aula19-clustering.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Aula 19 &mdash; Aprendizado não-supervisionado - Agrupamento\n",
    "\n",
    "Renato Vimieiro\n",
    "\n",
    "rv2 {em} cin.ufpe.br\n",
    "\n",
    "maio 2017"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Introdução\n",
    "\n",
    "Dedicamos as últimas aulas ao estudo de modelos para aprendizado supervisionado. No caso supervisionado, o objetivo do algoritmo de aprendizado é ajustar um modelo capaz de predizer um valor para uma variável dependente a partir de um conjunto de variáveis independentes. Vimos que esse problema está relacionado tanto a variáveis contínuas (regressão) quanto variáveis categóricas (classificação). No problema de aprendizado não-supervisionado, os algoritmos não levam em consideração a variável dependente (se ela existir). \n",
    "\n",
    "Os algoritmos de agrupamento ou clustering são classificados como algoritmos de aprendizado não-supervisionado. O objetivo desses algoritmos é encontrar subgrupos de exemplos que concentram mais características em comum entre si do que com exemplos em outros grupos. Essa tarefa pode, por exemplo, preceder o aprendizado supervisionado. Podemos querer identificar grupos de clientes para direcionar nossa campanha de marketing e, em seguida, podemos aplicar técnicas de aprendizado supervisionado, e.g. regressão linear, para ajustar um modelo para cada segmento. No exemplo que consideramos durante as aulas de regressão linear, isso significaria verificar a efetividade (aumento de vendas) de nossa campanha para cada grupo de clientes. Dessa forma, poderíamos constatar que entre um público mais jovem campanhas majoritariamente online e TV seriam mais efetivas que em rádio e jornal impresso. Por outro lado, campanhas direcionadas a idosos seriam o inverso. No caso de classificação, poderíamos encontrar grupos de pacientes com câncer de próstata similares e depois ajustar um modelo baseado em rótulos atribuídos a esses grupos.\n",
    "\n",
    "Os algoritmos de clustering dividem as amostras em grupos baseado em um função de (dis)similaridade/distância. A escolha da função de distância é dependente dos dados. As mais populares são: euclidiana, cosseno e correlação para dados numéricos; e manhattan e jaccard para dados categóricos. Os algoritmos podem buscar os grupos dividindo as amostras em subgrupos (o conjunto de todas é um único cluster), sendo chamados de **particionais**; ou podem reunir amostras mais comuns em um grupo (cada amostra é inicialmente um grupo), sendo chamados de **aglomerativos**. O representante mais conhecido do universo particional é o *K-means*, enquanto o aglomerativo mais conhecido é o *hierárquico*. Estudaremos ambos nessa aula."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Distância\n",
    "\n",
    "As distâncias mencionadas acima são formalizadas pelas fórmulas a seguir. Nesse contexto, imaginamos um conjunto de dados com $n$ amostras cada uma descrita por $p$ atributos. Seguem as distâncias:\n",
    "\n",
    "- Euclidiana\n",
    "$$\n",
    "d(x,z) = ||x-z||_2 = \\sqrt{\\sum_j^p (x_j - z_j)^2}\n",
    "$$\n",
    "\n",
    "- Cosseno\n",
    "$$\n",
    "d(x,z) = 1 - \\frac{x\\cdot z}{||x||_2 ||z||_2} = 1-\\frac{\\sum x_jz_j}{\\sqrt{\\sum x_j^2}\\sqrt{\\sum z_j^2}}\n",
    "$$\n",
    "\n",
    "- Baseada na correlação de Pearson\n",
    "$$\n",
    "d(x,z) = 1-\\frac{cov(x,z)}{\\sigma_x \\sigma_z} = 1-\\frac{\\sum (x_j-\\bar{x})(z_j-\\bar{z})}{\\sqrt{\\sum (x_j-\\bar{x})^2}\\sqrt{\\sum (z_j-\\bar{z})^2}}\n",
    "$$\n",
    "\n",
    "- Manhattan\n",
    "$$\n",
    "d(x,z) = ||x-z||_1 = \\sum_j^p |x_j - z_j|\n",
    "$$\n",
    "\n",
    "- Jaccard\n",
    "$$\n",
    "d(x,z) = 1 - \\frac{|x\\cap z|}{|x\\cup z|}\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## K-Means\n",
    "\n",
    "Como discutimos anteriormente, o K-Means é um algoritmo particional. O funcionamento do algoritmo é bastante simples. Inicialmente, o algoritmo gera aleatoriamente $k$ representantes, um para cada grupo. Em seguida, o algoritmo assinala as amostras aos grupos aos quais elas são mais próximas. A distância é avaliada com base no representante do grupo. Feito isso, novos representantes são escolhidos para cada grupo e o processo se repete. \n",
    "Os representantes são gerados com base nas médias dos atributos das amostras dos grupos, daí o nome do algoritmo. \n",
    "O processo segue até que os representantes (**centroids**) não mudem, ou que a diferença entre os centroids atuais e anteriores seja mínima.\n",
    "\n",
    "Formalmente, esse processo visa a minimizar a seguinte função objetivo\n",
    "\n",
    "$$\n",
    "SSE(C) = \\sum_i^k\\sum_{x_j\\in C_i}||x_j - \\mu_i||^2 = \\sum_i^k\\sum_{x_j\\in C_i}\\sum^p (x_{jl} - \\bar{x_{il}})^2\n",
    "$$\n",
    "\n",
    "Para ilustar o processo, vamos inicialmente gerar um conjunto de dados artificial dividio em três grupos."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "import numpy as np\n",
    "import seaborn as sns\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.axes._subplots.AxesSubplot at 0x10f94f780>"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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1eG3xnbjHokt5ZXGhwpt4E+7tncOiRQvYvn0k46iUBqeHpHyyuJDUtbr9WPR2GqHZRE15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\nhSRJSpXFhSRJStX/B1xxmK7V24jYAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10f9425f8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x1 = np.random.normal(loc=-1,scale=0.7,size=20)\n",
    "x2 = np.random.normal(loc=0,scale=1.,size=20)\n",
    "x3 = np.random.normal(loc=1,scale=0.5,size=20)\n",
    "\n",
    "y1 = np.random.normal(loc=1,scale=0.5,size=20)\n",
    "y2 = np.random.normal(loc=-2,scale=0.5,size=20)\n",
    "y3 = np.random.normal(loc=0,scale=0.5,size=20)\n",
    "\n",
    "df = pd.DataFrame(dict(x=np.concatenate([x1,x2,x3]),y=np.concatenate([y1,y2,y3]),c=np.repeat([1,2,3],20)))\n",
    "df.plot.scatter(x='x',y='y')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([-0.13682671, -0.20222997]),\n",
       " array([-1.65669336,  1.5451075 ]),\n",
       " array([ 0.81076057,  1.41638   ]))"
      ]
     },
     "execution_count": 85,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c1 = np.random.uniform(-2,2,size=2)\n",
    "c2 = np.random.uniform(-2,2,size=2)\n",
    "c3 = np.random.uniform(-2,2,size=2)\n",
    "c1,c2,c3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x111aad780>"
      ]
     },
     "execution_count": 86,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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de0afaQI26dqf15VancWF1ADfKCfPJl0H53WlVmdxITXAN8rJs0nXwXldqdVZ\nXEgN8I1y8pImXcs4bOAepj9616hD05TwulKrK+1fHRdUef/DmdpNZ+c05s6dfcAhVe1mKvLQ198/\nqpvliuWtsarfayKR1zxkcV3lNRfNZh6qhnMx6SEzi4s24Q9LwjxUjZWLdtsC6TVRZS4S5qGq3uLC\naRFJ+7gFUlIact9EK4RwKLAaeDlJMfQV4B0xxj9kGphUQONtgVztWSOSJqgVRi4+CZwAvAx4CXA8\n8KlMI5IKyi2QktKQ6+IihDALWAqcG2P8UYzxfuAdwKtCCIdkG51UPG6BlJSGXBcXwBDJdMgDI24r\nAR1AdyYRSQXmFkhJacj1mosY427g6/vdfD7QF2PckkFIUqElPSgYtQXyAntQSJqkzIuLEEIX8OQx\n7t4UY9w54ntXAsuAlzYjNulgirhtc9HChVzl4k1JDci8uABOBr4N1Fox9irgiwAhhLcBVwPnxxjv\nmOw/0tGR9xmgqVV5/eYhvTz0bezngx9dx6xjqkeHf/Cj67jkvGksOiH/b85eEwnzUGUuEuahqt4c\ntEQTrRDCu4ArgAtjjB+u4yny/yLVcv7u7RezafqfjtpdUS6XedLj9/LPH/lghpFJUqomvaI7DyMX\n4wohrAAuJxmx+Ei9z7Nt2y4GB9u301pHxzR6emaahxTzsGX745QOO3Db5pbte9m6daCh524Gr4mE\neagyFwnzUFXJxWTlurgIIcwFPgJcD6wNIRw54u7fxRgn/H99cHCo7du4gnmoSCMPsw8psadcPmDk\nYvYMWirHXhMJ81BlLhLmoX55n1B6CTAbWAH8dvjPpuG/j84wLsltm5I0hlyPXMQYPwt8Nus4pFrc\ntilJteW6uJAatf9W0Te+dimnv+Dk1J7fbZuSdCCLCxVW5YTP2fOrW0Xff81arurp4tj5x2UdniQV\nlsWFCqvWCZ+zjjmNj1+7lisuuyizuIrYeEuSRsr7gk6pbmOd8LltV3arvyujKVu6F/P44aewpXsx\nvWvW0dffn1lMkpQ2iwsV1lgnfPbMzO6yrzWaMnv+aVx/622ZxSRJabO4UGHV2iq686E7eesbz8gs\nprFGU3bstomspOKwuFBhJVtFl3HYwD1Mf/Qu5g1s4JLzzuCkE5+ZWUxjjaZ0d026u64k5ZYLOlVo\n+28V7ezMtp5eceaS6g6WUmlf460LVi7LNC5JSpPFhdRENt6S1A4sLqQms/GWpKJzzYUkSUqVxYUk\nSUqVxYVofwx6AAAKyUlEQVQkSUqVay4kTSnbnUvtx5ELSVPGdudSe7K4kDRlbHcutSeLC0lTxnbn\nUnuyuJA0ZWx3LrUnF3RKBZHHhZO2O5fakyMXUgHkdeFkrcPjVtnuXCo8Ry6kAhhv4eTqjFuN2+5c\naj+OXEgF4MJJSXlicSEVgAsnJeWJxYVUACvOXMLAg3fuKzAqCydXLF+ScWSS2pFrLqQCSBZOwg23\n3s723UN0d5W4wIWTkjJicSEVhAsnJeWF0yKSJClVjlxIypU8NgOTNDmOXEjKjbw2A5M0ORYXknLD\nU1SlYrC4kJQbNgOTisHiQlJu2AxMKgaLC0m5YTMwqRhyv1skhPAE4GPAi4GdwA3ARTHGoUwDk5Q6\nm4FJxZD74gK4CRgCTgYOB24Gfg/8ryyDkjQ1bAYmtb5cFxchhEOAh4FLY4y/BGIIYR1waraRSZKk\nseS6uIgxPga8vvJ1COEZwCuAT2QWlCRJGlfLLOgMIfwLsBHYSrIGQ5Ik5VDmIxchhC7gyWPcvSnG\nuHP4v98OzAXWALcCr5zMv9PR0TJ11JSovH7zYB4qzEXCPFSZi4R5qKo3B6X995Q3WwjhdODbQK1A\nXhVj/OJ+3/9s4F7gqTHGhyb4z9iBR5Kk+ky60UzmxcV4QghzgJfFGNeOuG0mMAA8J8Z43wSfqrxt\n2y4GB9t392pHxzR6emZiHsxDhblImIcqc5EwD1XDuZh0cZH5tMhBzAJuDSE8GGP83vBtzwH2Aj+f\nzBMNDg6xd297XyRgHirMQ5W5SJiHKnORMA/1y/WEUozxEeDzwJoQwokhhNOAfwauiTHuyDY6SZJU\nS66Li2HnAA8AXycpNL4EvCfTiCRJ0pjyPi1CjHE78LdZxyFJkiamFUYuJElSC7G4kCRJqbK4kCRJ\nqbK4kCRJqbK4kCRJqbK4kCRJqbK4kCRJqbK4kCRJqbK4kCRJqbK4kCRJqbK4kCRJqbK4kCRJqbK4\nkCRJqbK4kCRJqbK4kCRJqbK4kCRJqbK4kCRJqbK4kCRJqbK4kCRJqbK4kCRJqbK4kCRJqbK4kCRJ\nqbK4kCRJqbK4kCRJqbK4kCRJqbK4kCRJqbK4kCRJqbK4kCRJqbK4kCRJqbK4kCRJqbK4kCRJqbK4\nkCRJqWqp4iKE8LEQwrezjkOSJI2tZYqLEMLzgTcD5axjkSRJY2uJ4iKEMB34JHB31rFIkqTxtURx\nAfw98ADwzawDkSRJ48t9cRFCeDrwFuCCrGORJEkH15l1ACGELuDJY9y9iWQ65H0xxt+FEOr+dzo6\ncl9HTanK6zcP5qHCXCTMQ5W5SJiHqnpzkHlxAZwMfJvaCzX/HpgWY/zfDf4bpZ6emQ0+RTGYh4R5\nqDIXCfNQZS4S5qF+pXI5v5svQgjfAhYDe4dvOgToAHYCC2KMv8kqNkmSVFseRi7GczYwsnQ8H3gu\ncBbw20wikiRJ48p1cRFj3DTy6xDCFmBXjPFXGYUkSZIOwtUqkiQpVblecyFJklqPIxeSJClVFheS\nJClVFheSJClVFheSJClVFheSJClVue5zkZYQwhOAjwEvJunueQNwUYxxKNPAMhBCOBRYDbycpLj8\nCvCOGOMfMg0sQyGErwE3xRhvyDqWZgkhzCD5mVhK8jOxOsb4j9lGlZ3hfPwAODfG+K9Zx9NsIYSj\ngGuAF5JcD2uBv48xPpZpYBkIIfwx8FHgFGAzsCbGeFW2UWUrhPAV4JEY4zkTfUy7jFzcBMwhOcfk\n1cCZwLszjSg7nwROAF4GvAQ4HvhUphFlJIRQCiF8BHhR1rFk4CrgJODPgLcBl4QQlmYaUUaGC4tb\ngAVZx5KhzwNdJG+oy4G/Aj6QaUQZCCGUSD5wPQKcSHIi96oQwvJMA8vQ8Gt/2WQfV/jiIoRwCPAw\n8LaYuAtYB5yabWTNF0KYRfJJ9dwY449ijPcD7wBeNZyntjH8Se0OkhGc32ccTlMNXwdvBM6LMT4Q\nY7wduAJYmW1kzRdCOB64Bzg261iyEpLjpp8L/E2M8WfDvyPfR3LMQrs5EvgRyfvFv8cYv0rye6Lt\n3i8AQghzSX43fH+yjy38tMjwsN7rK1+HEJ4BvAL4RGZBZWeI5M30gRG3lUgOg+sGtmQRVEZOAh4C\nlgE/zDiWZnsmyc/+hhG3fRe4KJtwMnU6yZvHKpLpgHb0MPCyGOOjI24rAYdmFE9mYowPk4xsAxBC\nOAV4AckIRju6imQZwZMn+8DCFxcjhRD+heRC+QHJfHNbiTHuBr6+383nA30xxnYqLIgxfhn4MkDy\nwa2tPAl4NMa4d8RtjwBdIYTDYoybM4qr6WKM+z5ktOF1AMDweqt9vxeGpwZWAt/MLKgcCCH8GngK\nye+J9ZkGk4EQwp8Dp5FMo0/6w3ghiosQQhdjV1abYoyVTyRvB+YCa4BbgVc2IbymmkQuCCGsJPnk\n/tJmxNZMk8lDG5oF7NnvtsrXM5oci/LnSpL1Bs/JOpCMLQWeSPLG+k8kH8TawvA6pI+TTA/tqafw\nLkRxQbJQ89tArYNSXgV8ESDGuBEghPAG4N4QwjExxoeaFmVzTCgXIYS3AVcD58cY72heeE0zoTy0\nqd0cWERUvm7noqvthRAuB84Dzogx/jTreLIUY7wPIIRwAXBjCOHC/Ub7iuxS4AcxxrpHrwpRXMQY\nv8MYi1NDCHNCCGfEGNeOuPknw38fTjLvXhjj5aIihPAukkU6F8YY1zQlsCabSB7a2H8Ch4cQpo3Y\njv1EYFeMsa0Wt6pqeOfUm4GzY4y3ZR1PFkIIRwCLhxc5V/wEOATooX3Wpb0GODKEsH346xkAIYRl\nMcaeiTxBO/zynQXcGkI4ecRtzwH2Aj/PJqTshBBWAJeTjFh8OOt4lIn7gceB54247TTg3mzCUdZC\nCJcAbwJeE2P8XNbxZOhYYH0I4YkjbnsO8Ls2W5d2Oslai2cO//kicPvwf09IIUYuxhNjfCSE8Hlg\nTQjh70j6XfwzcE2McUe20TXX8LaijwDXA2tDCEeOuPt37dhUrB3FGHeFEG4APhFCOAc4GrgQ+JtM\nA1MmhrfjrgI+BNw98vdCjPGRzALLxr0kC/6vCyG8k6TYuBzozTSqJosx/sfIr4dHMMoxxl9N9Dna\nYeQC4ByS7ZdfJ2kW8yXgPZlGlI2XALOBFcBvh/9sGv776AzjylqtdRlF906SLbjfIik439uuQ+Ej\ntON1AMnW/GkkBcb+vxfayvAHrFcCA8DdJA0Gry7q9PFUKpXL7frzJEmSpkK7jFxIkqQmsbiQJEmp\nsriQJEmpsriQJEmpsriQJEmpsriQJEmpsriQJEmpsriQJEmpsriQJEmpsriQJEmpsriQJEmpsriQ\nJEmpsriQ1FQhhKUhhKEQwqtG3HZzCOFXIYRDs4xNUjo8FVVS04UQrgdeBCwA/hK4Hjg9xrgh08Ak\npaIz6wAktaWVwAPAtcCfA5dZWEjF4bSIpKaLMW4H3ggsBX4BfDDbiCSlyeJCUlaeA+wFAjA/41gk\npcjiQlLThRBOAC4D3gTcB3wm24gkpcniQlJThRCmAzcC34ox/h/gb4GTQgjvyTQwSamxuJDUbB8k\nmQb5O4AY478D7wMuDSEsyjIwSelwK6okSUqVIxeSJClVFheSJClVFheSJClVFheSJClVFheSJClV\nFheSJClVFheSJClVFheSJClVFheSJClVFheSJClVFheSJClV/x9lacFID3GLRwAAAABJRU5ErkJg\ngg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x111978860>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ax = df.plot.scatter(x='x',y='y')\n",
    "ax.scatter(*c1,color='red',marker='^',s=50)\n",
    "ax.scatter(*c2,marker='d',color='green',s=50)\n",
    "ax.scatter(*c3,marker='s',color='magenta',s=50)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>c1</th>\n",
       "      <th>c2</th>\n",
       "      <th>c3</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1.273081</td>\n",
       "      <td>1.258080</td>\n",
       "      <td>1.387498</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>1.562645</td>\n",
       "      <td>1.015393</td>\n",
       "      <td>1.500119</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>1.249776</td>\n",
       "      <td>1.066079</td>\n",
       "      <td>1.891891</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>1.762626</td>\n",
       "      <td>1.171655</td>\n",
       "      <td>1.300557</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>1.495759</td>\n",
       "      <td>1.740090</td>\n",
       "      <td>0.759979</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>2.093913</td>\n",
       "      <td>0.540350</td>\n",
       "      <td>2.666412</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>1.235466</td>\n",
       "      <td>1.081134</td>\n",
       "      <td>1.859949</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>0.549764</td>\n",
       "      <td>1.870729</td>\n",
       "      <td>1.493944</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>0.666184</td>\n",
       "      <td>1.712691</td>\n",
       "      <td>1.534769</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>3.012438</td>\n",
       "      <td>0.699721</td>\n",
       "      <td>2.951997</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>1.015130</td>\n",
       "      <td>1.323819</td>\n",
       "      <td>1.664219</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>2.439890</td>\n",
       "      <td>1.130602</td>\n",
       "      <td>1.699740</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>1.383605</td>\n",
       "      <td>1.552104</td>\n",
       "      <td>0.986979</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>1.039173</td>\n",
       "      <td>1.367367</td>\n",
       "      <td>2.136672</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>1.544684</td>\n",
       "      <td>0.775002</td>\n",
       "      <td>1.957129</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>1.963393</td>\n",
       "      <td>1.412444</td>\n",
       "      <td>1.123980</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>1.672082</td>\n",
       "      <td>0.655674</td>\n",
       "      <td>2.165343</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>0.251738</td>\n",
       "      <td>2.112985</td>\n",
       "      <td>1.959088</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18</th>\n",
       "      <td>2.315044</td>\n",
       "      <td>0.316554</td>\n",
       "      <td>2.692657</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>2.199114</td>\n",
       "      <td>0.957290</td>\n",
       "      <td>1.653515</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>20</th>\n",
       "      <td>1.124327</td>\n",
       "      <td>3.032667</td>\n",
       "      <td>2.987127</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>21</th>\n",
       "      <td>1.244120</td>\n",
       "      <td>3.558506</td>\n",
       "      <td>2.611616</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>22</th>\n",
       "      <td>1.604327</td>\n",
       "      <td>3.402580</td>\n",
       "      <td>3.464848</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>23</th>\n",
       "      <td>0.984871</td>\n",
       "      <td>2.562515</td>\n",
       "      <td>2.836247</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>24</th>\n",
       "      <td>1.213370</td>\n",
       "      <td>3.391375</td>\n",
       "      <td>2.932862</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25</th>\n",
       "      <td>2.700031</td>\n",
       "      <td>4.893671</td>\n",
       "      <td>4.225086</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>26</th>\n",
       "      <td>0.518245</td>\n",
       "      <td>2.760275</td>\n",
       "      <td>2.309506</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>27</th>\n",
       "      <td>1.618174</td>\n",
       "      <td>3.885007</td>\n",
       "      <td>3.129332</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>28</th>\n",
       "      <td>1.104069</td>\n",
       "      <td>3.137747</td>\n",
       "      <td>2.933055</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>29</th>\n",
       "      <td>1.689543</td>\n",
       "      <td>3.848683</td>\n",
       "      <td>3.363370</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>30</th>\n",
       "      <td>1.448419</td>\n",
       "      <td>3.756722</td>\n",
       "      <td>2.582407</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>31</th>\n",
       "      <td>1.310801</td>\n",
       "      <td>3.516953</td>\n",
       "      <td>2.982306</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>32</th>\n",
       "      <td>2.059328</td>\n",
       "      <td>3.948721</td>\n",
       "      <td>3.873746</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>33</th>\n",
       "      <td>1.034245</td>\n",
       "      <td>2.542632</td>\n",
       "      <td>2.876094</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>34</th>\n",
       "      <td>2.186509</td>\n",
       "      <td>3.784751</td>\n",
       "      <td>4.057023</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>35</th>\n",
       "      <td>2.295226</td>\n",
       "      <td>4.600447</td>\n",
       "      <td>3.225987</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>36</th>\n",
       "      <td>2.334269</td>\n",
       "      <td>4.517581</td>\n",
       "      <td>3.907930</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>37</th>\n",
       "      <td>1.903519</td>\n",
       "      <td>3.698791</td>\n",
       "      <td>3.752773</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>38</th>\n",
       "      <td>1.093510</td>\n",
       "      <td>3.207708</td>\n",
       "      <td>2.882379</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>39</th>\n",
       "      <td>2.380299</td>\n",
       "      <td>4.387815</td>\n",
       "      <td>4.116673</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>40</th>\n",
       "      <td>1.734930</td>\n",
       "      <td>2.673933</td>\n",
       "      <td>0.341047</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>41</th>\n",
       "      <td>1.877891</td>\n",
       "      <td>3.237492</td>\n",
       "      <td>0.945539</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>42</th>\n",
       "      <td>0.997870</td>\n",
       "      <td>2.076297</td>\n",
       "      <td>0.896565</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>43</th>\n",
       "      <td>1.354488</td>\n",
       "      <td>3.233747</td>\n",
       "      <td>1.442995</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>44</th>\n",
       "      <td>1.026695</td>\n",
       "      <td>2.733287</td>\n",
       "      <td>1.133988</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>45</th>\n",
       "      <td>0.152995</td>\n",
       "      <td>2.175458</td>\n",
       "      <td>1.778355</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>46</th>\n",
       "      <td>0.803642</td>\n",
       "      <td>2.927582</td>\n",
       "      <td>1.659591</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>47</th>\n",
       "      <td>1.003518</td>\n",
       "      <td>3.109336</td>\n",
       "      <td>1.692535</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>48</th>\n",
       "      <td>1.647714</td>\n",
       "      <td>2.971952</td>\n",
       "      <td>0.774555</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>49</th>\n",
       "      <td>1.502094</td>\n",
       "      <td>2.928736</td>\n",
       "      <td>0.856597</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>50</th>\n",
       "      <td>3.081027</td>\n",
       "      <td>4.829448</td>\n",
       "      <td>2.527240</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>51</th>\n",
       "      <td>1.873724</td>\n",
       "      <td>3.641340</td>\n",
       "      <td>1.554465</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>52</th>\n",
       "      <td>0.727406</td>\n",
       "      <td>2.730913</td>\n",
       "      <td>1.470608</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>53</th>\n",
       "      <td>1.650475</td>\n",
       "      <td>3.433325</td>\n",
       "      <td>1.437420</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>54</th>\n",
       "      <td>1.403474</td>\n",
       "      <td>2.521505</td>\n",
       "      <td>0.550494</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>55</th>\n",
       "      <td>1.210058</td>\n",
       "      <td>2.816349</td>\n",
       "      <td>1.025231</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>56</th>\n",
       "      <td>1.255723</td>\n",
       "      <td>3.519686</td>\n",
       "      <td>2.223126</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>57</th>\n",
       "      <td>0.774569</td>\n",
       "      <td>2.919624</td>\n",
       "      <td>1.688968</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>58</th>\n",
       "      <td>0.986192</td>\n",
       "      <td>3.284822</td>\n",
       "      <td>2.533706</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>59</th>\n",
       "      <td>0.533303</td>\n",
       "      <td>2.815461</td>\n",
       "      <td>2.263360</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "          c1        c2        c3\n",
       "0   1.273081  1.258080  1.387498\n",
       "1   1.562645  1.015393  1.500119\n",
       "2   1.249776  1.066079  1.891891\n",
       "3   1.762626  1.171655  1.300557\n",
       "4   1.495759  1.740090  0.759979\n",
       "5   2.093913  0.540350  2.666412\n",
       "6   1.235466  1.081134  1.859949\n",
       "7   0.549764  1.870729  1.493944\n",
       "8   0.666184  1.712691  1.534769\n",
       "9   3.012438  0.699721  2.951997\n",
       "10  1.015130  1.323819  1.664219\n",
       "11  2.439890  1.130602  1.699740\n",
       "12  1.383605  1.552104  0.986979\n",
       "13  1.039173  1.367367  2.136672\n",
       "14  1.544684  0.775002  1.957129\n",
       "15  1.963393  1.412444  1.123980\n",
       "16  1.672082  0.655674  2.165343\n",
       "17  0.251738  2.112985  1.959088\n",
       "18  2.315044  0.316554  2.692657\n",
       "19  2.199114  0.957290  1.653515\n",
       "20  1.124327  3.032667  2.987127\n",
       "21  1.244120  3.558506  2.611616\n",
       "22  1.604327  3.402580  3.464848\n",
       "23  0.984871  2.562515  2.836247\n",
       "24  1.213370  3.391375  2.932862\n",
       "25  2.700031  4.893671  4.225086\n",
       "26  0.518245  2.760275  2.309506\n",
       "27  1.618174  3.885007  3.129332\n",
       "28  1.104069  3.137747  2.933055\n",
       "29  1.689543  3.848683  3.363370\n",
       "30  1.448419  3.756722  2.582407\n",
       "31  1.310801  3.516953  2.982306\n",
       "32  2.059328  3.948721  3.873746\n",
       "33  1.034245  2.542632  2.876094\n",
       "34  2.186509  3.784751  4.057023\n",
       "35  2.295226  4.600447  3.225987\n",
       "36  2.334269  4.517581  3.907930\n",
       "37  1.903519  3.698791  3.752773\n",
       "38  1.093510  3.207708  2.882379\n",
       "39  2.380299  4.387815  4.116673\n",
       "40  1.734930  2.673933  0.341047\n",
       "41  1.877891  3.237492  0.945539\n",
       "42  0.997870  2.076297  0.896565\n",
       "43  1.354488  3.233747  1.442995\n",
       "44  1.026695  2.733287  1.133988\n",
       "45  0.152995  2.175458  1.778355\n",
       "46  0.803642  2.927582  1.659591\n",
       "47  1.003518  3.109336  1.692535\n",
       "48  1.647714  2.971952  0.774555\n",
       "49  1.502094  2.928736  0.856597\n",
       "50  3.081027  4.829448  2.527240\n",
       "51  1.873724  3.641340  1.554465\n",
       "52  0.727406  2.730913  1.470608\n",
       "53  1.650475  3.433325  1.437420\n",
       "54  1.403474  2.521505  0.550494\n",
       "55  1.210058  2.816349  1.025231\n",
       "56  1.255723  3.519686  2.223126\n",
       "57  0.774569  2.919624  1.688968\n",
       "58  0.986192  3.284822  2.533706\n",
       "59  0.533303  2.815461  2.263360"
      ]
     },
     "execution_count": 87,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def dist(x):\n",
    "    return pd.Series(dict(c1=np.linalg.norm(x-c1),c2=np.linalg.norm(x-c2),c3=np.linalg.norm(x-c3)))\n",
    "\n",
    "df[['x','y']].apply(dist,axis=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0    c2\n",
       "1    c2\n",
       "2    c2\n",
       "3    c2\n",
       "4    c3\n",
       "dtype: object"
      ]
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "clusters = df[['x','y']].apply(dist,axis=1).apply(lambda x: x.argmin(),axis=1)\n",
    "clusters.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x111be6320>"
      ]
     },
     "execution_count": 89,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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Q1bjT/T6GJisnRXeClbLL36BqOA8+9GCxRfifgFdTdqM2SVLlDBdqODOPnMnkl6bAK4Gn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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x111a772b0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ax = df[clusters=='c1'].plot.scatter(x='x',y='y',marker='^',color='red')\n",
    "df[clusters=='c2'].plot.scatter(x='x',y='y',marker='d', color='green',ax=ax)\n",
    "df[clusters=='c3'].plot.scatter(x='x',y='y',marker='s',color='magenta', ax=ax)\n",
    "ax.scatter(*c1,color='red',marker='^',s=50)\n",
    "ax.scatter(*c2,marker='d',color='green',s=50)\n",
    "ax.scatter(*c3,marker='s',color='magenta',s=50)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {},
   "outputs": [],
   "source": [
    "c = df.groupby(clusters)[['x','y']].mean()\n",
    "c1 = c.loc['c1',:].values\n",
    "c2 = c.loc['c2',:].values\n",
    "c3 = c.loc['c3',:].values"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x111d26860>"
      ]
     },
     "execution_count": 91,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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r+tH2ex+v3fuKpDvnmFuLS9qHxYU0AVk1cdI4TWXklt5twGzo2zz8eO1mHJWS0mJxIU1A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\nAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11197b588>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ax = df.plot.scatter(x='x',y='y')\n",
    "ax.scatter(*c1,color='red',marker='^',s=50)\n",
    "ax.scatter(*c2,marker='d',color='green',s=50)\n",
    "ax.scatter(*c3,marker='s',color='magenta',s=50)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x110583fd0>"
      ]
     },
     "execution_count": 92,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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NwmgvlG44VZ2bdI3N55XyznAhTYIvlBPnJl1j83mlvDNcSJNQ1BfKUqnE1z59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\n+v8xEhE0RgfGfwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x111d2ec18>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "clusters = df[['x','y']].apply(dist,axis=1).apply(lambda x: x.argmin(),axis=1)\n",
    "ax = df[clusters=='c1'].plot.scatter(x='x',y='y',marker='^',color='red')\n",
    "df[clusters=='c2'].plot.scatter(x='x',y='y',marker='d', color='green',ax=ax)\n",
    "df[clusters=='c3'].plot.scatter(x='x',y='y',marker='s',color='magenta', ax=ax)\n",
    "ax.scatter(*c1,color='red',marker='^',s=50)\n",
    "ax.scatter(*c2,marker='d',color='green',s=50)\n",
    "ax.scatter(*c3,marker='s',color='magenta',s=50)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Acima foram mostradas as duas primeiras iterações do K-means sobre o conjunto de dados sintético. Repare que os centroids são deslocados para o centro de massa dos clusters. Esse deslocamento modifica os clusters, atraindo mais amostras. Isso, por sua vez, desloca novamente os centroids até se estabilizarem. Portanto, a posição inicial dos centroids tem um impacto muito grande no resultado final. Em geral, as implementações executam o algoritmo com diversos centroids iniciais e reportam o melhor resultado.\n",
    "\n",
    "A seguir eu mostro o resultado obtido com `sklearn`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "KMeans(algorithm='auto', copy_x=True, init='k-means++', max_iter=300,\n",
       "    n_clusters=3, n_init=10, n_jobs=1, precompute_distances='auto',\n",
       "    random_state=None, tol=0.0001, verbose=0)"
      ]
     },
     "execution_count": 94,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sklearn.cluster import KMeans\n",
    "model = KMeans(n_clusters=3).fit(df[['x','y']].values)\n",
    "model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x112a7a390>"
      ]
     },
     "execution_count": 102,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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cHOSQ4w/htpfcVh0JVIGDfncQt/7zrY5RkFQk4+8VUEcWei7GFUI4DlhAtcfi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\nSVKiDBeSJClRhgtJkpSo/w/ev0y8+UMU/AAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x112a1bf28>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "clusters = np.asarray(['c{}'.format(a+1) for a in model.labels_],dtype='object')\n",
    "ax = df[clusters=='c1'].plot.scatter(x='x',y='y',marker='^',color='red')\n",
    "df[clusters=='c2'].plot.scatter(x='x',y='y',marker='d', color='green',ax=ax)\n",
    "df[clusters=='c3'].plot.scatter(x='x',y='y',marker='s',color='magenta', ax=ax)\n",
    "c1, c2, c3 = model.cluster_centers_\n",
    "ax.scatter(*c1,color='red',marker='^',s=50)\n",
    "ax.scatter(*c2,marker='d',color='green',s=50)\n",
    "ax.scatter(*c3,marker='s',color='magenta',s=50)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Agrupamento hierárquico \n",
    "\n",
    "O algoritmo de clustering hierárquico, diferentemente do K-means, começa com cada amostra sendo um cluster e, a cada iteração, tenta combinar as amostras/clusters mais próximas/similares em um único cluster. O algoritmo termina quando todas as amostras são combinadas em um único cluster. \n",
    "\n",
    "A grande vantagem do cluster hierárquico é que essa sequência de combinações (ou fusões) podem ser reconstruídas numa árvore, chamada de dendrograma. Essa estrutura permite tanto a visualização das amostras mais próximas, como também definir o número 'ideal' de clusters.\n",
    "\n",
    "A seguir vemos um exemplo dessa estrutura."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 122,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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n2nbq2bX93r340P4LMz8aiBtLAQAAAP0pT7fq75N0o6QnSCp1tjnolUF4ri3d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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x112c508d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.spatial.distance import pdist\n",
    "from scipy.cluster.hierarchy import linkage, dendrogram\n",
    "distancia = pdist(df[['x','y']].values)\n",
    "clusters = linkage(distancia,method='complete')\n",
    "plt.figure(figsize=(12,5))\n",
    "dn = dendrogram(clusters,leaf_font_size=9)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Nesse ponto surgem duas perguntas: \n",
    "\n",
    "1. Como são computadas as distâncias entre amostras e clusters, e entre clusters?\n",
    "2. O quê significa a altura em que os clusters são combinados? Ou melhor, como ler o dendrograma?\n",
    "\n",
    "A generalização do conceito de distância (primeira pergunta) pode ser feita de diversas formas, tornando-se, assim, um parâmetro do algoritmo. As formas mais comuns são:\n",
    "\n",
    "- **single link**: menor distância intercluster. A distância entre dois clusters A e B é a menor distância entre as amostras pertencentes a A, e as de B\n",
    "- **complete link**: maior distância intercluster. É o oposto da single.\n",
    "- **group average**: é a distância média entre as amostras dos dois clusters\n",
    "- **centroid**: distância entre os centroids dos dois clusters (computados como no K-means)\n",
    "- **Ward's method**: a distância é definida como o incremento na soma dos quadrados dos erros definido como no K-means.\n",
    "\n",
    "Naturalmente, a escolha de diferentes critérios de distância levam a diferentes resultados. Contudo, single link tende a resultar em clusters menos balanceados. Frequentemente, os clusters são formados agregando-se uma amostra por vez. Assim, ela se torna menos interessante na prática. \n",
    "\n",
    "Em relação à segunda pergunta, o dendrograma é interpretado como a seguir. O diagrama é lido de baixo para cima. As linhas horizontais indicam a combinação de dois clusters. A altura dessa combinação (linha vertical) indica a distância (como computada acima) entre os dois clusters. Assim, percebemos que as amostras 2 e 6 são mais próximas entre si que as amostras 25 e 36. Embora ambos os clusters tenham sido formado no 'primeiro nível'. Outra consideração importante se faz em relação à posição das amostras em relação ao eixo x. Não podemos tirar conclusões a partir dessas posições. Por exemplo, podemos tender a interpretar que a amostra 50 é mais próxima de 40 do que de 41. Contudo, essa conclusão é errada, uma vez que a análise é feita com base nas distância entre clusters.\n",
    "\n",
    "Podemos usar o dendrograma para particionar as amostras em grupos. Por exemplo, se podarmos a árvore na altura 3, obtemos os três clusters em verde, vermelho e azul claro. Esse corte pode ser feito como a seguir:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 125,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1,\n",
       "       1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 0, 2, 2, 0,\n",
       "       2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 1, 1])"
      ]
     },
     "execution_count": 125,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from scipy.cluster.hierarchy import cut_tree\n",
    "labels = cut_tree(clusters,height=3)\n",
    "labels.flatten()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Métricas de avaliação do agrupamento\n",
    "\n",
    "Assim como no caso de aprendizado supervisionado, existem inúmeras métricas para avaliar os resultados de um agrupamento. Algumas métricas avaliam a homogeneidade dos clusters, outras avaliam o grau de concordância entre os grupos e os rótulos iniciais.\n",
    "\n",
    "#### Adjusted Rand Index\n",
    "\n",
    "Podemos usar o Fleiss Kappa como métrica de concordância dos rótulos. Outra alternativa é o Índice de Rand, ou sua versão ajustada. O índice de Rand é dado pela seguinte fórmula:\n",
    "\n",
    "$$\n",
    "RI = \\frac{a+b}{a+b+c+d}\n",
    "$$\n",
    "\n",
    "onde $a,b,c,d$ são dados por\n",
    "\n",
    "- $a$ é o número de pares que pertencem ao mesmo grupo nas partições S e T\n",
    "- $b$ é o número de pares que pertencem a grupos distintos nas partições S e T\n",
    "- $c$ é o número de pares que pertencem ao mesmo grupo em S e grupos distintos em T\n",
    "- $d$ é o número de pares que pertencem ao mesmo grupo em T e grupos distintos em S\n",
    "\n",
    "O índice de Rand ajustado é o quanto o RI difere do valor esperado. O valor do índice de Rand ajustado é dado por:\n",
    "\n",
    "$$\n",
    "ARI = \\frac{\\sum_{ij}\\binom{|S_i\\cap T_j|}{2} - [\\sum\\binom{|S_i|}{2}\\sum\\binom{|T_j|}{2}]/\\binom{n}{2}}{\\frac{1}{2}[\\sum\\binom{|S_i|}{2}+\\sum\\binom{|T_j|}{2}]- [\\sum\\binom{|S_i|}{2}\\sum\\binom{|T_j|}{2}]/\\binom{n}{2}}\n",
    "$$\n",
    "\n",
    "Ambas as versões do índice variam entre 0 e 1, onde 0 indica nenhuma concordância e 1 completa concordância.\n",
    "\n",
    "Computando o ARI para nosso conjunto de dados sintético, temos:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 128,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.85426366756290029"
      ]
     },
     "execution_count": 128,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sklearn.metrics import adjusted_rand_score\n",
    "adjusted_rand_score(model.labels_,df.c)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 129,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.76784277919085198"
      ]
     },
     "execution_count": 129,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "adjusted_rand_score(labels.flatten(),df.c)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Silhueta\n",
    "\n",
    "A métrica da silhueta é uma medida de homogeneidade do agrupamento. Ela avalia, em média, o quão similar uma amostra é das demais amostras de seu grupo quando comparada às amostras do cluster mais próximo. O valor da métrica para uma dada amostra é:\n",
    "\n",
    "$$\n",
    "s(x) = \\frac{b(x)-a(x)}{max(b(x),a(x))}\n",
    "$$\n",
    "\n",
    "onde $b(x)$ é a menor distância média entre $x$ e as amostras de outros clusters, e $a(x)$ é a distância média de $x$  para as amostras dentro do cluster de $x$.\n",
    "\n",
    "O valor da métrica para o cluster é a média sobre todas as amostras. Ela varia entre -1 (quando a amostra é mais distante das amostras do cluster) e 1 (quando ela é mais distante das amostras fora do cluster).\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 132,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.55094839  0.59975346  0.61007481  0.53705848  0.2453906   0.57287273\n",
      "  0.60913245  0.20997806  0.34122234  0.49796695  0.54534328  0.48821488\n",
      "  0.38468948  0.51187653  0.63623312  0.43788456  0.62564506  0.12741325\n",
      "  0.56935257  0.54101476  0.58009586  0.32095295  0.62147863  0.36196189\n",
      "  0.59805905  0.49401579  0.29288657  0.53444108  0.59488347  0.62417385\n",
      "  0.13512402  0.5944502   0.62453798  0.34245002  0.57054492  0.1408177\n",
      "  0.55087135  0.62300985  0.59086818  0.5894282   0.42427742  0.60317254\n",
      "  0.08260327  0.64610776  0.59553057  0.00945097  0.46487972  0.49871715\n",
      "  0.60340607  0.62762909  0.40785854  0.61844411  0.51803303  0.65225402\n",
      "  0.44623994  0.62842982  0.11854829  0.43630536  0.38335413  0.24895565]\n",
      "0.474021912518\n"
     ]
    }
   ],
   "source": [
    "from sklearn.metrics import silhouette_score, silhouette_samples\n",
    "print(silhouette_samples(df[['x','y']].values,model.labels_))\n",
    "print(silhouette_score(df[['x','y']].values,model.labels_))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A silhueta é uma métrica muito usada também para definir o número de clusters no K-means. Podemos plotar os valores de silhueta de cada amostra por cluster e, assim, verificar visualmente a homogeneidade dos clusters obtidos. Se houverem muitos clusters com homogeneidade abaixo da média (ou essa for baixa), isso indica que devemos testar outros valores de $k$. A documentação de `sklearn` ilustra um exemplo disso http://scikit-learn.org/stable/auto_examples/cluster/plot_kmeans_silhouette_analysis.html#selecting-the-number-of-clusters-with-silhouette-analysis-on-kmeans-clustering\n",
    "\n",
    "Finalmente, `sklearn` possui uma boa documentação sobre métricas de avaliação do agrupamento no seguinte link http://scikit-learn.org/stable/modules/clustering.html#clustering-performance-evaluation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
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