notebooks/4-calculus-ii.ipynb

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        "<a href=\"https://colab.research.google.com/github/joanby/matematicas-ml/blob/master/4_calculus_ii.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
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    {
      "cell_type": "markdown",
      "metadata": {
        "id": "aTOLgsbN69-P"
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      "source": [
        "# Cálculo II: Derivadas parciales e integrales"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "yqUB9FTRAxd-"
      },
      "source": [
        "Esta clase, *Cálculo II: Derivadas Parciales e Integrales*, se basa en el cálculo de derivadas de una sola variable para introducir los gradientes y el cálculo integral. Los gradientes de aprendizaje, facilitados por el cálculo de derivadas parciales, son la base del entrenamiento con datos de la mayoría de los algoritmos de aprendizaje automático, es decir, el descenso de gradiente estocástico (SGD). Combinado con el principio de la regla de la cadena (también tratado en esta clase), el SGD permite al algoritmo de retropropagación entrenar redes neuronales profundas.\n",
        "\n",
        "El cálculo integral, por su parte, resulta útil para innumerables tareas relacionadas con el aprendizaje automático, como encontrar el área bajo la denominada «curva ROC», una métrica predominante para evaluar modelos de clasificación. El contenido de esta clase es fundamental para otras clases de la serie *Fundamentos del aprendizaje automático*, especialmente *Probabilidad y teoría de la información* y *Optimización*."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "d4tBvI88BheF"
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      "source": [
        "En el transcurso del estudio de este tema, usted:\n",
        "\n",
        "* Desarrollar una comprensión de lo que está pasando bajo el capó de los algoritmos de aprendizaje automático, incluidos los utilizados para el aprendizaje profundo.\n",
        "* Ser capaz de comprender los detalles de la derivada parcial, cálculo multivariante que es común en los documentos de aprendizaje automático, así como muchos en otros temas que subyacen ML, incluyendo la teoría de la información y algoritmos de optimización.\n",
        "* Utilizar el cálculo integral para determinar el área bajo cualquier curva dada, una tarea recurrente en ML aplicada, por ejemplo, para evaluar el rendimiento del modelo mediante el cálculo de la métrica ROC AUC."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Z68nQ0ekCYhF"
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      "source": [
        "**Tenga en cuenta que este cuaderno Jupyter no está diseñado para funcionar por sí solo. Es el código que acompaña a una clase o a los vídeos de la serie [Machine Learning Foundations](https://github.com/jonkrohn/ML-foundations) de Jon Krohn, que ofrecen detalles sobre lo siguiente:**\n",
        "\n",
        "*Segmento 1: Repaso de Cálculo Introductorio*.\n",
        "\n",
        "* El método Delta\n",
        "* Diferenciación con reglas\n",
        "* AutoDiff: Diferenciación Automática\n",
        "\n",
        "\n",
        "*Segmento 2: Gradientes de Aprendizaje Automático*\n",
        "\n",
        "* Derivadas Parciales de Funciones Multivariantes\n",
        "* La regla de la cadena de derivadas parciales\n",
        "* Coste Cuadrático\n",
        "* Gradientes\n",
        "* Gradient Descent\n",
        "* Backpropagation\n",
        "* Derivadas parciales de orden superior\n",
        "\n",
        "\n",
        "*Segmento 3: Integrales*\n",
        "\n",
        "* Clasificación binaria\n",
        "* La matriz de confusión\n",
        "* La curva ROC (Receiver-Operating Characteristic)\n",
        "* Cálculo manual de integrales\n",
        "* Integración numérica con Python\n",
        "* Hallar el área bajo la curva ROC\n",
        "* Recursos para profundizar en el cálculo"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "qMomz4o6DoeT"
      },
      "source": [
        "## Segmento 1: Repaso de Cálculo Introductorio"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "yeOe0DrkDoeT"
      },
      "source": [
        "Revisar el notebook de [*Regresión en Pytorch* ](https://github.com/joanby/matematicas-ml/blob/master/notebooks/regression-in-pytorch.ipynb)."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "9v7wHzEaDoeT"
      },
      "source": [
        "## Segmento 2: Gradientes aplicados al Machine Learning"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "-09IzaDRDoeU"
      },
      "source": [
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "import torch\n",
        "import math # para la constante PI"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "gmonXg8-DoeV"
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      "source": [
        "### Derivadas parciales de funciones multivariantes"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "jt2Fb7bmDoeV"
      },
      "source": [
        "Definimos la función $f(x, y)$ para $z = x^2 - y^2$:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Xb_f6pUYDoeV"
      },
      "source": [
        "def f(my_x, my_y):\n",
        "    return my_x**2 - my_y**2"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "2Yxx42_UDoeV"
      },
      "source": [
        "Traza $z$ con respecto a $x$ variando $x$..."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Toba6LVQDoeW"
      },
      "source": [
        "xs = np.linspace(-3, 3, 1000)"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Sazf_xu9DoeW"
      },
      "source": [
        "...mientras mantenemos $y$ constante (por ejemplo en $y = 0$):"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "9Z31rNZvDoeW"
      },
      "source": [
        "zs_wrt_x = f(xs, 0)"
      ],
      "execution_count": null,
      "outputs": []
    },
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          "base_uri": "https://localhost:8080/",
          "height": 279
        },
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        "outputId": "fc25527e-3fe2-46a6-fff1-d87a9d3239b7"
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      "source": [
        "fig, ax = plt.subplots()\n",
        "plt.axvline(x=0, color='lightgray')\n",
        "plt.axhline(y=0, color='lightgray')\n",
        "\n",
        "plt.xlabel('x')\n",
        "plt.ylabel('z', rotation=0)\n",
        "_ = ax.plot(xs, zs_wrt_x)"
      ],
      "execution_count": null,
      "outputs": [
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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "TctSbA-VDoeY"
      },
      "source": [
        "Para determinar la pendiente de $z$ respecto a $x$ en un punto dado de la curva, podemos utilizar la derivada parcial de las diapositivas: $$ \\frac{\\partial z}{\\partial x} = 2x$$"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "WM2UTR4DDoeY"
      },
      "source": [
        "def delz_delx(my_x, my_y): # y no es relevante para *esta* derivada parcial; a menudo lo será\n",
        "    return 2*my_x"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "NqarAEMODoeY"
      },
      "source": [
        "x_samples = [-2, -1, 0, 1, 2]"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "ExzFLbjODoeY"
      },
      "source": [
        "colors = ['red', 'orange', 'green', 'blue', 'purple']"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "rPeZ7Qf0DoeZ"
      },
      "source": [
        "def point_and_tangent_wrt_x(my_xs, my_x, my_y, my_f, fprime, col):\n",
        "\n",
        "    my_z = my_f(my_x, my_y) # z = f(x, y)\n",
        "    plt.scatter(my_x, my_z, c=col, zorder=3)\n",
        "\n",
        "    tangent_m = fprime(my_x, my_y) # La pendiente es la derivada parcial de f(x, y) con respecto a x\n",
        "    tangent_b = my_z - tangent_m*my_x # La recta es z=mx+b, por tanto b=z-mx\n",
        "    tangent_line = tangent_m*my_xs + tangent_b\n",
        "\n",
        "    plt.plot(my_xs, tangent_line, c=col,\n",
        "             linestyle='dashed', linewidth=0.7, zorder=3)"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 279
        },
        "id": "e3O2HIInDoeZ",
        "outputId": "c7cd75a8-888b-49a0-8262-e660ba93faa4"
      },
      "source": [
        "fig, ax = plt.subplots()\n",
        "plt.axvline(x=0, color='lightgray')\n",
        "plt.axhline(y=0, color='lightgray')\n",
        "\n",
        "for i, x in enumerate(x_samples):\n",
        "    point_and_tangent_wrt_x(xs, x, 0, f, delz_delx, colors[i])\n",
        "\n",
        "plt.ylim(-1, 9)\n",
        "plt.xlabel('x')\n",
        "plt.ylabel('z', rotation=0)\n",
        "_ = ax.plot(xs, zs_wrt_x)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "a8IJZYZfDoeZ"
      },
      "source": [
        "Demostrando así visualmente $\\frac{\\partial z}{\\partial x} = 2x$.\n",
        "\n",
        "Es decir, la pendiente de $z$ a lo largo del eje $x$ es *dos veces* el valor de $x$."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "3Vxr1W0IDoea"
      },
      "source": [
        "**Volver a las slides.**"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ZEVnit5gDoea"
      },
      "source": [
        "Ahora vamos a representar $z$ con respecto a $y$ variando $y$..."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "ZZF8285-Doea"
      },
      "source": [
        "ys = np.linspace(-3, 3, 1000)"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "CSzx8uhLDoea"
      },
      "source": [
        "...mientras mantenemos $x$ constantw (es decir, por ejemplo $x$ = 0):"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "D7L4yZ1aDoeb"
      },
      "source": [
        "zs_wrt_y = f(0, ys)"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 279
        },
        "id": "rGNORzJCDoeb",
        "outputId": "8a0a0131-f0ff-4826-c99f-5afcd248abfd"
      },
      "source": [
        "fig, ax = plt.subplots()\n",
        "plt.axvline(x=0, color='lightgray')\n",
        "plt.axhline(y=0, color='lightgray')\n",
        "\n",
        "plt.xlabel('y')\n",
        "plt.ylabel('z', rotation=0)\n",
        "_ = ax.plot(ys, zs_wrt_y)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "yftPl60zDoeb"
      },
      "source": [
        "Como en las diapositivas, la derivada parcial de $z$ respecto a $y$ resulta ser independiente de $x$ (igual que antes observamos que $x$ es independiente de $y$), por lo que mientras $z$ varía en función tanto de $x$ como de $y$, la pendiente de $z$ respecto a $y$ es la misma sea cual sea $x$: $$ \\frac{\\partial z}{\\partial y} = -2y $$"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "xX3gAdyWDoec"
      },
      "source": [
        "Así, por ejemplo, mantener $x$ constante en 2 en lugar de 0 aumenta $z$, pero no tiene ningún impacto en la pendiente de $z$ respecto a $y$:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "sdaSBdlrDoec"
      },
      "source": [
        "zs_wrt_y = f(2, ys)"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 279
        },
        "id": "5PW49KD7Doec",
        "outputId": "fc717b75-b0b0-4977-8baf-436526ae02a3"
      },
      "source": [
        "fig, ax = plt.subplots()\n",
        "plt.axvline(x=0, color='lightgray')\n",
        "plt.axhline(y=0, color='lightgray')\n",
        "\n",
        "plt.xlabel('y')\n",
        "plt.ylabel('z', rotation=0)\n",
        "_ = ax.plot(ys, zs_wrt_y)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "8vMax3uODoec"
      },
      "source": [
        "def delz_dely(my_x, my_y):\n",
        "    return -2*my_y"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "pkQ4GQyBDoed"
      },
      "source": [
        "y_samples = [-2, -1, 0, 1, 2]"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "YxsR448zDoed"
      },
      "source": [
        "def point_and_tangent_wrt_y(my_ys, my_x, my_y, my_f, fprime, col): # cambiamos my_xs por my_ys\n",
        "\n",
        "    my_z = my_f(my_x, my_y)\n",
        "    plt.scatter(my_y, my_z, c=col, zorder=3) # cambiamos my_x por my_y\n",
        "\n",
        "    tangent_m = fprime(my_x, my_y)\n",
        "    tangent_b = my_z - tangent_m*my_y # cambiamos my_x por my_y\n",
        "    tangent_line = tangent_m*my_ys + tangent_b # cambiamos my_xs por my_ys\n",
        "\n",
        "    plt.plot(my_ys, tangent_line, c=col,\n",
        "             linestyle='dashed', linewidth=0.7, zorder=3) # cambiamos my_xs por my_ys"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 279
        },
        "id": "_8EXlci4Doed",
        "outputId": "aa4c98c5-4bea-4ff3-9943-d6fa1973864f"
      },
      "source": [
        "fig, ax = plt.subplots()\n",
        "plt.axvline(x=0, color='lightgray')\n",
        "plt.axhline(y=0, color='lightgray')\n",
        "\n",
        "for i, y in enumerate(y_samples):\n",
        "    point_and_tangent_wrt_y(ys, 2, y, f, delz_dely, colors[i])\n",
        "\n",
        "plt.ylim(-5, 5)\n",
        "plt.xlabel('y')\n",
        "plt.ylabel('z', rotation=0)\n",
        "_ = ax.plot(xs, zs_wrt_y)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ZV2FTIU66UG-"
      },
      "source": [
        "Demostrando así visualmente $\\frac{\\partial z}{\\partial y} = -2y$.\n",
        "\n",
        "Es decir, la pendiente de $z$ a lo largo del eje $y$ es *el doble* del valor de $y$ *e invertida*, lo que da lugar a que la curva parabólica se abra hacia abajo."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "V7P12cWuDoef"
      },
      "source": [
        "**Ejercicios**: Usa lápiz y papel para determinar:\n",
        "\n",
        "* El valor de $z$,\n",
        "* La pendiente de $z$ con respecto a $x$,\n",
        "* Y la pendiente de $z$ con respecto a $y$\n",
        "\n",
        "...en los puntos donde:\n",
        "\n",
        "1. $x = 3, y = 0$\n",
        "2. $x = 2, y = 3$\n",
        "3. $x = -2, y = -3$"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "IwiI0uisDoeg"
      },
      "source": [
        "Determinar derivadas parciales a mano utilizando reglas es útil para entender cómo funciona el cálculo. En la práctica, sin embargo, la autodiferenciación nos permite hacerlo más fácilmente (sobre todo si hay un gran número de variables). Por ejemplo, utilicemos la biblioteca de diferenciación automática PyTorch para calcular la pendiente de $z$ con respecto a $x$ e $y$ en un punto cualquiera $(x, y, z)$:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "SHeFsRfrDoeg",
        "outputId": "df4dc534-155a-4092-d1cd-9aca6fd90c31"
      },
      "source": [
        "x = torch.tensor(0.).requires_grad_()\n",
        "x"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "tensor(0., requires_grad=True)"
            ]
          },
          "metadata": {},
          "execution_count": 20
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "WzWjTchwDoeg",
        "outputId": "163249ec-b6da-4823-ed21-135f07dec9bc"
      },
      "source": [
        "y = torch.tensor(0.).requires_grad_()\n",
        "y"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "tensor(0., requires_grad=True)"
            ]
          },
          "metadata": {},
          "execution_count": 21
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "B7TprOE-Doeh",
        "outputId": "d013b94c-f1cf-42de-8eb4-e2b8e9c6def2"
      },
      "source": [
        "z = f(x, y) # Forward pass\n",
        "z"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "tensor(0., grad_fn=<SubBackward0>)"
            ]
          },
          "metadata": {},
          "execution_count": 22
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "GyefZb67Doeh"
      },
      "source": [
        "z.backward() # Autodiff"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "_L7fUeBfDoeh"
      },
      "source": [
        "Como ya sabíamos por nuestros ejercicios anteriores, la pendiente del punto (0, 0, 0) es cero con respecto a los ejes $x$ e $y$:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "VCX2ngT9Doeh",
        "outputId": "559ab351-8d5c-4159-f5a3-0f705f42bba4"
      },
      "source": [
        "x.grad"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "tensor(0.)"
            ]
          },
          "metadata": {},
          "execution_count": 24
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "1DeU2IYODoei",
        "outputId": "6eceaacb-4f52-449d-e04a-a0ab8c401420"
      },
      "source": [
        "y.grad"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "tensor(-0.)"
            ]
          },
          "metadata": {},
          "execution_count": 25
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Eblx3YW6Doei"
      },
      "source": [
        "**Ejercicio**: Repite los ejercicios más recientes de lápiz y papel usando PyTorch (o TensorFlow, si lo prefieres)."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "LPxR9412Doei"
      },
      "source": [
        "**Volver a las slides aquí.**"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "3NqDfYGeDoej"
      },
      "source": [
        "### Derivadas parciales del volumen de un cilindro"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "8zOAmeOnDoej"
      },
      "source": [
        "El volumen de un cilindro se describe por $v = \\pi r^2 l$ donde:\n",
        "\n",
        "* $r$ es el radio del cilindro\n",
        "* $l$ es su altura"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Y345YQ2iDoej"
      },
      "source": [
        "def cylinder_vol(my_r, my_l):\n",
        "    return math.pi * my_r**2 * my_l"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "6eGolCTLDoej",
        "outputId": "943dca09-0bea-4cbb-b089-dcb5c06ae879"
      },
      "source": [
        "# Si por ejemplo el radio es de 3 metros...\n",
        "r = torch.tensor(3.).requires_grad_()\n",
        "r"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "tensor(3., requires_grad=True)"
            ]
          },
          "metadata": {},
          "execution_count": 27
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "sp3kUgQ6Doej",
        "outputId": "88edcfa6-3d3c-4c64-acac-bed2c4f9f017"
      },
      "source": [
        "# ...y la altura es de 5 metros:\n",
        "l = torch.tensor(5.).requires_grad_()\n",
        "l"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "tensor(5., requires_grad=True)"
            ]
          },
          "metadata": {},
          "execution_count": 28
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "5FBBkAM6Doek",
        "outputId": "ebdbb9be-32db-4ad0-c3c5-1fc3255d0843"
      },
      "source": [
        "# Entonces el volumen sería de 141.4 metros cúbicos:\n",
        "v = cylinder_vol(r, l)\n",
        "v"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "tensor(141.3717, grad_fn=<MulBackward0>)"
            ]
          },
          "metadata": {},
          "execution_count": 29
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "0iGlPQyjDoek"
      },
      "source": [
        "v.backward()"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "YvIA09YGDoek",
        "outputId": "167d0cba-4083-4b18-cf3c-685f4ae4799d"
      },
      "source": [
        "l.grad"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "tensor(28.2743)"
            ]
          },
          "metadata": {},
          "execution_count": 31
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "TYD2k4J3Doel"
      },
      "source": [
        "Y como hemos visto en las transparencias: $$\\frac{\\partial v}{\\partial l} = \\pi r^2$$"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "q6YRwpMvDoel",
        "outputId": "ad1513dc-9d12-421e-f678-768ea34b376a"
      },
      "source": [
        "math.pi * 3**2"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "28.274333882308138"
            ]
          },
          "metadata": {},
          "execution_count": 32
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "UavsjoQFDoel"
      },
      "source": [
        "Esto significa que con $r = 3$, un cambio en $l$ de una unidad corresponde a un cambio en $v$ de 28,27$\\text{m}^3$. Podemos demostrarlo nosotros mismos:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "H32PplygDoel",
        "outputId": "d8aa066e-845f-409e-fc5b-0effbfc89dfc"
      },
      "source": [
        "cylinder_vol(3, 6)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "169.64600329384882"
            ]
          },
          "metadata": {},
          "execution_count": 33
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "vmwxrip1Doem",
        "outputId": "01d33fa2-9643-4651-f8ac-2d477a0ce96b"
      },
      "source": [
        "cylinder_vol(3, 6) - cylinder_vol(3, 5)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "28.274333882308127"
            ]
          },
          "metadata": {},
          "execution_count": 34
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "wGuDzwwxDoem",
        "outputId": "d94c0263-8313-401d-cc38-7740ddd9465d"
      },
      "source": [
        "cylinder_vol(3, 7) - cylinder_vol(3, 6)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "28.274333882308156"
            ]
          },
          "metadata": {},
          "execution_count": 35
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "aDJydipwDoel"
      },
      "source": [
        "**Volver a las slides aquí.**"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "yC4GJ8iZDoem"
      },
      "source": [
        "Para cambios en $v$ con respecto a $r$ tenemos lo siguiente de las diapositivas: $$\\frac{\\parcial v}{\\parcial r} = 2 \\pi r l$$"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "cAVbIBiCDoen",
        "outputId": "d3617c80-b132-4d5e-ac41-ddd2656afcdc"
      },
      "source": [
        "r.grad"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "tensor(94.2478)"
            ]
          },
          "metadata": {},
          "execution_count": 36
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "UzHESN6XDoen",
        "outputId": "8b7cea78-2e75-42b3-e87a-099aee3ae11d"
      },
      "source": [
        "2 * math.pi * 3 * 5"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "94.24777960769379"
            ]
          },
          "metadata": {},
          "execution_count": 37
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "lWzx4YxuDoen"
      },
      "source": [
        "$r$ está incluida en la derivada parcial por lo que su ajuste afecta a la escala de su impacto sobre $v$. Aunque es nuestro primer ejemplo en este cuaderno, es típico en cálculo que la derivada sólo se aplique a un $\\Delta r$ infinitesimalmente pequeño. Cuanto más pequeño sea $\\Delta r$, más se acercará a la verdadera $\\frac{\\partial v}{\\partial r}$. Por ejemplo, en $\\Delta r = 1 \\times 10^{-6}$:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "JN0waPUmDoen"
      },
      "source": [
        "delta = 1e-6"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "BCk5RPL8Doeo",
        "outputId": "2f980253-6e39-4320-876e-757bdec33697"
      },
      "source": [
        "(cylinder_vol(3 + delta, 5) - cylinder_vol(3, 5)) / delta # Dividiendo por delta, restaura la escala"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "94.24779531741478"
            ]
          },
          "metadata": {},
          "execution_count": 39
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "v4U_lu-yDoeo"
      },
      "source": [
        "**Volver a las slides aquí.**"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "3CKfShVZDoeo"
      },
      "source": [
        "### Gradientes del Coste con respecto a los Parámetros del Modelo"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "XbAlp7N3Doeo"
      },
      "source": [
        "Ver los notebooks independientes:\n",
        "\n",
        "* [Gradiente de regresión de un punto](https://github.com/joanby/matematicas-ml/blob/master/notebooks/single-point-regression-gradient.ipynb)\n",
        "* [Gradiente de regresión por lotes](https://github.com/joanby/matematicas-ml/blob/master/notebooks/batch-regression-gradient.ipynb)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "8GkxmPtUDoeo"
      },
      "source": [
        "## Segmento 3: Integrales"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "7RNXGSU_Doep"
      },
      "source": [
        "from scipy.integrate import quad # \"cuadratura\" = integración numérica (opuesta a la simbólica)"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "0gcCUd4QDoep"
      },
      "source": [
        "De las slides: $$ \\int_1^2 \\frac{x}{2} dx = \\frac{3}{4} $$"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "lQSGdUlIDoep"
      },
      "source": [
        "def g(x):\n",
        "    return x/2"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "Ifu7x0iRDoep",
        "outputId": "cc0b40e2-6a61-4662-e835-16ddbbc3c378"
      },
      "source": [
        "quad(g, 1, 2)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "(0.75, 8.326672684688674e-15)"
            ]
          },
          "metadata": {},
          "execution_count": 42
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "HfwAbtc5Doep"
      },
      "source": [
        "La segunda salida de `quad` es una estimación del error absoluto de la integral, que en este caso es esencialmente cero."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "By-nY8jBDoeq"
      },
      "source": [
        "**Volver a las slides aquí.**"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "BXY-74QaDoeq"
      },
      "source": [
        "def h(x):\n",
        "    return 2*x"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "A26vlFzCDoeq",
        "outputId": "a1554c68-3084-4358-f1f7-322219ec2eb9"
      },
      "source": [
        "quad(h, 3, 4)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "(7.0, 7.771561172376096e-14)"
            ]
          },
          "metadata": {},
          "execution_count": 44
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Crw18QJv7oY3"
      },
      "source": [
        "**Volver a las slides aquí.**"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "AAmdDoskDoeq"
      },
      "source": [
        "### Área bajo la curva ROC"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "j_wFvlHEDoeq"
      },
      "source": [
        "Cuando no tenemos una función pero sí tenemos coordenadas $(x, y)$, podemos utilizar el método `auc()` de la librería scikit-learn, que utiliza una aproximación numérica (la [regla trapezoidal](https://en.wikipedia.org/wiki/Trapezoidal_rule)) para encontrar el área bajo la curva descrita por las coordenadas:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "_X0uiqKLDoeq"
      },
      "source": [
        "from sklearn.metrics import auc"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "XilqUs--Doer"
      },
      "source": [
        "A partir de las diapositivas, las coordenadas $(x, y)$ de nuestra curva ROC de detección de perritos calientes son:\n",
        "\n",
        "* (0, 0)\n",
        "* (0, 0.5)\n",
        "* (0.5, 0.5)\n",
        "* (0.5, 1)\n",
        "* (1, 1)"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "yu9wg0aoDoer"
      },
      "source": [
        "xs = [0, 0,   0.5, 0.5, 1]\n",
        "ys = [0, 0.5, 0.5, 1,   1]"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "BwnqDGueDoer",
        "outputId": "9a1ba72e-0bc6-41e9-9d2f-ee2583fffa3a"
      },
      "source": [
        "auc(xs, ys)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.75"
            ]
          },
          "metadata": {},
          "execution_count": 47
        }
      ]
    }
  ]
}