{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# K-means"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Na aprendizagem de máquina não supervisionada temos algoritmos que trabalham com dados que não possuem classificações prévias para treinamento. Assim, uma das formas de extrair informações desses dados é através da procura por grupos\n",
    "de dados que são semelhantes entre si. Chamamos esse tipo de tarefa de clusterização e os grupos encontrados de clusters."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "O K-means é um dos algoritmos de clusterização mais populares. Ele procura por um número predeterminado de clusters através da definição dos seus centroides (centros geométricos dos clusters). "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Dados"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Para entender o funcionamento do algoritmo, vamos utilizar o dataset apresentado a seguir. Nele temos exemplos com duas características, uma representada pelo eixo X e outra pelo eixo Y."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "    <div class=\"bk-root\">\n",
       "        <a href=\"https://bokeh.pydata.org\" target=\"_blank\" class=\"bk-logo bk-logo-small bk-logo-notebook\"></a>\n",
       "        <span id=\"a4883006-66d5-48e0-b81b-5a65d911c737\">Loading BokehJS ...</span>\n",
       "    </div>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/javascript": [
       "\n",
       "(function(root) {\n",
       "  function now() {\n",
       "    return new Date();\n",
       "  }\n",
       "\n",
       "  var force = true;\n",
       "\n",
       "  if (typeof (root._bokeh_onload_callbacks) === \"undefined\" || force === true) {\n",
       "    root._bokeh_onload_callbacks = [];\n",
       "    root._bokeh_is_loading = undefined;\n",
       "  }\n",
       "\n",
       "  var JS_MIME_TYPE = 'application/javascript';\n",
       "  var HTML_MIME_TYPE = 'text/html';\n",
       "  var EXEC_MIME_TYPE = 'application/vnd.bokehjs_exec.v0+json';\n",
       "  var CLASS_NAME = 'output_bokeh rendered_html';\n",
       "\n",
       "  /**\n",
       "   * Render data to the DOM node\n",
       "   */\n",
       "  function render(props, node) {\n",
       "    var script = document.createElement(\"script\");\n",
       "    node.appendChild(script);\n",
       "  }\n",
       "\n",
       "  /**\n",
       "   * Handle when an output is cleared or removed\n",
       "   */\n",
       "  function handleClearOutput(event, handle) {\n",
       "    var cell = handle.cell;\n",
       "\n",
       "    var id = cell.output_area._bokeh_element_id;\n",
       "    var server_id = cell.output_area._bokeh_server_id;\n",
       "    // Clean up Bokeh references\n",
       "    if (id != null && id in Bokeh.index) {\n",
       "      Bokeh.index[id].model.document.clear();\n",
       "      delete Bokeh.index[id];\n",
       "    }\n",
       "\n",
       "    if (server_id !== undefined) {\n",
       "      // Clean up Bokeh references\n",
       "      var cmd = \"from bokeh.io.state import curstate; print(curstate().uuid_to_server['\" + server_id + \"'].get_sessions()[0].document.roots[0]._id)\";\n",
       "      cell.notebook.kernel.execute(cmd, {\n",
       "        iopub: {\n",
       "          output: function(msg) {\n",
       "            var id = msg.content.text.trim();\n",
       "            if (id in Bokeh.index) {\n",
       "              Bokeh.index[id].model.document.clear();\n",
       "              delete Bokeh.index[id];\n",
       "            }\n",
       "          }\n",
       "        }\n",
       "      });\n",
       "      // Destroy server and session\n",
       "      var cmd = \"import bokeh.io.notebook as ion; ion.destroy_server('\" + server_id + \"')\";\n",
       "      cell.notebook.kernel.execute(cmd);\n",
       "    }\n",
       "  }\n",
       "\n",
       "  /**\n",
       "   * Handle when a new output is added\n",
       "   */\n",
       "  function handleAddOutput(event, handle) {\n",
       "    var output_area = handle.output_area;\n",
       "    var output = handle.output;\n",
       "\n",
       "    // limit handleAddOutput to display_data with EXEC_MIME_TYPE content only\n",
       "    if ((output.output_type != \"display_data\") || (!output.data.hasOwnProperty(EXEC_MIME_TYPE))) {\n",
       "      return\n",
       "    }\n",
       "\n",
       "    var toinsert = output_area.element.find(\".\" + CLASS_NAME.split(' ')[0]);\n",
       "\n",
       "    if (output.metadata[EXEC_MIME_TYPE][\"id\"] !== undefined) {\n",
       "      toinsert[toinsert.length - 1].firstChild.textContent = output.data[JS_MIME_TYPE];\n",
       "      // store reference to embed id on output_area\n",
       "      output_area._bokeh_element_id = output.metadata[EXEC_MIME_TYPE][\"id\"];\n",
       "    }\n",
       "    if (output.metadata[EXEC_MIME_TYPE][\"server_id\"] !== undefined) {\n",
       "      var bk_div = document.createElement(\"div\");\n",
       "      bk_div.innerHTML = output.data[HTML_MIME_TYPE];\n",
       "      var script_attrs = bk_div.children[0].attributes;\n",
       "      for (var i = 0; i < script_attrs.length; i++) {\n",
       "        toinsert[toinsert.length - 1].firstChild.setAttribute(script_attrs[i].name, script_attrs[i].value);\n",
       "      }\n",
       "      // store reference to server id on output_area\n",
       "      output_area._bokeh_server_id = output.metadata[EXEC_MIME_TYPE][\"server_id\"];\n",
       "    }\n",
       "  }\n",
       "\n",
       "  function register_renderer(events, OutputArea) {\n",
       "\n",
       "    function append_mime(data, metadata, element) {\n",
       "      // create a DOM node to render to\n",
       "      var toinsert = this.create_output_subarea(\n",
       "        metadata,\n",
       "        CLASS_NAME,\n",
       "        EXEC_MIME_TYPE\n",
       "      );\n",
       "      this.keyboard_manager.register_events(toinsert);\n",
       "      // Render to node\n",
       "      var props = {data: data, metadata: metadata[EXEC_MIME_TYPE]};\n",
       "      render(props, toinsert[toinsert.length - 1]);\n",
       "      element.append(toinsert);\n",
       "      return toinsert\n",
       "    }\n",
       "\n",
       "    /* Handle when an output is cleared or removed */\n",
       "    events.on('clear_output.CodeCell', handleClearOutput);\n",
       "    events.on('delete.Cell', handleClearOutput);\n",
       "\n",
       "    /* Handle when a new output is added */\n",
       "    events.on('output_added.OutputArea', handleAddOutput);\n",
       "\n",
       "    /**\n",
       "     * Register the mime type and append_mime function with output_area\n",
       "     */\n",
       "    OutputArea.prototype.register_mime_type(EXEC_MIME_TYPE, append_mime, {\n",
       "      /* Is output safe? */\n",
       "      safe: true,\n",
       "      /* Index of renderer in `output_area.display_order` */\n",
       "      index: 0\n",
       "    });\n",
       "  }\n",
       "\n",
       "  // register the mime type if in Jupyter Notebook environment and previously unregistered\n",
       "  if (root.Jupyter !== undefined) {\n",
       "    var events = require('base/js/events');\n",
       "    var OutputArea = require('notebook/js/outputarea').OutputArea;\n",
       "\n",
       "    if (OutputArea.prototype.mime_types().indexOf(EXEC_MIME_TYPE) == -1) {\n",
       "      register_renderer(events, OutputArea);\n",
       "    }\n",
       "  }\n",
       "\n",
       "  \n",
       "  if (typeof (root._bokeh_timeout) === \"undefined\" || force === true) {\n",
       "    root._bokeh_timeout = Date.now() + 5000;\n",
       "    root._bokeh_failed_load = false;\n",
       "  }\n",
       "\n",
       "  var NB_LOAD_WARNING = {'data': {'text/html':\n",
       "     \"<div style='background-color: #fdd'>\\n\"+\n",
       "     \"<p>\\n\"+\n",
       "     \"BokehJS does not appear to have successfully loaded. If loading BokehJS from CDN, this \\n\"+\n",
       "     \"may be due to a slow or bad network connection. Possible fixes:\\n\"+\n",
       "     \"</p>\\n\"+\n",
       "     \"<ul>\\n\"+\n",
       "     \"<li>re-rerun `output_notebook()` to attempt to load from CDN again, or</li>\\n\"+\n",
       "     \"<li>use INLINE resources instead, as so:</li>\\n\"+\n",
       "     \"</ul>\\n\"+\n",
       "     \"<code>\\n\"+\n",
       "     \"from bokeh.resources import INLINE\\n\"+\n",
       "     \"output_notebook(resources=INLINE)\\n\"+\n",
       "     \"</code>\\n\"+\n",
       "     \"</div>\"}};\n",
       "\n",
       "  function display_loaded() {\n",
       "    var el = document.getElementById(\"a4883006-66d5-48e0-b81b-5a65d911c737\");\n",
       "    if (el != null) {\n",
       "      el.textContent = \"BokehJS is loading...\";\n",
       "    }\n",
       "    if (root.Bokeh !== undefined) {\n",
       "      if (el != null) {\n",
       "        el.textContent = \"BokehJS \" + root.Bokeh.version + \" successfully loaded.\";\n",
       "      }\n",
       "    } else if (Date.now() < root._bokeh_timeout) {\n",
       "      setTimeout(display_loaded, 100)\n",
       "    }\n",
       "  }\n",
       "\n",
       "\n",
       "  function run_callbacks() {\n",
       "    try {\n",
       "      root._bokeh_onload_callbacks.forEach(function(callback) { callback() });\n",
       "    }\n",
       "    finally {\n",
       "      delete root._bokeh_onload_callbacks\n",
       "    }\n",
       "    console.info(\"Bokeh: all callbacks have finished\");\n",
       "  }\n",
       "\n",
       "  function load_libs(js_urls, callback) {\n",
       "    root._bokeh_onload_callbacks.push(callback);\n",
       "    if (root._bokeh_is_loading > 0) {\n",
       "      console.log(\"Bokeh: BokehJS is being loaded, scheduling callback at\", now());\n",
       "      return null;\n",
       "    }\n",
       "    if (js_urls == null || js_urls.length === 0) {\n",
       "      run_callbacks();\n",
       "      return null;\n",
       "    }\n",
       "    console.log(\"Bokeh: BokehJS not loaded, scheduling load and callback at\", now());\n",
       "    root._bokeh_is_loading = js_urls.length;\n",
       "    for (var i = 0; i < js_urls.length; i++) {\n",
       "      var url = js_urls[i];\n",
       "      var s = document.createElement('script');\n",
       "      s.src = url;\n",
       "      s.async = false;\n",
       "      s.onreadystatechange = s.onload = function() {\n",
       "        root._bokeh_is_loading--;\n",
       "        if (root._bokeh_is_loading === 0) {\n",
       "          console.log(\"Bokeh: all BokehJS libraries loaded\");\n",
       "          run_callbacks()\n",
       "        }\n",
       "      };\n",
       "      s.onerror = function() {\n",
       "        console.warn(\"failed to load library \" + url);\n",
       "      };\n",
       "      console.log(\"Bokeh: injecting script tag for BokehJS library: \", url);\n",
       "      document.getElementsByTagName(\"head\")[0].appendChild(s);\n",
       "    }\n",
       "  };var element = document.getElementById(\"a4883006-66d5-48e0-b81b-5a65d911c737\");\n",
       "  if (element == null) {\n",
       "    console.log(\"Bokeh: ERROR: autoload.js configured with elementid 'a4883006-66d5-48e0-b81b-5a65d911c737' but no matching script tag was found. \")\n",
       "    return false;\n",
       "  }\n",
       "\n",
       "  var js_urls = [\"https://cdn.pydata.org/bokeh/release/bokeh-0.13.0.min.js\", \"https://cdn.pydata.org/bokeh/release/bokeh-widgets-0.13.0.min.js\", \"https://cdn.pydata.org/bokeh/release/bokeh-tables-0.13.0.min.js\", \"https://cdn.pydata.org/bokeh/release/bokeh-gl-0.13.0.min.js\"];\n",
       "\n",
       "  var inline_js = [\n",
       "    function(Bokeh) {\n",
       "      Bokeh.set_log_level(\"info\");\n",
       "    },\n",
       "    \n",
       "    function(Bokeh) {\n",
       "      \n",
       "    },\n",
       "    function(Bokeh) {\n",
       "      console.log(\"Bokeh: injecting CSS: https://cdn.pydata.org/bokeh/release/bokeh-0.13.0.min.css\");\n",
       "      Bokeh.embed.inject_css(\"https://cdn.pydata.org/bokeh/release/bokeh-0.13.0.min.css\");\n",
       "      console.log(\"Bokeh: injecting CSS: https://cdn.pydata.org/bokeh/release/bokeh-widgets-0.13.0.min.css\");\n",
       "      Bokeh.embed.inject_css(\"https://cdn.pydata.org/bokeh/release/bokeh-widgets-0.13.0.min.css\");\n",
       "      console.log(\"Bokeh: injecting CSS: https://cdn.pydata.org/bokeh/release/bokeh-tables-0.13.0.min.css\");\n",
       "      Bokeh.embed.inject_css(\"https://cdn.pydata.org/bokeh/release/bokeh-tables-0.13.0.min.css\");\n",
       "    }\n",
       "  ];\n",
       "\n",
       "  function run_inline_js() {\n",
       "    \n",
       "    if ((root.Bokeh !== undefined) || (force === true)) {\n",
       "      for (var i = 0; i < inline_js.length; i++) {\n",
       "        inline_js[i].call(root, root.Bokeh);\n",
       "      }if (force === true) {\n",
       "        display_loaded();\n",
       "      }} else if (Date.now() < root._bokeh_timeout) {\n",
       "      setTimeout(run_inline_js, 100);\n",
       "    } else if (!root._bokeh_failed_load) {\n",
       "      console.log(\"Bokeh: BokehJS failed to load within specified timeout.\");\n",
       "      root._bokeh_failed_load = true;\n",
       "    } else if (force !== true) {\n",
       "      var cell = $(document.getElementById(\"a4883006-66d5-48e0-b81b-5a65d911c737\")).parents('.cell').data().cell;\n",
       "      cell.output_area.append_execute_result(NB_LOAD_WARNING)\n",
       "    }\n",
       "\n",
       "  }\n",
       "\n",
       "  if (root._bokeh_is_loading === 0) {\n",
       "    console.log(\"Bokeh: BokehJS loaded, going straight to plotting\");\n",
       "    run_inline_js();\n",
       "  } else {\n",
       "    load_libs(js_urls, function() {\n",
       "      console.log(\"Bokeh: BokehJS plotting callback run at\", now());\n",
       "      run_inline_js();\n",
       "    });\n",
       "  }\n",
       "}(window));"
      ],
      "application/vnd.bokehjs_load.v0+json": "\n(function(root) {\n  function now() {\n    return new Date();\n  }\n\n  var force = true;\n\n  if (typeof (root._bokeh_onload_callbacks) === \"undefined\" || force === true) {\n    root._bokeh_onload_callbacks = [];\n    root._bokeh_is_loading = undefined;\n  }\n\n  \n\n  \n  if (typeof (root._bokeh_timeout) === \"undefined\" || force === true) {\n    root._bokeh_timeout = Date.now() + 5000;\n    root._bokeh_failed_load = false;\n  }\n\n  var NB_LOAD_WARNING = {'data': {'text/html':\n     \"<div style='background-color: #fdd'>\\n\"+\n     \"<p>\\n\"+\n     \"BokehJS does not appear to have successfully loaded. If loading BokehJS from CDN, this \\n\"+\n     \"may be due to a slow or bad network connection. Possible fixes:\\n\"+\n     \"</p>\\n\"+\n     \"<ul>\\n\"+\n     \"<li>re-rerun `output_notebook()` to attempt to load from CDN again, or</li>\\n\"+\n     \"<li>use INLINE resources instead, as so:</li>\\n\"+\n     \"</ul>\\n\"+\n     \"<code>\\n\"+\n     \"from bokeh.resources import INLINE\\n\"+\n     \"output_notebook(resources=INLINE)\\n\"+\n     \"</code>\\n\"+\n     \"</div>\"}};\n\n  function display_loaded() {\n    var el = document.getElementById(\"a4883006-66d5-48e0-b81b-5a65d911c737\");\n    if (el != null) {\n      el.textContent = \"BokehJS is loading...\";\n    }\n    if (root.Bokeh !== undefined) {\n      if (el != null) {\n        el.textContent = \"BokehJS \" + root.Bokeh.version + \" successfully loaded.\";\n      }\n    } else if (Date.now() < root._bokeh_timeout) {\n      setTimeout(display_loaded, 100)\n    }\n  }\n\n\n  function run_callbacks() {\n    try {\n      root._bokeh_onload_callbacks.forEach(function(callback) { callback() });\n    }\n    finally {\n      delete root._bokeh_onload_callbacks\n    }\n    console.info(\"Bokeh: all callbacks have finished\");\n  }\n\n  function load_libs(js_urls, callback) {\n    root._bokeh_onload_callbacks.push(callback);\n    if (root._bokeh_is_loading > 0) {\n      console.log(\"Bokeh: BokehJS is being loaded, scheduling callback at\", now());\n      return null;\n    }\n    if (js_urls == null || js_urls.length === 0) {\n      run_callbacks();\n      return null;\n    }\n    console.log(\"Bokeh: BokehJS not loaded, scheduling load and callback at\", now());\n    root._bokeh_is_loading = js_urls.length;\n    for (var i = 0; i < js_urls.length; i++) {\n      var url = js_urls[i];\n      var s = document.createElement('script');\n      s.src = url;\n      s.async = false;\n      s.onreadystatechange = s.onload = function() {\n        root._bokeh_is_loading--;\n        if (root._bokeh_is_loading === 0) {\n          console.log(\"Bokeh: all BokehJS libraries loaded\");\n          run_callbacks()\n        }\n      };\n      s.onerror = function() {\n        console.warn(\"failed to load library \" + url);\n      };\n      console.log(\"Bokeh: injecting script tag for BokehJS library: \", url);\n      document.getElementsByTagName(\"head\")[0].appendChild(s);\n    }\n  };var element = document.getElementById(\"a4883006-66d5-48e0-b81b-5a65d911c737\");\n  if (element == null) {\n    console.log(\"Bokeh: ERROR: autoload.js configured with elementid 'a4883006-66d5-48e0-b81b-5a65d911c737' but no matching script tag was found. \")\n    return false;\n  }\n\n  var js_urls = [\"https://cdn.pydata.org/bokeh/release/bokeh-0.13.0.min.js\", \"https://cdn.pydata.org/bokeh/release/bokeh-widgets-0.13.0.min.js\", \"https://cdn.pydata.org/bokeh/release/bokeh-tables-0.13.0.min.js\", \"https://cdn.pydata.org/bokeh/release/bokeh-gl-0.13.0.min.js\"];\n\n  var inline_js = [\n    function(Bokeh) {\n      Bokeh.set_log_level(\"info\");\n    },\n    \n    function(Bokeh) {\n      \n    },\n    function(Bokeh) {\n      console.log(\"Bokeh: injecting CSS: https://cdn.pydata.org/bokeh/release/bokeh-0.13.0.min.css\");\n      Bokeh.embed.inject_css(\"https://cdn.pydata.org/bokeh/release/bokeh-0.13.0.min.css\");\n      console.log(\"Bokeh: injecting CSS: https://cdn.pydata.org/bokeh/release/bokeh-widgets-0.13.0.min.css\");\n      Bokeh.embed.inject_css(\"https://cdn.pydata.org/bokeh/release/bokeh-widgets-0.13.0.min.css\");\n      console.log(\"Bokeh: injecting CSS: https://cdn.pydata.org/bokeh/release/bokeh-tables-0.13.0.min.css\");\n      Bokeh.embed.inject_css(\"https://cdn.pydata.org/bokeh/release/bokeh-tables-0.13.0.min.css\");\n    }\n  ];\n\n  function run_inline_js() {\n    \n    if ((root.Bokeh !== undefined) || (force === true)) {\n      for (var i = 0; i < inline_js.length; i++) {\n        inline_js[i].call(root, root.Bokeh);\n      }if (force === true) {\n        display_loaded();\n      }} else if (Date.now() < root._bokeh_timeout) {\n      setTimeout(run_inline_js, 100);\n    } else if (!root._bokeh_failed_load) {\n      console.log(\"Bokeh: BokehJS failed to load within specified timeout.\");\n      root._bokeh_failed_load = true;\n    } else if (force !== true) {\n      var cell = $(document.getElementById(\"a4883006-66d5-48e0-b81b-5a65d911c737\")).parents('.cell').data().cell;\n      cell.output_area.append_execute_result(NB_LOAD_WARNING)\n    }\n\n  }\n\n  if (root._bokeh_is_loading === 0) {\n    console.log(\"Bokeh: BokehJS loaded, going straight to plotting\");\n    run_inline_js();\n  } else {\n    load_libs(js_urls, function() {\n      console.log(\"Bokeh: BokehJS plotting callback run at\", now());\n      run_inline_js();\n    });\n  }\n}(window));"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "  <div class=\"bk-root\" id=\"984a704b-069b-4ecf-896c-6eec21ca84e5\"></div>\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/javascript": [
       "(function(root) {\n",
       "  function embed_document(root) {\n",
       "    \n",
       "  var docs_json = 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Application\",\"version\":\"0.13.0\"}};\n",
       "  var render_items = [{\"docid\":\"da36779d-8129-46f1-98b5-07113aadf7eb\",\"notebook_comms_target\":\"0b759e33-ea06-4108-9fca-a3ddbe038117\",\"roots\":{\"9d38c1a5-27d5-45fb-802a-ca9edb3d3a29\":\"984a704b-069b-4ecf-896c-6eec21ca84e5\"}}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        embed_document(root);\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "      attempts++;\n",
       "      if (attempts > 100) {\n",
       "        console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\")\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "9d38c1a5-27d5-45fb-802a-ca9edb3d3a29"
      }
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<p><code>&lt;Bokeh Notebook handle for <strong>In[1]</strong>&gt;</code></p>"
      ],
      "text/plain": [
       "<bokeh.io.notebook.CommsHandle at 0x111cf5860>"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sklearn.datasets.samples_generator import make_blobs\n",
    "from bokeh.io import show, output_notebook\n",
    "from bokeh.plotting import figure\n",
    "output_notebook()\n",
    "\n",
    "X, _ = make_blobs(n_samples=300, centers=4, random_state=20)\n",
    "\n",
    "x = X[:, 0]\n",
    "y = X[:, 1]\n",
    "\n",
    "p = figure(plot_width=600, plot_height=360)\n",
    "p.circle(x, y, radius=0.1)\n",
    "show(p, notebook_handle=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "No gráfico acima, todos os pontos são da cor azul, ou seja, não existe identificação nos dados que separem os exemplos em grupos. Entretanto, note que visualmente é possível enxergar quatro diferentes agrupamentos de pontos. Assim, mesmo todos os pontos sendo iguais (cor azul) a semelhança de acordo com as características nos traz uma informação potencialmente valiosa para o entendimento dos dados."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Algoritmo"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "O primeiro passo para execução do algoritmo K-means é definir quantos clusters deverão ser descobertos. Usando os dados apresentados anteriormente, vamos definir a procura por 4 clusters.\n",
    "Cada cluster é representado pelo seu centroide, assim, teremos 4 centroides. No início do algoritmo os centroides são posicionados aleatoriamente no espaço onde estão representados os dados de treinamento.\n",
    "\n",
    "O gráfico a seguir mostra os dados com 4 centroides posicionados aleatoriamente. Os centroides são representados por triângulos."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "  <div class=\"bk-root\" id=\"723e15d6-d7d9-4b6a-8369-a7f37ea391a2\"></div>\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/javascript": [
       "(function(root) {\n",
       "  function embed_document(root) {\n",
       "    \n",
       "  var docs_json = 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Application\",\"version\":\"0.13.0\"}};\n",
       "  var render_items = [{\"docid\":\"0b59a132-4c14-402d-a149-29b196ac1228\",\"notebook_comms_target\":\"2daeb086-e4d1-4bfe-b37f-766a22572b07\",\"roots\":{\"17fe1cf2-a7f1-4ec8-bead-c4b068fe6163\":\"723e15d6-d7d9-4b6a-8369-a7f37ea391a2\"}}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        embed_document(root);\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "      attempts++;\n",
       "      if (attempts > 100) {\n",
       "        console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\")\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "17fe1cf2-a7f1-4ec8-bead-c4b068fe6163"
      }
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<p><code>&lt;Bokeh Notebook handle for <strong>In[2]</strong>&gt;</code></p>"
      ],
      "text/plain": [
       "<bokeh.io.notebook.CommsHandle at 0x11524c4e0>"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import numpy as np\n",
    "\n",
    "np.random.seed(1)  # definindo o seed para tornar os resultados do notebook sempre iguais\n",
    "centroids_x = np.random.uniform(-10, 10, 4)\n",
    "\n",
    "np.random.seed(3)  # definindo o seed para tornar os resultados do notebook sempre iguais\n",
    "centroids_y = np.random.uniform(-2, 10, 4)\n",
    "\n",
    "centroids = []\n",
    "for i in range(len(centroids_x)):\n",
    "    centroids.append((centroids_x[i], centroids_y[i]))\n",
    "\n",
    "p = figure(plot_width=600, plot_height=360)\n",
    "p.circle(x, y, radius=0.1)\n",
    "p.triangle(centroids[0][0], centroids[0][1], size=12, color=\"red\", line_color=\"black\")\n",
    "p.triangle(centroids[1][0], centroids[1][1], size=12, color=\"green\", line_color=\"black\")\n",
    "p.triangle(centroids[2][0], centroids[2][1], size=12, color=\"orange\", line_color=\"black\")\n",
    "p.triangle(centroids[3][0], centroids[3][1], size=12, color=\"purple\", line_color=\"black\")\n",
    "show(p, notebook_handle=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Com os centroides inicializados, no passo número 2 do algoritmo, verifica-se para cada ponto, qual o centroide mais próximo.\n",
    "Assim, é definido que o ponto pertence ao grupo representado pelo centroide mais próximo."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Para calcular a proximidade entre um ponto e um centroide, basta calcular a distância euclidiana entre eles.\n",
    "Para pontos em duas dimensões como os representados no nosso exemplo, a fórmula da distância euclidiana é:\n",
    "\n",
    "$$d(a, b) = \\sqrt{(a_x - b_x)^2 + (a_y - b_y)^2}$$\n",
    "\n",
    "Onde $a$ e $b$ são dois pontos, $a_x$ é a coordenada X do ponto $a$ e $a_y$ é a coordenada Y do ponto $a$, e $b_x$ é a coordenada X do ponto $b$ e $b_y$ é a coordenada Y do ponto $b$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A função a seguir calcula a distância entre um ponto e um centroide:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10.862127336783487"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import math\n",
    "\n",
    "def distance(centroid, point):\n",
    "    x_diff = centroid[0] - point[0]\n",
    "    y_diff = centroid[1] - point[1]\n",
    "    \n",
    "    return math.sqrt(x_diff**2 + y_diff**2)\n",
    "\n",
    "distance(centroids[0], X[0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A seguir temos uma função que itera sobre os dados de treinamento para definir o grupo de cada ponto de acordo com centroide mais próximo."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1, 1, 1, 1, 1, 3, 1, 0, 1, 1, 1, 3, 0, 1, 1, 1, 2, 2, 1, 2, 3, 2, 3, 3, 2, 2, 2, 0, 3, 1, 0, 2, 2, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 3, 1, 1, 3, 1, 2, 1, 2, 2, 2, 1, 1, 3, 2, 1, 1, 3, 1, 0, 2, 3, 1, 2, 1, 3, 2, 1, 1, 3, 3, 2, 1, 0, 0, 3, 1, 0, 1, 1, 1, 2, 2, 3, 1, 1, 0, 3, 3, 2, 3, 1, 3, 3, 1, 0, 1, 3, 1, 1, 3, 1, 3, 3, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 3, 2, 2, 1, 0, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 0, 1, 1, 1, 2, 3, 3, 1, 1, 2, 1, 3, 2, 0, 2, 3, 2, 2, 0, 3, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 3, 1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 1, 2, 2, 2, 0, 1, 1, 3, 1, 2, 0, 0, 3, 2, 1, 2, 1, 3, 0, 1, 0, 3, 1, 2, 1, 3, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 1, 2, 1, 1, 1, 1, 3, 1, 1, 2, 3, 3, 2, 1, 1, 1, 1, 2, 2, 1, 1, 3, 0, 1, 3, 3, 1, 1, 3, 1, 0, 3, 1, 2, 2, 1, 2, 3, 0, 3, 1, 0, 2, 0, 3, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 2, 1, 3, 2, 1, 1, 2, 3, 1, 3, 1, 1, 1, 1, 3, 0, 3]\n"
     ]
    }
   ],
   "source": [
    "def get_groups(centroids, points):\n",
    "    groups = []\n",
    "\n",
    "    for x_i in points:\n",
    "        distances = []\n",
    "\n",
    "        for c_i in centroids:\n",
    "            distances.append(distance(c_i, x_i))\n",
    "\n",
    "        groups.append(np.argmin(distances))\n",
    "        \n",
    "    return groups\n",
    "    \n",
    "groups = get_groups(centroids, X)\n",
    "print(groups)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A lista `groups` contém a informação sobre a que centroide os pontos foram atribuídos. Cada centroide possui um número, sendo o vermelho o 0, o verde o 1, o laranja o 2 e o roxo o 3. \n",
    "\n",
    "Assim, um número $J$ na posição $I$ da lista representa que o ponto $I$ foi atribuído ao grupo do centroide $J$. Por exemplo, no resultado acima, a posição 0 da lista tem o valor 1, ou seja, o ponto 0 do dataset faz parte do cluster 1."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Com esses dados em mãos, vamos agora mostrar o gráfico com os pontos coloridos de acordo com os seus centroides."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
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       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
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       "  var render_items = [{\"docid\":\"d5605295-9f8c-4700-aa35-8da918b924c7\",\"notebook_comms_target\":\"c249f029-32fa-4004-aaef-8727b81900de\",\"roots\":{\"1c05756f-0736-4533-9130-a2ff0d65ce53\":\"eb4a14c4-7774-4c0c-9466-eb7a5873d88e\"}}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        embed_document(root);\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "      attempts++;\n",
       "      if (attempts > 100) {\n",
       "        console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\")\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "1c05756f-0736-4533-9130-a2ff0d65ce53"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "def plot_chart(points, centroids, groups):\n",
    "    colors = [\"red\", \"green\", \"orange\", \"purple\"]\n",
    "    chart_colors = [colors[x] for x in groups]\n",
    "    \n",
    "    x = points[:, 0]\n",
    "    y = points[:, 1]\n",
    "\n",
    "    p = figure(plot_width=600, plot_height=360)\n",
    "    p.circle(x, y, radius=0.1, color=chart_colors)\n",
    "    p.triangle(centroids[0][0], centroids[0][1], size=12, color=\"red\", line_color=\"black\")\n",
    "    p.triangle(centroids[1][0], centroids[1][1], size=12, color=\"green\", line_color=\"black\")\n",
    "    p.triangle(centroids[2][0], centroids[2][1], size=12, color=\"orange\", line_color=\"black\")\n",
    "    p.triangle(centroids[3][0], centroids[3][1], size=12, color=\"purple\", line_color=\"black\")\n",
    "    show(p, notebook_handle=True)\n",
    "    \n",
    "plot_chart(X, centroids, groups)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "O passo número 3 do algoritmo é o reposicionamento dos centroides.\n",
    "Para cada centroide, vamos calcular a média dos pontos que fazem parte do grupo que ele determina. O valor encontrado é a nova posição do centroide.\n",
    "\n",
    "O código a seguir calcula a nova posição dos centroides. A função `points_mean` calcula a média dos pontos e a função `move_centroids` itera sobre os dados de treinamento e calcula a nova posição dos centroides."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "def points_mean(points):\n",
    "    sum_x = 0\n",
    "    sum_y = 0\n",
    "    total = len(points)\n",
    "\n",
    "    for point in points:\n",
    "        sum_x += point[0]\n",
    "        sum_y += point[1]\n",
    "\n",
    "    return (sum_x/total, sum_y/total)\n",
    "\n",
    "\n",
    "def move_centroids(centroids, points, groups):\n",
    "    points_in_groups = [[] for i in range(len(centroids))]\n",
    "\n",
    "    for index, group in enumerate(groups):\n",
    "        points_in_groups[group].append(points[index])\n",
    "    \n",
    "    moved_centroids = []\n",
    "    for p in points_in_groups:\n",
    "        moved_centroids.append(points_mean(p))\n",
    "    \n",
    "    return moved_centroids\n",
    "\n",
    "centroids = move_centroids(centroids, X, groups)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "O gráfico a seguir mostra os centroides reposicionados:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "  <div class=\"bk-root\" id=\"8a799a05-fc3e-472d-b07a-4adea751c719\"></div>\n"
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     },
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     "output_type": "display_data"
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       "  function embed_document(root) {\n",
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Application\",\"version\":\"0.13.0\"}};\n",
       "  var render_items = [{\"docid\":\"ac7debce-62e7-4c68-8cdb-05a037919801\",\"notebook_comms_target\":\"8498cb3c-93ca-42ef-9394-413adf68e70e\",\"roots\":{\"47db7bac-b663-46b5-84ff-e27c03252d83\":\"8a799a05-fc3e-472d-b07a-4adea751c719\"}}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        embed_document(root);\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "      attempts++;\n",
       "      if (attempts > 100) {\n",
       "        console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\")\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "47db7bac-b663-46b5-84ff-e27c03252d83"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_chart(X, centroids, groups)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "O resultado não está muito bom, os grupos parecem misturados, entretanto a execução do algoritmo ainda não terminou.\n",
    "\n",
    "O K-means continua sua execução repetindo o passo 2 (atribuição dos grupos) e o passo 3 (reposicionamento dos centroides), até que no passo 2 não existam mudanças nas atribuições dos grupos.\n",
    "\n",
    "Vamos repetir o passo 2:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "  <div class=\"bk-root\" id=\"48998c1d-3232-42fd-88a5-abab861bb894\"></div>\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/javascript": [
       "(function(root) {\n",
       "  function embed_document(root) {\n",
       "    \n",
       "  var docs_json = 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Application\",\"version\":\"0.13.0\"}};\n",
       "  var render_items = [{\"docid\":\"979be737-63b6-4243-831a-ff82eee8ff36\",\"notebook_comms_target\":\"4e15a839-0123-4a02-999d-eccaf119c6cb\",\"roots\":{\"5ed6b03b-d482-4afd-b473-820fb12e6dc1\":\"48998c1d-3232-42fd-88a5-abab861bb894\"}}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        embed_document(root);\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "      attempts++;\n",
       "      if (attempts > 100) {\n",
       "        console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\")\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "5ed6b03b-d482-4afd-b473-820fb12e6dc1"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "groups = get_groups(centroids, X)\n",
    "plot_chart(X, centroids, groups)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note que alguns pontos mudaram de cor, ou seja, eles foram atribuídos a grupos diferentes. Com isso, os centroides também mudarão de posição."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "  <div class=\"bk-root\" id=\"f26a13c7-882c-446e-9784-3258b5d3d2b0\"></div>\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/javascript": [
       "(function(root) {\n",
       "  function embed_document(root) {\n",
       "    \n",
       "  var docs_json = 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Application\",\"version\":\"0.13.0\"}};\n",
       "  var render_items = [{\"docid\":\"5ee07304-1a75-4c57-bffb-8a24637453fe\",\"notebook_comms_target\":\"5b42439f-ecd6-45bd-b04d-ae6cec98fd8a\",\"roots\":{\"7401fcdd-8401-4fd5-8348-4540cf798cf7\":\"f26a13c7-882c-446e-9784-3258b5d3d2b0\"}}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        embed_document(root);\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "      attempts++;\n",
       "      if (attempts > 100) {\n",
       "        console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\")\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "7401fcdd-8401-4fd5-8348-4540cf798cf7"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "centroids = move_centroids(centroids, X, groups)\n",
    "plot_chart(X, centroids, groups)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Executando os passos 2 e 3 mais 4 vezes, temos:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "  <div class=\"bk-root\" id=\"9757de1b-62e3-4d1d-9941-d6cd0d818ea3\"></div>\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/javascript": [
       "(function(root) {\n",
       "  function embed_document(root) {\n",
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Application\",\"version\":\"0.13.0\"}};\n",
       "  var render_items = [{\"docid\":\"06fef1e7-56d9-4f08-bc54-c5d87c266397\",\"roots\":{\"6b4aa635-dbce-481d-9b87-fcdc8d92fdd4\":\"9757de1b-62e3-4d1d-9941-d6cd0d818ea3\"}}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        embed_document(root);\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "      attempts++;\n",
       "      if (attempts > 100) {\n",
       "        console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\")\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "6b4aa635-dbce-481d-9b87-fcdc8d92fdd4"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from bokeh.layouts import gridplot\n",
    "\n",
    "plots_grid = [[]]\n",
    "for i in range(4):\n",
    "    groups = get_groups(centroids, X)\n",
    "    centroids = move_centroids(centroids, X, groups)\n",
    "    \n",
    "    colors = [\"red\", \"green\", \"orange\", \"purple\"]\n",
    "    chart_colors = [colors[x] for x in groups]\n",
    "    \n",
    "    p = figure(plot_width=450, plot_height=270)\n",
    "    p.circle(x, y, radius=0.1, color=chart_colors)\n",
    "    p.triangle(centroids[0][0], centroids[0][1], size=12, color=\"red\", line_color=\"black\")\n",
    "    p.triangle(centroids[1][0], centroids[1][1], size=12, color=\"green\", line_color=\"black\")\n",
    "    p.triangle(centroids[2][0], centroids[2][1], size=12, color=\"orange\", line_color=\"black\")\n",
    "    p.triangle(centroids[3][0], centroids[3][1], size=12, color=\"purple\", line_color=\"black\")\n",
    "    \n",
    "    if (len(plots_grid[-1]) == 2):\n",
    "        plots_grid.append([])\n",
    "    \n",
    "    plots_grid[-1].append(p)\n",
    "    \n",
    "p = gridplot(plots_grid)\n",
    "show(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Nos últimos dois gráficos, chegamos a um ponto de convergência. Note que os grupos não mudam nos dois gráficos de baixo. Com isso, o K-means pode ser terminado e os quatro clusters foram encontrados."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Limitações"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Inicialização aleatória"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "O resultado final do K-means é dependente da inicialização aleatória dos centroides. Por isso, o seu resultado pode variar a cada execução.\n",
    "No exemplo anterior, os quatro clusters encontrados são coerentes, mas não podemos garantir que esses quatro clusters sempre serão encontrados.\n",
    "\n",
    "Vamos inicializar os centroides de uma maneira diferente:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "  <div class=\"bk-root\" id=\"9dc60ede-9afe-470e-812a-4485919ba952\"></div>\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/javascript": [
       "(function(root) {\n",
       "  function embed_document(root) {\n",
       "    \n",
       "  var docs_json = 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Application\",\"version\":\"0.13.0\"}};\n",
       "  var render_items = [{\"docid\":\"d1e88dd6-b2b6-40f6-a126-1ec287358da9\",\"notebook_comms_target\":\"0fd59e92-e35a-478d-a706-873c9b18ca70\",\"roots\":{\"a29cba59-cd6c-4886-a286-7392559d2d4c\":\"9dc60ede-9afe-470e-812a-4485919ba952\"}}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        embed_document(root);\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "      attempts++;\n",
       "      if (attempts > 100) {\n",
       "        console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\")\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "a29cba59-cd6c-4886-a286-7392559d2d4c"
      }
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<p><code>&lt;Bokeh Notebook handle for <strong>In[11]</strong>&gt;</code></p>"
      ],
      "text/plain": [
       "<bokeh.io.notebook.CommsHandle at 0x11541cf98>"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.random.seed(4)\n",
    "centroids_x = np.random.uniform(-10, 10, 4)\n",
    "\n",
    "np.random.seed(10)\n",
    "centroids_y = np.random.uniform(-2, 10, 4)\n",
    "\n",
    "centroids2 = []\n",
    "for i in range(len(centroids_x)):\n",
    "    centroids2.append((centroids_x[i], centroids_y[i]))\n",
    "\n",
    "p = figure(plot_width=600, plot_height=360)\n",
    "p.circle(x, y, radius=0.1)\n",
    "p.triangle(centroids2[0][0], centroids2[0][1], size=12, color=\"red\", line_color=\"black\")\n",
    "p.triangle(centroids2[1][0], centroids2[1][1], size=12, color=\"green\", line_color=\"black\")\n",
    "p.triangle(centroids2[2][0], centroids2[2][1], size=12, color=\"orange\", line_color=\"black\")\n",
    "p.triangle(centroids2[3][0], centroids2[3][1], size=12, color=\"purple\", line_color=\"black\")\n",
    "show(p, notebook_handle=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Abaixo temos a evolução dos clusters a cada iteração do algoritmo:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "  <div class=\"bk-root\" id=\"03bc6642-182b-497e-8896-cdabed53fad9\"></div>\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/javascript": [
       "(function(root) {\n",
       "  function embed_document(root) {\n",
       "    \n",
       "  var docs_json = 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Application\",\"version\":\"0.13.0\"}};\n",
       "  var render_items = [{\"docid\":\"63f5c80c-bffb-417f-908a-6884d4497485\",\"roots\":{\"8da5be1b-9a15-48c6-af56-be2938036688\":\"03bc6642-182b-497e-8896-cdabed53fad9\"}}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        embed_document(root);\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "      attempts++;\n",
       "      if (attempts > 100) {\n",
       "        console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\")\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "8da5be1b-9a15-48c6-af56-be2938036688"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plots_grid = [[]]\n",
    "for i in range(4):\n",
    "    groups2 = get_groups(centroids2, X)\n",
    "    centroids2 = move_centroids(centroids2, X, groups2)\n",
    "    \n",
    "    colors = [\"red\", \"green\", \"orange\", \"purple\"]\n",
    "    chart_colors = [colors[x] for x in groups2]\n",
    "    \n",
    "    p = figure(plot_width=450, plot_height=270)\n",
    "    p.circle(x, y, radius=0.1, color=chart_colors)\n",
    "    p.triangle(centroids2[0][0], centroids2[0][1], size=12, color=\"red\", line_color=\"black\")\n",
    "    p.triangle(centroids2[1][0], centroids2[1][1], size=12, color=\"green\", line_color=\"black\")\n",
    "    p.triangle(centroids2[2][0], centroids2[2][1], size=12, color=\"orange\", line_color=\"black\")\n",
    "    p.triangle(centroids2[3][0], centroids2[3][1], size=12, color=\"purple\", line_color=\"black\")\n",
    "    \n",
    "    if (len(plots_grid[-1]) == 2):\n",
    "        plots_grid.append([])\n",
    "    \n",
    "    plots_grid[-1].append(p)\n",
    "    \n",
    "p = gridplot(plots_grid)\n",
    "show(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Perceba que foi criado um grande grupo verde os grupos vermelho e laranja passaram a dividir os pontos do que na execução anterior era um grupo sozinho."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Para resolver o problema da dependência da inicialização dos centroides, uma estratégia utilizada é executar o K-means várias vezes. Após o fim de todas as execuções é possível medir a qualidade do resultado final através da equação de distorção. Quanto menor a distorção, melhores foram os agrupamentos."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Dado que $x^{(i)}$ é o i-ésimo ponto do dataset, que $c_j^{(i)}$ é o centróide $j$ atribuído ao i-ésimo ponto e que existem $m$ pontos no dataset. A distorção é definida por:\n",
    "\n",
    "$$distorção = \\frac{1}{m} \\sum_{i=1}^{m} d(x^{(i)}, c_j^{(i)})^2$$\n",
    "\n",
    "Lembrando que $d(a, b)$ é a distância entre os pontos $a$ e $b$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Dados os pontos, os centroides e a informação sobre quais pontos fazem parte de quais grupos, a função abaixo calcula a distorção:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "def distortion(centroids, points, groups):\n",
    "    result = 0\n",
    "    for index, group in enumerate(groups):\n",
    "        result += (distance(points[index], centroids[group]))**2\n",
    "        \n",
    "    return result/len(points)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Os centroides e os grupos atribuídos aos pontos do nosso primeiro exemplo estão armazenados nas variáveis `centroids` e `groups` respectivamente. Os centroides e grupos do segundo exemplo estão armazenados nas variáveis `centroids2` e `groups2` respectivamente.\n",
    "\n",
    "Calculando a distorção para os dois exemplos temos:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.888723131110457\n",
      "9.215098905694967\n"
     ]
    }
   ],
   "source": [
    "print(distortion(centroids, X, groups))\n",
    "print(distortion(centroids2, X, groups2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Confirmando o que foi analisado visualmente, a distorção do primeiro exemplo é menor que a do segundo. Com isso podemos concluir que os agrupamentos do primeiro exemplo são melhores que o do segundo."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Fronteiras lineares"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ao definir os clusters, o K-means utiliza fronteiras lineares para separar os pontos. Assim, dependendo da organização dos pontos de um dataset, pode ser impossível encontrar grupos de pontos de forma satisfatória.\n",
    "\n",
    "Abaixo temos um conjunto de pontos, no qual, visualmente, podemos identificar dois grupos. Entretanto, esses grupos não podem ser separados linearmente."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
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       "\n",
       "\n",
       "\n",
       "\n",
       "  <div class=\"bk-root\" id=\"12dd3ddd-fb0d-4f57-96eb-d9cfbada5b41\"></div>\n"
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     },
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     "output_type": "display_data"
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Application\",\"version\":\"0.13.0\"}};\n",
       "  var render_items = [{\"docid\":\"8463d71a-22a1-49de-8734-80df98cbbc33\",\"notebook_comms_target\":\"3d3473d4-8ac6-4e2c-b7cf-a42a39348a59\",\"roots\":{\"801db8eb-9332-401b-8d2f-56154064ddd5\":\"12dd3ddd-fb0d-4f57-96eb-d9cfbada5b41\"}}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        embed_document(root);\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "      attempts++;\n",
       "      if (attempts > 100) {\n",
       "        console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\")\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "801db8eb-9332-401b-8d2f-56154064ddd5"
      }
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<p><code>&lt;Bokeh Notebook handle for <strong>In[15]</strong>&gt;</code></p>"
      ],
      "text/plain": [
       "<bokeh.io.notebook.CommsHandle at 0x10b528940>"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sklearn.datasets.samples_generator import make_moons\n",
    "\n",
    "X, _ = make_moons(n_samples=300, noise=0.07, random_state=0)\n",
    "\n",
    "x = X[:, 0]\n",
    "y = X[:, 1]\n",
    "\n",
    "p = figure(plot_width=600, plot_height=360)\n",
    "p.circle(x, y, radius=0.015)\n",
    "show(p, notebook_handle=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Inicializando dois centroides aleatoriamente, temos:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "  <div class=\"bk-root\" id=\"fd5c94e2-8bb7-40ae-b8fa-4289450ff1f0\"></div>\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/javascript": [
       "(function(root) {\n",
       "  function embed_document(root) {\n",
       "    \n",
       "  var docs_json = 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       "  var render_items = [{\"docid\":\"6644200d-f4e7-4341-80b7-4a2f88c60939\",\"notebook_comms_target\":\"86a34ecf-e0e3-4ffd-863a-91b598f71f46\",\"roots\":{\"016d9aa8-03d5-4ea1-925c-2475aabbd445\":\"fd5c94e2-8bb7-40ae-b8fa-4289450ff1f0\"}}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        embed_document(root);\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "      attempts++;\n",
       "      if (attempts > 100) {\n",
       "        console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\")\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "016d9aa8-03d5-4ea1-925c-2475aabbd445"
      }
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<p><code>&lt;Bokeh Notebook handle for <strong>In[16]</strong>&gt;</code></p>"
      ],
      "text/plain": [
       "<bokeh.io.notebook.CommsHandle at 0x1155115c0>"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.random.seed(0)\n",
    "centroids_x = np.random.uniform(-1, 2, 2)\n",
    "\n",
    "np.random.seed(0)\n",
    "centroids_y = np.random.uniform(-0.5, 1, 2)\n",
    "\n",
    "centroids3 = []\n",
    "for i in range(len(centroids_x)):\n",
    "    centroids3.append((centroids_x[i], centroids_y[i]))\n",
    "\n",
    "p = figure(plot_width=600, plot_height=360)\n",
    "p.circle(x, y, radius=0.015)\n",
    "p.triangle(centroids3[0][0], centroids3[0][1], size=12, color=\"red\", line_color=\"black\")\n",
    "p.triangle(centroids3[1][0], centroids3[1][1], size=12, color=\"green\", line_color=\"black\")\n",
    "show(p, notebook_handle=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Executando quatro iterações do algoritmo, temos a seguinte evolução dos grupos encontrados:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "  <div class=\"bk-root\" id=\"1daf77fd-afc1-4c61-905e-214cfd796975\"></div>\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/javascript": [
       "(function(root) {\n",
       "  function embed_document(root) {\n",
       "    \n",
       "  var docs_json = 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CDSView\"}},\"id\":\"39f45f3f-5140-4b4d-b107-43490106421b\",\"type\":\"GlyphRenderer\"},{\"attributes\":{\"data_source\":{\"id\":\"53b9b2e0-8ae8-4dca-98eb-eed7db916d86\",\"type\":\"ColumnDataSource\"},\"glyph\":{\"id\":\"2442ca42-c588-44bf-824d-c4d0824587eb\",\"type\":\"Circle\"},\"hover_glyph\":null,\"muted_glyph\":null,\"nonselection_glyph\":{\"id\":\"a51ab7ea-c0c9-4bac-a80e-0b97b8572906\",\"type\":\"Circle\"},\"selection_glyph\":null,\"view\":{\"id\":\"b667f219-7b83-4860-98c1-ba72f073a532\",\"type\":\"CDSView\"}},\"id\":\"6ffdbeb7-f449-4f59-b272-7e67ddcb1028\",\"type\":\"GlyphRenderer\"},{\"attributes\":{\"source\":{\"id\":\"453d5032-d85f-42f6-8d55-210b19083d08\",\"type\":\"ColumnDataSource\"}},\"id\":\"67461ca9-28d1-4158-81bc-00872117c3e2\",\"type\":\"CDSView\"},{\"attributes\":{\"below\":[{\"id\":\"8b34630f-b165-4fbf-91de-4d1266f4a0a1\",\"type\":\"LinearAxis\"}],\"left\":[{\"id\":\"71951c2d-42ca-4440-a9dc-ddfbf223a90a\",\"type\":\"LinearAxis\"}],\"plot_height\":270,\"plot_width\":450,\"renderers\":[{\"id\":\"8b34630f-b165-4fbf-91de-4d1266f4a0a1\",\"type\":\"LinearAxis\"},{\"id\":\"6562e671-dbb1-410e-b19c-592cb103e70a\",\"type\":\"Grid\"},{\"id\":\"71951c2d-42ca-4440-a9dc-ddfbf223a90a\",\"type\":\"LinearAxis\"},{\"id\":\"83926987-79a3-41ce-b2c1-c62e2d7dd001\",\"type\":\"Grid\"},{\"id\":\"b2baf9ce-4114-4eca-a183-cbf81e4a0723\",\"type\":\"BoxAnnotation\"},{\"id\":\"92bbc7ad-9de3-4427-929f-8815e9ab7d4d\",\"type\":\"GlyphRenderer\"},{\"id\":\"e737cbd6-a67d-4c81-ae4f-c38848d0bec5\",\"type\":\"GlyphRenderer\"},{\"id\":\"35f81ac2-c5bf-412e-a62d-0034d4462d23\",\"type\":\"GlyphRenderer\"}],\"title\":{\"id\":\"be31731d-a7a3-4aa6-b9e6-59c2b5429870\",\"type\":\"Title\"},\"toolbar\":{\"id\":\"baceea72-be00-4741-9cd5-b617bd0e2fe0\",\"type\":\"Toolbar\"},\"toolbar_location\":null,\"x_range\":{\"id\":\"e97c6898-4cfc-41e0-89c6-1b53a4c29e2e\",\"type\":\"DataRange1d\"},\"x_scale\":{\"id\":\"0e4404f6-d9e4-4a51-b239-9c328443f979\",\"type\":\"LinearScale\"},\"y_range\":{\"id\":\"fcb77b0a-a1c1-4a3c-b44e-b0f7de3f1c7a\",\"type\":\"DataRange1d\"},\"y_scale\":{\"id\":\"7d09093d-7163-4f9e-aee9-ade6edc4cc3c\",\"type\":\"LinearScale\"}},\"id\":\"c3cb1b0a-f785-4c27-af02-8cb09812eee6\",\"subtype\":\"Figure\",\"type\":\"Plot\"},{\"attributes\":{\"active_drag\":\"auto\",\"active_inspect\":\"auto\",\"active_multi\":null,\"active_scroll\":\"auto\",\"active_tap\":\"auto\",\"tools\":[{\"id\":\"66943088-4b80-4e6b-9033-e948b3349b21\",\"type\":\"PanTool\"},{\"id\":\"18c5944d-4608-4d65-a41f-5f1e4de961d6\",\"type\":\"WheelZoomTool\"},{\"id\":\"60c31263-1f11-47ca-a3e8-1a5d77329fe1\",\"type\":\"BoxZoomTool\"},{\"id\":\"36eb2e80-1b7d-4842-8f4a-1a4e488a365d\",\"type\":\"SaveTool\"},{\"id\":\"9900d889-f037-4d03-a9ea-75a7c1467d31\",\"type\":\"ResetTool\"},{\"id\":\"fbb592e4-d16c-4f9d-b4b0-ef20f5ec41c9\",\"type\":\"HelpTool\"}]},\"id\":\"baceea72-be00-4741-9cd5-b617bd0e2fe0\",\"type\":\"Toolbar\"},{\"attributes\":{\"fill_color\":{\"value\":\"red\"},\"size\":{\"units\":\"screen\",\"value\":12},\"x\":{\"value\":0.1411738438063256},\"y\":{\"value\":0.30278817501498684}},\"id\":\"bf3ca0e8-4450-41a3-96be-968a532ce481\",\"type\":\"Triangle\"},{\"attributes\":{\"callback\":null},\"id\":\"e97c6898-4cfc-41e0-89c6-1b53a4c29e2e\",\"type\":\"DataRange1d\"},{\"attributes\":{},\"id\":\"7d09093d-7163-4f9e-aee9-ade6edc4cc3c\",\"type\":\"LinearScale\"},{\"attributes\":{\"callback\":null},\"id\":\"fcb77b0a-a1c1-4a3c-b44e-b0f7de3f1c7a\",\"type\":\"DataRange1d\"},{\"attributes\":{},\"id\":\"0e4404f6-d9e4-4a51-b239-9c328443f979\",\"type\":\"LinearScale\"}],\"root_ids\":[\"08cc8cc3-dbfb-435a-8fa5-dfe1082d526f\"]},\"title\":\"Bokeh Application\",\"version\":\"0.13.0\"}};\n",
       "  var render_items = [{\"docid\":\"0c0250f7-398a-4407-9ab8-e00ee7ccd94e\",\"roots\":{\"08cc8cc3-dbfb-435a-8fa5-dfe1082d526f\":\"1daf77fd-afc1-4c61-905e-214cfd796975\"}}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        embed_document(root);\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "      attempts++;\n",
       "      if (attempts > 100) {\n",
       "        console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\")\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "08cc8cc3-dbfb-435a-8fa5-dfe1082d526f"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plots_grid = [[]]\n",
    "for i in range(4):\n",
    "    groups3 = get_groups(centroids3, X)\n",
    "    centroids3 = move_centroids(centroids3, X, groups3)\n",
    "    \n",
    "    colors = [\"red\", \"green\"]\n",
    "    chart_colors = [colors[x] for x in groups3]\n",
    "    \n",
    "    p = figure(plot_width=450, plot_height=270)\n",
    "    p.circle(x, y, radius=0.015, color=chart_colors)\n",
    "    p.triangle(centroids3[0][0], centroids3[0][1], size=12, color=\"red\", line_color=\"black\")\n",
    "    p.triangle(centroids3[1][0], centroids3[1][1], size=12, color=\"green\", line_color=\"black\")\n",
    "    \n",
    "    if (len(plots_grid[-1]) == 2):\n",
    "        plots_grid.append([])\n",
    "    \n",
    "    plots_grid[-1].append(p)\n",
    "    \n",
    "p = gridplot(plots_grid)\n",
    "show(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note que os clusters encontrados não são exatamente o que era esperado. O K-means limitou-se a dividir os conjuntos através de uma reta. Assim, dependendo do dataset, o K-means pode não ser adequado para encontrar os grupos desejados."
   ]
  }
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