQ0FHZ2duaM\nQggh5H/M2vhzcnLQsmVL1bKvry9ycnLMuUkrw8QOIBATO4BATOwARmBiBxCIiR1AICZ2AIGYSaro\n/OSuMZ6+hDGNqVOnQi6XAwDc3NwQHBwMhUIB4OmpowFYbPnvna9rGTqu17y+ufLrn+eCnnkrl59u\ng/a3+rJheS7okbdy2Vr2t6HLf2c3ZV7z72/z5v97G4bm0bTMAKz737Ic2ph1xp+amoro6GikpKQA\nAGJiYmBjY4NFixbVWPeDDz6Ak5MTzfjr0BzRJJUpt6bqlLt6ZZrxa6tu+Rl/aGgoMjIykJmZibKy\nMmzduhVDhw7VuK61NHZCCKnrzNr4bW1tERcXh/DwcHTo0AFjx45FYGAg4uPjER8fDwD4888/0bJl\nS3z++ef46KOP4Ofnh+LiYnPGsiAmdgCBmNgBBGJiBzACEzuAQEzsAAIxsQMIxExSxawzfgAYNGgQ\nBg0apHZZVFSU6t/NmjVDdna2uWMQQgj5H7PO+E2FZvwmqEy5NVWn3NUrU25tW5BodhFm/IQQQqwP\nNX6zYmIHEIiJHUAgJnYAIzCxAwjExA4gEBM7gEDMJFWo8RNCSD1DM34D0QxUY3XKXb0y5dZUXaK5\nAelmpxk/IYQQUOM3MyZ2AIGY2AEEYmIHMAITO4BATOwAAjGxAwjETFKFGj8hhNQzNOM3EM1ANVan\n3NUrU25N1SWaG5BudprxE0IIATV+M2NiBxCIiR1AICZ2ACMwsQMIxMQOIBATO4BAzCRVqPETQkg9\nQzN+A9EMVGN1yl29MuXWVF2iuQHpZqcZPyGEEFDjNzMmdgCBmNgBBGJiBzACEzuAQEzsAAIxsQMI\nxExShRo/IYTUMzTjNxDNQDVWp9zVK1NuTdUlmhuQbnaa8RNCCAE1fjNjYgcQiIkdQCAmdgAjMLED\nCMTEDiAQEzuAQMwkVajxE0JIPUMzfgPRDFRjdcpdvTLl1lRdorkB6WanGT8hhBCYufGnpKQgICAA\nbdu2RWxsrMZ1Xn/9dbRt2xZBQUE4f/68OeOIgIkdQCAmdgCBmNgBjMDEDiAQEzuAQEzsAAIxk1Qx\nW+NXKpWYN28eUlJSkJ6ejqSkJFy+fFltnX379uGPP/5ARkYGEhIS8Oqrr5orjkguiB1AIMpteVLN\nTrktyzS5zdb409LS4O/vD7lcDjs7O0RGRiI5OVltnV27dmHKlCkAgO7du+PBgwe4e/euuSKJ4IHY\nAQSi3JYn1eyU27JMk9tsjT8nJwctW7ZULfv6+iInJ0fnOrdu3TJXJEIIITBj43/6TrVu1d9x1vd2\n0pApdgCBMsUOIFCm2AGMkCl2AIEyxQ4gUKbYAQTKNEkVW5NU0cDHxwfZ2dmq5ezsbPj6+ta6zq1b\nt+Dj41OjVlBQkJX9QTAky7eGVTbr/aTcNSpbTW7AkOyUW2N1A9eX6mNF/9xBQUEaLzdb4w8NDUVG\nRgYyMzPRokULbN26FUlJSWrrDB06FHFxcYiMjERqairc3NzQtGnTGrUuXJDqGzGEEGJ9zNb4bW1t\nERcXh/DwcCiVSsyYMQOBgYGIj48HAERFRWHw4MHYt28f/P394ejoiLVr15orDiGEkP+RxCd3CSGE\nmA59ctdELl++jB9//BHFxcVql6ekpIiUSD8nTpxAeno6AIAxhmXLluHHH38UOVX9cfz4cSxfvhwH\nDx4UO0qtUlNTUVBQAAAoLS3FP//5T0RERGDRokWqy63RypUr1d5HlIrHjx/j22+/xeHDhwEAmzZt\nwty5c7Fq1So8efLE6Pr0jN8EVq5ciVWrViEwMBDnz5/HF198geHDhwMAunTpYrWfSF68eDGOHj0K\npVKJ/v3746effsLLL7+MQ4cOYciQIXjrrbfEjqi3yZMnY/369WLH0Klbt25IS0sDACQmJmLVqlUY\nMWIEDh48iIiICCxevFjkhJp16NABFy9ehK2tLWbNmgVHR0e88sorOHz4MC5evIidO3eKHVEjV1dX\nODg4oE2bNhg/fjxGjx4Nb29vsWPpNH78eCiVSpSWlsLNzQ3FxcUYOXKk6g/Bt98a9sZ0DZwY7dln\nn+VFRUWcc85v3LjBQ0JC+Oeff8455zw4OFjMaLUKDAzkT5484SUlJdzJyYk/ePCAc855aWkp79Sp\nk8jptIuIiOBDhgzhERERqh8HBwfV5das6uMhJCSE37t3j3POeXFxMX/22WfFiqVTQECA6t9dunRR\nu65z586WjqO34OBgrlQq+YEDB/i0adO4l5cXDw8P5+vWreOFhYVix9OqY8eOnHPOnzx5wr29vfmT\nJ08455xXVFSorjOG2d7crU8453BycgIAyOVyMMYwatQoZGVlWc1ZRTWxt7eHra0tbG1t0aZNG7i6\nugIAGjduDBsb650C3rp1Cx06dMDMmTNhY2MDzjnOnDmDhQsXih1NJ6VSiby8PHDOoVQqVc8+HR0d\nYWtrvb+Ozz77LNasWYPp06cjKCgIv/zyC7p27YqrV6/C3t5e7Hi1srGxwYABAzBgwACUlZVh//79\nSEpKwoIFC5Cbmyt2PI0qKirw+PFjlJaW4uHDhygoKICnpycePXqEiooKo+tb72+3hDRp0kTtkFMn\nJyfs2bMHf/31Fy5evChisto1bNgQpaWlAIBz586pLn/w4IFVN/4zZ84gJCQEH3/8MVxcXKBQKNCo\nUSP069cP/fr1EzterQoLCxESEoLQ0FAUFBTgzp07AICioiKRk9Xum2++wbFjx9C6dWukp6ejV69e\neOaZZzBz5kx88803YsfTm729PYYNG4YtW7YgKytL7DhaTZw4EYGBgejRoweWL1+O559/HjNnzkTX\nrl1Vp7kxBs34TSA7Oxt2dnZo1qyZ2uWcc5w8eRJ9+vQRKVntHj16hEaNGtW4PDc3F3fu3EGnTp1E\nSKW/W7du4c0330STJk2wa9cuSb6JV6m0tBR3797FM888I3aUWhUUFODGjRsoLy+Hr69vjce8tbly\n5Qrat28vdgxBMjMz4eLiAg8PD1y7dg1nzpxBQECA1g9lGYIaP5G8PXv24NSpU/jkk0/EjkKIJFDj\nJ4SQesZ6B7mEEELMgho/IYTUM9T4CSGknqHGT4gAmZmZVn/UEyHaUOMnhJB6hho/IUa6fv06nnvu\nOZw9e1bsKIToxXo/I06IBFy5cgXjxo3Dt99+S6MfIhnU+AkR6N69exg+fDi+//57BAQEiB2HEL3R\nqIcQgdzc3NCqVSscP35c7CiEGISe8RMikL29PXbu3Inw8HA4OTlh3LhxYkciRC/U+AkRSCaTwcHB\nAXv27EFYWBicnZ0REREhdixCdKJz9RBCSD1DM35CCKlnqPETQkg9Q42fEELqGWr8hBBSz1DjJ4SQ\neoYaPyGE1DPU+AkhpJ75f43uqGjLl8N0AAAAAElFTkSuQmCC\n",
 623        "text": [
 624         "<matplotlib.figure.Figure at 0x10e531e50>"
 625        ]
 626       }
 627      ],
 628      "prompt_number": 33
 629     },
 630     {
 631      "cell_type": "markdown",
 632      "metadata": {},
 633      "source": [
 634       "So the optimal value for $k$ appears to be $k=4$."
 635      ]
 636     },
 637     {
 638      "cell_type": "heading",
 639      "level": 3,
 640      "metadata": {},
 641      "source": [
 642       "Exercise 22.8 (Wholesale data)"
 643      ]
 644     },
 645     {
 646      "cell_type": "markdown",
 647      "metadata": {},
 648      "source": [
 649       "As a final exercise in this notebook, we will consider some data about commercial wholesale data, available from the UCI machine learning repository. The data is held in csv format, so we will use the `read_csv` function from pandas to extract it."
 650      ]
 651     },
 652     {
 653      "cell_type": "code",
 654      "collapsed": false,
 655      "input": [
 656       "import pandas as pd \n",
 657       "\n",
 658       "wholesale = pd.read_csv('https://archive.ics.uci.edu/ml/machine-learning-databases/00292/Wholesale%20customers%20data.csv',\n",
 659       "                        delimiter=',')"
 660      ],
 661      "language": "python",
 662      "metadata": {},
 663      "outputs": [],
 664      "prompt_number": 34
 665     },
 666     {
 667      "cell_type": "markdown",
 668      "metadata": {},
 669      "source": [
 670       "Each row of the Wholesale data represents a customer\u2019s annual purchases (in monetary units) of six different product categories, along with the purchasing channel and region.\n",
 671       "\n",
 672       "To see the first five rows of the data and the column headings, use the command `wholesale.head()`:"
 673      ]
 674     },
 675     {
 676      "cell_type": "code",
 677      "collapsed": false,
 678      "input": [
 679       "wholesale.head()"
 680      ],
 681      "language": "python",
 682      "metadata": {},
 683      "outputs": [
 684       {
 685        "html": [
 686         "<div style=\"max-height:1000px;max-width:1500px;overflow:auto;\">\n",
 687         "<table border=\"1\" class=\"dataframe\">\n",
 688         "  <thead>\n",
 689         "    <tr style=\"text-align: right;\">\n",
 690         "      <th></th>\n",
 691         "      <th>Channel</th>\n",
 692         "      <th>Region</th>\n",
 693         "      <th>Fresh</th>\n",
 694         "      <th>Milk</th>\n",
 695         "      <th>Grocery</th>\n",
 696         "      <th>Frozen</th>\n",
 697         "      <th>Detergents_Paper</th>\n",
 698         "      <th>Delicassen</th>\n",
 699         "    </tr>\n",
 700         "  </thead>\n",
 701         "  <tbody>\n",
 702         "    <tr>\n",
 703         "      <th>0</th>\n",
 704         "      <td> 2</td>\n",
 705         "      <td> 3</td>\n",
 706         "      <td> 12669</td>\n",
 707         "      <td> 9656</td>\n",
 708         "      <td> 7561</td>\n",
 709         "      <td>  214</td>\n",
 710         "      <td> 2674</td>\n",
 711         "      <td> 1338</td>\n",
 712         "    </tr>\n",
 713         "    <tr>\n",
 714         "      <th>1</th>\n",
 715         "      <td> 2</td>\n",
 716         "      <td> 3</td>\n",
 717         "      <td>  7057</td>\n",
 718         "      <td> 9810</td>\n",
 719         "      <td> 9568</td>\n",
 720         "      <td> 1762</td>\n",
 721         "      <td> 3293</td>\n",
 722         "      <td> 1776</td>\n",
 723         "    </tr>\n",
 724         "    <tr>\n",
 725         "      <th>2</th>\n",
 726         "      <td> 2</td>\n",
 727         "      <td> 3</td>\n",
 728         "      <td>  6353</td>\n",
 729         "      <td> 8808</td>\n",
 730         "      <td> 7684</td>\n",
 731         "      <td> 2405</td>\n",
 732         "      <td> 3516</td>\n",
 733         "      <td> 7844</td>\n",
 734         "    </tr>\n",
 735         "    <tr>\n",
 736         "      <th>3</th>\n",
 737         "      <td> 1</td>\n",
 738         "      <td> 3</td>\n",
 739         "      <td> 13265</td>\n",
 740         "      <td> 1196</td>\n",
 741         "      <td> 4221</td>\n",
 742         "      <td> 6404</td>\n",
 743         "      <td>  507</td>\n",
 744         "      <td> 1788</td>\n",
 745         "    </tr>\n",
 746         "    <tr>\n",
 747         "      <th>4</th>\n",
 748         "      <td> 2</td>\n",
 749         "      <td> 3</td>\n",
 750         "      <td> 22615</td>\n",
 751         "      <td> 5410</td>\n",
 752         "      <td> 7198</td>\n",
 753         "      <td> 3915</td>\n",
 754         "      <td> 1777</td>\n",
 755         "      <td> 5185</td>\n",
 756         "    </tr>\n",
 757         "  </tbody>\n",
 758         "</table>\n",
 759         "</div>"
 760        ],
 761        "metadata": {},
 762        "output_type": "pyout",
 763        "prompt_number": 35,
 764        "text": [
 765         "   Channel  Region  Fresh  Milk  Grocery  Frozen  Detergents_Paper  Delicassen\n",
 766         "0        2       3  12669  9656     7561     214              2674        1338\n",
 767         "1        2       3   7057  9810     9568    1762              3293        1776\n",
 768         "2        2       3   6353  8808     7684    2405              3516        7844\n",
 769         "3        1       3  13265  1196     4221    6404               507        1788\n",
 770         "4        2       3  22615  5410     7198    3915              1777        5185"
 771        ]
 772       }
 773      ],
 774      "prompt_number": 35
 775     },
 776     {
 777      "cell_type": "markdown",
 778      "metadata": {},
 779      "source": [
 780       "For the `Channel` column, `1` represents a purchase made in a hotel, a restaurant or a caf\u00e9 and `2` represents a purchase made through a retail outlet. For the `Region` column, `1` represents Lisbon, `2` represents Porto, and `3` represents Other. "
 781      ]
 782     },
 783     {
 784      "cell_type": "markdown",
 785      "metadata": {},
 786      "source": [
 787       "Suppose a senior manager of this wholesale company wants to group customers who are in Lisbon based on their purchasing statistics. What is a suitable (optimal) number of clusters of Lisbon customers? (Use the Silhouette method.)"
 788      ]
 789     },
 790     {
 791      "cell_type": "heading",
 792      "level": 4,
 793      "metadata": {},
 794      "source": [
 795       "Lisbon:"
 796      ]
 797     },
 798     {
 799      "cell_type": "markdown",
 800      "metadata": {},
 801      "source": [
 802       "As before, we will run the clustering algorithm on a range of values of $k$, and use the silhouette function to determine which value of $k$ gives the most successful cluster analysis."
 803      ]
 804     },
 805     {
 806      "cell_type": "code",
 807      "collapsed": false,
 808      "input": [
 809       "import sklearn.cluster as sc\n",
 810       "\n",
 811       "# convert data into array\n",
 812       "x = array(wholesale)\n",
 813       "\n",
 814       "# run k-means for k = 2, 3, 4, 5, 6, 7, 8\n",
 815       "s = zeros(6)\n",
 816       "c = 0\n",
 817       "for i in range(2,8):\n",
 818       "    kmeans = sc.KMeans(n_clusters=i)\n",
 819       "    x_lisbon = x[find(equal(x[:,1],1)),:] # consider only Lisbon data\n",
 820       "    assigned_clusters = kmeans.fit(x_lisbon).labels_\n",
 821       "    s[c] = average(silhouette(x_lisbon, assigned_clusters))\n",
 822       "    c = c + 1\n",
 823       "    "
 824      ],
 825      "language": "python",
 826      "metadata": {},
 827      "outputs": [
 828       {
 829        "metadata": {},
 830        "output_type": "display_data",
 831        "png": 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 832        "text": [
 833         "<matplotlib.figure.Figure at 0x10f528e50>"
 834        ]
 835       },
 836       {
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 840        "text": [
 841         "<matplotlib.figure.Figure at 0x10f528ad0>"
 842        ]
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 848        "text": [
 849         "<matplotlib.figure.Figure at 0x10edca450>"
 850        ]
 851       },
 852       {
 853        "metadata": {},
 854        "output_type": "display_data",
 855        "png": 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 856        "text": [
 857         "<matplotlib.figure.Figure at 0x10f6c1f50>"
 858        ]
 859       },
 860       {
 861        "metadata": {},
 862        "output_type": "display_data",
 863        "png": 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 864        "text": [
 865         "<matplotlib.figure.Figure at 0x10fae6850>"
 866        ]
 867       },
 868       {
 869        "metadata": {},
 870        "output_type": "display_data",
 871        "png": 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 872        "text": [
 873         "<matplotlib.figure.Figure at 0x10ed8e5d0>"
 874        ]
 875       }
 876      ],
 877      "prompt_number": 36
 878     },
 879     {
 880      "cell_type": "markdown",
 881      "metadata": {},
 882      "source": [
 883       "Again, we make a bar plot of $s$ to evaluate the final results:"
 884      ]
 885     },
 886     {
 887      "cell_type": "code",
 888      "collapsed": false,
 889      "input": [
 890       "# display result on a bar chart\n",
 891       "bar(range(2,8), s)\n",
 892       "ylabel('Average Silhouette')\n",
 893       "xlabel('k')"
 894      ],
 895      "language": "python",
 896      "metadata": {},
 897      "outputs": [
 898       {
 899        "metadata": {},
 900        "output_type": "pyout",
 901        "prompt_number": 37,
 902        "text": [
 903         "<matplotlib.text.Text at 0x10fd31510>"
 904        ]
 905       },
 906       {
 907        "metadata": {},
 908        "output_type": "display_data",
 909        "png": 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qKkobNmzQlClTFBMTo9tuuy3YbQFnhUtJA4DBWBMAAIMRAgBgMEIAAAxGCACA\nwQgBADAYIQAABiMEAMBghAAAGOz/AQnqiJ73DLAWAAAAAElFTkSuQmCC\n",
 910        "text": [
 911         "<matplotlib.figure.Figure at 0x10faa67d0>"
 912        ]
 913       }
 914      ],
 915      "prompt_number": 37
 916     },
 917     {
 918      "cell_type": "markdown",
 919      "metadata": {},
 920      "source": [
 921       "In this case, the optimum value appears to be $k=3$."
 922      ]
 923     }
 924    ],
 925    "metadata": {}
 926   }
 927  ]
 928 }