only_numpy_pca.ipynb

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        "<a href=\"https://colab.research.google.com/github/mashaan14/only-numpy/blob/main/only_numpy_pca.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",
      "source": [
        "# Coding Principal Component Analysis (PCA) using numpy functions"
      ],
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        "id": "-YBWNHOo1aKL"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "import numpy as np\n",
        "import pandas as pd\n",
        "from matplotlib import pyplot as plt"
      ],
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        "id": "wx2BYCWY1nxA"
      },
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      "outputs": []
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        "np.random.seed(0)\n",
        "X = np.concatenate((np.ones((5, 2)), 5*np.ones((5, 2))), axis=0)\n",
        "X = X+np.random.normal(0, 0.5, size=X.shape)\n",
        "display(pd.DataFrame(X))"
      ],
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        "plt.scatter(X[:,0], X[:,1]);\n",
        "plt.tick_params(axis='both',which='both',bottom=False,top=False,left=False,right=False,\n",
        "                labelbottom=False,labeltop=False,labelleft=False,labelright=False);\n",
        "plt.savefig('data.png', bbox_inches='tight', dpi=600)"
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      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 640x480 with 1 Axes>"
            ],
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          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Step 1: Standardize the data\n",
        "The dataset should have a zero mean an unit variance. This process is called standardization. *italicized text*"
      ],
      "metadata": {
        "id": "eJPEfdDB17Be"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "XNormed = (X - X.mean())/(X.std())"
      ],
      "metadata": {
        "id": "YcfOAi2P1_A_"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Step 2: Compute the covariance matrix\n",
        "The diagonal elements in the covariance matrix represent the variance of the features, while the off-diagonal elements represent the covariance between each pair of features.\n",
        "\n"
      ],
      "metadata": {
        "id": "rqKfCWUU2YBP"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "cov_mat = np.cov(XNormed.T)\n",
        "display(pd.DataFrame(cov_mat))"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 112
        },
        "id": "wujIs9tW2ewl",
        "outputId": "673fed95-5a8e-4c9f-a98a-b7d5473ce483"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
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              "    async function quickchart(key) {\n",
              "      const quickchartButtonEl =\n",
              "        document.querySelector('#' + key + ' button');\n",
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              "            'suggestCharts', [key], {});\n",
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            ]
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Step 3: Perform eigendecomposition on the covariance matrix\n",
        "The eigenvectors represent the new space with a magnitude equal to their corresponding eigenvalues."
      ],
      "metadata": {
        "id": "EGqt3BsB5SNI"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "eigenvalues, eigenvectors = np.linalg.eig(cov_mat)"
      ],
      "metadata": {
        "id": "CsMC-t-A53kI"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "display(pd.DataFrame(eigenvalues))\n",
        "display(pd.DataFrame(eigenvectors))"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 207
        },
        "id": "RDu4ixV26h3e",
        "outputId": "0c8f1d35-3638-4ce4-a836-55dd640f3e92"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
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              "    0% {\n",
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              "      border-bottom-color: var(--fill-color);\n",
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              "      border-left-color: var(--fill-color);\n",
              "      border-top-color: var(--fill-color);\n",
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              "      border-left-color: var(--fill-color);\n",
              "      border-top-color: var(--fill-color);\n",
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              "    40% {\n",
              "      border-color: transparent;\n",
              "      border-right-color: var(--fill-color);\n",
              "      border-top-color: var(--fill-color);\n",
              "    }\n",
              "    60% {\n",
              "      border-color: transparent;\n",
              "      border-right-color: var(--fill-color);\n",
              "    }\n",
              "    80% {\n",
              "      border-color: transparent;\n",
              "      border-right-color: var(--fill-color);\n",
              "      border-bottom-color: var(--fill-color);\n",
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              "    90% {\n",
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              "      border-bottom-color: var(--fill-color);\n",
              "    }\n",
              "  }\n",
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              "\n",
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    },
    {
      "cell_type": "markdown",
      "source": [
        "## Step 4: Order the eigenvectors"
      ],
      "metadata": {
        "id": "PSplfADC-wBB"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# Order the eigenvectors in a decreasing order based on the magnitude of their corresponding eigenvalues\n",
        "eigenvectors_sorted = eigenvectors[:,eigenvalues.argsort()[::-1]]\n",
        "display(pd.DataFrame(eigenvectors_sorted))"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 112
        },
        "id": "xrPDFV87Cf-2",
        "outputId": "78b096e7-2377-4fc1-a147-1fb4ede06631"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "          0         1\n",
              "0 -0.689559 -0.724230\n",
              "1 -0.724230  0.689559"
            ],
            "text/html": [
              "\n",
              "  <div id=\"df-3a0dd0ca-ce6a-4080-9bfc-c728c4ed0653\" class=\"colab-df-container\">\n",
              "    <div>\n",
              "<style scoped>\n",
              "    .dataframe tbody tr th:only-of-type {\n",
              "        vertical-align: middle;\n",
              "    }\n",
              "\n",
              "    .dataframe tbody tr th {\n",
              "        vertical-align: top;\n",
              "    }\n",
              "\n",
              "    .dataframe thead th {\n",
              "        text-align: right;\n",
              "    }\n",
              "</style>\n",
              "<table border=\"1\" class=\"dataframe\">\n",
              "  <thead>\n",
              "    <tr style=\"text-align: right;\">\n",
              "      <th></th>\n",
              "      <th>0</th>\n",
              "      <th>1</th>\n",
              "    </tr>\n",
              "  </thead>\n",
              "  <tbody>\n",
              "    <tr>\n",
              "      <th>0</th>\n",
              "      <td>-0.689559</td>\n",
              "      <td>-0.724230</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>1</th>\n",
              "      <td>-0.724230</td>\n",
              "      <td>0.689559</td>\n",
              "    </tr>\n",
              "  </tbody>\n",
              "</table>\n",
              "</div>\n",
              "    <div class=\"colab-df-buttons\">\n",
              "\n",
              "  <div class=\"colab-df-container\">\n",
              "    <button class=\"colab-df-convert\" onclick=\"convertToInteractive('df-3a0dd0ca-ce6a-4080-9bfc-c728c4ed0653')\"\n",
              "            title=\"Convert this dataframe to an interactive table.\"\n",
              "            style=\"display:none;\">\n",
              "\n",
              "  <svg xmlns=\"http://www.w3.org/2000/svg\" height=\"24px\" viewBox=\"0 -960 960 960\">\n",
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              "  </svg>\n",
              "    </button>\n",
              "\n",
              "  <style>\n",
              "    .colab-df-container {\n",
              "      display:flex;\n",
              "      gap: 12px;\n",
              "    }\n",
              "\n",
              "    .colab-df-convert {\n",
              "      background-color: #E8F0FE;\n",
              "      border: none;\n",
              "      border-radius: 50%;\n",
              "      cursor: pointer;\n",
              "      display: none;\n",
              "      fill: #1967D2;\n",
              "      height: 32px;\n",
              "      padding: 0 0 0 0;\n",
              "      width: 32px;\n",
              "    }\n",
              "\n",
              "    .colab-df-convert:hover {\n",
              "      background-color: #E2EBFA;\n",
              "      box-shadow: 0px 1px 2px rgba(60, 64, 67, 0.3), 0px 1px 3px 1px rgba(60, 64, 67, 0.15);\n",
              "      fill: #174EA6;\n",
              "    }\n",
              "\n",
              "    .colab-df-buttons div {\n",
              "      margin-bottom: 4px;\n",
              "    }\n",
              "\n",
              "    [theme=dark] .colab-df-convert {\n",
              "      background-color: #3B4455;\n",
              "      fill: #D2E3FC;\n",
              "    }\n",
              "\n",
              "    [theme=dark] .colab-df-convert:hover {\n",
              "      background-color: #434B5C;\n",
              "      box-shadow: 0px 1px 3px 1px rgba(0, 0, 0, 0.15);\n",
              "      filter: drop-shadow(0px 1px 2px rgba(0, 0, 0, 0.3));\n",
              "      fill: #FFFFFF;\n",
              "    }\n",
              "  </style>\n",
              "\n",
              "    <script>\n",
              "      const buttonEl =\n",
              "        document.querySelector('#df-3a0dd0ca-ce6a-4080-9bfc-c728c4ed0653 button.colab-df-convert');\n",
              "      buttonEl.style.display =\n",
              "        google.colab.kernel.accessAllowed ? 'block' : 'none';\n",
              "\n",
              "      async function convertToInteractive(key) {\n",
              "        const element = document.querySelector('#df-3a0dd0ca-ce6a-4080-9bfc-c728c4ed0653');\n",
              "        const dataTable =\n",
              "          await google.colab.kernel.invokeFunction('convertToInteractive',\n",
              "                                                    [key], {});\n",
              "        if (!dataTable) return;\n",
              "\n",
              "        const docLinkHtml = 'Like what you see? Visit the ' +\n",
              "          '<a target=\"_blank\" href=https://colab.research.google.com/notebooks/data_table.ipynb>data table notebook</a>'\n",
              "          + ' to learn more about interactive tables.';\n",
              "        element.innerHTML = '';\n",
              "        dataTable['output_type'] = 'display_data';\n",
              "        await google.colab.output.renderOutput(dataTable, element);\n",
              "        const docLink = document.createElement('div');\n",
              "        docLink.innerHTML = docLinkHtml;\n",
              "        element.appendChild(docLink);\n",
              "      }\n",
              "    </script>\n",
              "  </div>\n",
              "\n",
              "\n",
              "<div id=\"df-d186fb48-47ec-415d-b27d-8dcde381955a\">\n",
              "  <button class=\"colab-df-quickchart\" onclick=\"quickchart('df-d186fb48-47ec-415d-b27d-8dcde381955a')\"\n",
              "            title=\"Suggest charts.\"\n",
              "            style=\"display:none;\">\n",
              "\n",
              "<svg xmlns=\"http://www.w3.org/2000/svg\" height=\"24px\"viewBox=\"0 0 24 24\"\n",
              "     width=\"24px\">\n",
              "    <g>\n",
              "        <path d=\"M19 3H5c-1.1 0-2 .9-2 2v14c0 1.1.9 2 2 2h14c1.1 0 2-.9 2-2V5c0-1.1-.9-2-2-2zM9 17H7v-7h2v7zm4 0h-2V7h2v10zm4 0h-2v-4h2v4z\"/>\n",
              "    </g>\n",
              "</svg>\n",
              "  </button>\n",
              "\n",
              "<style>\n",
              "  .colab-df-quickchart {\n",
              "      --bg-color: #E8F0FE;\n",
              "      --fill-color: #1967D2;\n",
              "      --hover-bg-color: #E2EBFA;\n",
              "      --hover-fill-color: #174EA6;\n",
              "      --disabled-fill-color: #AAA;\n",
              "      --disabled-bg-color: #DDD;\n",
              "  }\n",
              "\n",
              "  [theme=dark] .colab-df-quickchart {\n",
              "      --bg-color: #3B4455;\n",
              "      --fill-color: #D2E3FC;\n",
              "      --hover-bg-color: #434B5C;\n",
              "      --hover-fill-color: #FFFFFF;\n",
              "      --disabled-bg-color: #3B4455;\n",
              "      --disabled-fill-color: #666;\n",
              "  }\n",
              "\n",
              "  .colab-df-quickchart {\n",
              "    background-color: var(--bg-color);\n",
              "    border: none;\n",
              "    border-radius: 50%;\n",
              "    cursor: pointer;\n",
              "    display: none;\n",
              "    fill: var(--fill-color);\n",
              "    height: 32px;\n",
              "    padding: 0;\n",
              "    width: 32px;\n",
              "  }\n",
              "\n",
              "  .colab-df-quickchart:hover {\n",
              "    background-color: var(--hover-bg-color);\n",
              "    box-shadow: 0 1px 2px rgba(60, 64, 67, 0.3), 0 1px 3px 1px rgba(60, 64, 67, 0.15);\n",
              "    fill: var(--button-hover-fill-color);\n",
              "  }\n",
              "\n",
              "  .colab-df-quickchart-complete:disabled,\n",
              "  .colab-df-quickchart-complete:disabled:hover {\n",
              "    background-color: var(--disabled-bg-color);\n",
              "    fill: var(--disabled-fill-color);\n",
              "    box-shadow: none;\n",
              "  }\n",
              "\n",
              "  .colab-df-spinner {\n",
              "    border: 2px solid var(--fill-color);\n",
              "    border-color: transparent;\n",
              "    border-bottom-color: var(--fill-color);\n",
              "    animation:\n",
              "      spin 1s steps(1) infinite;\n",
              "  }\n",
              "\n",
              "  @keyframes spin {\n",
              "    0% {\n",
              "      border-color: transparent;\n",
              "      border-bottom-color: var(--fill-color);\n",
              "      border-left-color: var(--fill-color);\n",
              "    }\n",
              "    20% {\n",
              "      border-color: transparent;\n",
              "      border-left-color: var(--fill-color);\n",
              "      border-top-color: var(--fill-color);\n",
              "    }\n",
              "    30% {\n",
              "      border-color: transparent;\n",
              "      border-left-color: var(--fill-color);\n",
              "      border-top-color: var(--fill-color);\n",
              "      border-right-color: var(--fill-color);\n",
              "    }\n",
              "    40% {\n",
              "      border-color: transparent;\n",
              "      border-right-color: var(--fill-color);\n",
              "      border-top-color: var(--fill-color);\n",
              "    }\n",
              "    60% {\n",
              "      border-color: transparent;\n",
              "      border-right-color: var(--fill-color);\n",
              "    }\n",
              "    80% {\n",
              "      border-color: transparent;\n",
              "      border-right-color: var(--fill-color);\n",
              "      border-bottom-color: var(--fill-color);\n",
              "    }\n",
              "    90% {\n",
              "      border-color: transparent;\n",
              "      border-bottom-color: var(--fill-color);\n",
              "    }\n",
              "  }\n",
              "</style>\n",
              "\n",
              "  <script>\n",
              "    async function quickchart(key) {\n",
              "      const quickchartButtonEl =\n",
              "        document.querySelector('#' + key + ' button');\n",
              "      quickchartButtonEl.disabled = true;  // To prevent multiple clicks.\n",
              "      quickchartButtonEl.classList.add('colab-df-spinner');\n",
              "      try {\n",
              "        const charts = await google.colab.kernel.invokeFunction(\n",
              "            'suggestCharts', [key], {});\n",
              "      } catch (error) {\n",
              "        console.error('Error during call to suggestCharts:', error);\n",
              "      }\n",
              "      quickchartButtonEl.classList.remove('colab-df-spinner');\n",
              "      quickchartButtonEl.classList.add('colab-df-quickchart-complete');\n",
              "    }\n",
              "    (() => {\n",
              "      let quickchartButtonEl =\n",
              "        document.querySelector('#df-d186fb48-47ec-415d-b27d-8dcde381955a button');\n",
              "      quickchartButtonEl.style.display =\n",
              "        google.colab.kernel.accessAllowed ? 'block' : 'none';\n",
              "    })();\n",
              "  </script>\n",
              "</div>\n",
              "    </div>\n",
              "  </div>\n"
            ]
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "legend_label = ['$1^{st}$ principal component', '$2^{nd}$ principal component', 'data']\n",
        "new_colors = ['#ff7f0e', '#2ca02c', '#1f77b4']\n",
        "for i in range(eigenvectors_sorted.shape[1]):\n",
        "  plt.quiver(3, 3 , eigenvectors_sorted[0,i], eigenvectors_sorted[1,i], scale=7, label=legend_label[i], color=new_colors[i])\n",
        "\n",
        "plt.scatter(X[:,0], X[:,1], label=legend_label[-1], c=new_colors[-1]);\n",
        "plt.tick_params(axis='both',which='both',bottom=False,top=False,left=False,right=False,\n",
        "                labelbottom=False,labeltop=False,labelleft=False,labelright=False);\n",
        "plt.legend()\n",
        "plt.savefig('eigenvectors_sorted.png', bbox_inches='tight', dpi=600)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 406
        },
        "id": "mfnETzDK7pZg",
        "outputId": "db480e77-a527-4c21-9ca1-8b7e5b1dd38f"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 640x480 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Step 5: Compute the transformation"
      ],
      "metadata": {
        "id": "mRIOP9g5F4qU"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "X_transformed = np.dot(XNormed, eigenvectors_sorted)"
      ],
      "metadata": {
        "id": "dgOCz0rtF_r-"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "plt.scatter(X_transformed[:,0], X_transformed[:,1]);\n",
        "plt.tick_params(axis='both',which='both',bottom=False,top=False,left=False,right=False,\n",
        "                labelbottom=False,labeltop=False,labelleft=False,labelright=False);\n",
        "plt.savefig('X_transformed.png', bbox_inches='tight', dpi=600)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 406
        },
        "id": "J6qS91qBGhLw",
        "outputId": "94418a04-03c1-4d05-fae4-b20d98cd4bbb"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 640x480 with 1 Axes>"
            ],
            "image/png": "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\n"
          },
          "metadata": {}
        }
      ]
    }
  ]
}