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<!DOCTYPE html>
<html lang="en">
  <head>
    <meta charset="utf-8" />
    <link rel="icon" crossorigin="anonymous" href="https://cdn.jsdelivr.net/npm/@marimo-team/frontend@0.23.3/dist/favicon.ico" />
    <!-- Preload is necessary because we show these images when we disconnect from the server,
    but at that point we cannot load these images from the server -->
    <link rel="preload" crossorigin="anonymous" href="https://cdn.jsdelivr.net/npm/@marimo-team/frontend@0.23.3/dist/assets/gradient-yHQUC_QB.png" as="image" />
    <link rel="preload" crossorigin="anonymous" href="https://cdn.jsdelivr.net/npm/@marimo-team/frontend@0.23.3/dist/assets/noise-60BoTA8O.png" as="image" />
    <!-- Preload the fonts -->
    <link rel="preload" crossorigin="anonymous" href="https://cdn.jsdelivr.net/npm/@marimo-team/frontend@0.23.3/dist/assets/Lora-VariableFont_wght-CZceb_kH.woff2" as="font" type="font/woff2" crossorigin="anonymous" />
    <link rel="preload" crossorigin="anonymous" href="https://cdn.jsdelivr.net/npm/@marimo-team/frontend@0.23.3/dist/assets/PTSans-Regular-Bam3NpBI.woff2" as="font" type="font/woff2" crossorigin="anonymous" />
    <link rel="preload" crossorigin="anonymous" href="https://cdn.jsdelivr.net/npm/@marimo-team/frontend@0.23.3/dist/assets/PTSans-Bold-C_DwAp7Z.woff2" as="font" type="font/woff2" crossorigin="anonymous" />
    <link rel="preload" crossorigin="anonymous" href="https://cdn.jsdelivr.net/npm/@marimo-team/frontend@0.23.3/dist/assets/FiraMono-Regular-CEsLFVD9.woff2" as="font" type="font/woff2" crossorigin="anonymous" />
    <link rel="preload" crossorigin="anonymous" href="https://cdn.jsdelivr.net/npm/@marimo-team/frontend@0.23.3/dist/assets/FiraMono-Medium-D5MAsWEG.woff2" as="font" type="font/woff2" crossorigin="anonymous" />
    <link rel="preload" crossorigin="anonymous" href="https://cdn.jsdelivr.net/npm/@marimo-team/frontend@0.23.3/dist/assets/FiraMono-Bold-C6PDArdf.woff2" as="font" type="font/woff2" crossorigin="anonymous" />

    <meta name="viewport" content="width=device-width, initial-scale=1" />
    <meta name="theme-color" content="#000000" />
    <meta name="description" content="a marimo app" />
    <link rel="apple-touch-icon" crossorigin="anonymous" href="https://cdn.jsdelivr.net/npm/@marimo-team/frontend@0.23.3/dist/apple-touch-icon.png" />
    <link rel="manifest" crossorigin="anonymous" href="https://cdn.jsdelivr.net/npm/@marimo-team/frontend@0.23.3/dist/manifest.json" />

    <script data-marimo="true">
      function __resizeIframe(obj) {
        const scrollbarHeight = 20; // Max between windows, mac, and linux

        function setHeight() {
          // Guard against race condition where iframe isn't ready
          if (!obj.contentWindow?.document?.documentElement) {
            return;
          }
          const element = obj.contentWindow.document.documentElement;
          // If there is no vertical scrollbar, we don't need to resize the iframe
          if (element.scrollHeight === element.clientHeight) {
            return;
          }

          // Create a new height that includes the scrollbar height if it's visible
          const hasHorizontalScrollbar = element.scrollWidth > element.clientWidth;
          const newHeight = element.scrollHeight + (hasHorizontalScrollbar ? scrollbarHeight : 0);

          // Only update the height if it's different from the current height
          if (obj.style.height !== `${newHeight}px`) {
            obj.style.height = `${newHeight}px`;
          }
        }

        // Resize the iframe to the height of the content and bottom scrollbar height
        setHeight();

        // Resize the iframe when the content changes
        const resizeObserver = new ResizeObserver((_entries) => {
          setHeight();
        });
        // Only observe if iframe content is ready
        if (obj.contentWindow?.document?.body) {
          resizeObserver.observe(obj.contentWindow.document.body);
        }
      }
    </script>
    <marimo-filename hidden>notes.py</marimo-filename>
    <!-- TODO(Trevor): Legacy, required by VS Code plugin. Remove when plugin is updated (see marimo/server/_templates/template.py) -->
    <marimo-version data-version="{{ version }}" hidden></marimo-version>
    <marimo-user-config data-config="{{ user_config }}" hidden></marimo-user-config>
    <marimo-server-token data-token="{{ server_token }}" hidden></marimo-server-token>
    <!-- /TODO -->
    <title>notes</title>
    <script type="module" crossorigin crossorigin="anonymous" src="https://cdn.jsdelivr.net/npm/@marimo-team/frontend@0.23.3/dist/assets/index-BjiE1T38.js"></script>
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<script data-marimo="true">
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            notebookCode: "import marimo\n\n__generated_with = \"0.23.3\"\napp = marimo.App()\n\n\n@app.cell\ndef _():\n    import marimo as mo\n    import numpy as np\n\n    # Remove and warn if matplotlib is missing.\n    try:\n        import matplotlib.pyplot as plt\n    except ModuleNotFoundError:\n        plt = None\n        mo.notification(\"Matplotlib is not installed. Plots will be skipped or limited.\", kind=\"warning\")\n\n    from sklearn.datasets import make_regression\n    from sklearn.linear_model import LinearRegression\n\n    return LinearRegression, make_regression, mo, np, plt\n\n\n@app.cell\ndef _(mo):\n    mo.md(\"# Linear Regression: A Beginner's Guide\")\n    mo.md(\n        \"Linear regression is a fundamental technique in machine learning used to model the relationship between a dependent variable and one or more independent variables.\"\n    )\n    mo.md(\n        \"In this notebook, we'll explore what linear regression is, how it works mathematically, how to fit a model using scikit-learn, and how to implement it from scratch using gradient descent.\"\n    )\n    return\n\n\n@app.cell\ndef _(LinearRegression, make_regression, mo, np, plt):\n    mo.md(\"## What is Linear Regression?\")\n    mo.md(\n        \"Linear regression aims to find the 'best fit' line through data points. This line is defined by weights (coefficients) and a bias term.\"\n    )\n\n    # Create sample data\n    X, y = make_regression(n_samples=100, n_features=1, noise=10, random_state=42)\n\n    # Fit sklearn linear regression\n    lin_reg = LinearRegression()\n    lin_reg.fit(X, y)\n\n    # Calculate predictions\n    y_pred = lin_reg.predict(X)\n\n    if plt is not None:\n        # Sort the data for a prettier line plot\n        sorted_idx = np.argsort(X[:, 0])\n        X_sorted = X[sorted_idx]\n        y_pred_sorted = y_pred[sorted_idx]\n        y_sorted = y[sorted_idx]\n\n        # Plot\n        fig, ax = plt.subplots(figsize=(10, 6))\n        ax.scatter(X, y, alpha=0.6, label=\"Data points\")\n        ax.plot(X_sorted, y_pred_sorted, color=\"red\", linewidth=2, label=\"Fitted line\")\n        ax.set_xlabel(\"X\")\n        ax.set_ylabel(\"y\")\n        ax.legend()\n        ax.grid(True, alpha=0.3)\n        ax.set_title(\"Linear Regression Example\")\n\n        mo.display(fig)\n    else:\n        mo.notification(\"Skipping plot: matplotlib is not available.\", kind=\"error\")\n    return X, lin_reg, y, y_pred\n\n\n@app.cell\ndef _(lin_reg, mo):\n    mo.md(\"## Mathematical Foundation\")\n    mo.md(\"The linear regression model can be expressed as:\")\n    mo.md(\"$$y = Xw + b$$\")\n    mo.md(\"Where:\")\n    mo.md(\"- $y$ is the target variable\")\n    mo.md(\"- $X$ is the matrix of input features\")\n    mo.md(\"- $w$ is the vector of weights\")\n    mo.md(\"- $b$ is the bias term\")\n\n    mo.md(\"In our example:\")\n    mo.md(f\"- Weights (w): {lin_reg.coef_[0]:.2f}\")\n    mo.md(f\"- Bias (b): {lin_reg.intercept_:.2f}\")\n    return\n\n\n@app.cell\ndef _(mo, np, y, y_pred):\n    mo.md(\"## Loss Function: Mean Squared Error (MSE)\")\n    mo.md(\n        \"The goal of linear regression is to minimize the error between predicted and actual values. The most common loss function is Mean Squared Error:\"\n    )\n    mo.md(\"$$\\\\text{MSE} = \\\\frac{1}{n} \\\\sum_{i=1}^{n} (y_i - \\\\hat{y}_i)^2$$\")\n    mo.md(\"Where:\")\n    mo.md(\"- $y_i$ is the actual value\")\n    mo.md(\"- $\\\\hat{y}_i$ is the predicted value\")\n    mo.md(\"- $n$ is the number of samples\")\n\n    # Calculate MSE\n    mse = np.mean((y - y_pred) ** 2)\n    mo.md(f\"In our example, the MSE is approximately: {mse:.2f}\")\n    return\n\n\n@app.cell\ndef _(X, mo, np, y):\n    mo.md(\"## How Does It Work? Gradient Descent\")\n    mo.md(\n        \"To minimize the MSE, we can use an optimization algorithm called gradient descent. This algorithm iteratively adjusts the weights and bias to reduce the loss function.\"\n    )\n\n    # Simple implementation of gradient descent\n    def gradient_descent(X, y, learning_rate=0.01, iterations=1000):\n        m = len(X)\n        w = np.random.randn(1)\n        b = 0.0\n\n        # Reshape X\n        x_vec = X.ravel()\n        for i in range(iterations):\n            y_pred = x_vec * w + b\n\n            # Compute gradients\n            dw = (-2 / m) * np.sum(x_vec * (y - y_pred))\n            db = (-2 / m) * np.sum(y - y_pred)\n\n            w -= learning_rate * dw\n            b -= learning_rate * db\n\n        return w, b\n\n    w_gd, b_gd = gradient_descent(X, y)\n    y_pred_gd = X.ravel() * w_gd + b_gd\n\n    mse_gd = np.mean((y - y_pred_gd) ** 2)\n\n    mo.md(\"Using gradient descent (learning rate=0.01, 1000 iterations):\")\n    mo.md(f\"- Weights (w): {w_gd[0]:.2f}\")\n    mo.md(f\"- Bias (b): {b_gd:.2f}\")\n    mo.md(f\"- MSE: {mse_gd:.2f}\")\n    return (y_pred_gd,)\n\n\n@app.cell\ndef _(X, mo, np, plt, y, y_pred, y_pred_gd):\n    mo.md(\"## Visualizing the Process\")\n    if plt is not None:\n        fig, ax = plt.subplots(figsize=(10, 6))\n        ax.scatter(X, y, alpha=0.6, label=\"Data points\")\n\n        # For clean lines, sort the X for both predictions\n        sorted_idx = np.argsort(X[:, 0])\n        X_sorted = X[sorted_idx]\n        y_pred_sorted = y_pred[sorted_idx]\n        y_pred_gd_sorted = y_pred_gd[sorted_idx]\n\n        ax.plot(X_sorted, y_pred_sorted, color=\"red\", linewidth=2, label=\"Sklearn fit\")\n        ax.plot(X_sorted, y_pred_gd_sorted, color=\"green\", linewidth=2, linestyle=\"--\", label=\"Gradient descent\")\n        ax.set_xlabel(\"X\")\n        ax.set_ylabel(\"y\")\n        ax.legend()\n        ax.grid(True, alpha=0.3)\n        ax.set_title(\"Comparison: Sklearn vs Gradient Descent\")\n        mo.display(fig)\n    else:\n        mo.notification(\"Skipping plot: matplotlib is not available.\", kind=\"error\")\n    return\n\n\n@app.cell\ndef _(mo):\n    mo.md(\"## Key Takeaways\")\n    mo.md(\"1. Linear regression models the relationship between variables using a linear equation.\")\n    mo.md(\"2. The parameters (weights and bias) are learned by minimizing the loss function.\")\n    mo.md(\"3. Mean Squared Error (MSE) is a common loss function.\")\n    mo.md(\"4. Gradient descent is commonly used to optimize parameters.\")\n    mo.md(\"5. Even simple models like linear regression can be very effective!\")\n\n    mo.md(\"---\")\n    mo.md(\"**Try modifying the data, learning rate, or number of iterations above to see how the model changes!**\")\n    return\n\n\nif __name__ == \"__main__\":\n    app.run()\n",
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            "notebook": {"cells": [{"code": "import marimo as mo\nimport numpy as np\n\n# Remove and warn if matplotlib is missing.\ntry:\n    import matplotlib.pyplot as plt\nexcept ModuleNotFoundError:\n    plt = None\n    mo.notification(\"Matplotlib is not installed. Plots will be skipped or limited.\", kind=\"warning\")\n\nfrom sklearn.datasets import make_regression\nfrom sklearn.linear_model import LinearRegression", "code_hash": "6311d2d636bfb7b8a25160e81c78d21d", "config": {"column": null, "disabled": false, "hide_code": false}, "id": "Hbol", "name": "_"}, {"code": "mo.md(\"# Linear Regression: A Beginner's Guide\")\nmo.md(\n    \"Linear regression is a fundamental technique in machine learning used to model the relationship between a dependent variable and one or more independent variables.\"\n)\nmo.md(\n    \"In this notebook, we'll explore what linear regression is, how it works mathematically, how to fit a model using scikit-learn, and how to implement it from scratch using gradient descent.\"\n)", "code_hash": "ae6176c366881e7b334d536a6a4860a5", "config": {"column": null, "disabled": false, "hide_code": false}, "id": "MJUe", "name": "_"}, {"code": "mo.md(\"## What is Linear Regression?\")\nmo.md(\n    \"Linear regression aims to find the 'best fit' line through data points. This line is defined by weights (coefficients) and a bias term.\"\n)\n\n# Create sample data\nX, y = make_regression(n_samples=100, n_features=1, noise=10, random_state=42)\n\n# Fit sklearn linear regression\nlin_reg = LinearRegression()\nlin_reg.fit(X, y)\n\n# Calculate predictions\ny_pred = lin_reg.predict(X)\n\nif plt is not None:\n    # Sort the data for a prettier line plot\n    sorted_idx = np.argsort(X[:, 0])\n    X_sorted = X[sorted_idx]\n    y_pred_sorted = y_pred[sorted_idx]\n    y_sorted = y[sorted_idx]\n\n    # Plot\n    fig, ax = plt.subplots(figsize=(10, 6))\n    ax.scatter(X, y, alpha=0.6, label=\"Data points\")\n    ax.plot(X_sorted, y_pred_sorted, color=\"red\", linewidth=2, label=\"Fitted line\")\n    ax.set_xlabel(\"X\")\n    ax.set_ylabel(\"y\")\n    ax.legend()\n    ax.grid(True, alpha=0.3)\n    ax.set_title(\"Linear Regression Example\")\n\n    mo.display(fig)\nelse:\n    mo.notification(\"Skipping plot: matplotlib is not available.\", kind=\"error\")", "code_hash": "170ff6c75442add92b65232c51558154", "config": {"column": null, "disabled": false, "hide_code": false}, "id": "vblA", "name": "_"}, {"code": "mo.md(\"## Mathematical Foundation\")\nmo.md(\"The linear regression model can be expressed as:\")\nmo.md(\"$$y = Xw + b$$\")\nmo.md(\"Where:\")\nmo.md(\"- $y$ is the target variable\")\nmo.md(\"- $X$ is the matrix of input features\")\nmo.md(\"- $w$ is the vector of weights\")\nmo.md(\"- $b$ is the bias term\")\n\nmo.md(\"In our example:\")\nmo.md(f\"- Weights (w): {lin_reg.coef_[0]:.2f}\")\nmo.md(f\"- Bias (b): {lin_reg.intercept_:.2f}\")", "code_hash": "59a3865c53fa126bd4cf2f338bb8a1d3", "config": {"column": null, "disabled": false, "hide_code": false}, "id": "bkHC", "name": "_"}, {"code": "mo.md(\"## Loss Function: Mean Squared Error (MSE)\")\nmo.md(\n    \"The goal of linear regression is to minimize the error between predicted and actual values. The most common loss function is Mean Squared Error:\"\n)\nmo.md(\"$$\\\\text{MSE} = \\\\frac{1}{n} \\\\sum_{i=1}^{n} (y_i - \\\\hat{y}_i)^2$$\")\nmo.md(\"Where:\")\nmo.md(\"- $y_i$ is the actual value\")\nmo.md(\"- $\\\\hat{y}_i$ is the predicted value\")\nmo.md(\"- $n$ is the number of samples\")\n\n# Calculate MSE\nmse = np.mean((y - y_pred) ** 2)\nmo.md(f\"In our example, the MSE is approximately: {mse:.2f}\")", "code_hash": "de9208afe44eff71d520ff697cad4770", "config": {"column": null, "disabled": false, "hide_code": false}, "id": "lEQa", "name": "_"}, {"code": "mo.md(\"## How Does It Work? Gradient Descent\")\nmo.md(\n    \"To minimize the MSE, we can use an optimization algorithm called gradient descent. This algorithm iteratively adjusts the weights and bias to reduce the loss function.\"\n)\n\n# Simple implementation of gradient descent\ndef gradient_descent(X, y, learning_rate=0.01, iterations=1000):\n    m = len(X)\n    w = np.random.randn(1)\n    b = 0.0\n\n    # Reshape X\n    x_vec = X.ravel()\n    for i in range(iterations):\n        y_pred = x_vec * w + b\n\n        # Compute gradients\n        dw = (-2 / m) * np.sum(x_vec * (y - y_pred))\n        db = (-2 / m) * np.sum(y - y_pred)\n\n        w -= learning_rate * dw\n        b -= learning_rate * db\n\n    return w, b\n\nw_gd, b_gd = gradient_descent(X, y)\ny_pred_gd = X.ravel() * w_gd + b_gd\n\nmse_gd = np.mean((y - y_pred_gd) ** 2)\n\nmo.md(\"Using gradient descent (learning rate=0.01, 1000 iterations):\")\nmo.md(f\"- Weights (w): {w_gd[0]:.2f}\")\nmo.md(f\"- Bias (b): {b_gd:.2f}\")\nmo.md(f\"- MSE: {mse_gd:.2f}\")", "code_hash": "df993a47fd4bb8dc97b0a17617769ede", "config": {"column": null, "disabled": false, "hide_code": false}, "id": "PKri", "name": "_"}, {"code": "mo.md(\"## Visualizing the Process\")\nif plt is not None:\n    fig, ax = plt.subplots(figsize=(10, 6))\n    ax.scatter(X, y, alpha=0.6, label=\"Data points\")\n\n    # For clean lines, sort the X for both predictions\n    sorted_idx = np.argsort(X[:, 0])\n    X_sorted = X[sorted_idx]\n    y_pred_sorted = y_pred[sorted_idx]\n    y_pred_gd_sorted = y_pred_gd[sorted_idx]\n\n    ax.plot(X_sorted, y_pred_sorted, color=\"red\", linewidth=2, label=\"Sklearn fit\")\n    ax.plot(X_sorted, y_pred_gd_sorted, color=\"green\", linewidth=2, linestyle=\"--\", label=\"Gradient descent\")\n    ax.set_xlabel(\"X\")\n    ax.set_ylabel(\"y\")\n    ax.legend()\n    ax.grid(True, alpha=0.3)\n    ax.set_title(\"Comparison: Sklearn vs Gradient Descent\")\n    mo.display(fig)\nelse:\n    mo.notification(\"Skipping plot: matplotlib is not available.\", kind=\"error\")", "code_hash": "306bc2687366587cb56a645ff49a1445", "config": {"column": null, "disabled": false, "hide_code": false}, "id": "Xref", "name": "_"}, {"code": "mo.md(\"## Key Takeaways\")\nmo.md(\"1. Linear regression models the relationship between variables using a linear equation.\")\nmo.md(\"2. The parameters (weights and bias) are learned by minimizing the loss function.\")\nmo.md(\"3. Mean Squared Error (MSE) is a common loss function.\")\nmo.md(\"4. Gradient descent is commonly used to optimize parameters.\")\nmo.md(\"5. Even simple models like linear regression can be very effective!\")\n\nmo.md(\"---\")\nmo.md(\"**Try modifying the data, learning rate, or number of iterations above to see how the model changes!**\")", "code_hash": "87605417fa79512c613421afe7a9f671", "config": {"column": null, "disabled": false, "hide_code": false}, "id": "SFPL", "name": "_"}], "metadata": {"marimo_version": "0.23.3"}, "version": "1"},
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