aula-16-minimos-quadrados.ipynb

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    "# Ajuste de Curvas por Mínimos Quadrados\n",
    "\n",
    "## Regressão Linear\n",
    "\n",
    "O exemplo mais simples de aproximação por mínimos quadrados é ajustar uma reta a um conjunto de pares de observações $(x_1, y_1), (x_2, y_2), \\dots, (x_n, y_n)$. A expressão matemática do ajuste por uma reta é\n",
    "\n",
    "$$ y = a_0 + a_1 x + e \\qquad (1) $$\n",
    "\n",
    "onde $a_0$ e $a_1$ são coeficientes representando a intersecção com o eixo $y$ e a inclinação, respectivamente, e $e$ é o erro ou resíduo entre o modelo e a observação, o qual pode ser representado, depois de se reorganizar a equação acima, por\n",
    "\n",
    "$$ e = y − a_0 − a_1 x $$\n",
    "\n",
    "Portanto, o erro ou resíduo é a discrepância entre o valor verdadeiro de $y$ e o valor aproximado, $a_0 + a_1 x$, previsto pela equação linear.\n",
    "\n",
    "### Critério para um Melhor Ajuste\n",
    "\n",
    "Uma estratégia para ajustar uma melhor reta pelos dados é minimizar a soma dos quadrados dos resíduos entre o $y$ medido e o $y$ calculado com o modelo linear\n",
    "\n",
    "$$ S_r = \\sum _{i = 1}^{n} e_i^2 = \\sum _{i = 1}^{n} (y_{i,medido} - y_{i,modelo})^2 = \\sum _{i = 1}^{n} (y_i - a_0 - a_1 x_i)^2 \\qquad (2) $$\n",
    "\n",
    "Esse critério tem diversas vantagens, incluindo o fato de que ele fornece uma única reta para um dado conjunto de dados.\n",
    "\n",
    "### Ajuste por Mínimos Quadrados por uma Reta\n",
    "\n",
    "Para determinar os valores de $a_0$ e $a_1$, a Equação (2) é derivada com relação a cada coeficiente\n",
    "\n",
    "$$\n",
    "\\dfrac{\\partial S_r}{\\partial a_0} = - 2 \\sum (y_i - a_0 - a_1 x_i) \\\\\n",
    "\\dfrac{\\partial S_r}{\\partial a_1} = - 2 \\sum [(y_i - a_0 - a_1 x_i) x_i]\n",
    "$$\n",
    "\n",
    "Igualando essas derivadas a zero será obtido um $S_r$ mínimo\n",
    "\n",
    "$$\n",
    "0 = \\sum y_i - \\sum a_0 - \\sum a_1 x_i \\\\\n",
    "0 = \\sum y_i x_i - \\sum a_0 x_i - \\sum a_1 x_i^2\n",
    "$$\n",
    "\n",
    "Agora, pode-se expressar essas equações como um conjunto de duas equações lineares simultâneas em duas variáveis ($a_0$ e $a_1$)\n",
    "\n",
    "$$\n",
    "n a_0 + \\left( \\sum x_i \\right) a_1 = \\sum y_i \\\\\n",
    "\\left( \\sum x_i \\right) a_0 + \\left( \\sum x_i^2 \\right) a_1 = \\sum x_i y_i\n",
    "$$\n",
    "\n",
    "Essas são as chamadas equações normais. Elas podem ser resolvidas simultaneamente para obter-se\n",
    "\n",
    "$$\n",
    "a_1 = \\dfrac{n \\sum x_i y_i - \\sum x_i \\sum y_i}{n \\sum x_i^2 - (\\sum x_i)^2} \\qquad (3) \\\\\n",
    "a_0 = \\overline{y} - a_1 \\overline{x} \\qquad (4)\n",
    "$$\n",
    "\n",
    "onde $\\overline{y}$ e $\\overline{x}$ são as médias de $y$ e $x$, respectivamente.\n",
    "\n",
    "### Quantificação do Erro da Regressão Linear\n",
    "\n",
    "Um “desvio padrão” para a reta de regressão pode ser determinado por\n",
    "\n",
    "$$ s_{y/x} = \\sqrt{\\dfrac{S_r}{n-2}} $$\n",
    "\n",
    "onde $s_{y/x}$ é chamado erro padrão da estimativa. O subscrito “$y/x$” indica que o erro é para um valor previsto de $y$ correspondente a um valor particular de $x$. Além disso, observe que agora estamos dividindo por $n − 2$ porque duas estimativas provenientes dos dados —$a_0$ e $a_1$ — foram usadas para calcular $S_r$; portanto, perdemos dois graus de liberdade.  \n",
    "\n",
    "Exatamente como no caso do desvio padrão, o erro padrão da estimativa quantifica a dispersão dos dados. Entretanto, erro padrão da estimativa quantifica a dispersão em torno da reta de regressão, em contraste com o desvio padrão original que quantificava a dispersão em torno da média.\n",
    "\n",
    "A fim de quantificar o \"quão bom\" é um ajuste de reta, dois conceitos utilizados são os coeficiente de determinação ($r^2$) e o coeficiente de correlação ($r = \\sqrt{r^2}$).\n",
    "\n",
    "$$ r^2 = \\dfrac{S_t - S_r}{S_t} $$\n",
    "\n",
    "$$ S_t = \\sum _{i = 1}^n (y_i − \\overline{y})^2 $$\n",
    "\n",
    "Para um ajuste perfeito, $S_r = 0$ e $r = r_2 = 1$, significando que a reta explica 100\\% da variação dos dados. Para $r = r_2 = 0$, $S_r = St$ e o ajuste não representa nenhuma melhora. Uma formulação alternativa para $r$ que é mais conveniente para implementação computacional é\n",
    "\n",
    "$$ r = \\dfrac{n \\sum x_i y_i - (\\sum x_i)(\\sum y_i)}{\\sqrt{n \\sum x_i^2 - (\\sum x_i)^2} \\sqrt{n \\sum y_i^2 - (\\sum y_i)^2}} \\qquad (5) $$\n",
    "\n",
    "## Regressão Polinomial\n",
    "\n",
    "O procedimento dos mínimos quadrados pode ser prontamente estendido para ajustar dados por um polinômio de grau mais alto. Por exemplo, suponha que se queira ajustar um polinômio de segundo grau ou quadrático\n",
    "\n",
    "$$ y = a_0 + a_1 x + a_2 x_2 + e \\qquad (6) $$\n",
    "\n",
    "Nesse caso, a soma dos quadrados dos resíduos é \n",
    "\n",
    "$$ S_r = \\sum _{i = 1}^n (y_i − a_0 − a_1 x_i − a_2 x_i^2)^2 \\qquad (7) $$\n",
    "\n",
    "Seguindo um procedimento análogo ao anterior, toma-se a derivada da Equação (7) com relação a cada um dos coeficientes desconhecidos do polinômio, como em\n",
    "\n",
    "$$\n",
    "\\dfrac{\\partial S_r}{\\partial a_0} = - 2 \\sum (y_i - a_0 - a_1 x_i - a_2 x_i^2) \\\\\n",
    "\\dfrac{\\partial S_r}{\\partial a_1} = - 2 \\sum [(y_i - a_0 - a_1 x_i - a_2 x_i^2) x_i] \\\\\n",
    "\\dfrac{\\partial S_r}{\\partial a_2} = - 2 \\sum [(y_i - a_0 - a_1 x_i - a_2 x_i^2) x_i^2]\n",
    "$$\n",
    "\n",
    "Essas equações podem ser igualadas a zero e reorganizadas para determinar o seguinte conjunto de equações normais\n",
    "\n",
    "$$\n",
    "n a_0 + \\left( \\sum x_i \\right) a_1 + \\left( \\sum x_i^2 \\right) a_2 = \\sum y_i \\\\\n",
    "\\left( \\sum x_i \\right) a_0 + \\left( \\sum x_i^2 \\right) a_1 + \\left( \\sum x_i^3 \\right) a_2 = \\sum x_i y_i \\\\\n",
    "\\left( \\sum x_i^2 \\right) a_0 + \\left( \\sum x_i^3 \\right) a_1 + \\left( \\sum x_i^4 \\right) a_2 = \\sum x_i^2 y_i\n",
    "$$\n",
    "\n",
    "Nesse caso, vê-se que o problema de determinar o polinômio de segundo grau por mínimos quadrados é equivalente a resolver um sistema de três equações lineares simultâneas. Na forma matricial, temos\n",
    "\n",
    "$$\n",
    "\\begin{bmatrix}\n",
    "n & \\sum x_i & \\sum x_i^2 \\\\\n",
    "\\sum x_i & \\sum x_i^2 & \\sum x_i^3 \\\\\n",
    "\\sum x_i^2 & \\sum x_i^3 & \\sum x_i^4\n",
    "\\end{bmatrix}\n",
    "\\begin{bmatrix}\n",
    "a_0 \\\\\n",
    "a_1 \\\\\n",
    "a_2\n",
    "\\end{bmatrix} =\n",
    "\\begin{bmatrix}\n",
    "\\sum y_i \\\\\n",
    "\\sum x_i y_i \\\\\n",
    "\\sum x_i^2 y_i\n",
    "\\end{bmatrix} \\qquad (8)\n",
    "$$\n",
    "\n",
    "O caso bidimensional pode ser facilmente estendido para um polinômio de grau $m$.\n",
    "\n",
    "$$ y = a_0 + a_1 x + a_2 x^2 + \\dots + a_m x^m + e $$"
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    "## Nota\n",
    "\n",
    "O presente texto (conteúdo teórico) é um resumo baseado no livro Métodos Numéricos para Engenharia (Chapra e Canale)."
   ]
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    "## Motivação: Evolução da População Paraibana\n",
    "\n",
    "Este exemplo lê um arquivo CSV, plota o gráfico de dispersão, compara modelos e formata os dados para visualização."
   ]
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     "text": [
      "População obtida por interpolação linear em: 1975 = 1225580; 1982 = 1552017; \n"
     ]
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       "    console.debug(\"Bokeh: BokehJS loaded, going straight to plotting\");\n",
       "    run_inline_js();\n",
       "  } else {\n",
       "    load_libs(css_urls, js_urls, function() {\n",
       "      console.debug(\"Bokeh: BokehJS plotting callback run at\", now());\n",
       "      run_inline_js();\n",
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       "  }\n",
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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    const el = document.getElementById(\"1217\");\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  function run_callbacks() {\n    try {\n      root._bokeh_onload_callbacks.forEach(function(callback) {\n        if (callback != null)\n          callback();\n      });\n    } finally {\n      delete root._bokeh_onload_callbacks\n    }\n    console.debug(\"Bokeh: all callbacks have finished\");\n  }\n\n  function load_libs(css_urls, js_urls, callback) {\n    if (css_urls == null) css_urls = [];\n    if (js_urls == null) js_urls = [];\n\n    root._bokeh_onload_callbacks.push(callback);\n    if (root._bokeh_is_loading > 0) {\n      console.debug(\"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.debug(\"Bokeh: BokehJS not loaded, scheduling load and callback at\", now());\n    root._bokeh_is_loading = css_urls.length + js_urls.length;\n\n    function on_load() {\n      root._bokeh_is_loading--;\n      if (root._bokeh_is_loading === 0) {\n        console.debug(\"Bokeh: all BokehJS libraries/stylesheets loaded\");\n        run_callbacks()\n      }\n    }\n\n    function on_error(url) {\n      console.error(\"failed to load \" + url);\n    }\n\n    for (let i = 0; i < css_urls.length; i++) {\n      const url = css_urls[i];\n      const element = document.createElement(\"link\");\n      element.onload = on_load;\n      element.onerror = on_error.bind(null, url);\n      element.rel = \"stylesheet\";\n      element.type = \"text/css\";\n      element.href = url;\n      console.debug(\"Bokeh: injecting link tag for BokehJS stylesheet: \", url);\n      document.body.appendChild(element);\n    }\n\n    for (let i = 0; i < js_urls.length; i++) {\n      const url = js_urls[i];\n      const element = document.createElement('script');\n      element.onload = on_load;\n      element.onerror = on_error.bind(null, url);\n      element.async = false;\n      element.src = url;\n      console.debug(\"Bokeh: injecting script tag for BokehJS library: \", url);\n      document.head.appendChild(element);\n    }\n  };\n\n  function inject_raw_css(css) {\n    const element = document.createElement(\"style\");\n    element.appendChild(document.createTextNode(css));\n    document.body.appendChild(element);\n  }\n\n  const js_urls = [\"https://cdn.bokeh.org/bokeh/release/bokeh-2.4.3.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-gl-2.4.3.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-widgets-2.4.3.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-tables-2.4.3.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-mathjax-2.4.3.min.js\"];\n  const css_urls = [];\n\n  const inline_js = [    function(Bokeh) {\n      Bokeh.set_log_level(\"info\");\n    },\nfunction(Bokeh) {\n    }\n  ];\n\n  function run_inline_js() {\n    if (root.Bokeh !== undefined || force === true) {\n          for (let i = 0; i < inline_js.length; i++) {\n      inline_js[i].call(root, root.Bokeh);\n    }\nif (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      const cell = $(document.getElementById(\"1217\")).parents('.cell').data().cell;\n      cell.output_area.append_execute_result(NB_LOAD_WARNING)\n    }\n  }\n\n  if (root._bokeh_is_loading === 0) {\n    console.debug(\"Bokeh: BokehJS loaded, going straight to plotting\");\n    run_inline_js();\n  } else {\n    load_libs(css_urls, js_urls, function() {\n      console.debug(\"Bokeh: BokehJS plotting callback run at\", now());\n      run_inline_js();\n    });\n  }\n}(window));"
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     "metadata": {
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       "id": "1158"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import pandas as pd\n",
    "from bokeh.io import output_notebook\n",
    "from bokeh.plotting import figure, show\n",
    "from bokeh.models import PrintfTickFormatter, Range1d\n",
    "\n",
    "from scipy import stats\n",
    "from scipy import interpolate\n",
    "\n",
    "# tabela de dados\n",
    "df = pd.read_csv(\"file-pop-pb.csv\")\n",
    "x = df['ano']\n",
    "y = df['pop. urbana']\n",
    "\n",
    "# grafico de dispersao\n",
    "f = figure(plot_width=400, plot_height=400)\n",
    "f.scatter(x,y,fill_color=\"blue\",radius=1,alpha=1)\n",
    "\n",
    "# formatacao\n",
    "f.background_fill_color = \"blue\"\n",
    "f.background_fill_alpha = 0.02\n",
    "f.x_range=Range1d(1960, 2020)\n",
    "e = 5.e5\n",
    "f.y_range=Range1d(y.min() - e,y.max() + e)\n",
    "f.xaxis.axis_label = \"Ano\"\n",
    "f.yaxis.axis_label = \"Pop. urbana - PB\"\n",
    "f.yaxis[0].formatter = PrintfTickFormatter(format=\"%1.2e\")\n",
    "\n",
    "# interpolacao\n",
    "\n",
    "f.line(x,y,color='green')\n",
    "\n",
    "# estimando o valor da populacao em 1975 e 1982 por interpolacao\n",
    "p = interpolate.interp1d(x, y)\n",
    "\n",
    "xx = [1975,1982]\n",
    "yy = p(xx)\n",
    "print(\"População obtida por interpolação linear em: 1975 = {:d}; 1982 = {:d}; \".format(int(yy[0]),int(yy[1])))\n",
    "f.square(xx,yy,color='black',line_width=4)\n",
    "\n",
    "# ajuste por minimos quadrados\n",
    "\n",
    "slope, intercept, r_value, p_value, std_err = stats.linregress(x,y)\n",
    "px = intercept + slope*x\n",
    "f.line(x,px,color='red')\n",
    "\n",
    "output_notebook()\n",
    "show(f)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.988997710545606e-06"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>pop. urbana</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>ano</th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>1970</th>\n",
       "      <td>1002156</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1980</th>\n",
       "      <td>1449004</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1991</th>\n",
       "      <td>2015576</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2000</th>\n",
       "      <td>2447212</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2010</th>\n",
       "      <td>2902044</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "      pop. urbana\n",
       "ano              \n",
       "1970      1002156\n",
       "1980      1449004\n",
       "1991      2015576\n",
       "2000      2447212\n",
       "2010      2902044"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.set_index('ano')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exemplo\n",
    "\n",
    "Vamos resolver um problema de regressão linear por meio das equações normais passo a passo. Em primeiro lugar, importemos os módulos de computação numérica e de plotagem. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "# importação de módulos \n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import sympy as sy\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consideremos a simples tabela abaixo de um experimento fictício que busca correlacionar densidade e diâmetro médio de grãos em alimentos.\n",
    "\n",
    "densidade | diâmetro médio de grão\n",
    "-|-\n",
    "1.0|0.5\n",
    "2.1|2.5\n",
    "3.3|2.0\n",
    "4.5|4.2\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Vamos escrever os dados como _arrays_."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "# tabela de dados\n",
    "x = np.array([1,2,3,4]) # densidade\n",
    "y = np.array([0.5,2.5,2.0,4.0]) # diâmetro"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Agora, calculamos os coeficientes linear $\\alpha_0$ e angular $\\alpha_1$ pelas fórmulas das equações normais vistas em aula."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "m = np.size(x)\n",
    "alpha1 = (m*np.dot(x,y) - np.sum(x)*np.sum(y))/(m*np.dot(x,x)-np.sum(x)**2)\n",
    "alpha0 = np.mean(y) - alpha1*np.mean(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Podemos agora escrever a equação da reta de regressão usando o _array_ `x` como abscissa. Este será o nosso _modelo de ajuste_."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "y2 = alpha0 + alpha1*x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Enfim, plotamos o gráfico de dispersão dos valores _medidos_ juntamente com o _modelo de ajuste_ da seguinte forma:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "mod = plt.plot(x,y2,'r:'); # modelo\n",
    "med = plt.scatter(x,y,c='b'); # medição\n",
    "plt.legend({'modelo de ajuste':mod, 'medição':med}); # legenda\n",
    "#plt.legend({'medição':med}); # legenda\n",
    "\n",
    "# esta linha adiciona a equação de ajuste ao gráfico na posição (x,y) = (2.8,0.8)\n",
    "# com fonte tamanho 14 e cor RGB = [0.4,0.5,0.4].\n",
    "plt.annotate('y= {0:.2f} + {1:.2f}x'.format(alpha0,alpha1),(2.8,0.8),fontsize=14,c=[0.4,0.5,0.4]);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Na prática, podemos calcular regressão linear usando o módulo `scipy.stats`. Vide _Code session 7_."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
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