{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Regressão Linear Simples com Descida de Gradiente\n", "\n", "A Regressão Linear é um método de aprendizagem de máquina supervisionada para prever rótulos contínuos em dados que seguem uma tendência linear.\n", "\n", "Por exemplo, a seguir temos representados num gráfico os preços de casas (eixo Y) de acordo com o seu tamanho (eixo X). Os preços são mostrados em dólares americanos e estão divididos por mil. Os tamanhos estão em \"square feet\" e também estão divididos por mil. Esses dados foram coletados em 1977, assim, os preços das casas são muito inferiores aos preços atuais.\n", "\n", "Tais dados são os dados de treinamento. Eles foram extraídos do link: http://people.sc.fsu.edu/~jburkardt/datasets/regression/x27.txt" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "scrolled": true }, "outputs": [ { "data": { "text/html": [ "\n", "
\n", " \n", " Loading BokehJS ...\n", "
" ] }, "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", " \"
\\n\"+\n", " \"

\\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", " \"

\\n\"+\n", " \"\\n\"+\n", " \"\\n\"+\n", " \"from bokeh.resources import INLINE\\n\"+\n", " \"output_notebook(resources=INLINE)\\n\"+\n", " \"\\n\"+\n", " \"
\"}};\n", "\n", " function display_loaded() {\n", " var el = document.getElementById(\"d4a243a1-0af5-41e4-a906-25e8739bdf9a\");\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(\"d4a243a1-0af5-41e4-a906-25e8739bdf9a\");\n", " if (element == null) {\n", " console.log(\"Bokeh: ERROR: autoload.js configured with elementid 'd4a243a1-0af5-41e4-a906-25e8739bdf9a' 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(\"d4a243a1-0af5-41e4-a906-25e8739bdf9a\")).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 \"
\\n\"+\n \"

\\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 \"

\\n\"+\n \"\\n\"+\n \"\\n\"+\n \"from bokeh.resources import INLINE\\n\"+\n \"output_notebook(resources=INLINE)\\n\"+\n \"\\n\"+\n \"
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" ], "text/plain": [ "" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from bokeh.io import show, output_notebook\n", "from bokeh.plotting import figure\n", "output_notebook()\n", "\n", "sizes = [0.998, 1.500, 1.175, 1.232, 1.121, 0.988, 1.240, 1.501, 1.225, 1.552, 0.975, 1.121, 1.020,\n", " 1.501, 1.664, 1.488, 1.376, 1.500, 1.256, 1.690, 1.820, 1.652, 1.777, 1.504, 1.831, 1.200 ]\n", "prices = [25.9, 29.5, 27.9, 25.9, 29.9, 29.9, 30.9, 28.9, 35.9, 31.5, 31.0, 30.9, 30.0, 28.9, 36.9,\n", " 41.9, 40.5, 43.9, 37.5, 37.9, 44.5, 37.9, 38.9, 36.9, 45.8, 41.0]\n", "\n", "p = figure(plot_width=700, plot_height=250)\n", "p.circle(sizes, prices)\n", "show(p, notebook_handle=True)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Utilizando os dados, como saber qual deve ser o preço de uma casa com o tamanho 1400 square feet?\n", "\n", "Uma das formas de responder essa pergunta é traçar uma reta seguindo a tendência linear dos dados de treinamento e usá-la como guia para prever o valor de um ponto que não se encontra no gráfico." ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", "\n", "\n", "\n", "
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" ], "text/plain": [ "" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p.line([1,2], [29.3015, 43.3078], line_width=2)\n", "show(p, notebook_handle=True)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Visualmente podemos perceber que o preço apontado pela reta para uma casa de 1400 square feet é aproximadamente 35 mil." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Uma reta que segue a tendência linear de um conjunto de pontos em duas dimensões pode ser gerada usando Regressão Linear Simples. A Regressão Linear Simples modela o relacionamento entre as variáveis através de uma reta." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Na matemática, uma reta pode ser definida como uma função linear: $f(x) = w_1x + w_0$, onde $w_0$ e $w_1$ são os coeficientes da função. $w_0$ especifica o deslocamento vertical da reta e $w_1$ a sua inclinação. No gráfico acima a função da reta é: $f(x) = 14.0063x + 15.2952$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "A reta ideal encontrada pela Regressão Linear Simples é a reta na qual a média da distância de todos os pontos para a reta é a menor possível. Ou seja, a reta que minimiza o valor de $\\frac{1}{2m} \\sum_{i=1}^{m} (f(x_i) - y_i)^2$, onde m é a quantidade de pontos nos dados de treinamento, e ($x_i$, $y_i$) define um ponto.\n", "\n", "Analisando o somatório, temos que $(f(x_i) - y_i)$ mede a distância entre um ponto $(x_i, y_i)$ e a reta, já que $y_i$ é o valor do ponto no eixo Y e $f(x_i)$ é o valor estimado pela reta.\n", "A diferença é elevada ao quadrado para que as distâncias sejam sempre positivas, não fazendo diferença se o ponto está acima ou abaixo da reta.\n", "Por fim, a média é calculada dividindo o somatório por $2m$. O número 2 aparece para facilitar cálculos que serão realizados em passos seguintes.\n", "\n", "Podemos escrever o somatório anterior como uma função de $w_0$ e $w_1$. Assim, temos:\n", "\n", "\\begin{equation*}\n", "J(w_0, w_1) = \\frac{1}{2m} \\sum_{i=1}^{m} (w_1x + w_0 - y_i)^2\n", "\\end{equation*}\n", "\n", "Essa função é chamada de Erro Médio Quadrático." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Vamos definir uma função em Python para calcular o Erro Médio Quadrático:" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "def mean_squared_error(xs, ys, w0, w1):\n", " total = 0\n", "\n", " for i in range(len(xs)):\n", " x, y = xs[i], ys[i]\n", " fx = w1*x + w0\n", " error = (fx - y)**2\n", " total += error\n", "\n", " return total/(2*len(xs))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Calculando o Erro Médio Quadrático para a função da reta exibida anteriormente, onde $w_0 = 15.2952$ e $w_1 = 14.0063$, temos:" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "O Erro Médio Quadrático é: 10.553925770613349\n" ] } ], "source": [ "print(\"O Erro Médio Quadrático é: {}\".format(mean_squared_error(sizes, prices, 15.2952, 14.0063)))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Para encontrar o valor mínimo de $J(w_0, w_1)$ utilizaremos o algoritmo da descida de gradiente.\n", "\n", "O algoritmo inicia com dois valores aleatórios para $w_0$ e $w_1$ e em seguida, passo a passo, ele ajusta os valores de $w_0$ e $w_1$ para que $J(w_0, w_1)$ seja minimizada. O tamanho desses ajustes em cada passo é determinado por um parâmetro $\\alpha$ chamado de taxa de aprendizagem. Se $\\alpha$ for muito pequeno o algoritmo pode demorar a convergir, se $\\alpha$ for um valor alto os ajustes podem ser grandes de uma forma que o algoritmo nunca consiga convergir.\n", "\n", "Para descobrir a direção que $w_0$ e $w_1$ devem ser ajustados para diminuir o valor de $J(w_0, w_1)$ vamos calcular o gradiente da função, ou seja, vamos calcular as derivadas parciais de $J(w_0, w_1)$.\n", "\n", "- $\\frac{\\partial}{\\partial w_0} J(w_0, w_1) = \\frac{1}{m} \\sum_{i=1}^{m} (f(x_i) - y_i)$\n", "- $\\frac{\\partial}{\\partial w_1} J(w_0, w_1) = \\frac{1}{m} \\sum_{i=1}^{m} ((f(x_i) - y_i)x_i)$\n", "\n", "As derivadas parciais são calculadas pelas funções abaixo:" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [], "source": [ "def d_w0(xs, ys, w0, w1):\n", " total = 0\n", "\n", " for i in range(len(xs)):\n", " x, y = xs[i], ys[i]\n", " fx = w1*x + w0\n", " error = (fx - y)\n", " total += error\n", "\n", " return total/(len(xs))\n", "\n", "\n", "def d_w1(xs, ys, w0, w1):\n", " total = 0\n", "\n", " for i in range(len(xs)):\n", " x, y = xs[i], ys[i]\n", " fx = w1*x + w0\n", " error = (fx - y)\n", " total += error*x\n", "\n", " return total/(len(xs))" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "Com isso, o algoritmo se resume a repetir até a convergência os seguintes passos:\n", "\n", "- $w_0 := w_0 - \\alpha \\frac{\\partial}{\\partial w_0} J(w_0, w_1)$\n", "- $w_1 := w_1 - \\alpha \\frac{\\partial}{\\partial w_1} J(w_0, w_1)$\n", "\n", "É importante que as atualizações de $w_0$ e $w_1$ sejam feitas de forma simultânea." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "A cada iteração gravaremos os valores de $w_0$, $w_1$ e do Erro Médio Quadrático para analisarmos as mudanças na reta posteriormente. Além disso, o algoritmo terminará quando o Erro Médio Quadrático não diminuir pelo menos 0.00001." ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Mean squared error: 10.556045385998793\n", "f(x) = 14.251632539788755x + 14.948262924183263\n" ] } ], "source": [ "import random\n", "\n", "w0 = random.uniform(-2, 2)\n", "w1 = random.uniform(-2, 2)\n", "alpha = 0.1\n", "\n", "coefficients = [(w0, w1)]\n", "errors = [mean_squared_error(sizes, prices, w0, w1)]\n", "\n", "while True:\n", " w0i = w0 - alpha*d_w0(sizes, prices, w0, w1)\n", " w1i = w1 - alpha*d_w1(sizes, prices, w0, w1)\n", "\n", " w0 = w0i\n", " w1 = w1i\n", " \n", " coefficients.append((w0, w1))\n", " errors.append(mean_squared_error(sizes, prices, w0, w1))\n", " \n", " if (errors[-2] - errors[-1] < 0.00001):\n", " break\n", "\n", "print(\"Mean squared error: {}\".format(mean_squared_error(sizes, prices, w0, w1)))\n", "print(\"f(x) = {}x + {}\".format(w1, w0))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Usando a função encontrada, vamos visualizar a reta iniciando em $x=1$ e terminando em $x=2$:" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", "\n", "\n", "\n", "
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" ], "text/plain": [ "" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p_final = figure(plot_width=700, plot_height=250)\n", "p_final.circle(sizes, prices)\n", "p_final.line([1,2], [w1 + w0, w1*2 + w0], line_width=2)\n", "show(p_final, notebook_handle=True)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Como gravamos a evolução dos valores de $w_0$ e $w_1$ podemos plotar gráficos para visualizarmos quais foram as retas geradas nas iterações. A função a seguir exibe as retas nas 12 primeiras iterações." ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", "\n", "\n", "\n", "
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Application\",\"version\":\"0.13.0\"}};\n", " var render_items = [{\"docid\":\"65440c8f-bda7-4cbe-aa55-8cfbedf1e2b2\",\"roots\":{\"a25da3b0-4232-475e-996d-707dc3d9ab56\":\"66edba3e-7c04-45ba-8857-8425606957be\"}}];\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": "a25da3b0-4232-475e-996d-707dc3d9ab56" } }, "output_type": "display_data" } ], "source": [ "from bokeh.layouts import gridplot\n", "\n", "plots_grid = [[]]\n", "for i in range(12):\n", " w0, w1 = coefficients[i]\n", " \n", " p = figure(plot_width=250, plot_height=200)\n", " p.circle(sizes, prices)\n", " p.line([1,2], [w1 + w0, w1*2 + w0], line_width=2)\n", " \n", " if (len(plots_grid[-1]) == 3):\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": [ "Logo nas primeiras iterações a reta já é muito semelhante ao resultado final.\n", "\n", "Outra forma de analisar a evolução da solução ao longo das iterações é através dos valores dos erros. Vamos plotar uma curva com todos os seus valores." ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", "\n", "\n", "\n", "
\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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[[], []]\n", "for i, error in enumerate(errors):\n", " error_points[0].append(i)\n", " error_points[1].append(error)\n", "\n", "error_chart = figure(plot_width=800, plot_height=300)\n", "error_chart.line(error_points[0], error_points[1], line_width=2)\n", "show(error_chart)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "O Erro Médio Quadrático cai rapidamente nas primeiras iterações, confirmando o fato das retas nas primeiras iterações já serem muito parecidas com o resultado final." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.7.2" } }, "nbformat": 4, "nbformat_minor": 1 }