{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "cXY3yC_9YofV"
      },
      "source": [
        "<style>\n",
        "  .justified {\n",
        "    text-align: justify;\n",
        "    text-justify: inter-word;\n",
        "  }\n",
        "</style>\n",
        "\n",
        "<div class=\"justified\">\n",
        "\n",
        "# Elementos de redes neuronales\n",
        "\n",
        "```{margin}\n",
        "*Como prerrequisitos para esta lección, sugiero leer la [lección anterior](../alfa/programacion.ipynb) y el artículo de Khan Academy sobre la [pendiente matemática](https://es.khanacademy.org/math/cc-eighth-grade-math/cc-8th-linear-equations-functions/8th-slope/a/slope-from-two-points).*\n",
        "```\n",
        "\n",
        "Este ensayo está inspirado en [micrograd](https://github.com/karpathy/micrograd) de Andrej Karpathy, así como su texto sobre redes neuronales {cite}`KarpathyNets`. A continuación, construiremos una red neuronal desde cero, explicando a detalle cada uno de sus elementos."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "RbSUc-NKiP2c"
      },
      "source": [
        "### La derivada demostrada según el orden geométrico\n",
        "\n",
        "Las redes neuronales son gigantescas funciones matemáticas. La característica distintiva de estas funciones, que catalizan gran parte de lo que hoy se denomina «inteligencia artificial» (aunque, más estrictamente, deberíamos hablar de «aprendizaje profundo» o *deep learning*), es que pueden «aprender» a refinar, optimizar y generalizar las «reglas» (parámetros) que las componen. Tal proceso de aprendizaje consiste en la iteración de operaciones matemáticas propias del cálculo diferencial y vectorial. Por ello, para poder formular una red neuronal, procuraremos primero entender intuitivamente a la derivada: esa médula matemática del aprendizaje."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "iBfYkpuaE_-c"
      },
      "source": [
        "Para tal propósito, construiremos la derivada desde cero, utilizando apenas unas cuantas nociones básicas de geometría.\n",
        "\n",
        "En geometría, denominamos «pendiente» al valor de la inclinación de una recta. Conceptualmente, la pendiente puede ser interpretada como la proporción del cambio que hay entre las variables $x$ y $y$ de la recta. En otras palabras, la pendiente mide qué tanto influye $x$ en el valor de $y$, o qué tanto cambia $y$ cuando avanzamos en $x$. Supongamos, por ejemplo, la recta de una función que podemos definir como $f(x, b) = x \\cdot b$, donde:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "2JOJFKA84lh7"
      },
      "outputs": [],
      "source": [
        "!pip install matplotlib --upgrade"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "Xog7ax-r4mjy"
      },
      "outputs": [],
      "source": [
        "import matplotlib.pyplot as plt\n",
        "import numpy as np\n",
        "import pandas as pd"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 390
        },
        "id": "rcKHWKTxGt6X",
        "outputId": "72991426-5bc6-46d7-d241-3fa803d5c80b"
      },
      "outputs": [
        {
          "data": {
            "text/html": [
              "<style type=\"text/css\">\n",
              "</style>\n",
              "<table id=\"T_aa72e_\" class=\"dataframe\">\n",
              "  <thead>\n",
              "    <tr>\n",
              "      <th class=\"col_heading level0 col0\" >Valores de x</th>\n",
              "      <th class=\"col_heading level0 col1\" >Valores de b</th>\n",
              "      <th class=\"col_heading level0 col2\" >Valores de y</th>\n",
              "    </tr>\n",
              "  </thead>\n",
              "  <tbody>\n",
              "    <tr>\n",
              "      <td id=\"T_aa72e_row0_col0\" class=\"data row0 col0\" >0</td>\n",
              "      <td id=\"T_aa72e_row0_col1\" class=\"data row0 col1\" >5</td>\n",
              "      <td id=\"T_aa72e_row0_col2\" class=\"data row0 col2\" >0</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <td id=\"T_aa72e_row1_col0\" class=\"data row1 col0\" >2</td>\n",
              "      <td id=\"T_aa72e_row1_col1\" class=\"data row1 col1\" >5</td>\n",
              "      <td id=\"T_aa72e_row1_col2\" class=\"data row1 col2\" >10</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <td id=\"T_aa72e_row2_col0\" class=\"data row2 col0\" >4</td>\n",
              "      <td id=\"T_aa72e_row2_col1\" class=\"data row2 col1\" >5</td>\n",
              "      <td id=\"T_aa72e_row2_col2\" class=\"data row2 col2\" >20</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <td id=\"T_aa72e_row3_col0\" class=\"data row3 col0\" >6</td>\n",
              "      <td id=\"T_aa72e_row3_col1\" class=\"data row3 col1\" >5</td>\n",
              "      <td id=\"T_aa72e_row3_col2\" class=\"data row3 col2\" >30</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <td id=\"T_aa72e_row4_col0\" class=\"data row4 col0\" >8</td>\n",
              "      <td id=\"T_aa72e_row4_col1\" class=\"data row4 col1\" >5</td>\n",
              "      <td id=\"T_aa72e_row4_col2\" class=\"data row4 col2\" >40</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <td id=\"T_aa72e_row5_col0\" class=\"data row5 col0\" >10</td>\n",
              "      <td id=\"T_aa72e_row5_col1\" class=\"data row5 col1\" >5</td>\n",
              "      <td id=\"T_aa72e_row5_col2\" class=\"data row5 col2\" >50</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <td id=\"T_aa72e_row6_col0\" class=\"data row6 col0\" >12</td>\n",
              "      <td id=\"T_aa72e_row6_col1\" class=\"data row6 col1\" >5</td>\n",
              "      <td id=\"T_aa72e_row6_col2\" class=\"data row6 col2\" >60</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <td id=\"T_aa72e_row7_col0\" class=\"data row7 col0\" >14</td>\n",
              "      <td id=\"T_aa72e_row7_col1\" class=\"data row7 col1\" >5</td>\n",
              "      <td id=\"T_aa72e_row7_col2\" class=\"data row7 col2\" >70</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <td id=\"T_aa72e_row8_col0\" class=\"data row8 col0\" >16</td>\n",
              "      <td id=\"T_aa72e_row8_col1\" class=\"data row8 col1\" >5</td>\n",
              "      <td id=\"T_aa72e_row8_col2\" class=\"data row8 col2\" >80</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <td id=\"T_aa72e_row9_col0\" class=\"data row9 col0\" >18</td>\n",
              "      <td id=\"T_aa72e_row9_col1\" class=\"data row9 col1\" >5</td>\n",
              "      <td id=\"T_aa72e_row9_col2\" class=\"data row9 col2\" >90</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <td id=\"T_aa72e_row10_col0\" class=\"data row10 col0\" >20</td>\n",
              "      <td id=\"T_aa72e_row10_col1\" class=\"data row10 col1\" >5</td>\n",
              "      <td id=\"T_aa72e_row10_col2\" class=\"data row10 col2\" >100</td>\n",
              "    </tr>\n",
              "  </tbody>\n",
              "</table>\n"
            ],
            "text/plain": [
              "<pandas.io.formats.style.Styler at 0x7f175a8453d0>"
            ]
          },
          "execution_count": 47,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "x = np.arange(0, 21, 2)\n",
        "b = 5\n",
        "def f(x, b): return x*b\n",
        "y = f(x, b)\n",
        "\n",
        "d = {'Valores de x': x, 'Valores de b': b, 'Valores de y': y}\n",
        "tabla = pd.DataFrame(data=d)\n",
        "tabla = tabla.style.hide_index()\n",
        "tabla"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "jo2jWqSnIA2b"
      },
      "source": [
        "En términos más simples, la función es una multiplicación entre $x$ (números del $0$ al $20$ en intervalos de $2$) y $b$ (es decir, $5$). Podemos visualizarla de la siguiente manera:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 442
        },
        "id": "yn4n6M1KF_MA",
        "outputId": "0c88ba2c-e7d1-4e12-f094-9670299e7e81"
      },
      "outputs": [
        {
          "data": {
            "image/png": 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",
            "text/plain": [
              "<Figure size 720x504 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          },
          "output_type": "display_data"
        }
      ],
      "source": [
        "fig, ax = plt.subplots(figsize=(10, 7))\n",
        "ax.plot(x, y, marker='o', color='blue') \n",
        "plt.xticks(ticks=x)\n",
        "plt.yticks(ticks=y)\n",
        "ax.set_facecolor('black')\n",
        "ax.set_xlabel('Valores de x')\n",
        "ax.set_ylabel('Valores de y')\n",
        "ax.hlines(y=10, xmin=0, xmax=2, linewidth=1, color='white', linestyles='dashed')\n",
        "ax.vlines(x=2, ymin=0, ymax=10, linewidth=1, color='white', linestyles='dashed')\n",
        "plt.show()"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "S7-ur-FJIEXb"
      },
      "source": [
        "Ahora, ¿cómo puedo saber cuánto está influyendo cada valor de $x$ en cada valor de $y$? Es decir, si paso de $x=2$ a $x=4$, ¿qué le pasa a $y$? $y$ avanza desde $10$ hasta $20$, es decir, cinco veces $x$: $5 \\times 2 = 10$. Y si paso de $x=4$ a $x=6$, es decir, si avanzo dos «pasos» en $x$, $y$ nuevamente avanza cinco veces lo que avancé en $x$, pasando de $20$ a $30$.\n",
        "\n",
        "En ese sentido, la proporción que hay entre $x$ y $y$ es de $5$, $5$ es el número que determina cada valor de $y$ a partir de $x$ o, dicho de otra forma, $5$ es la «fuerza» con la que $x$ influye en $y$. Esta intuición es clave, pero al mismo tiempo resulta evidente porque la definición de mi función —o sea, cada valor de $y$— indica que debo multiplicar $x$ por $5$. Al calcular la pendiente, es como si no supiéramos este número y tuviéramos que deducirlo matemáticamente.\n",
        "\n",
        "La fórmula de la pendiente nos ofrece un método general para calcular esta proporción o fuerza de la que hablamos:\n",
        "\n",
        "```{margin}\n",
        "Los símbolos $\\Delta, \\delta$ son letras del alfabeta griego llamadas «delta», y generalmente se utilizan en matemáticas para representar o leer un cambio: «cambio en $y$ sobre cambio en $x$».\n",
        "```\n",
        "\n",
        "$$\n",
        "Pendiente = \\frac{y_2 - y_1}{x_2 - x_1} = \\frac{\\Delta y}{\\Delta x}\n",
        "$$\n",
        "\n",
        "Entonces la fórmula solo nos dice que debemos tomar dos puntos cualesquiera de $x$ y $y$ para calcular la pendiente. Por ejemplo, si consideramos los valores del punto denotado por las líneas punteadas en la gráfica anterior, de tal manera que $x_1 = 2, y_1 = 10$, podemos tomar el punto siguiente para calcular la pendiente o inclinación de la recta, de manera que:\n",
        "\n",
        "$$\n",
        "Pendiente = \\frac{20 - 10}{4 - 2} = \\frac{10}{2} = 5\n",
        "$$\n",
        "\n",
        "Pero las funciones lineales —es decir, las rectas—, por definición tienen la misma inclinación en todos sus puntos o segmentos. Comprobémoslo cambiando nuestras $x_2, y_2$:\n",
        "\n",
        "$$\n",
        "Pendiente = \\frac{y_2 - y_1}{x_2 - x_1} = \\frac{60 - 10}{12 - 2} = \\frac{50}{10} = 5\n",
        "$$\n",
        "\n",
        "Naturalmente, la pendiente sigue siendo la misma. Ahora reiteremos, ¿exactamente qué quiere decir el número cinco? Que por cada paso que damos en $x$, $y$ avanza cinco veces más. Dicho de otra forma, $y$ es el resultado de multiplicar $x$ por cinco; o $x$ influye con una «fuerza» de cinco veces sí misma en $y$; o $x$ transforma a $y$ con una fuerza de cinco veces sí misma.\n",
        "\n",
        "Visualicemos la pendiente en color rojo:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 459
        },
        "id": "U1wltsUJLD85",
        "outputId": "83da3b81-2455-4a7b-8f34-d5d6eeb1f852"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "Valor de la pendiente: 5.0\n"
          ]
        },
        {
          "data": {
            "image/png": 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",
            "text/plain": [
              "<Figure size 720x504 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          },
          "output_type": "display_data"
        }
      ],
      "source": [
        "pendiente = (y[6] - y[3]) / (x[6] - x[3])\n",
        "\n",
        "print(f'Valor de la pendiente: {pendiente}')\n",
        "fig, ax = plt.subplots(figsize=(10, 7))\n",
        "ax.plot(x, y, marker='o', color='blue', label='recta de la función') \n",
        "plt.xticks(ticks=x)\n",
        "ax.set_facecolor('black')\n",
        "ax.set_xlabel('Valores de x')\n",
        "ax.set_ylabel('Valores de y') \n",
        "ax.axline((x[6], y[6]), slope=pendiente, color='red', label='pendiente') \n",
        "ax.hlines(y=10, xmin=0, xmax=2, linewidth=1, color='white', linestyles='dashed')\n",
        "ax.vlines(x=2, ymin=0, ymax=10, linewidth=1, color='white', linestyles='dashed')\n",
        "ax.legend()\n",
        "plt.show()"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "JO9L9CXULZdh"
      },
      "source": [
        "La pendiente es idéntica a la recta porque, como decíamos, la recta tiene la misma inclinación —pendiente— en todos sus puntos. Ahora, el problema con la pendiente es que esta propiedad es al mismo tiempo una limitación fundamental: la pendiente solo es válida para una recta o función lineal.\n",
        "\n",
        "Dado que las funciones no lineales no tienen la misma inclinación en todos sus puntos, no podemos hablar de la «inclinación» ni de la «pendiente de una función no lineal». Visualicémoslo con una función cuadrática, donde mis valores de $x$ ahora van del $0$ al $5.99$. Digamos también que me interesa saber la inclinación o «pendiente» de la función cuando $x=5$:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 442
        },
        "id": "eDkt6KGsLAI5",
        "outputId": "8707edd1-0712-4e1b-e54b-f2082d6ced6a"
      },
      "outputs": [
        {
          "data": {
            "image/png": 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",
            "text/plain": [
              "<Figure size 720x504 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          },
          "output_type": "display_data"
        }
      ],
      "source": [
        "x = np.arange(0, 6, 0.0001)\n",
        "def nolineal(x): return x**2\n",
        "y = nolineal(x)\n",
        "\n",
        "pendiente = (5**2 - 2**2) / (5 - 2)\n",
        "\n",
        "fig, ax = plt.subplots(figsize=(10, 7))\n",
        "ax.plot(x, y, color='blue', label='gráfica de la función') \n",
        "ax.set_facecolor('black')\n",
        "ax.set_xticks(np.arange(-5, 6, 1))\n",
        "ax.set_xlabel('Valores de x')\n",
        "ax.set_ylabel('Valores de y')\n",
        "ax.set_ylim(min(y), max(y))\n",
        "ax.set_xlim(min(x), max(x))\n",
        "ax.axline((2, 4), slope=pendiente, color='red', label='pendiente?')\n",
        "ax.hlines(y=4, xmin=0, xmax=2, linewidth=1, color='white', linestyles='dashed')\n",
        "ax.hlines(y=25, xmin=0, xmax=5, linewidth=1, color='white', linestyles='dashed')\n",
        "ax.vlines(x=2, ymin=0, ymax=4, linewidth=1, color='white', linestyles='dashed')\n",
        "ax.vlines(x=5, ymin=0, ymax=25, linewidth=1, color='white', linestyles='dashed')\n",
        "ax.legend()\n",
        "plt.show()"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "gq2fVE3sMHT5"
      },
      "source": [
        "Para calcular la inclinación de la función (es decir, la curva azul) cuando $x=5$, he utilizado la fórmula de la pendiente tomando como referencia los puntos $x_1=2, x_2=5$, de tal manera que:\n",
        "\n",
        "$$\n",
        "Pendiente = \\frac{y_2 - y_1}{x_2 - x_1} = \\frac{25 - 4}{5 - 2} = \\frac{21}{3} = 7\n",
        "$$\n",
        "\n",
        "Pero la realidad es que mi función no cambió en esa proporción con respecto al punto $x=5$: esa pendiente de $7$ corresponde a la secante[^1] que atraviesa la función, no a la «pendiente» del punto $x=5$. La gráfica evidencia que el resultado es inexacto: queda claro que la función tiene una inclinación distinta a la de la recta roja.\n",
        "```{margin}\n",
        "Tengamos en mente lo siguiente: nuestra función es curva porque su inclinación siempre cambia, y su inclinación siempre cambia porque cada valor de $x$ influye en $y$ de manera distinta. Intuitivamente, cuando $x=4$, $y$ resulta de $4^2$, es decir, $y$ resulta de multiplicar $4 \\times 4$; pero cuando $x=5$, $y$ aumenta en una proporción diferente: $5 \\times 5$. Por contra, las rectas o funciones lineales siempre cambian en la misma proporción —anteriormente, siempre se multiplicaba cada $x$ por $5$—, de manera que su inclinación es única y constante.\n",
        "```\n",
        "Además de que la función tiene múltiples «pendientes» —porque su inclinación siempre cambia—, estas «pendientes» son inexactas, puesto que no miden con precisión el impacto (o inclinación) de un solo punto en la función, sino que la fórmula de la pendiente necesariamente nos ofrece la inclinación de una recta trazada entre dos valores de $x$ y dos valores de $y$. Puesto que la función con la que lidiamos ahora es curva, los puntos de $x$ y $y$ que tomamos no son satisfactorios porque denotan una línea, no una curva. En ese sentido, el resultado está sesgado.\n",
        "\n",
        "Entonces, ¿cómo puedo hacer para encontrar el verdadero impacto de $x$ en $y$? Podemos pensar en el siguiente truco: si la pendiente de la secante, aunque errónea, se acerca de alguna forma a nuestro valor deseado, ¿no podríamos hacer una secante más pequeña, tal que fuera más similar al punto que nos interesa? Es decir, si tomáramos la pendiente de la distancia entre dos puntos de $x$ más cercanos entre sí, eso tendría que darnos una mejor aproximación al resultado correcto, ¿no? Veámoslo así:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 475
        },
        "id": "Dg_T-qzrhMAj",
        "outputId": "6412ad3b-8f0b-48b4-f4ff-804443784914"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "Valor de la pendiente con una secante más pequeña: 9.0\n"
          ]
        },
        {
          "data": {
            "image/png": 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",
            "text/plain": [
              "<Figure size 720x504 with 2 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          },
          "output_type": "display_data"
        }
      ],
      "source": [
        "pendiente = (5**2 - 4**2) / (5 - 4)\n",
        "\n",
        "print(f'Valor de la pendiente con una secante más pequeña: {pendiente}')\n",
        "fig, ax = plt.subplots(ncols=2, figsize=(10, 7))\n",
        "ax[0].plot(x, y, color='blue', label='gráfica de la función') \n",
        "ax[0].set_facecolor('black')\n",
        "ax[0].set_xticks(np.arange(0, 6, 1))\n",
        "ax[0].set_xlabel('Valores de x')\n",
        "ax[0].set_ylabel('Valores de y / f(x)')\n",
        "ax[0].axline((5, 25), slope=pendiente, color='red', label='pendiente?')\n",
        "ax[0].hlines(y=16, xmin=0, xmax=4, linewidth=1, color='white', linestyles='dashed')\n",
        "ax[0].hlines(y=25, xmin=0, xmax=5, linewidth=1, color='white', linestyles='dashed')\n",
        "ax[0].vlines(x=4, ymin=0, ymax=16, linewidth=1, color='white', linestyles='dashed')\n",
        "ax[0].vlines(x=5, ymin=0, ymax=25, linewidth=1, color='white', linestyles='dashed')\n",
        "ax[0].legend()\n",
        "ax[1].plot(x, y, color='blue', label='gráfica de la función') \n",
        "ax[1].set_facecolor('black')\n",
        "ax[1].set_xticks(np.arange(-5, 6, 1))\n",
        "ax[1].set_xlabel('Valores de x')\n",
        "ax[1].set_ylabel('Valores de y / f(x)')\n",
        "ax[1].set_title('Zoom')\n",
        "ax[1].set_ylim(12, 30)\n",
        "ax[1].set_xlim(2, 7)\n",
        "ax[1].axline((5, 25), slope=pendiente, color='red', label='pendiente?')\n",
        "ax[1].hlines(y=16, xmin=0, xmax=4, linewidth=1, color='white', linestyles='dashed')\n",
        "ax[1].hlines(y=25, xmin=0, xmax=5, linewidth=1, color='white', linestyles='dashed')\n",
        "ax[1].vlines(x=4, ymin=0, ymax=16, linewidth=1, color='white', linestyles='dashed')\n",
        "ax[1].vlines(x=5, ymin=0, ymax=25, linewidth=1, color='white', linestyles='dashed')\n",
        "ax[1].legend()\n",
        "plt.show()"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "okWoLUBFjVIQ"
      },
      "source": [
        "Gráficamente podemos comprobar que hemos obtenido un mucho mejor resultado: la secante que trazamos tiene una inclinación similar a la de la curva de la función. Pero las líneas punteadas —las cuales denotan los puntos que hemos tomado como referencia para calcular la pendiente— nos indican que todavía podemos mejorar, tomando como referencia coordenadas más cercanas entre sí.\n",
        "\n",
        "Bien, pues antes de continuar experimentando, adelantemos ya que esta línea de razonamiento es precisamente la que dio origen a la derivada. El truco es este: podemos calcular la pendiente de líneas secantes cada vez más pequeñas y similares a nuestro punto, obteniendo cada vez mejores resultados. Es más, podemos llegar a calcular la pendiente de una línea infinitamente pequeña, tan pequeña que sería casi idéntica al punto, y la pendiente de esa línea infinitamente pequeña sería igual a la pendiente del punto. En ese mismo sentido, podríamos decir incluso —y de hecho se hace así en el ámbito matemático— que estamos calculando la pendiente de una línea tangente al punto, puesto que su pendiente sería igual a la de una secante infinitamente pequeña que se confunde con el punto.\n",
        "\n",
        "```{margin}\n",
        "Hasta donde sé, fue Leibniz quien inventó la derivada razonando de una manera [casi idéntica](https://youtu.be/5M2RWtD4EzI) a la nuestra.\n",
        "```\n",
        "\n",
        "```{figure} ../../img/derivada.png\n",
        "---\n",
        "width: 90%\n",
        "name: derivada\n",
        "---\n",
        "Como en la imagen, queremos obtener la pendiente de una secante cada vez más pequeña —más cercana al punto que nos interesa—, es decir, con una distancia $h$ cada vez menor. Podemos hacer esta línea tan infinitamente pequeña que se identifique con el punto; por tanto, su pendiente sería igual a la pendiente de una línea tangente al punto.\n",
        "```\n",
        "Utilizando la fórmula de la pendiente para poner en práctica nuestra intuición, podemos conseguir un resultado como este:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 475
        },
        "id": "BaCeXFEPhjiM",
        "outputId": "36b38fff-95a0-4864-8529-1b96e0939cbd"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "Valor de la pendiente con una secante más pequeña: 9.990000000000023\n"
          ]
        },
        {
          "data": {
            "image/png": 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",
            "text/plain": [
              "<Figure size 720x504 with 2 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          },
          "output_type": "display_data"
        }
      ],
      "source": [
        "pendiente = (5**2 - 4.99**2) / (5 - 4.99)\n",
        "\n",
        "print(f'Valor de la pendiente con una secante más pequeña: {pendiente}')\n",
        "fig, ax = plt.subplots(ncols=2, figsize=(10, 7))\n",
        "ax[0].plot(x, y, color='blue', label='gráfica de la función') \n",
        "ax[0].set_facecolor('black')\n",
        "ax[0].set_xticks(np.arange(0, 6, 1))\n",
        "ax[0].set_xlabel('Valores de x')\n",
        "ax[0].set_ylabel('Valores de y / f(x)')\n",
        "ax[0].axline((5, 25), slope=pendiente, color='red', label='pendiente?')\n",
        "ax[0].hlines(y=4.99**2, xmin=0, xmax=4.99, linewidth=1, color='white', linestyles='dashed')\n",
        "ax[0].hlines(y=25, xmin=0, xmax=5, linewidth=1, color='white', linestyles='dashed')\n",
        "ax[0].vlines(x=4.99, ymin=0, ymax=4.99**2, linewidth=1, color='white', linestyles='dashed')\n",
        "ax[0].vlines(x=5, ymin=0, ymax=25, linewidth=1, color='white', linestyles='dashed')\n",
        "ax[0].legend()\n",
        "ax[1].plot(x, y, color='blue', label='gráfica de la función') \n",
        "ax[1].set_facecolor('black')\n",
        "ax[1].set_xticks(np.arange(-5, 6, 1))\n",
        "ax[1].set_xlabel('Valores de x')\n",
        "ax[1].set_ylabel('Valores de y / f(x)')\n",
        "ax[1].set_title('Zoom')\n",
        "ax[1].set_ylim(22, 28)\n",
        "ax[1].set_xlim(2, 7)\n",
        "ax[1].axline((5, 25), slope=pendiente, color='red', label='pendiente?')\n",
        "ax[1].hlines(y=4.99**2, xmin=0, xmax=4.99, linewidth=1, color='white', linestyles='dashed')\n",
        "ax[1].hlines(y=25, xmin=0, xmax=5, linewidth=1, color='white', linestyles='dashed')\n",
        "ax[1].vlines(x=4.99, ymin=0, ymax=4.99**2, linewidth=1, color='white', linestyles='dashed')\n",
        "ax[1].vlines(x=5, ymin=0, ymax=25, linewidth=1, color='white', linestyles='dashed')\n",
        "ax[1].legend()\n",
        "plt.show()"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "1z8brlkej30l"
      },
      "source": [
        "En esencia, hemos minimizado las distancias entre $x_2$ y $x_1$ al utilizar los valores $x_2=5, x_1=4.99$ para obtener un resultado más riguroso. Gráficamente, corroboramos que nuestro resultado es más exacto: la línea que podemos trazar con la pendiente obtenida es casi idéntica a la función, al punto que la opaca cuando la miramos desde cerca. Suponiendo, como en nuestro dibujo anterior, que llamemos $h$ a la distancia entre las $x$, nuestra fórmula actual para la pendiente de esta tangente o secante infinitamente pequeña luce así:\n",
        "\n",
        "$$\n",
        "Pendiente = \\frac{y_2 - y_1}{x_2 - x_1} = \\frac{f(x+h) - f(x)}{(x+h) - x} = \\frac{f(x+h) - f(x)}{h}\n",
        "$$\n",
        "\n",
        "Como decía, idealmente queremos que la distancia $h$ sea lo más pequeña posible, es decir, lo más cercana a $0$ posible (sin ser $0$, puesto que necesitamos una distancia entre dos $x$ para tener una línea y así ser capaces de obtener su pendiente, puesto que un punto no tiene pendiente). Para expresar esta idea, emplearemos la expresión $\\lim _{h \\rightarrow 0}$, es decir, «cuando $h$ es tan pequeña que se acerca a $0$» o, más estrictamente, «el límite de la función cuando $h$ tiende a $0$». De tal manera que:\n",
        "\n",
        "$$\n",
        "\\lim _{h \\rightarrow 0} \\frac{f(x+h) - f(x)}{h}\n",
        "$$\n",
        "```{margin}\n",
        "Sobre límites se puede leer [este artículo](https://es.khanacademy.org/math/ap-calculus-ab/ab-limits-new/ab-1-2/a/limits-intro?modal=1); sobre derivadas, [son recomendables también estos videos](https://es.khanacademy.org/math/ap-calculus-ab/ab-differentiation-1-new).\n",
        "```\n",
        "Pero ahora ya no hablamos de una pendiente exactamente, sino que la estamos alterando para conseguir un resultado distinto. A este nuevo concepto lo denominaremos «derivada»:\n",
        "\n",
        "$$\n",
        "Derivada = \\lim _{h \\rightarrow 0} \\frac{f(x+h) - f(x)}{h}\n",
        "$$\n",
        "\n",
        "En la práctica también suele escribirse como $\\frac{dy}{dx}$, o $f'(x)$.\n",
        "\n",
        "```{margin}\n",
        "Bien vista, la expresión $\\frac{dy}{dx}$ ilustra a la perfección el concepto: estamos midiendo la diferencia en el valor de $y$ ($dy$) que provoca un cambio infinitamente pequeño en la variable $x$, es decir, una pequeña diferencia —denotada por $h$— en $x$: $dx$. Finalmente, para averiguar la proporción que guardan entre ellos, los dividimos entre sí, midiendo la derivada —diferencia— de $y$ con respecto a $x$ —es decir, con respecto a una pequeña diferencia en el valor de $x$. El nombre «cálculo diferencial» viene precisamente de la medición de esas pequeñas diferencias en los valores de las variables.\n",
        "```\n",
        "\n",
        "Aplicada a nuestro problema, podemos utilizarla así:\n",
        "\n",
        "$$\n",
        "\\frac{dy}{dx} =\\lim _{h \\rightarrow 0} \\frac{f(x+h) - f(x)}{h} = \\frac{(5+0.001)^2 - 5^2}{0.001} = 10.00\n",
        "$$\n",
        "\n",
        "Programáticamente:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "zDULvEePlU4m",
        "outputId": "86234c8f-c227-4f78-b237-3478afc28fc8"
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "10.001000000002591"
            ]
          },
          "execution_count": 36,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "def derivada(f, x):\n",
        "  h = 0.001\n",
        "  return (f(x+h) - f(x)) / h\n",
        "\n",
        "derivada(nolineal, 5)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "VJhiu4YHE8ny"
      },
      "source": [
        "Finalmente hemos descubierto que cuando $x=5$, $x$ «influye» en $y$ con una «fuerza» de $10$. Analíticamente, podemos comprobarlo de esta forma:\n",
        "\n",
        "$$\n",
        "\\begin{gather*}\n",
        "\\frac{dy}{dx} =\\lim _{h \\rightarrow 0} \\frac{(x+h)^2 - x^2}{h} = \\lim _{h \\rightarrow 0} \\frac{(x+h)(x+h) - x^2}{h} = \\lim _{h \\rightarrow 0} \\frac{x^2 + xh + hx + h^2 - x^2}{h} \\\\\n",
        "= \\lim _{h \\rightarrow 0} \\frac{2xh + h^2}{h} = \\lim _{h \\rightarrow 0} \\frac{h(2x + h)}{h} = \\lim _{h \\rightarrow 0} (2x + h) = 2x + 0 = 2x\n",
        "\\end{gather*}\n",
        "$$\n",
        "\n",
        "Ahora, podemos generalizar esta expresión bajo la fórmula $nx^{n-1}$, es decir, cuando $f(x) = x^2$ y $x=5$, entonces:\n",
        "\n",
        "$$\n",
        "\\begin{gather*}\n",
        "2x = 2(5) = 10 \\\\\n",
        "nx^{n-1} = 2(5)^{2-1} = 10\n",
        "\\end{gather*}\n",
        "$$\n",
        "\n",
        "En suma, $10$ es efectivamente la proporción o fuerza con la que $x$ influye en $y$, $10$ es la derivada de $y$ con respecto a $x$ cuando $x=5$. Y, según hemos visto, podemos obtener este resultado de formas distintas: aplicando la fórmula, programáticamente, geométricamente y analítica o algebraicamente. En general, toda fórmula o expresión matemática tiene un trasfondo racional como los que hemos llevado a cabo, aunque las más de las veces se requieren algunos desarrollos lógico-matemáticos ulteriores para generalizarlas.\n",
        "\n",
        "Volviendo a lo nuestro, podemos concluir que, conceptualmente, la derivada —igual que la pendiente— mide la **proporción o magnitud del cambio** que una variable provoca en el resultado de una función. Digamos que la derivada mide la fuerza con la que una variable influye —en un determinado punto— en el resultado de una función.\n",
        "```{margin}\n",
        "En rigor, la derivada no solo se diferencia de la pendiente por referirse a funciones no lineales, sino también porque es útil para medir el impacto que diferentes variables tienen en una función compuesta (es decir, en una función de más de dos dimensiones o variables). Esto lo veremos más adelante, sin embargo, es importante ser conscientes de que nuestra formulación geométrica de la derivada es una manera intuitiva —y legítima en términos históricos— de entenderla, pero su concepto puede ampliarse a problemas, funciones y dimensiones más complejas que son difíciles de visualizar.\n",
        "```"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "kBc49tIHuz9c"
      },
      "source": [
        "Pongamos otro ejemplo: si los valores de la función nunca cambiaran, entonces su derivada sería $0$. Geométricamente, no habría nada de inclinación en la tangente a cualquier punto de la función; conceptualmente, la definición de la función indicaría que su resultado nunca cambia: ninguna variable tendría influencia en la función, de manera que el «cambio» siempre sería nulo, cero.\n",
        "\n",
        "Supongamos que quiero medir la distancia que recorro en $20$ segundos: si con el paso del tiempo no avanzo nada o, lo que es lo mismo, si la variable tiempo no influye en la función distancia, entonces la función, la pendiente y la tangente a cualquier punto son $0$, por lo que lucen así:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 442
        },
        "id": "wKdEfzmPiPqe",
        "outputId": "17342f35-20f9-4abc-c454-af77a9154cb7"
      },
      "outputs": [
        {
          "data": {
            "image/png": 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",
            "text/plain": [
              "<Figure size 720x504 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          },
          "output_type": "display_data"
        }
      ],
      "source": [
        "tiempo = np.arange(0, 21, 1)\n",
        "def f(t): return t*0\n",
        "distancia = f(tiempo)\n",
        "\n",
        "fig, ax = plt.subplots(figsize=(10, 7))\n",
        "ax.plot(tiempo, distancia, marker='o', color='blue', label='recta de la función') \n",
        "ax.set_facecolor('black')\n",
        "ax.set_xlabel('Tiempo (segundos)')\n",
        "ax.set_ylabel('Distancia (metros)')\n",
        "ax.axline((tiempo[0], distancia[0]), slope=0, color='red', label='derivada / pendiente')\n",
        "ax.legend()\n",
        "plt.show()"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "SJafUmOpiQUa"
      },
      "source": [
        "Veamos un último ejemplo, ahora con la función suma: mi variable $x$ tendrá valores del $0$ al $12$ con intervalos de $2$, mientras que mi función es $x+6$. Mi tabla de valores entonces debería quedar así:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 266
        },
        "id": "r9F_uMZg9b4n",
        "outputId": "d16ff9c6-73ea-435c-f9a8-b1da1783017a"
      },
      "outputs": [
        {
          "data": {
            "text/html": [
              "<style type=\"text/css\">\n",
              "</style>\n",
              "<table id=\"T_c3ccb_\" class=\"dataframe\">\n",
              "  <thead>\n",
              "    <tr>\n",
              "      <th class=\"col_heading level0 col0\" >Valores de x</th>\n",
              "      <th class=\"col_heading level0 col1\" >Valores de y</th>\n",
              "    </tr>\n",
              "  </thead>\n",
              "  <tbody>\n",
              "    <tr>\n",
              "      <td id=\"T_c3ccb_row0_col0\" class=\"data row0 col0\" >0</td>\n",
              "      <td id=\"T_c3ccb_row0_col1\" class=\"data row0 col1\" >6</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <td id=\"T_c3ccb_row1_col0\" class=\"data row1 col0\" >2</td>\n",
              "      <td id=\"T_c3ccb_row1_col1\" class=\"data row1 col1\" >8</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <td id=\"T_c3ccb_row2_col0\" class=\"data row2 col0\" >4</td>\n",
              "      <td id=\"T_c3ccb_row2_col1\" class=\"data row2 col1\" >10</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <td id=\"T_c3ccb_row3_col0\" class=\"data row3 col0\" >6</td>\n",
              "      <td id=\"T_c3ccb_row3_col1\" class=\"data row3 col1\" >12</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <td id=\"T_c3ccb_row4_col0\" class=\"data row4 col0\" >8</td>\n",
              "      <td id=\"T_c3ccb_row4_col1\" class=\"data row4 col1\" >14</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <td id=\"T_c3ccb_row5_col0\" class=\"data row5 col0\" >10</td>\n",
              "      <td id=\"T_c3ccb_row5_col1\" class=\"data row5 col1\" >16</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <td id=\"T_c3ccb_row6_col0\" class=\"data row6 col0\" >12</td>\n",
              "      <td id=\"T_c3ccb_row6_col1\" class=\"data row6 col1\" >18</td>\n",
              "    </tr>\n",
              "  </tbody>\n",
              "</table>\n"
            ],
            "text/plain": [
              "<pandas.io.formats.style.Styler at 0x7f175a3a3190>"
            ]
          },
          "execution_count": 45,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "x = np.arange(0, 13, 2)\n",
        "def f(x): return x+6\n",
        "y = f(x)\n",
        "\n",
        "z = {'Valores de x': x, 'Valores de y': y}\n",
        "tabla = pd.DataFrame(data=z)\n",
        "tabla = tabla.style.hide_index()\n",
        "tabla"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "1Rc_6XsjZm-t"
      },
      "source": [
        "Ahora preguntémonos: ¿en qué proporción cambia $y$ cuando aumentamos $x$? Si paso de mi primer valor de $x$ al segundo, es decir, si paso de $0$ a $2$, ¿cuánto estoy aumentando a mi primer valor de $y$? ¡También dos! Es decir, que la proporción de los cambios entre $x$ y $y$ es 1: si aumento dos a $x$, también aumento $2$ a $y$. En nuestro primer ejemplo, si agregábamos $2$ a $x$, nuestro valor de $y$ aumentaba cinco veces $x$.\n",
        "\n",
        "Esto tiene sentido porque la multiplicación aumenta nuestros valores con mayor velocidad y en diferente proporción que la suma: por ejemplo, $2 \\times 4 = 8$, pero $2 + 4 = 6$, y conforme tomamos números más grandes, nuestra multiplicación continúa creciendo a mayor velocidad que la suma: $2 \\times 10 = 20$, pero $2 + 10 = 12$.\n",
        "\n",
        "Grafiquemos nuestra nueva función de suma:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 442
        },
        "id": "fzrKL4kXXkyV",
        "outputId": "10f3c76b-db2f-4693-8e3f-e038c0c12122"
      },
      "outputs": [
        {
          "data": {
            "image/png": 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aa1i0aBFLliypugz57LPP8qtf/QqAgw46iN/97ndV57rqqquYPn06EcF//vMf3n///S/c6mry5Mkceuihdf5uAwcOpF27drRr147+/fvzwgsvMGPGDJ566imOPfZYAJYsWcLs2bPp3LkzXbt2ZbvtcvNBtt9+e+bPn8+iRYtYtGhR1Y3MDzroIJ544okvfNZTTz3FpEmTuOWWWwD47LPPePvt3JD1rrvuSvv27QHYeuutefvtt6vuJ7rC008/zcsvv8z3vve9qvdvuOGGdf5ekqTC0ZftGckGfIcppHzIfwz/4dkiC/k3RBYjZ18DXk4pzanrxYhYF2iTUlqUf7w/cGFTfPCqRrha2vLly2nfvn3VyE5ta6+9dp37Bw4cyFVXXcXChQt56aWX6N8/N1/iwgsv5LLLLqNHjx5MmDCBioqKVX7+Pffcw4cffsjNN99MeXk5hx12GJ9//vkXjnvhhRdqNJDV1R5liwhSShx//PF8+9vfrvHavHnzaNu2bdXzNm3asGzZslXWWF1KiUsvvZStttqqxv7nn3+etdZaq+p5WVlZnedNKXHwwQdz+umn1/szJUnZCOAA+nEOy6pW8v81e+dX8v/iX+BLTbNNCIiI24DJwHYRMSciTsq/dDS1LmlGRNeIuDv/dFPgsYh4BpgC/DOldG9z1dnc+vbtyyOPPMKnn37K4sWLefTRR4HcJcquXbvywAMPALnm4dV6XHJdZ5116NmzJ2PGjGHvvfeuGl1bvHgxnTp1orKyknvv/e/XtfPOO3P//fcD1Nj/8ccfs+GGG1JeXs7UqVOZP/+L1+lff/11ttpqq5Xmsh555BE+++wzFixYQEVFBT179mSPPfZg/PjxfPLJJ0Du8uwHH9Q5LwSADh060KFDB2bMmPGFGqvbfffd+fOf/0xKuSvXr7zyykrPuUJ5eTmVlZVAbnTtoYceqqpl4cKFdf7OkqTs1Az5V5RUyL8hmm3kLKU0ZCX7j69j3zzgoPzjN4DezVVXS9t+++3Zb7/9OPbYY9lggw3o2bNn1Wu/+MUvuOSSSxg3bhyVlZXst99+9OjRY7Xn3G+//Tj33HO55pprqvadcsopnHDCCay//vr06tWLxYsXAzBixAh++tOf8sc//rFG4P/AAw/khz/8IUcffTQ77LDDF0akAJ544gn22GOPldax7bbbcuqpp7JgwQJOOukkNt54YzbeeGNmzZrFiSeeCOSayQsvvJA2bVb+94Cf/exnVRMCVjYr9KSTTuLXv/41Q4YMYfny5XTr1o3LL7985V8S8K1vfYshQ4aw3XbbcdFFF3HKKadw+umnk1KivLycH/3oR3Tp0mWV55AkNb/WEPJviFgxElEK6poQcM8993whR6X6+Z//+R9+/vOf1/n9jR07lrXXXpvjjjsug8qaz3vvvceBBx6YdRmS1CrUDPkv5j76MpqUD/m3ChV1refq7Zu0Ur///e+zLkGSVIJaY8i/IWzOtEaGDSvopeUkSQVmRch/JMv4KjP4iA5czl5cwautIuTfECXfnC1fvjzrElRE/PdFkprWWrTlGHZjJPPZkQrm0JmRDOQ6pvNRca6S1eya896aBWHmzJlVM/akVamsrGTmzOK7B5skFaL16ciPGcQsNuQGHqOSMr7LnmzDe4xhEh+xKOsSC1bJTwjYYIMNGDVqFN27d1/ljEG1bsuXL2fmzJmMGjWKDz/8MOtyJKlobZEP+Q/Nh/zvpy+Xta6Qf0PUOSGg5JszSZLU/GqG/KNayH9lt9EWztaUJElNaWUh/yt5lTmG/NeYzZkkSWqQXMh/ACN525B/M7A5kyRJ9bI+HRlOH87kZbryOM/kV/L/M1NY2kpurdQSbM4kSdIq1Qz5P8L99OX7dM2H/Fd/X2g1jM2ZJEmqUx+25xxX8m9xNmeSJKmGA+jPOVQa8s+IzZkkSaoK+Y/gHXox1ZB/hmzOJElqxdajI6fUCvkfx57cYcg/MzZnkiS1QlvQlbPpzslMo0M+5H883ZjINAz5Z8vmTJKkVqQP23EOG+ZD/u9yOwMYzXuG/AuIzZkkSa3AAfTjnBor+e/NlbxiyL8A2ZxJklSiaob8a6/k/0jW5WklbM4kSSoxhvyLW5usC5AkSWtqCDALWAbMYgu+z68ZyGyWcwmP8Dzd2J++7MKr3MITLKUy43pVH46cSZJUlIYA1wHr0odpjGQ0R/JnEsHtDGAM7/GMIf+iFCmlrGtoMhFROr+MJEmr9AYH8DIjGc1gHuIjOjCWYVzBkcxht6yLU/1UpJT6197pyJkkSUVkLdoyhAGM5FB68QJz6MY5/IqxDOMj1gOWZ12iGsnmTJKkIrAeHRlOH87Kh/yfpRfH8Ufu4CiWsla1I9/KrEY1DZszSZIKWO2V/CfSJ7+Sfw/g21CjMVsM/CSbQtVkbM4kSSpAfdiOkWzIkdVW8s+F/Kfnj5gGBHAxsAW5EbOfALdlVLGaihMCJEkqIAfQj5EsZzDT8yH/PlzBq8xxfbJS5IQASZIK0X9D/v9dyf8cBjKW6XzEpKzLUwuzOZMkKSNfDPlv60r+sjmTJKmlrTzkPw14LevylDGbM0mSWsjqQ/6SzZkkSc2udsj/N+yVD/k/kXVpKkA2Z5IkNYO2tOUYBjCCd9jJkL8awOZMkqQmtCLkfyav0C0f8v8ee3K7IX/Vk82ZJElNYAu6chbbMpSKqpD/CXQ15K8GszmTJKkRvhjy340x/MeQv9aYzZkkSWvgiyH/vfMh/8ezLk1FzuZMkqR6aks5x7BbHSH/GXzEI1mXpxLRprlOHBHjIuLdiHi+2r5RETE3Imbkt4NW8t4DIuKViJgZEec2V42SJNXHenTkRwxiFhtxI4+TCL7HnmzDe4xmEh/xUdYlqoQ0243PI2Ig8DHwx5RSr/y+UcDHKaXRq3hfGfAqsB8wB3gaGJJSerEen+mNzyVJTSYX8u/OUKbRgY+ZSB9GE9zPtKxLU2lo2Rufp5QmRcRWa/DWAcDMlNIbABFxO/ANYLXNmSRJTWEXtmMkG3EUTwHvcDu7MdqV/NVCmu2y5iqcHhHP5i97blDH692A2dWez8nvq1NEDIuIqRExtakLlSS1Ll+nHxPpw3Re4TCe5Qr2Yhs24Tie4Blezbo8tRIt3ZxdDXwZ2AWYD4xp7AlTSmNTSv3rGhaUJGl12lLO99iLZ+nOvVSwA/M4h4FsThtGMonZzM+6RLUyLdqcpZTeSSktSyktB64jdwmztrnA5tWeb5bfJ0lSk6ke8r+pWsh/a95nNJNYaMhfGWnR5iwiulR7+i3g+ToOexrYNiK2joi1gKOB8S1RnySp9G1BV8YwkNks51Ie4UW68nX60pvXuJknWEpl1iWqlWu2CQERcRuwD9ApIuYAFwD7RMQuQALeBIbnj+0K/CGldFBKqTIiTgfuA8qAcSmlF5qrTklS67ALPRhJJ0P+KnjNtpRGFlxKQ5JU29fpy0gSX2M6i2jPWPpyBa+ZJVMhaNmlNCRJykpbyhnCbozkHXZiGnPZlHMYxHVMZyGTsi5PWqUsltKQJKlZrEdHzmHgSkL+jxjyV1Fw5EySVPQ2pytnV63kP4mJ9OFEuuVX8n8t6/KkBrE5kyQVrbpC/mN4jxmG/FXEbM4kSUWnZsh/HlewVz7k/0TWpUmNZnMmSSoKdYX8f8RAxjLDkL9KihMCJEkFrXbIH+D77MXWvM9lruSvEuTImSSpIG1OF85m26qQ/wP04UQ2434qgJlZlyc1G5szSVJBqRnyf9eQv1odmzNJUkH4Ysh/b67gVUP+anXMnEmSWsgQYBawLP9zCG0p53vsxbN0516msQPz+BED2Zw2jOQRb7GkVsnmTJLUAoYA1wFbAW1Yj/U5h17MopMhf6kWL2tKklrAxcC6bM5bnMUVDOU6OrKIB9iXk+jKfUzDkL+UY3MmSWp2u/A+Izifo7kdgDs4itGMZAa9gbJsi5MKjM2ZJKnZ7E8/zmE5X6M/i2jPlZzJbzib2WyRP+LNLMuTCpLNmSSpSa1YyX8E77IzFfmV/I9mLGNYSNdqRy4GfpJVmVLBckKAJKlJ1F7JP0jVQv63s5CR5EbKlud/DgVuy7BiqTBFSinrGppMRJTOLyNJRWJzunAW2zKU6fmQfx9GE/mQv6RVqEgp9a+908uakqQ1sgs9GEEnjs6v5H8HuzGa913JX2okmzNJUoPsT1/OqbaS/5XsxW+Y6Ur+UhOxOZMkrVZbyjmaAYzkP+zMNObSmR8xiLFMZyGTsi5PKik2Z5KklepIB4bTh7N4hW48wXN05/vsxW08xVLezro8qSTZnEmSvmAzOnM2PfIh/0k8QB9Oopsr+UstwOZMklSlNz0YSSeOYgrBf7iD3RjD+0w35C+1GJszSRL705eRJPbLh/x/y55cwUzeMuQvtTibM0lqpWqH/OexKT9mENca8pcy5R0CJKmV6UiHqpX8/8gTVSv5b8X7/IpHWMhHWZcotWqOnElSK2HIXyoONmeSVOIM+UvFxeZMkkqUIX+pONmcSVIJMeQvFT8nBEhSCehIB0YykDfoxB95gjYkjjfkLxUlR84kqYhtRmfOogfD8iH/B+nDyXQ15C8VMZszSSpCvenBCDpxdD7k/2cGMJoPDPlLJcDmTJKKyH705Zw6Q/6Tsy5NUhOxOZOkAreykP9YprPAkL9UcpwQIEkFamUh/63zIf8FhvylkuTImSQVGEP+UutmcyZJBaI32zKCjQ35S61cszVnETEOOAR4N6XUK7/vMuBQ4HPgdeCElNKCOt77JrAIWAZUppT6N1edkpS1miH/+Yb8pVauOTNnNwIH1No3EeiVUtoZeBU4bxXv3zeltIuNmaRS1JZyvsuezKAH9zONHZnHjxnEFrRhBJN4i3lZlygpI83WnKWUJgEf1Np3f0qpMv/0SWCz5vp8SSpE1UP+N/MEZSzPh/w/MOQvCch2tuaJwD0reS0B90dERUQMW9VJImJYREyNiKlNXqEkNZHN6MxlDGQ2cBmTeIUuHEg/dmImN/E4n7M06xIlFYhMJgRExP8ClcCtKzlk75TS3IjYBJgYES/nR+K+IKU0FhibP29qloIlaQ3tzLaMrBXyH8OHTDPkL2klWrw5i4jjyU0UGJxSqrOZSinNzf98NyLuBAaAKy1KKh770ZeRwP5M42Pm8bt8yP/fhvwlrUaLXtaMiAOAHwGHpZQ+Wckx60ZEhxWPgf2B51uuSklaM7VD/r2Yy48ZxOaU8UMm8W9D/pLqodmas4i4DZgMbBcRcyLiJOB3QAdylypnRMQ1+WO7RsTd+bduCjwWEc8AU4B/ppTuba46JamxDPlLakqxkiuLRcnMmaSWVHMl/0U8SB9G04Z7qci6NEnFoaKuJcO8Q4AkNZAhf0nNyRufS9JKDQFmkbtZyRvsx/HcR1+e4TW+xTP8jj3pTmeOZTLTeDnjWiWVCkfOJKlOQ4DraEtbjuJWRjKa3jzLPDpzLoO4lukscBK5pGZgcyZJdejITxjK1ZzNb9iMubxAT05gHH9iLz5nu6zLk1TCbM4kqZrN2JQz2Y7h7ElHFvEQ+zKU67iXA4AAlmddoqQSZ3MmSdQO+T/OX/gWozmXafSrdeRbmdQnqfWwOZPUqu1HH0YStVbyf51/0xbYvtbRi4GfZFClpNbE5kxSq9OWMo5iN0byHr2Zzjw2rSPkf1v+58XAFuRGzH5Sbb8kNQ8XoZXUanSkPUPpy9m8yma8zQt8mdF05k9M4XOWZl2epNbHRWgltU7/DflPpyOTeIhdGEq3/Er+r2ddniTVYHMmqWR9MeQ/gNF8yDRmZF2aJK2UzZmkkvM1+nBOnSH/yVmXJkmrZXMmqSSUU8bRDKgK+c9nE1fyl1SUvLempKLWgfaMYCBvsAk3M5lylnMCe7EVH3Ipj7CAj7IuUZIaxJEzSUWpG505ix4MYwbr5UP+w+nKvVSQDPlLKmI2Z5KKys50ZwSbMIQpBP/hLwxgDB9QYchfUomwOZNUFL7GLpxDm3zIfz6/Zw9+wxuG/CWVHJszSQWrZsh/Rq2Q/6NZlydJzcIJAZIKjiF/Sa2ZI2eSCoYhf0myOZNUAOoO+X9oyF9Sq2RzJikzX2MXRtKGr1eF/PfkN67kL6mVszmT1KLKKeModmMk77FLPuR/HoO4xpX8JQlwQoCkFtKB9vwwH/K/hSdoy7KqkP8lhvwlqYojZ5KaVTc6cybbMZzprMck/kVvQ/6StAo2Z5KaRfWQfxve5c/slg/5P5N1aZJU0GzOJDUpQ/6S1Dg2Z5IaLRfyH8BI3jfkL0mN5IQASWusZsh/siF/SWoCjpxJarAvhvxdyV+SmorNmaR624nujKwz5D8j69IkqWTYnElarcH04RzCkL8ktQCbM0l1qhnyn14V8r+WGXxoyF+Sms1qJwRExJiI2LElipGUvdoh/7Wo5MRqIf8PWZh1iZJU0uozcvYSMDYiyoEbgNtSSv7fWSox3diUM9m+Rsj/FLpyDxUk3si6PElqNSKlVL8DI7YDTgCGAI8D16WU/tWMtTVYRNTvl5FUpWbIfzl/YQCjWUAFL2ddmiSVuoqUUv/aO+uVOYuIMmD7/PYe8Azww4gYnlI6uknLlNQi6gr5X8FM3uTJrEuTpFZttSNnEXE5cAjwEHB9SmlKtddeSSlt17wl1p8jZ9Kq1Qz5v8p8NuFKdsiH/E0rSFILW+ORs2eB81NKi+t4bUCjy5LU7DrQnqH05WxeY3Mm8yLbcCJ7cStT+JxHsi5PklTNamdrppRuWEljxuomBkTEuIh4NyKer7Zvw4iYGBGv5X9usJL3fj9/zGsR8f3V1SlpCDALWJb/OYRubMqlDGI2bRjDJGayKQfTn168wQ08zucszbZkSdIXNPe9NW8EDqi171zgwZTStsCD+ec1RMSGwAXAbuRG5y5YWRMnCXKN2XXAVkAbduIjbqSMWbzPCB7lHnakP9vzVWZwN1Px+r8kFa5mbc5SSpOAD2rt/gZwU/7xTcA363jr14GJKaUPUkofAhP5YpMnqcrFwDoM5gHu4QCepTeHcydXcTLd6cIQJjv7UpKKRL2as4jYOyJOyD/eOCK2bsRnbppSmp9//DawaR3HdANmV3s+J7+vrtqGRcTUiJjaiJrq7YILLiClVLX17duXvn371th3wQUXADB37tyqfVOn5sq79tpraxzbpUsXDjnkkBr7hg4dClBj3/jx4wEYP358jf0AQ4cOrbHvkEMOoUuXLjX2XXvttQBMnTq1at/cuXP9nUrmd9qKlIIHLniMvS/4Gj/hl2zBW5zN73mTuU3wb74kqcVU/599XRu5y4sTgFfzz7sCj6/ufdXevxXwfLXnC2q9/mEd7xlJbhLCiuc/BUbW47OSm1tr2TrQPv2AgenfdE0J0gvskE7g+pRSSrBim5V5nW5ubm5uK92m1tXP1Gfk7FvAYcBigJTSPKBDPd63Mu9ERBeA/M936zhmLrB5teeb5fdJrV5XNuVSBjKb4NdM4g025mCOpxdPcQMnMmHCiiMXAz/JsFJJ0pqoT3P2ee5v4iSAiFi3kZ85Hvh+/vH3gbvqOOY+YP+I2CA/EWD//D6p1dqJ7tzInrzJ+4zgMe6hF7uyA/vyDHdzI4nhwJscdthy4E1gKHBbpjVLktZAPS4VjgSuBd4g93/7ycAZ9bykeRswH1hKLjd2ErARuVmarwEPABvmj+0P/KHae08EZua3E+r5eVkPT7q5Nfk2mF3SPfRLCdIi1k2/YWDaim4rPX78+PGZ1+zm5ubmVq+tzsua9bq3ZkTsR270KoD7UkoTV/umDHiHAJWKcso4kgGM5AP68ApvszFX0pNr6rGSf0qJiGihSiVJjVDnHQLqfePzYmBzpmLXgfacTF/OZiZbMI8X2YbRdMmv5F+/BWNtziSpaDTs9k0RsYjckFudUkodm6gwqdXryqacxXYMZzrrMYmH6c2pdOUeppJ4I+vyJEktaKXNWUqpA0BE/IJcbuxmcpc1jwW6tEh1UonrRXdGsgnHMIU2PMZf2I0xLGAqz6zxOR01k6TiVp/ZmoellK5KKS1KKX2UUrqa3Cr/ktbQYHbhHvrxHDM5ghlcxZ5VK/lP5aVGnXvF4riSpOK02sxZRDwB/B64ndxlziHA/6SU9mz+8hrGzJkKWWNC/g1h5kySikadmbP6jJwdAxwJvJPfvpPfJ6keOtCeHzCQ19mUW5lMO5ZyEnuzFQv5fzzSpI2ZJKn4OVtTaiY1Q/6LeJjejKYtdzN15TNtmoAjZ5JUNBo2W1PSmmmOkH9DHHrooS3yOZKk5mFzJjWRwezCSMo4gAoWM4+r2JPf8DpvMrlF66ioqGjRz5MkNS0va0qN0FIh/4bwsqYkFY01mxAQEWdFRMfIuT4ipkXE/s1To1QcOrCuIX9JUrOoz2zNE1NKH5G7t+YGwHHAJc1alVSgurIplzCItyjj10ziDTbmEPqzI28wjsf4jM+zLlGSVOTqkzlbcX3kIODmlNIL4TUTtTK96M4INuUYplDGo/yVAYxmYYuF/Bti7NixWZcgSWqE+ixCewPQDdga6A2UAQ+nlPo1f3kNY+ZMTe2r+ZD/gVSwmHW4nv5czuu8ydysS5MkFb86M2f1ac7aALsAb6SUFkTERkC3lNKzzVJmI9icqSmUU8Z38iH/vgUS8m+IqVOn0r//F/5blyQVnjW+Q0ACegJn5p+vC7RrwsKkgtC+Wsj/T0xm7SIN+ffrV3CD2pKkBqjPyNnVwHLgqymlHSJiA+D+lNKuLVFgQzhypjXRlU05k+0ZznTW56MWW8m/ubiUhiQVjTW+Q8BuKaW+ETEdIKX0YUSs1eTlSS2srpD/GBbydAGG/Bti3rx5WZcgSWqE+jRnSyOijNzlTSJiY3IjaVJRqhnyn8c17JEP+T+ZdWlNolu3blmXIElqhPpkzq4E7gQ2iYhfAo8BFzdrVVITK6eMIexBBdvxIDPow1v8hEFsTlvOYlJJzb684IILsi5BktQIq8yc5Wdq7g58AAwmt+bZgymll1qmvIYxc6ba2rMuJ9OPs3mdLZnLS2zDaLpyK1NKdsFYM2eSVDTWeCmN6SmlPs1WVhOyOdMKpRbybwibM0kqGms8IeDBiDgc+HsqpbukqyTtyJcZSeeSC/lLklqP+jRnw4EfAssi4tP8vpRS6th8ZUkNUzPkP59r2IPf8AazSiTk3xCucyZJxW21zVlKqUNLFCI1VM2V/GfwNhvzvwziGp7hAyZlXZ4kSWtktZkzgIg4DBiYf/pwSukfzVrVGjJz1jrUFfIfQ1duKeGQf0OYOZOkorHGEwIuAXYFbs3vGgJMTSmd1+QlNpLNWWmrHfJ/hN5cxlrczdMlH/JvCJszSSoaa9ycPQvsklJann9eBkxPKe3cLGU2gs1ZaaoZ8l/GX9mNMSzgaQpyRZfM2ZxJUtFY49maAOuTW+sMYL2mqkhalZWH/CdnXVpBGzVqVNYlSJIaoT4jZ0OAS4B/kVuEdiBwbkrpjuYvr2EcOSt+NUP+r/A2G/NbeuZD/guyLk+SpKa0Zpc1ASKiC7ncGcCUlNLbTVxck7A5KyZDyN0FbAvgLdrzU07mLUP+TWDu3LneX1OSikPDmrOI6Luqs6WUpjVRYU3G5qxYDAGuA9alC/M4kys5hWtYn4U8Qm9Gsxb/NOS/xsycSVLRaHBz9q9VnCyllL7aVJU1FZuzYjGLHfmYEYzhWG6ljGX8jcMZzfE8zcFZF1f0bM4kqWg0bEJASmnf5q1HrdG+7MI5nMqB3Mti1uFahnM5P2AW2wDLsy6vJFRUVGRdgiSpEeqbOesF9ATardiXUvpjM9a1Rhw5K0y1Q/7vsAlXcibXcAofsFG1I98Ets6oSkmSWlydI2dtVveuiLgA+G1+2xf4FXBYk5enktOedTmbgcykM39iMuvwOSezN1uyHxdzdq3GbDHwk6xKLSnXXntt1iVIkhojpbTKDXiOXBP3TP75psDE1b0viw1IbtlvXdgk/T8GpQ/pmBKkh+mdDmHXFDWOG5JgVoJl+Z9DMq+7VLaU+4/Bzc3Nza3wt6l19TP1WYR2SUppeURURkRH4F1g83q8T63MjnyZEXTmWKZQxqP8jQGM5iOe5pk6jr4tv0mSpOrq05xNjYj1ya19UAF8DC7Rrv/Khfz/u5L/tezB5bzBLJ7MujRJkorOqpbS+D3wp5TS49X2bQV0TCk9u8YfGLEdUP3uAtsAP0sp/abaMfsAdwGz8rv+nlK6sB7nrvuXUZMrp4wjGMBIPqQfL/MOnbiSHV3JvwB06dKF+fPnZ12GJGn1GnxvzVeB0fm7A/wZuC2lNL2xVaSUXgF2gaqbqM8F7qzj0EdTSoc09vPUtNqzLifTL7+S/2ReZmtOZu/8Sv6PZF2egH79+vGPf/wj6zIkSWtopbM1U0pXpJT2AAYB7wPjIuLliLggIno00ecPBl5PKf27ic6nZtKFTfh/DGI2ZVzOJN6kE4eyKz2ZxfU85i2WCsiECROyLkGS1Aj1Wues6uCIPsA4YOeUUlmjPzxiHDAtpfS7Wvv3Af4GzAHmASNTSi+s5BzDgGH5p/0aW5NqqhnyX8bfGMAYPmIKL2ZdmlbCOwRIUtFYsxufR0Q5cCBwNLmRrofJXeK8qzHVRMRa5BqvHVNK79R6rSOwPKX0cUQcBFyRUtq2Huc0c9ZE9mUXRlLOQUxlMeswjv78hlm8weysS9Nq2JxJUtFoWOYsIvYjd4fqg4ApwO3AsJTS4iYq6EByo2bv1H4hpfRRtcd3R8RVEdEppfReE3226lAz5D+Dd+jE+Qziap7hAyZlXZ7qadiwYas/SJJUsFY1W/Mh4E/A31JKHzb5B0fcDtyXUrqhjtc6A++klFJEDAD+CmyZVjPM58jZmqkZ8p/Ly2zNGLpxM1PMkkmS1HzW7LJmc4iIdYG3gG1SSgvz+04BSCldExGnA6cClcAS4IcppSfqcV6bswbowiacyQ6cwnTW5yMeoTejWYt/8jR+kcXLy5qSVDQKpzlrLjZn9WPIv7TZnElS0WjwOmcqMTVD/rmV/HMhf1fylySpUNiclbgyyviOIf9WxXXOJKm4eVmzRLVnXU6iHz8w5C9JUqGq87LmSu8QoOLUhU24mEG8RTm/YRL/ZqOqlfz/4Er+rcL48eOzLkGS1AiOnJWInnyZEXThuzxFGcv4OwMYbci/VXJCgCQVDScElKLqIf9PmGfIX5KkImdzVoTKaMN32M2QvyRJJcjMWRFpz7qcxUBepwu3MZl1+Yyh7M2WfMQveYQPWJB1iSoAXtKUpOJmc1YEDPmrIYYOHZp1CZKkRnBCQAHryTb5kP8UQ/6qNycESFLRcEJAsdiH3pxD23zI35X8JUlqTWzOCkQZbTiC3TiHD+nHM4b8JUlqpcycZWxFyH8mXbjdkL+awKGHHpp1CZKkRnDkLCOd2YQz2YFTmMEGTGISO3MmXfkHT5OYlXV5KmIVFRVZlyBJagQnBLQwQ/5qbk4IkKSi4YSALO1Db0bSloPzIf+x7M7lvGnIX5Ik1WBz1oxqh/zfZSN+yiCu4hk+4NGsy5MkSQXICQHNoO6Q/1fYko+5yJC/mtnYsWOzLkGS1AhmzppQZzbmTHrmQ/4LmcTOjOZL+ZC/JElSDXVmzhw5awI92Ybr2Yt/s4Af8SgPsAO7sSODeJYJNmZqYVOnTs26BElSIzhyVm9DgIuBLYC3gPPYh5eqhfzXZhz98yH/2c1XhrQaztaUpKLhbM01NwS4DliXMio5gicZyWv0N+QvSZKamJc16+Vi1iVxJlcwk+7czhA6sIhhjDHkr4Izb968rEuQJDWClzVXozNwBudyKtewAQt4lL25jHP4B4fks2RlTf2RkiSpdXBCQEP0BK4H3gTO5VIeZDC7M5mBPMoEDiPRhlz2TCosF1xwQdYlSJIawZGzWvYBRgIHA58A44DLOYQ3uB1Yt9qRi4GhwG2N/UipSTkhQJKKhiNnK1MGHAU8DfwL2BX4Kbl5mWcAb/APco3Ym8Dy/E8bM0mS1PRa9cjZusBJwA+ArYBXgDHAzcCnTV2c1EIcOZOkouFSGivkQv5wKrAB8ChwJvAPcMFYFb1+/fplXYIkqRFaVXO2A7k82bFAW+DvwGjgqSyLkiRJqqZVXNYcBJxD7ZA/vNFypUktxsuaklQ0WtdlzTLgCHIjZf2Bd8mF/K8G3s+wLkmSpFUpueasrpD/MAz5S5Kk4lBSzVlX4Hn+G/I/C5iAIX+1LqNGjcq6BElSI5RU5qx/RDoXQ/6SJKkolP4itM8D38HGTK3b3Llzsy5BktQIJdWcfZZ1AVIB6Nq1a9YlSJIaoaSaM0mSpGJncyaVmIqKiqxLkCQ1QmYTAiLiTWARsAyorB2Ii9wqmlcAB5FbO/b4lNK01ZyzdGY3SJKkUleQEwL2TSntUldhwIHAtvltGLn1YyWtxrXXXpt1CZKkRsi6OVuVbwB/TDlPAutHRJesi5IK3bBhw7IuQZLUCFk2Zwm4PyIqIqKuP026AbOrPZ+T31dDRAyLiKkRMbWZ6pQkSWoxWd4hYO+U0tyI2ASYGBEvp5QmNfQkKaWxwFgwcyZJkopfZiNnKaW5+Z/vAncCA2odMhfYvNrzzfL7JK2C65xJUnHLpDmLiHUjosOKx8D+5Bb4r2488L3I2R1YmFKa38KlSkWnX79+WZcgSWqErC5rbgrcmVstg3LgTymleyPiFICU0jXA3eSW0ZhJbimNEzKqVSoqEyZMIP/fliSpCJXUjc/NnEmQUrI5k6TiUJDrnEmSJKkamzOpxLjOmSQVNy9rSpIkZcPLmlJrUEp/4ZKk1sjmTJIkqYDYnEmSJBUQmzOpxEyYMCHrEiRJjeCEAEmSpGw4IUBqDcaPH591CZKkRnDkTCox3iFAkoqGI2eSJEmFzuZMkiSpgNicSSXGS5qSVNxszqQSM3To0KxLkCQ1ghMCpBLjhABJKhpOCJAkSSp0NmeSJEkFxOZMKjGHHnpo1iVIkhrB5kwqMRUVFVmXIElqBJszqcTMmzcv6xIkSY1gcyZJklRAbM4kSZIKiM2ZVGLGjh2bdQmSpEZwEVpJkqRsuAit1BpMnTo16xIkSY3gyJlUYrx9kyQVDUfOJEmSCp3NmVRiXOdMkoqbzZlUYrp165Z1CZKkRrA5k0rMBRdckHUJkqRGcEKAVGKcECBJRcMJAZIkSYXO5kySJKmA2JxJJaZfv35ZlyBJagSbM0mSpALihACpxDghQJKKhhMCJEmSCp3NmSRJUgGxOZNKzKhRo7IuQZLUCC2eOYuIzYE/ApsCCRibUrqi1jH7AHcBs/K7/p5SurAe5zZzJkmSikWdmbPyDAqpBEaklKZFRAegIiImppRerHXcoymlQzKoTypqc+fO9f6aklTEWvyyZkppfkppWv7xIuAlwD9JpCbStWvXrEuQJDVCppmziNgK6AM8VcfLe0TEMxFxT0TsuIpzDIuIqRExtbnqlCRJailZXNYEICLaA38Dzk4pfVTr5WnAlimljyPiIOD/gG3rOk9KaSwwNn9OM2dq9SoqKrIuQZLUCJksQhsRbYF/APellH5dj+PfBPqnlN5bzXE2Z5IkqVgUxiK0kVu6/HrgpZU1ZhHROX8cETGAXJ3vt1yVUvG69tprsy5BktQIWSylsTfwKPAcsDy/+yfAFgAppWsi4nTgVHIzO5cAP0wpPVGPcztyplbP2zdJUtGoc+TMe2tKJcbmTJKKRmFc1pQkSdLK2ZxJJcZ1ziSpuNmcSSWmX79+WZcgSWoEM2dSiTFzJklFw8yZJElSobM5kyRJKiA2Z1KJGTZsWNYlSJIawcyZJElSNsycSa1BKf2FS5JaI5szSZKkAmJzJkmSVEBszqQSM2HChKxLkCQ1ghMCJEmSsuGEAKk1GD9+fNYlSJIawZEzqcR4+yZJKhqOnEmSJBU6mzNJkqQCYnMmlRgvaUpScbM5k0rM0KFDsy5BktQITgiQSowTAiSpaDghQJIkqdDZnEmSJBUQmzOpxBx66KFZlyBJagSbM6nEVFRUZF2CJKkRbM6kEjNv3rysS5AkNYLNmSRJUgGxOZMkSSogNmdSiRk7dmzWJUiSGsFFaCVJkrLhIrRSazB16tSsS5AkNYIjZ1KJ8fZNklQ0HDmTJEkqdDZnUolxnTNJKm42Z1KJ6datW9YlSJIaweZMKjEXXHBB1iVIkhrBCQFSiXFCgCQVDScESJIkFTqbM0mSpAJicyaVmH79+mVdgiSpETJpziLigIh4JSJmRsS5dbz+pYi4I//6UxGxVQZlSpIktbgWb84iogz4PXAg0BMYEhE9ax12EvBhSqk7cDlwactWKRWvioqKrEuQJDVCFiNnA4CZKaU3UkqfA7cD36h1zDeAm/KP/woMDqefSZKkViCL5qwbMLva8zn5fXUek1KqBBYCG7VIdZIkSRkqz7qAxoqIYcCwrOuQCsWoUaOyLkGS1AhZjJzNBTav9nyz/L46j4mIcmA94P26TpZSGptS6l/XIm5Sa/Tzn/886xIkSY2QRXP2NLBtRGwdEWsBRwPjax0zHvh+/vERwEOplG5lIEmStBItflkzpVQZEacD9wFlwLiU0gsRcSEwNaU0HrgeuDkiZgIfkGvgJEmSSp731pQkScqG99aUJEkqdDZnkiRJBcTmTJIkqYDYnEmSJBUQmzNJkqQCYnMmSZJUQGzOJEmSCojNmSRJUgGxOZMkSSogNmeSJEkFxOZMkiSpgNicSZIkFZDyrAtoYu8B/27mz+iU/xw1Db/Ppud32vT8TpuW32fT8zttWi31fW5Z185IKbXAZ5eOiJha1x3ktWb8Ppue32nT8zttWn6fTc/vtGll/X16WVOSJKmA2JxJkiQVEJuzhhubdQElxu+z6fmdNj2/06bl99n0/E6bVqbfp5kzSZKkAuLImSRJUgGxOZMkSSogNmf1FBEHRMQrETEzIs7Nup5iFxGbR8S/IuLFiHghIs7KuqZSEBFlETE9Iv6RdS2lICLWj4i/RsTLEfFSROyRdU3FLiJ+kP9v/vmIuC0i2mVdUzGJiHER8W5EPF9t34YRMTEiXsv/3CDLGovNSr7Ty/L/3T8bEXdGxPotWZPNWT1ERBnwe+BAoCcwJCJ6ZltV0asERqSUegK7A//jd9okzgJeyrqIEnIFcG9KaXugN363jRIR3YAzgf4ppV5AGXB0tlUVnRuBA2rtOxd4MKW0LfBg/rnq70a++J1OBHqllHYGXgXOa8mCbM7qZwAwM6X0Rkrpc+B24BsZ11TUUkrzU0rT8o8XkftDr1u2VRW3iNgMOBj4Q9a1lIKIWA8YCFwPkFL6PKW0INOiSkM5sHZElAPrAPMyrqeopJQmAR/U2v0N4Kb845uAb7ZkTcWuru80pXR/Sqky//RJYLOWrMnmrH66AbOrPZ+DjUSTiYitgD7AUxmXUux+A/wIWJ5xHaVia+A/wA35S8V/iIh1sy6qmKWU5gKjgbeA+cDClNL92VZVEjZNKc3PP34b2DTLYkrQicA9LfmBNmfKVES0B/4GnJ1S+ijreopVRBwCvJtSqsi6lhJSDvQFrk4p9QEW4+WiRslnob5BrvHtCqwbEd/NtqrSknLrY7lGVhOJiP8lF8O5tSU/1+asfuYCm1d7vll+nxohItqSa8xuTSn9Pet6itxewGER8Sa5y+5fjYhbsi2p6M0B5qSUVozo/pVcs6Y19zVgVkrpPymlpcDfgT0zrqkUvBMRXQDyP9/NuJ6SEBHHA4cAx6YWXhTW5qx+nga2jYitI2ItcgHW8RnXVNQiIshleV5KKf0663qKXUrpvJTSZimlrcj9+/lQSskRiUZIKb0NzI6I7fK7BgMvZlhSKXgL2D0i1sn/P2AwTrJoCuOB7+cffx+4K8NaSkJEHEAuJnJYSumTlv58m7N6yIcCTwfuI/c/kj+nlF7ItqqitxdwHLkRnhn57aCsi5JqOQO4NSKeBXYBLs62nOKWH4X8KzANeI7cn0HedqgBIuI2YDKwXUTMiYiTgEuA/SLiNXKjk5dkWWOxWcl3+jugAzAx/+fTNS1ak7dvkiRJKhyOnEmSJBUQmzNJkqQCYnMmSZJUQGzOJEmSCojNmSRJUgGxOZNUFCLiXxHx9Vr7zo6Iq1fxnocjon/zV1fnZ3+cxedKKn42Z5KKxW3kFtit7uj8/iYREWVNdS5JWlM2Z5KKxV+Bg/N36SAitiJ3f8ZHI+LqiJgaES9ExM/renNEDImI5yLi+Yi4tNr+jyNiTEQ8A+wREd+NiCn5hSevjYiy/HZj/r3PRcQP6jj/1hExOf/6RbVeOycino6IZ+uqLyK2jIjXIqJTRLSJiEcjYv9GfVuSipbNmaSikFL6AJgCHJjfdTS5u3Uk4H9TSv2BnYFBEbFz9fdGRFfgUuCr5Fb63zUivpl/eV3gqZRSb+B94Chgr5TSLsAy4Nj8e7qllHqllHYCbqijxCvI3SR9J2B+tc/eH9gWGJA/T7+IGFjrd/t3vr6rgRHAiyml+xvw9UgqITZnkopJ9Uub1S9pHhkR04DpwI5Az1rv2xV4OH/D7UrgVmBFg7QM+Fv+8WCgH/B0RMzIP98GeAPYJiJ+m7/n3kd11LZXtXpurrZ///w2ndxti7Yn16zVkFL6A9AROAUYufKvQFKpK8+6AElqgLuAyyOiL7BOSqkiIrYm18zsmlL6MCJuBNo14JyfppSW5R8HcFNK6bzaB0VEb+Dr5JqnI4ET6zhXXffDC+D/pZSuXVUREbEOsFn+aXtgUf3Kl1RqHDmTVDRSSh8D/wLG8d9Rqo7AYmBhRGzKfy97VjeF3OXOTvnQ/xDgkTqOexA4IiI2AYiIDfN5sE5Am5TS34Dzgb51vPdx/juqd2y1/fcBJ0ZE+/w5u604fy2XkhvR+xlwXZ1fgKRWwZEzScXmNuBO8o1QSumZiJgOvAzMJtck1ZBSmh8R55Jr7AL4Z0rprjqOezEizgfuj4g2wFLgf4AlwA35fQBfGFkDzgL+FBE/JjfCt+Kc90fEDsDkiAD4GPgu8O6KYyJiELlLr3ullJZFxOERcUJKqa5sm6QSF7ksrSRJkgqBlzUlSZIKiM2ZJElSAbE5kyRJKiA2Z5IkSQXE5kySJKmA2JxJkiQVEJszSZKkAvL/AX9qbHQZ1fcHAAAAAElFTkSuQmCC",
            "text/plain": [
              "<Figure size 720x504 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          },
          "output_type": "display_data"
        }
      ],
      "source": [
        "derivada = (y[2] - y[1]) / (x[2] - x[1])\n",
        "\n",
        "fig, ax = plt.subplots(figsize=(10, 7))\n",
        "ax.plot(x, y, marker='o', color='blue', label='recta de la función') \n",
        "ax.set_facecolor('black')\n",
        "ax.set_xlabel('Valores de x')\n",
        "ax.set_ylabel('Valores de y')\n",
        "ax.axline((x[1], y[1]), slope=derivada, color='red', label='derivada / pendiente')\n",
        "ax.hlines(y=10, xmin=0, xmax=4, linewidth=1, color='white', linestyles='dashed')\n",
        "ax.vlines(x=4, ymin=0, ymax=10, linewidth=1, color='white', linestyles='dashed')\n",
        "ax.legend()\n",
        "plt.show()"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "eH3YenH1u6Hf"
      },
      "source": [
        "La inclinación de la recta es menor porque $y$ aumenta lentamente en proporción a $x$, de $2$ en $2$. Una vez más: únicamente estamos averiguando la proporción que hay entre los valores de $x$ y de $y$. Si cada valor de $x$ aumenta significativamente cada valor correspondiente de la función, entonces nuestra derivada será alta y la inclinación de la recta será empinada. \n",
        "\n",
        "Por otra parte, cuando cada valor de la función aumenta poco conforme hacemos más grande nuestra variable de entrada, entonces la derivada será pequeña y la inclinación será poca."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "hcBGYQuhVo4W"
      },
      "source": [
        "### Pero ¿para qué sirve la derivada?\n",
        "\n",
        "Imaginemos que tenemos dos variables: $x = -2$ y $y = 3$. La función $f(x, y)$ las multiplica y su resultado es $-6$. Sin embargo, queremos alterar ese resultado para que sea $0$ y tenemos una restricción: la única manera de hacerlo es a través de las variables de entrada. Para ello, podríamos pensar en aumentar o disminuir las entradas en pasos pequeños y probar aleatoriamente hasta conseguir el resultado.\n",
        "\n",
        "```{margin}\n",
        "Ojo: disminuir el $3$ aumenta el resultado, puesto que está siendo multiplicado por un número negativo. Ejemplo: $3-1 = 2, 2 \\times -2 = -4$, y $-4$ es mayor que $-6$, el resultado inicial.\n",
        "```\n",
        "Supongamos que nuestro cambio aplicado será $0.01$: disminuiremos $y$ en esa cantidad y aumentaremos $x$ de la misma forma: $x = -1.99 \\times y = 2.99 = -5.95$. Bien, $-5.95$ es más cercano a $0$ que $-6$. Podríamos continuar este proceso hasta llegar a $0$, pero nuestro método no parece muy efectivo.\n",
        "\n",
        "Si recordamos la lección aprendida en las derivadas, se nos puede ocurrir utilizarlas para resolver este problema: dado que la derivada me dice el impacto que tiene una variable de entrada en la función, ¿será posible utilizarla para alterar el resultado de la función? Es decir, si la derivada me dice la magnitud del impacto que una varible tiene en el resultado, entonces debería poder utilizar esa información para influir en el resultado de manera más eficiente. Por ejemplo, si la derivada me indica que una variable influye positivamente en la función, pero yo quiero disminuirla, entonces sabré más o menos la magnitud en que debo disminuir esa variable, y así sucesivamente."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "PNrFqDOydgjv"
      },
      "source": [
        "### Pequegrad\n",
        "\n",
        "Para entender mejor la posible solución planteada, crearemos programáticamente la clase `Numero`, es decir, formularemos una estructura que nos permita definir, modificar y operar con números. De momento, cada `Numero` tendrá las siguientes propiedades: un `valor`, un par de valores `previos` para el caso de que el `valor` haya sido generado mediante una operación (por ejemplo, $2$ y $2$ en el caso de que multiplicados hayan generado el número $4$), la `operación` que generó dicho valor (en nuestro ejemplo, la multiplicación) y una etiqueta en caso de que queramos asociar nuestro `valor` a una variable. Por ahora, únicamente tendremos las operaciones de suma, multiplcación, resta y división:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "_Yze9yvclvhq"
      },
      "outputs": [],
      "source": [
        "class Numero:\n",
        "  def __init__(self, valor, _previos=(), _op='', etiqueta=''):\n",
        "    self.valor = valor\n",
        "    self._previos = set(_previos)\n",
        "    self._op = _op\n",
        "    self.etiqueta = etiqueta\n",
        "\n",
        "  def __add__(self, otro): # adición\n",
        "    otro = otro if isinstance(otro, Numero) else Numero(otro) # nos cercioramos de que el otro valor sea un Número\n",
        "    resultado = Numero(self.valor + otro.valor, (self, otro), '+')\n",
        "    return resultado\n",
        "\n",
        "  def __radd__(self, otro): \n",
        "    return self + otro\n",
        "\n",
        "  def __mul__(self, otro): # multiplicación\n",
        "    otro = otro if isinstance(otro, Numero) else Numero(otro)\n",
        "    resultado = Numero(self.valor * otro.valor, (self, otro), '*')\n",
        "    return resultado\n",
        "\n",
        "  def __rmul__(self, otro):\n",
        "    return self * otro\n",
        "\n",
        "  def __sub__(self, otro): # resta, substracción\n",
        "    return self + (-otro)\n",
        "\n",
        "  def __truediv__(self, otro): # división\n",
        "    return self * otro**-1 #dividir es lo mismo que multiplicar por el dividendo elevado a la menos 1\n",
        "  \n",
        "  def __rtruediv__(self, otro):\n",
        "    return otro * self**-1\n",
        "\n",
        "  def __neg__(self): # volver negativo un número\n",
        "    return self * -1\n",
        "\n",
        "  def __repr__(self):\n",
        "    return f'Valor={self.valor}' # esta función determina cómo se representa nuestro número"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "VDwSqPE72Dne"
      },
      "source": [
        "Utilizaremos un problema más desafiante que una simple multiplicación:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "EElKJC6DTSHR",
        "outputId": "b921f54b-39fe-45e3-b755-b82affc9d11f"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "Ejemplo: Valor de c: Valor=-6.0 | Valores previos: {Valor=3.0, Valor=-2.0} | Operación realizada para generar c: *\n"
          ]
        }
      ],
      "source": [
        "# Definimos la función L con los siguientes números y operaciones:\n",
        "\n",
        "a = Numero(-2.0, etiqueta='a')\n",
        "b = Numero(3.0, etiqueta='b')\n",
        "c = a*b; c.etiqueta = 'c' # definimos c, que es el resultado de multiplicar a y b\n",
        "d = Numero(10.0, etiqueta='d')\n",
        "e = c + d; e.etiqueta = 'e'\n",
        "f = Numero(-3.0); f.etiqueta = 'f'\n",
        "L = f * e; L.etiqueta= 'L'\n",
        "\n",
        "print(f'Ejemplo: Valor de c: {c} | Valores previos: {c._previos} | Operación realizada para generar c: {c._op}')"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "FLvY71CbeIHO"
      },
      "source": [
        "```{margin}\n",
        "El código para graficar no tiene que ver con las redes neuronales, de manera que queda fuera de nuestro enfoque explicarlo.\n",
        "```\n",
        "\n",
        "También crearemos una función para graficar nuestras operaciones:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "66jT2SWidSW2"
      },
      "outputs": [],
      "source": [
        "from graphviz import Digraph\n",
        "\n",
        "def rastreo(origen):\n",
        "  # construye un conjunto de todos los nodos en un gráfico\n",
        "  nodos, lineas = set(), set()\n",
        "  def construir(v):\n",
        "    if v not in nodos:\n",
        "      nodos.add(v)\n",
        "      for parte in v._previos:\n",
        "        lineas.add((parte, v))\n",
        "        construir(parte)\n",
        "  construir(origen)\n",
        "  return nodos, lineas\n",
        "\n",
        "def graficar(origen):\n",
        "  grafica = Digraph(format='svg', graph_attr={'rankdir': 'LR'}) # left to right, izquierda a derecha\n",
        "\n",
        "  nodos, lineas = rastreo(origen)\n",
        "  for n in nodos:\n",
        "    uid = str(id(n))\n",
        "    # por cada valor en la gráfica, crea un nodo rectangular ('record') para él\n",
        "    grafica.node(name=uid, label='{ %s | valor %.4f}' % (n.etiqueta, n.valor), shape='record')\n",
        "    if n._op:\n",
        "      # si el valor es resultado de una operación, crea un nodo para la operación\n",
        "      grafica.node(name = uid + n._op, label = n._op)\n",
        "      # conecta los nodos\n",
        "      grafica.edge(uid + n._op, uid)\n",
        "\n",
        "  for n1, n2 in lineas:\n",
        "    # conecta n1 al nodo operación de n2\n",
        "    grafica.edge(str(id(n1)), str(id(n2)) + n2._op)\n",
        "\n",
        "  return grafica"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "POesBWAy2P6P"
      },
      "source": [
        "Finalmente, visualizamos la función[^2]. El problema a resolver consiste en influir en los valores de la función $L$ para llevarla a $0$."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 192
        },
        "id": "2niNn1wpdXMm",
        "outputId": "043f6675-129c-4120-91d7-531063a97dc0"
      },
      "outputs": [
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          },
          "execution_count": 63,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "graficar(L)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "89o4mqtSx6vd"
      },
      "source": [
        "Como decíamos, intuitivamente creemos que la derivada puede ayudarnos: con la derivada sabemos qué impacto tiene cada variable en el resultado final $L$. En ese sentido, lo que tendremos que hacer primero es calcular la derivada de $L$ con respecto a cada variable[^3].\n",
        "\n",
        "En principio, la derivada de $L$ con respecto a sí misma es $1$: aunque suene absurdo y evidente a la vez, el cambio en $L$ es proporcional (idéntico) a sí mismo. Ahora, la derivada de $L$ con respecto a $f$ y $e$ la podemos averiguar con la fórmula que veníamos utilizando:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "fHdnQm5033ZI",
        "outputId": "e28b6f47-c684-4c30-8fce-81985e507b2b"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "Valor=-2.9999999995311555\n"
          ]
        }
      ],
      "source": [
        "def derivada():\n",
        "  h = 0.000001\n",
        "\n",
        "  # Función original\n",
        "  a = Numero(-2.0, etiqueta='a')\n",
        "  b = Numero(3.0, etiqueta='b')\n",
        "  c = a*b; c.etiqueta = 'c'\n",
        "  d = Numero(10.0, etiqueta='d')\n",
        "  e = c + d; e.etiqueta = 'e'\n",
        "  f = Numero(-3.0); f.etiqueta = 'f'\n",
        "  L1 = f * e; L.etiqueta= 'L1'\n",
        "\n",
        "  # Función con incremento h\n",
        "  a = Numero(-2.0, etiqueta='a')\n",
        "  b = Numero(3.0, etiqueta='b')\n",
        "  c = a*b; c.etiqueta = 'c'\n",
        "  d = Numero(10.0, etiqueta='d')\n",
        "  e = c + d; e.etiqueta = 'e'\n",
        "  e.valor += h # aumento de h para obtener la derivada de L con respecto a e\n",
        "  f = Numero(-3.0); f.etiqueta = 'f'\n",
        "  L2 = f * e; L.etiqueta= 'L2'\n",
        "\n",
        "  print((L2 - L1) / h) # fórmula de la derivada\n",
        "\n",
        "derivada()"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "BBnTUgPJ6BM9"
      },
      "source": [
        "La derivada de la función $L$ con respecto a $e$ es aproximadamente $-3$. A estas alturas, ya habremos notado un patrón: cuando derivamos una multiplicación, la derivada parcial con respecto al multiplicando es el multiplicador. En este caso, $L$ es el resultado de multiplicar $e$ con $f$. Vimos que la derivada con respecto a $e$ es $-3$, o sea $f$. Podemos inducir, empíricamente y por simetría, que la derivada con respecto a $f$ es $e$, o sea 4.\n",
        "\n",
        "Quizá también sea una buena oportunidad para demostrar analíticamente lo que venimos diciendo. Utilizando nuestra fórmula matemática, tenemos que:\n",
        "\n",
        "$$\n",
        "\\frac{\\partial L(e, f)}{\\partial e}=\\lim _{h \\rightarrow 0}{\\frac{L(e + h, f)-L(e, f)}{h}} = {\\frac{(e + h) \\cdot f - e \\cdot f}{h}} = {\\frac{ef + hf - ef}{h}} = {\\frac{hf}{h}} = f\n",
        "$$\n",
        "\n",
        "Y aunque la notación aparente complejidad, en realidad solo estamos sumando, multiplicando, restando y dividiendo.\n",
        "\n",
        "Ahora que ya tenemos esta información, queremos continuar al nodo anterior: las derivadas con respecto a $c$ y $d$. Pero aquí hay una sutileza que, bien entendida, nos dará la clave de las redes neuronales: debemos obtener la derivada de $L$ con respecto a $c$ y $d$, no la derivada de $e$ con respecto a ellas. Entonces, ¿cómo podemos averiguar el impacto que $c$ y $d$ tienen en $L$ a través de $e$? Para hacer este cálculo solo hace falta una intuición bastante simple.\n",
        "\n",
        "```{margin}\n",
        "Para abundar en la historia de las redes neuronales y el aprendizaje profundo en general, y de la propagación hacia atrás en particular, véase el trabajo de Schmidhuber {cite}`JSHistory` y el de Yuxi Liu {cite}`Backstory`.\n",
        "```\n",
        "(backprop)=\n",
        "### La regla de la cadena: propagación hacia atrás\n",
        "Tomemos prestada la analogía de George Simmons: si una bicicleta es dos veces más rápida que una persona corriendo, y un automóvil cuatro veces más rápido que una bicicleta, entonces el automóvil es $2 \\times 4 = 8$ veces más rápido que una persona corriendo.\n",
        "\n",
        "De igual forma, si queremos saber la influencia que $c$ tiene en $L$, solo debemos obtener la derivada de $e$ con respecto a $c$, y multiplicarla por la derivada de $L$ con respecto a $e$. Es decir, debemos multiplicar la fuerza de $c$ en $e$ y la de $e$ en $L$ para saber con cuánta fuerza influye $c$ en $L$. Lo mismo vale para $d$ y todas las demás; a esta regla se le llama «regla de la cadena» (y mediante ella se realiza la «propagación hacia atrás» en el campo de la inteligencia artificial).\n",
        "\n",
        "Ahora, para obtener la derivada de $e$ con respecto a $c$, recordemos otro patrón que vimos: en una suma, la derivada nos daba como resultado $1$ porque la función avanzaba en la misma proporción que avanzaba la variable. Verifiquémoslo:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "XbVZ79MhAiaX",
        "outputId": "85689724-dd47-4e58-a69c-3950552b8196"
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "0.9999999999621422"
            ]
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          "execution_count": 65,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "def deriv_e(c, d):\n",
        "  return (((c+0.00001)+d) - (c+d)) / 0.00001\n",
        "\n",
        "deriv_e(-6, 10)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Oqci9P96Bv9m"
      },
      "source": [
        "En efecto, el resultado es aproximadamente $1$. Por simetría nuevamente, entendemos que la derivada de $e$ con respecto a $d$ también es $1$. Y para el lector exigente, analíticamente:\n",
        "\n",
        "$$\n",
        "\\frac{\\partial e(c, d)}{\\partial d}=\\lim _{h \\rightarrow 0}{\\frac{e(c, d+h)-e(c, d)}{h}} = {\\frac{(c + d + h) - (c + d)}{h}} = {\\frac{(c+d) + h - (c+d)}{h}} = {\\frac{h}{h}} = 1\n",
        "$$\n",
        "\n",
        "Y ahora que ya tenemos ambas derivadas parciales, podemos multiplicarlas siguiendo la regla de la cadena: $1 \\times -3 = -3$. Así pues, la derivada de $L$ con respecto a $c$ y $d$ es $-3$. En términos matemáticos, hemos hecho algo equivalente a:\n",
        "\n",
        "$$\n",
        "\\frac{dz}{dx} = \\frac{dz}{dy} \\cdot \\frac{dy}{dx}\n",
        "$$\n",
        "\n",
        "Donde $z$ depende de $y$ y, a su vez, $y$ depende de $x$.\n",
        "\n",
        "Comprobémoslo programáticamente:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "9Mse5RqJADZu",
        "outputId": "623d297b-71f8-43d3-8464-d40fab04ed6b"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "Valor=-2.9999999995311555\n"
          ]
        }
      ],
      "source": [
        "def derivada():\n",
        "  h = 0.000001\n",
        "\n",
        "  # Función original\n",
        "  a = Numero(-2.0, etiqueta='a')\n",
        "  b = Numero(3.0, etiqueta='b')\n",
        "  c = a*b; c.etiqueta = 'c'\n",
        "  d = Numero(10.0, etiqueta='d')\n",
        "  e = c + d; e.etiqueta = 'e'\n",
        "  f = Numero(-3.0); f.etiqueta = 'f'\n",
        "  L1 = f * e; L.etiqueta= 'L1'\n",
        "\n",
        "  # Función con incremento h\n",
        "  a = Numero(-2.0, etiqueta='a')\n",
        "  b = Numero(3.0, etiqueta='b')\n",
        "  c = a*b; c.etiqueta = 'c'\n",
        "  c.valor += h # aumento de h para obtener la derivada de L con respecto a c\n",
        "  d = Numero(10.0, etiqueta='d')\n",
        "  e = c + d; e.etiqueta = 'e'\n",
        "  f = Numero(-3.0); f.etiqueta = 'f'\n",
        "  L2 = f * e; L.etiqueta= 'L2'\n",
        "\n",
        "  print((L2 - L1) / h) # fórmula de la derivada\n",
        "\n",
        "derivada()"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Hwg2Fb7GCQDt"
      },
      "source": [
        "Finalmente, debemos hacer lo mismo para obtener las derivadas con respecto a $a$ y $b$. Entonces, dado el patrón que habíamos descubierto, la derivada de $c$ con respecto a $a$ es $b$, o sea $3$, y viceversa: la derivada con respecto a $b$ es $a$, o sea $-2$. Pero recordemos: estas derivadas «locales» son de la función $c$, y a nosotros nos interesa la derivada de $L$, es decir, la magnitud de la influencia que $a$ y $b$ tienen en $L$. Para saberlo, nuevamente debemos aplicar la regla de la cadena y multiplicar las derivadas que tenemos por la derivada de $L$ con respecto a $c$. Entonces, la derivada parcial de $L$ con respecto a $a$ es $3 \\times -3 = -9$, mientras que la derivada parcial con respecto a $b$ es $-2 \\times -3 = 6$.\n",
        "```{margin}\n",
        "A la derivada global, es decir, a la que se da con respecto al resultado final y no al local, se le denomina «gradiente». Las fórmulas que utilizamos para derivar pueden leerse [aquí](https://es.wikipedia.org/wiki/Derivada#Cálculo_de_la_derivada); algunas otras fórmulas que utilizaremos a continuación, en {cite}`WoodANN`.\n",
        "```\n",
        "\n",
        "Ahora que ya tenemos estos valores, podemos optimizar nuestro código para contemplarlos y visualizarlos de mejor manera. Añadiremos también algunas funciones que después detallaremos. En la práctica, nadie calcula las derivadas manualmente como lo hicimos, puesto que sería una labor eterna; pero ya hemos aprendido los patrones para calcularlas, así que podemos implementarlos en nuestro código para que se calculen automáticamente:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "bMCGlNT4r9zr"
      },
      "outputs": [],
      "source": [
        "import math\n",
        "\n",
        "class Numero:\n",
        "  def __init__(self, valor, _previos=(), _op='', etiqueta=''):\n",
        "    self.valor = valor\n",
        "    self.grad = 0.0 # gradiente, comienza en 0\n",
        "    self._propagar = lambda: None # el valor predeterminado de la propagación hacia atrás es nulo\n",
        "    self._previos = set(_previos)\n",
        "    self._op = _op\n",
        "    self.etiqueta = etiqueta\n",
        "\n",
        "  def __add__(self, otro): \n",
        "    otro = otro if isinstance(otro, Numero) else Numero(otro) \n",
        "    resultado = Numero(self.valor + otro.valor, (self, otro), '+')\n",
        "\n",
        "    def _propagar():\n",
        "      self.grad += 1.0 * resultado.grad # la derivada de una suma es 1; después, se multiplica por la «derivada global»\n",
        "      otro.grad += 1.0 * resultado.grad\n",
        "    resultado._propagar = _propagar\n",
        "\n",
        "    return resultado\n",
        "\n",
        "  def __radd__(self, otro): \n",
        "    return self + otro\n",
        "\n",
        "  def __mul__(self, otro):\n",
        "    otro = otro if isinstance(otro, Numero) else Numero(otro)\n",
        "    resultado = Numero(self.valor * otro.valor, (self, otro), '*')\n",
        "\n",
        "    def _propagar():\n",
        "      self.grad += otro.valor * resultado.grad # la derivada de la multiplicación es igual al multiplicando por la derivada global\n",
        "      otro.grad += self.valor * resultado.grad\n",
        "    resultado._propagar = _propagar\n",
        "\n",
        "    return resultado\n",
        "\n",
        "  def __rmul__(self, otro):\n",
        "    return self * otro\n",
        "\n",
        "  def __pow__(self, otro): # potenciación\n",
        "    assert isinstance(otro, (int, float))\n",
        "    resultado = Numero(self.valor**otro, (self,), f'**{otro}')\n",
        "\n",
        "    def _propagar():\n",
        "      self.grad += otro * (self.valor ** (otro - 1)) * resultado.grad # recordemos la fórmula: yx**y-1\n",
        "    resultado._propagar = _propagar\n",
        "\n",
        "    return resultado\n",
        "\n",
        "  def __sub__(self, otro): \n",
        "    return self + (-otro)\n",
        "\n",
        "  def __rsub__(self, otro):\n",
        "    return otro + (-self)\n",
        "\n",
        "  def __neg__(self): \n",
        "    return self * -1\n",
        "\n",
        "  def __truediv__(self, otro): \n",
        "    return self * otro**-1 \n",
        "  \n",
        "  def __rtruediv__(self, otro):\n",
        "    return otro * self**-1\n",
        "\n",
        "  def ReLU(self): \n",
        "    resultado = Numero((0 if self.valor < 0 else self.valor), (self,), 'ReLU')\n",
        "\n",
        "    def _propagar():\n",
        "      self.grad += (1 if resultado.valor > 0 else 0) * resultado.grad # la derivada es 1 si x > 0, 0 si x < 0\n",
        "    resultado._propagar = _propagar\n",
        "    \n",
        "    return resultado\n",
        "\n",
        "  def tanh(self): # tangente hiperbólica\n",
        "    t = (math.exp(2*self.valor) - 1) / (math.exp(2*self.valor) + 1)\n",
        "    resultado = Numero(t, (self,), 'tanh')\n",
        "\n",
        "    def _propagar():\n",
        "      self.grad += (1 - t**2) * resultado.grad # según la fórmula 1-tanh**2x\n",
        "    resultado._propagar = _propagar\n",
        "\n",
        "    return resultado\n",
        "\n",
        "  def sigmoide(self):\n",
        "    s = 1/(1 + math.exp(self.valor))\n",
        "    resultado = Numero(s, (self,), 'sigmoide')\n",
        "\n",
        "    def _propagar():\n",
        "      self.grad += (s * (1 - s)) * resultado.grad # la derviada es s(x)(1-s(x))\n",
        "    resultado._propagar = _propagar\n",
        "\n",
        "    return resultado\n",
        "\n",
        "  def exp(self): # exponenciación\n",
        "    resultado = Numero(math.exp(self.valor), (self,), 'exp')\n",
        "\n",
        "    def _propagar():\n",
        "      self.grad += self.valor * resultado.grad\n",
        "    resultado._propagar = _propagar\n",
        "\n",
        "    return resultado\n",
        "\n",
        "  def log(self): # logaritmo natural\n",
        "    resultado = Numero(math.log(self.valor), (self,), 'log')\n",
        "\n",
        "    def _propagar():\n",
        "      self.grad += 1/self.valor * resultado.grad\n",
        "    resultado._propagar = _propagar\n",
        "\n",
        "    return resultado\n",
        "\n",
        "  def propagar(self):\n",
        "    # ordenamiento topológico\n",
        "    topo = []\n",
        "    visitados = set()\n",
        "    def construir_topo(v):\n",
        "      if v not in visitados:\n",
        "        visitados.add(v)\n",
        "        for previo in v._previos:\n",
        "          construir_topo(previo)\n",
        "        topo.append(v)\n",
        "    construir_topo(self)\n",
        "\n",
        "    self.grad = 1 # asignamos la derivada del valor final con respecto a sí mismo: 1\n",
        "    for nodo in reversed(topo): # comenzamos desde adelante hacia atrás\n",
        "      nodo._propagar()\n",
        "\n",
        "  def __repr__(self):\n",
        "    return f'Valor={self.valor}'"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "MuvOWSnTJMZI"
      },
      "outputs": [],
      "source": [
        "from graphviz import Digraph\n",
        "\n",
        "def rastreo(origen):\n",
        "  nodos, lineas = set(), set()\n",
        "  def construir(v):\n",
        "    if v not in nodos:\n",
        "      nodos.add(v)\n",
        "      for parte in v._previos:\n",
        "        lineas.add((parte, v))\n",
        "        construir(parte)\n",
        "  construir(origen)\n",
        "  return nodos, lineas\n",
        "\n",
        "def graficar(origen):\n",
        "  grafica = Digraph(format='svg', graph_attr={'rankdir': 'LR'})\n",
        "\n",
        "  nodos, lineas = rastreo(origen)\n",
        "  for n in nodos:\n",
        "    uid = str(id(n))\n",
        "    grafica.node(name=uid, label='{ %s | valor %.4f | grad %.4f}' % (n.etiqueta, n.valor, n.grad), shape='record')\n",
        "    if n._op:\n",
        "      grafica.node(name = uid + n._op, label = n._op)\n",
        "      grafica.edge(uid + n._op, uid)\n",
        "\n",
        "  for n1, n2 in lineas:\n",
        "    grafica.edge(str(id(n1)), str(id(n2)) + n2._op)\n",
        "\n",
        "  return grafica"
      ]
    },
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        "a = Numero(-2.0, etiqueta='a')\n",
        "b = Numero(3.0, etiqueta='b')\n",
        "c = a*b; c.etiqueta = 'c'\n",
        "d = Numero(10.0, etiqueta='d')\n",
        "e = c + d; e.etiqueta = 'e'\n",
        "f = Numero(-3.0); f.etiqueta = 'f'\n",
        "L = f * e; L.etiqueta= 'L'\n",
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        "graficar(L)"
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        "id": "KLt7qd_YTUwD"
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      "source": [
        "Ahora obtendremos las derivadas, pero programática en vez de manualmente:"
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        "L.propagar()\n",
        "graficar(L)"
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        "Bien. Ahora pongamos en práctica nuestra intuición: dado que el gradiente (o derivada global, o la colección de derivadas parciales de $L$ con respecto a cada variable) nos dice la «fuerza» con la que cada número aumenta el resultado final de la función, podemos utilizar esa misma fuerza para alterar nuestros valores y, con ello, aumentar el valor de $L$. La bondad de utilizar el gradiente es que nos indica la «dirección» y la magnitud en la que debemos modificar cada valor para aumentar el resultado final, es decir: aumentar el valor de cada variable en la misma dirección del gradiente, contribuye a elevar el valor de la función global. Incluso si el gradiente tiene un valor negativo, eso indicaría que el valor de la variable debe disminuir para incrementar la función. De igual forma, si lo que queremos es disminuir el valor de una función, solo debemos ir en la dirección opuesta —*i. e.*, negativa— al gradiente.\n",
        "\n",
        "Las siguientes imágenes ilustran con flechas azules la dirección del gradiente, y en blanco y negro a los valores de la función (el negro indica los valores más altos). Como digo, el gradiente indica la dirección en la que la función aumenta su valor:\n",
        "\n",
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        "\n",
        "Solo hace falta un detalle: al emplear esta estrategia, debemos ser cuidadosos de no excedernos en el aumento del resultado, puesto que queremos llegar a $0$ sin superarlo. Para ello, podemos atenuar la fuerza de los gradientes, multiplicándolos por un número pequeño como $0.01$ o $0.1$ para luego sumar (en caso de querer aumentar el resultado final de la función) o restar ese valor a la variable. Probemos un ejemplo con la variable $a$:"
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        "\n",
        "a = Numero(na.valor, etiqueta='a') # asignamos el nuevo valor a la variable\n",
        "b = Numero(3.0, etiqueta='b')\n",
        "c = a*b; c.etiqueta = 'c'\n",
        "d = Numero(10.0, etiqueta='d')\n",
        "e = c + d; e.etiqueta = 'e'\n",
        "f = Numero(-3.0); f.etiqueta = 'f'\n",
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    {
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      "metadata": {
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      "source": [
        "Bien, parece que nuestro método funciona. Sin embargo, comprobamos que quizá nuestra fuerza es elevada, puesto que disminuyó mucho el resultado. Probablemente tengamos un mejor resultado si hacemos esto con todas nuestras variables, pero con una fuerza más atenuada ($0.01$):"
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      "source": [
        "na = a + 0.01 * a.grad\n",
        "nb = b + 0.01 * b.grad\n",
        "nd = d + 0.01 * d.grad\n",
        "nf = f + 0.01 * f.grad\n",
        "\n",
        "a = Numero(na.valor, etiqueta='a')\n",
        "b = Numero(nb.valor, etiqueta='b')\n",
        "c = a*b; c.etiqueta = 'c'\n",
        "d = Numero(nd.valor, etiqueta='d')\n",
        "e = c + d; e.etiqueta = 'e'\n",
        "f = Numero(nf.valor); f.etiqueta = 'f'\n",
        "L = f * e; L.etiqueta= 'L'\n",
        "\n",
        "L.propagar()\n",
        "\n",
        "graficar(L)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "9iBO9yFndGFe"
      },
      "source": [
        "Aunque no hayamos conseguido un resultado de exactamente 0, nos hemos aproximado significativamente con nuestro nuevo método programático. Pronto explotaremos el verdadero poder detrás de esta nueva herramienta nuestra, aunque desde ya podamos proclamar —con Françoise Hardy— ¡voilà! Esta es la esencia del aprendizaje en una red neuronal.\n",
        "\n",
        "<iframe style=\"border-radius:12px\" src=\"https://open.spotify.com/embed/track/4YIZP4nlw3jV5DkXGB56iG?utm_source=generator\" width=\"100%\" height=\"152\" frameBorder=\"0\" allowfullscreen=\"\" allow=\"autoplay; clipboard-write; encrypted-media; fullscreen; picture-in-picture\" loading=\"lazy\"></iframe>\n",
        "\n",
        "---"
      ]
    },
    {
      "cell_type": "markdown",
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        "id": "3tujku9cKI3S"
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      "source": [
        "```{margin}\n",
        "*Bottomless wonders spring from simple rules, which are repeated without end*.\n",
        "Benoît Mandelbrot\n",
        "```\n",
        "\n",
        "### Perceptrón multicapa\n",
        "\n",
        "Anteriormente, construimos una especie de neurona; sin embargo, el poder de una red neuronal consiste en que tiene millones de neuronas, todas optimizando sus valores para darnos el resultado que deseamos (en nuestro ejemplo: $0$).\n",
        "\n",
        "A continuación, construiremos una red neuronal llamada «perceptrón multicapa» (*Multilayer Perceptron* o MLP en inglés) que nos permitirá resolver problemas prácticos. Como ejemplo, la entrenaremos para que identifique sarcasmo en reseñas de películas.\n",
        "\n",
        "```{margin}\n",
        "El ejemplo está inspirado en aquel formulado por [Sriraman Madhavan](https://qr.ae/pvKSx5).\n",
        "```"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "OKjErlMKgpXL"
      },
      "source": [
        "Para detectar sarcasmo en las reseñas que los usuarios dan sobre una película, podríamos tomar como base dos variables: el sentimiento (con valores del $1$ al $5$, donde $5$ es un sentimiento positivo y $1$ uno negativo) y la calificación asignada (también con valores del $1$ al $5$). El resultado tendría que ser $1$ para sarcasmo y $0$ para ausencia de sarcasmo. Por ejemplo: \n",
        "\n",
        "- *Genial, me encanta cuando veo una película de terror con la típica trama de siempre*. Calificación: $2/5$. \n",
        "\n",
        "En este caso, el sentimiento es positivo (utiliza expresiones como «genial», «me encanta»), pero la reseña es negativa, de manera que podemos establecer una relación de proporcionalidad inversa entre ambas variables para detectar el sarcasmo: si la calificación es baja pero el sentimiento «alto», entonces hay sarcasmo (es decir, sarcasmo $=1$).\n",
        "\n",
        "```{margin}\n",
        "Para visualizar de mejor manera los siguientes gráficos, recomiendo dar clic derecho en el gráfico y abrirlo en una nueva pestaña.\n",
        "```"
      ]
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        "outputId": "2f8b3242-6218-4933-becc-4c1e03718386"
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        "#entradas\n",
        "sentimiento = Numero(5.0, etiqueta='sentimiento')\n",
        "calificacion = Numero(5.0, etiqueta='calificacion')\n",
        "\n",
        "# pesos (weights)\n",
        "w1 = Numero(-3.0, etiqueta='w1')\n",
        "w2 = Numero(1.0, etiqueta='w2')\n",
        "\n",
        "# sesgo (bias)\n",
        "b = Numero(12, etiqueta='b')\n",
        "\n",
        "# x1*w1 + x2*w2 + b\n",
        "x1w1 = sentimiento*w1; x1w1.etiqueta='x1w1'\n",
        "x2w2 = calificacion*w2; x2w2.etiqueta='x2w2'\n",
        "sumatoria = x1w1 + x2w2; sumatoria.etiqueta='s'\n",
        "n = sumatoria + b; n.etiqueta='n'\n",
        "\n",
        "# activación\n",
        "sarcasmo = n.tanh(); sarcasmo.etiqueta= 'sarcasmo'\n",
        "\n",
        "sarcasmo.propagar()\n",
        "\n",
        "graficar(sarcasmo)"
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        "id": "KEigPdKGiZhX"
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      "source": [
        "Por el momento, nuestros valores son aleatorios y todo funciona mal. La calificación y el sentimiento que asignamos son $5$, pero la neurona indica que la reseña es sarcástica porque da $\\approx1$ como resultado. Al contrario, queremos que el sarcasmo sea $0$ en este caso. Podemos optimizarla programáticamente ahora que ya conocemos la propagación hacia atrás.\n",
        "\n",
        "Primero, crearemos la clase `Neurona` para crear una estructura como la anterior. Al pasar nuestras variables por la `Neurona`, obtendremos un `peso` y un `sesgo` por cada una. Hecho esto, realizaremos operaciones con el peso, el sesgo y las entradas para transformar las entradas y, con ello, obtengamos al final el resultado deseado. Por ejemplo: si mis entradas son $5$ y $5$ —como en nuestro ejemplo—, nosotros queremos encontrar unos pesos y sesgos tales que, multiplicados y sumados por $5$ y $5$, nos den como resultado un $0$, puesto que $5$ y $5$ significa ausencia de sarcasmo en este caso. Al mismo tiempo, queremos que esos mismos pesos y sesgos, por ejemplo al ser multiplicados y sumados por un sentimiento de $5$ y una calificación de $1$, nos arrojen $1$ como resultado, puesto que eso indicaría sarcasmo.\n",
        "\n",
        "Para obtener como resultado una probabilidad de sarcasmo del $0$ al $1$, utilizaremos la función tangente hiperbólica, la cual es muy similar a las que vimos con anterioridad (sigmoide, ReLU, softmax): cuando nuestros valores la atraviesen, ella nos dará como resultado un valor del $-1$ al $1$[^4]."
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    {
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      "execution_count": null,
      "metadata": {
        "id": "99ULoZpQjGd7"
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      "source": [
        "import random\n",
        "\n",
        "class Neurona:\n",
        "  def __init__(self, nentradas):\n",
        "    self.peso = [Numero(random.uniform(-1,1)) for i in range(nentradas)] # cada entrada tendrá un peso, que será un número aleatorio\n",
        "    self.sesgo = Numero(random.uniform(-1,1)) # un sesgo con valor aleatorio del -1 al 1\n",
        "\n",
        "  def __call__(self, x):\n",
        "    # peso * x + sesgo\n",
        "    activacion = sum((peso_i*x_i for peso_i, x_i in zip(self.peso, x)), self.sesgo) #multiplicamos, sumamos\n",
        "    resultado = activacion.tanh() # aplicamos la función tangente hiperbólica\n",
        "    return resultado\n",
        "\n",
        "  def parametros(self):\n",
        "    return self.peso + [self.sesgo]"
      ]
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        "#entradas\n",
        "sentimiento = Numero(5.0, etiqueta='sentimiento')\n",
        "calificacion = Numero(5.0, etiqueta='calificacion')\n",
        "\n",
        "N = Neurona(2) # creamos una neurona\n",
        "\n",
        "neuron = N([sentimiento, calificacion]) # damos a las variables como valores de entrada\n",
        "\n",
        "neuron.propagar()\n",
        "\n",
        "graficar(neuron)"
      ]
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    {
      "cell_type": "markdown",
      "metadata": {
        "id": "KexOMQj0k8ZR"
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      "source": [
        "Nuestros pesos y sesgos tienen valores aleatorios, de manera que esto sigue sin funcionar. Para optimizarlos apropiadamente, será mejor crear una arquitectura más robusta. Entendámonos: será difícil que encontremos un par de pesos tales que, multiplicados por nuestras dos entradas, reflejen apropiadamente el sarcasmo como queremos. Si tenemos más pesos, probablemente sea más fácil encontrar una combinación de números que arroje los resultados que queremos.\n",
        "\n",
        "Para ello, crearemos una red neuronal. Lo único que nos falta es una forma de conectar varias neuronas entre sí, así podremos tener más pesos y sesgos que entrenar, lo cual se traduce en mejor desempeño. \n",
        "\n",
        "Construiremos, pues, una «capa», que no es más que una forma programática de definir cuántas neuronas queremos para nuestras entradas, es decir, de generar una serie de neuronas; y finalmente conectaremos todo con un «perceptrón multicapa», que consiste en unir varias capas entre sí:"
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      "execution_count": null,
      "metadata": {
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      "source": [
        "class Capa: # Layer\n",
        "  def __init__(self, nentradas, nsalidas):\n",
        "    self.neuronas = [Neurona(nentradas) for _ in range(nsalidas)]\n",
        "\n",
        "  def __call__(self, x):\n",
        "    resultado = [n(x) for n in self.neuronas]\n",
        "    return resultado[0] if len(resultado) == 1 else resultado\n",
        "\n",
        "  def parametros(self):\n",
        "    return [parametro for n in self.neuronas for parametro in n.parametros()]\n",
        "\n",
        "class MLP: # Perceptrón multicapa\n",
        "  def __init__(self, nentrada, nsalidas):\n",
        "    tamaño = [nentrada] + nsalidas\n",
        "    self.capas = [Capa(tamaño[i], tamaño[i+1]) for i in range(len(nsalidas))]\n",
        "\n",
        "  def __call__(self, x):\n",
        "    for capa in self.capas:\n",
        "      x = capa(x)\n",
        "    return x\n",
        "\n",
        "  def parametros(self):\n",
        "    return [parametro for capa in self.capas for parametro in capa.parametros()]"
      ]
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      "cell_type": "markdown",
      "metadata": {
        "id": "j_6S7vRWn_dQ"
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      "source": [
        "Veamos una capa como ejemplo:"
      ]
    },
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        "x = [1.0, 2.0, 3.0]\n",
        "C = Capa(4, 1)\n",
        "graficar(C(x))"
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        "Ahora veamos una red neuronal de la especie perceptrón multicapa:\n"
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        "x = [3.0, 2.0]\n",
        "RN = MLP(4, [4, 4, 1])\n",
        "\n",
        "graficar(RN(x))"
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        "Como vemos, únicamente estamos interconectando y creando más neuronas para que nuestro modelo tenga más potencia. Al mismo tiempo, vemos que una neurona interconecta las entradas entre sí a través de pesos y sesgos. Los pesos y sesgos, además de realizar operaciones diferenciables (es decir, que podemos derivar), nos ayudarán a ir midiendo la relevancia de cada entrada para, con base en el gradiente, ajustarse en proporción a dicha relevancia. De igual forma, los pesos y sesgos articularán el patrón de números que en este caso necesitamos para convertir cada combinación de entradas en una probabilidad de sarcasmo.\n",
        "\n",
        "Ahora manos a la obra. Daremos a nuestro modelo cuatro entradas de entrenamiento (es decir, serán ejemplos para que la red neuronal aprenda): cada entrada tendrá el sentimiento y la calificación asignada, así como el valor de sarcasmo objetivo que deseamos:"
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        "outputId": "862a33dd-8f14-4c11-955a-efaacc100d68"
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "[Valor=0.3669362864564352,\n",
              " Valor=0.06860246608538911,\n",
              " Valor=0.3213103120229461,\n",
              " Valor=0.3486194765977301]"
            ]
          },
          "execution_count": 110,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "entradas = [\n",
        "    [5.0, 5.0], # no sarcasmo\n",
        "    [5.0, 1.0,], # sarcasmo\n",
        "    [5.0, 2.0], # sarcasmo\n",
        "    [4.0, 5.0], # no sarcasmo\n",
        "]\n",
        "\n",
        "objetivos = [0.0, 1.0, 1.0, 0.0]\n",
        "predicciones = [RN(x) for x in entradas]\n",
        "predicciones"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "mge0i3B5qPvK"
      },
      "source": [
        "Nuestras predicciones de sarcasmo fueron generadas con valores aleatorios, de manera que todas son erróneas. Ahora, debemos cuantificar qué tan alejadas están de su objetivo. Para ello, crearemos una función que mida la diferencia entre el objetivo y la predicción. Se trata de una simple resta, pero la elevaremos al cuadrado para obtener solo números positivos:\n",
        "\n",
        "```{margin}\n",
        "A esta función se le llama «error cuadrático medio» (o *mean squared error*, MSE en inglés); matemáticamente podemos definirla como $\\frac{1}{n} \\sum_{i=1}^n (y_i - \\hat{y}_i)^2$, aunque nosotros no hemos utilizado el promedio de la pérdida (o sea, $\\frac{1}{n}$) por conveniencia para nuestro problema. Esta expresión matemática es la «función de pérdida» (*loss function*) más simple que hay, pues solo estamos haciendo una sumatoria con la resta entre objetivos y predicciones. Digamos que esta operación mide la distancia entre el objetivo y la predicción.\n",
        "```"
      ]
    },
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      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "pqmWoTicMRqe",
        "outputId": "930e3756-6449-4221-c0d0-6bf69c44d98e"
      },
      "outputs": [
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          "data": {
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              "[Valor=0.13464223831843908,\n",
              " Valor=0.8675013661822187,\n",
              " Valor=0.46061969256639074,\n",
              " Valor=0.1215355394632753]"
            ]
          },
          "execution_count": 111,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "fn_perdida = [(pred - obj)**2 for pred, obj in zip(predicciones, objetivos)] #Mean Squared Error\n",
        "fn_perdida"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "54d_DuJ7rbHQ"
      },
      "source": [
        "Ahora, sumaremos los valores entre sí para obtener una sola cifra que nos permita cuantificar el desempeño general de nuestro modelo:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "l05ZxZAruJF0",
        "outputId": "b65dad37-1cef-4231-acd5-8e489c384bf6"
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "Valor=1.5842988365303239"
            ]
          },
          "execution_count": 112,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "perdida = sum(fn_perdida)\n",
        "perdida"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Hhtmm_bUrmpR"
      },
      "source": [
        "Luego, con base en la función de pérdida, haremos propagación hacia atrás: obtendremos los gradientes y, con ello, sabremos en qué dirección modificar nuestros pesos y sesgos para que la pérdida (o el error, o la distancia) sea $0$, es decir, que nuestras predicciones sean idénticas a nuestros objetivos:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "33nSkhd9uhrS"
      },
      "outputs": [],
      "source": [
        "perdida.propagar()"
      ]
    },
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          "execution_count": 114,
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      ],
      "source": [
        "graficar(perdida)"
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    },
    {
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      "metadata": {
        "id": "Tf7UDrAVsAm7"
      },
      "source": [
        "Esto ya luce mucho más como una red neuronal. Calcular manualmente todos los gradientes sería engorroso; pero la programación nos permite apalancarnos de los algoritmos, las matemáticas y las computadoras para explotar su potencial a gran escala.\n",
        "\n",
        "Echemos un vistazo a nuestros parámetros (o sea, pesos y sesgos):"
      ]
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              " Valor=0.25222074387643856,\n",
              " Valor=-0.636707051411459,\n",
              " Valor=-0.08679161950639269,\n",
              " Valor=0.26446327705831196,\n",
              " Valor=0.21401883070848138,\n",
              " Valor=0.3809439666539909,\n",
              " Valor=-0.010349121884670964,\n",
              " Valor=-0.47937169114088274,\n",
              " Valor=-0.7563496531288552,\n",
              " Valor=-0.690391901144549,\n",
              " Valor=0.9515007803442808,\n",
              " Valor=0.6590463877459278,\n",
              " Valor=0.21212402329507696]"
            ]
          },
          "execution_count": 115,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "RN.parametros()"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "iOZiAeAxsnPS"
      },
      "source": [
        "Aunque para nosotros puedan no significar nada, lo cierto es que con base en el gradiente, nuestro modelo aprenderá a configurarlos de tal manera que representen un patrón cuya función será arrojar el resultado que nosotros queremos.\n",
        "\n",
        "Ahora, dado que nuestra función de pérdida es positiva, queremos disminuirla. Si los gradientes nos indican la dirección en la que podemos aumentar el resultado de una función, entonces en este caso queremos ir en dirección opuesta al gradiente, pues eso tendría como resultado una disminución del resultado de la función. Para ello, sumaremos valor a las variables en el sentido **inverso** al gradiente (o, visto de otra forma, les restaremos valor según nos indique el gradiente). \n",
        "\n",
        "Para optimizar de mejor manera, haremos este mismo paso $30$ veces, puesto que los cambios serán diminutos:\n",
        "\n",
        "```{margin}\n",
        "Al proceso de «propagar hacia atrás» las derivadas parciales de cada neurona se le denomina «descenso de gradiente estocástico» (*stochastic gradient descent*) porque, como decíamos, disminuimos la función de pérdida con base en el gradiente. «Estocástico» significa «aleatorio» y utilizamos este término porque en un inicio nuestros parámetros $w, b$ eran aleatorios. En suma, utilizamos parámetros —sesgos, pesos— aleatorios para optimizar nuestra red neuronal, disminuyendo la pérdida a través de ellos con base en el gradiente.\n",
        "```"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "rziOa0tyIem3",
        "outputId": "e86e4652-9d50-4da5-df70-67ce0acb071a"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "0 1.5842988365303239\n",
            "1 1.2529558664951095\n",
            "2 0.587216087017959\n",
            "3 0.4191023147078547\n",
            "4 0.5752410954325351\n",
            "5 1.1191852453113773\n",
            "6 1.784366523080085\n",
            "7 1.652128457073363\n",
            "8 1.224657076741451\n",
            "9 0.26771058028648226\n",
            "10 0.41855267707767796\n",
            "11 0.707877061951523\n",
            "12 0.15361704203665727\n",
            "13 0.12954214337233716\n",
            "14 0.12316318950192393\n",
            "15 0.12964905806896287\n",
            "16 0.1261090038316882\n",
            "17 0.1336944727568814\n",
            "18 0.11078544833502019\n",
            "19 0.102441253791397\n",
            "20 0.08312278356688153\n",
            "21 0.0723339169867922\n",
            "22 0.06107490601393822\n",
            "23 0.053290880990198526\n",
            "24 0.04659821460517871\n",
            "25 0.04138965501323435\n",
            "26 0.03716069258091919\n",
            "27 0.03363478739730862\n",
            "28 0.030798690241934046\n",
            "29 0.02833061619126015\n"
          ]
        }
      ],
      "source": [
        "for k in range(30):\n",
        "\n",
        "  # paso hacia delante\n",
        "  preds = [RN(x) for x in entradas]\n",
        "  perdida = sum([(pred - obj)**2 for pred, obj in zip(preds, objetivos)])\n",
        "\n",
        "  # propagación hacia atrás, Stochastic Gradient Descent\n",
        "  for p in RN.parametros():\n",
        "    p.grad = 0.0\n",
        "  perdida.propagar()\n",
        "\n",
        "  # actualizar\n",
        "  for p in RN.parametros():\n",
        "    p.valor += -0.07 * p.grad\n",
        "\n",
        "  print(k, perdida.valor)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "72lITXbzvM3X"
      },
      "source": [
        "Al parecer, nuestra función de pérdida disminuyó casi a cero. Comprobemos que nuestras predicciones ahora sean más similares a nuestros objetivos:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "pFPNWNgHJoGD",
        "outputId": "4f3c940c-f980-454b-f6c0-acee985db17d"
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "[Valor=0.1016585786227638,\n",
              " Valor=0.9171470012496584,\n",
              " Valor=0.8984522443847082,\n",
              " Valor=0.02862836899357634]"
            ]
          },
          "execution_count": 117,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "preds"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "wr8eiFhbKAk_",
        "outputId": "af7e831a-90d8-4804-a315-7ae5faab29fa"
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "[0.0, 1.0, 1.0, 0.0]"
            ]
          },
          "execution_count": 118,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "objetivos"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "PLV-ppGIvZxW"
      },
      "source": [
        "En efecto, nuestro modelo ya es mucho más apto ahora que antes para detectar sarcasmo. ¿Qué tal si hacemos un test con una nueva calificación? El sentimiento será de $5$ y la calificación $1.5$:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "0cfYK1jVvl5P",
        "outputId": "3a4e5f6d-afd1-4e4a-8d4d-1c2662ba2f45"
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "Valor=0.9022088768249822"
            ]
          },
          "execution_count": 119,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "test = [5.0, 1.5]\n",
        "\n",
        "prediccion = RN(test)\n",
        "prediccion"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "oZmA9MRFv150"
      },
      "source": [
        "Nuestra red neuronal considera que existe un $\\approx90 \\%$ de probabilidad de que sea sarcasmo. Nada mal."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "LAftr5hQ2QSm"
      },
      "source": [
        "[^1]: La palabra «secante» significa \u001b«cortar» en latín (*secare*), de manera que la línea secante es aquella que corta una figura cuando la toca en dos puntos. Por otro lado, la palabra «tangente» significa «tocar» en latín (*tangere*), de manera que la tangente es una línea que apenas toca en un punto a otra figura. La palabra «tangible» viene de la misma raíz y de ahí su significado también.\n",
        "\n",
        "[^2]: A operar con varias funciones concatenadas entre sí se le denomina «composición de funciones». La composición o concatenación de funciones consiste en tomar la salida de una función para pasarla como entrada de otra, y así sucesivamente. Según el esquema de nuestra primera lección, podríamos expresarlo como $x \\mapsto f(x) \\mapsto g(f(x))$, etcétera. Una visualización luciría más o menos así: <img src='https://upload.wikimedia.org/wikipedia/commons/6/65/FunctionMachinesComposition.svg' width=300>\n",
        "\n",
        "[^3]: El nombre técnico de esta derivada es «derivada parcial», puesto que la función $L$ está compuesta de múltiples variables que, a su vez, están concatenadas entre sí mediante otras funciones (suma y multiplicación). En ese sentido, $L$ es una función compuesta, y deseamos saber la derivada (parcial) de dicha función con respecto a cada variable que la compone o, en otras palabras, queremos determinar de qué forma está influyendo cada variable en el resultado de la función. En síntesis, si tenemos solo una variable en la función, entonces hablamos de derivada; pero si tenemos más de una variable y estimamos la derivada con respecto a cada variable individual, entonces hablamos de derivada parcial. La derivada parcial utiliza la notación $\\frac{\\partial y(x, z)}{\\partial x}$ con una letra «d» cursiva ($\\partial$) para distinguirse. Debemos ser cuidadosos de no confundirla con delta, la «d» griega ($\\delta$).\n",
        "\n",
        "[^4]: La elección de una función no lineal (tanh, ReLU, etc.) es una cuestión técnica que tiene que ver con el problema a tratar y con cómo fluyen a través de ellas los gradientes. Dado que nuestro problema es relativamente simple, podemos usar cualquiera. Por otro lado, aunque suene extraño que nuestro modelo pueda arrojar probabilidades negativas, eso no sucederá siempre y cuando nuestras entradas estén en el rango de valores que seleccionamos para el entrenamiento (*i. e.*, 0 a 5 en ambas variables). Si, por ejemplo, dijéramos que la calificación es 30 y el sentimiento 20, nuestro modelo interpretaría esto como una probabilidad de sarcasmo negativa. Es obligado mencionar [el paper](https://cds.cern.ch/record/154856/files/pre-27827.pdf/) en el que el gran Feynman explica qué sentido tiene hablar de probabilidades negativas."
      ]
    }
  ],
  "metadata": {
    "colab": {
      "collapsed_sections": [],
      "provenance": []
    },
    "kernelspec": {
      "display_name": "Python 3",
      "name": "python3"
    },
    "language_info": {
      "name": "python"
    }
  },
  "nbformat": 4,
  "nbformat_minor": 0
}