lectures/cap06_2_cosine_similarity.ipynb

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  "metadata": {
    "colab": {
      "name": "cap06_2_cosine_similarity.ipynb",
      "provenance": [],
      "collapsed_sections": [],
      "authorship_tag": "ABX9TyP8f2s15XeIK006hRvBQGUU",
      "include_colab_link": true
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
    },
    "language_info": {
      "name": "python"
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
      },
      "source": [
        "<a href=\"https://colab.research.google.com/github/viniciusrpb/cic0269_natural_language_processing/blob/main/cap06_2_cosine_similarity.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "# Capítulo 6 - Similaridade entre Textos\n",
        "\n",
        "Nesta seção vamos estudar estratégias para comparar dois documentos de texto que estejam representados vetores numéricos. Assume-se que esses vetores possuem a mesma dimensão $d$.\n",
        "\n",
        "# 6.2. Similaridade Cosseno"
      ],
      "metadata": {
        "id": "cc2tpB79he-o"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "É uma métrica que calcula a similaridade entre dois vetores numéricos a partir da normalização do produto interno entre eles. Em Processamento de Linguagem Natural, é comumente empregada para comparar a similaridade entre dois documentos representados no modelo espaço vetorial (por exemplo, Bag-of-Words, Term Frequency-Inverse Document Frequency (TF-IDF)).\n",
        "\n",
        "### Teoria\n",
        "\n",
        "Sabe-se que dois vetores $\\vec{v_1}$ e $\\vec{v_2}$ representando documentos de texto são vistos no espaço geométrico plano da seguinte maneira:"
      ],
      "metadata": {
        "id": "nYXy3GNQCcHn"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "![cos1.png](data:image/png;base64,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)"
      ],
      "metadata": {
        "id": "LE7HeGtw7UQ3"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "em que $||\\vec{v}||$ denota o comprimento de um vetor $\\vec{v}$ qualquer.\n",
        "\n",
        "Percebe-se que $||\\vec{v_1}||$ e $||\\vec{v_2}||$ possuem tamanhos diferentes e, por isso, não são apropriados para comparar dois documentos de textos. O que fazer nessa situação? Podemos comparar esses documentos por meio do valor do ângulo $\\theta$ entre eles!\n",
        "\n",
        "Vamos relembrar a Lei dos Cossenos:\n",
        "\n",
        "\n",
        "\n",
        "\n"
      ],
      "metadata": {
        "id": "MAt6QADA7fcP"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        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)"
      ],
      "metadata": {
        "id": "IlFrCe4M9g9u"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Por conveniência, colocamos aqui Lei dos Cossenos:\n",
        "\n",
        "\\begin{equation}\n",
        "c^2 = a^2 + b^2 - 2 \\cdot a \\cdot b \\cdot \\cos \\theta\n",
        "\\end{equation}\n",
        "\n",
        "Substituindo as variáveis $a$, $b$ e $c$ pelos comprimentos dos vetores correspondentes $||\\vec{v_1}||$, $||\\vec{v_2}||$ e $||\\vec{v_1} - \\vec{v_2}||$:\n",
        "\n",
        "\\begin{equation}\n",
        "||\\vec{v_1} - \\vec{v_2}||^2 = ||\\vec{v_1}||^2 + ||\\vec{v_2}||^2 - 2 \\cdot ||\\vec{v_1}|| \\cdot ||\\vec{v_2}|| \\cdot \\cos \\theta\n",
        "\\end{equation}\n",
        "\n",
        "desenvolvendo a esquerda da igualdade e com uso da propriedade comutativa entre vetores, temos que:\n",
        "\n",
        "\\begin{equation}\n",
        "||\\vec{v_1} - \\vec{v_2}||^2 = ||\\vec{v_1}||^2 - 2 \\cdot \\vec{v_1} \\cdot \\vec{v_2} + ||\\vec{v_2}||^2\n",
        "\\end{equation}\n",
        "\n",
        "substituindo na primeira equação, tem-se que:\n",
        "\n",
        "\\begin{equation}\n",
        "||\\vec{v_1}||^2 - 2 \\cdot \\vec{v_1} \\cdot \\vec{v_2} + ||\\vec{v_2}||^2 = ||\\vec{v_1}||^2 + ||\\vec{v_2}||^2 - 2 \\cdot ||\\vec{v_1}|| \\cdot ||\\vec{v_2}|| \\cdot \\cos \\theta\n",
        "\\end{equation}\n",
        "\n",
        "fazendo-se alguns cortes nos termos, obteremos:\n",
        "\n",
        "\\begin{equation}\n",
        " - 2 \\cdot \\vec{v_1} \\cdot \\vec{v_2} = - 2 \\cdot ||\\vec{v_1}|| \\cdot ||\\vec{v_2}|| \\cdot \\cos \\theta\n",
        "\\end{equation}\n",
        "\n",
        "Isolando o cosseno\n",
        "\n",
        "\\begin{equation}\n",
        "\\cos \\theta = \\frac{\\vec{v_1} \\cdot \\vec{v_2}}{||\\vec{v_1}|| \\cdot ||\\vec{v_2}||} \n",
        "\\end{equation}\n",
        "\n",
        "### Abordagem Computacional\n",
        "\n",
        "A equação da similaridade cosseno adaptada para o contexto em que $v_1$ e $v_2$ são dois vetores computacionais é fornecida a seguir:\n",
        "\n",
        "\\begin{equation}\n",
        "cos(v_1,v_2) = \\frac{v_1 \\cdot v_2}{||v_1|| ||v_2||},\n",
        "\\end{equation}\n",
        "\n",
        "em que o produto interno é calculado como um produto escalar:\n",
        "\n",
        "\\begin{equation}\n",
        "v_1 \\cdot v_2 = \\sqrt{ \\sum_{i=1}^d v_1[i] \\cdot v_2[i]}\n",
        "\\end{equation}\n",
        "\n",
        "em que $d$ é a dimensão dos vetores $v_1$ e $v_2$. $||v||$ é a norma L2 do vetor $v$, calculada como: \n",
        "\n",
        "\\begin{equation}\n",
        "||v|| = \\sqrt{\\sum_{i=1}^d v[i]^2}.\n",
        "\\end{equation}\n",
        "\n",
        "### Implementação"
      ],
      "metadata": {
        "id": "h6tvw7hV7HjR"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "import math\n",
        "\n",
        "def cosine_similarity(vetor1,vetor2):\n",
        "\n",
        "    prod_interno = 0\n",
        "\n",
        "    for i,valor in enumerate(vetor1):\n",
        "        prod_interno += valor*vetor2[i]\n",
        "    \n",
        "    norma_vetor1 = 0\n",
        "    for x in vetor1:\n",
        "        norma_vetor1 += x*x\n",
        "    norma_vetor1 = norma_vetor1**0.5\n",
        "\n",
        "    #norma_vetor1 = math.sqrt(sum([x**2 for x in vetor1])) \n",
        "\n",
        "    norma_vetor2 = 0\n",
        "    for x in vetor2:\n",
        "        norma_vetor2 += x*x\n",
        "    norma_vetor2 = norma_vetor2**0.5\n",
        "\n",
        "    return prod_interno/(norma_vetor1*norma_vetor2)"
      ],
      "metadata": {
        "id": "kuftbkIjCeFV"
      },
      "execution_count": 1,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "Verificando se a implementação está correta com o seguinte exemplo:"
      ],
      "metadata": {
        "id": "9SPJvItWyufR"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "cosine_dissimilarity([0,0,1,1],[1,0,1,0])"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "XE3Id8G4DKox",
        "outputId": "64a55f2a-480e-4810-d707-e964a4f3e0a4"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.4999999999999999"
            ]
          },
          "metadata": {},
          "execution_count": 54
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "### Aplicação: Recuperação de Textos\n",
        "\n",
        "Aula do dia 30/06"
      ],
      "metadata": {
        "id": "xsZe9uZJynOr"
      }
    }
  ]
}