Update pages/12_Logistic_Regression.py

pages/12_Logistic_Regression.py

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+import streamlit as st
+
+st.set_page_config(page_title="Logistic Regression", page_icon="🤖", layout="wide")
+
+st.markdown("""
+    <style>
+        .stApp {
+            background-color: #1e1e1e;
+            color: white;
+        }
+        h1, h2, h3 {
+            color: #FF4C60;
+        }
+        .sidebar .sidebar-content {
+            background-color: #1e1e1e;
+        }
+        a {
+            color: #58a6ff;
+            text-decoration: none;
+        }
+        a:hover {
+            color: #1f78d1;
+        }
+    </style>
+""", unsafe_allow_html=True)
+
+st.sidebar.title("🤖 Logistic Regression")
+st.sidebar.markdown("Explore the theory behind Logistic Regression step-by-step.")
+st.sidebar.markdown("---")
+
+st.markdown("<h1 style='text-align: center;'>📈 Logistic Regression (Theory)</h1>", unsafe_allow_html=True)
+
+with st.expander("📘 What is Logistic Regression?"):
+    st.write("""
+    Logistic Regression is a **supervised learning algorithm** used for **classification tasks**.
+    - Mostly used for **binary classification**
+    - Can be extended to **multi-class** using techniques like **Softmax** or **One-vs-Rest (OvR)**
+
+    🔍 The goal is to find a **hyperplane** that linearly separates the classes.
+    ⚠️ Assumption: The data should be **linearly separable** or nearly so.
+    """)
+
+with st.expander("⚙️ Step vs Sigmoid Function"):
+    st.write("""
+    Initially, people used **step function** for classification:
+
+    ```text
+    Step Function Output:
+    if z >= 0 → Class 1
+    if z <  0 → Class 0
+    ```
+
+    ❌ But it's not differentiable → can't optimize via gradient descent.  
+    ✅ Instead, we use the **Sigmoid Function**:
+    $$
+    \\sigma(z) = \\frac{1}{1 + e^{-z}} \\quad \\text{where } z = w^T x + b
+    $$
+    Properties:
+    - \\( z >> 0 \\) → output ≈ 1  
+    - \\( z << 0 \\) → output ≈ 0  
+    - \\( z = 0 \\) → output = 0.5
+    """)
+
+with st.expander("🧮 Loss Function in Logistic Regression"):
+    st.write("""
+    Logistic Regression uses **Log Loss** (or Binary Cross Entropy) for optimization:
+    $$
+    \\mathcal{L} = - \\left[y \\log(\\hat{y}) + (1 - y) \\log(1 - \\hat{y}) \\right]
+    $$
+    - \\( y \\) is the actual class (0 or 1)  
+    - \\( \\hat{y} \\) is the predicted probability from sigmoid  
+    - Minimize this loss using **Gradient Descent**
+    """)
+
+with st.expander("🔁 Gradient Descent & Learning Rate"):
+    st.write("""
+    Gradient Descent updates weights to minimize loss:
+    $$
+    w = w - \\alpha \\cdot \\frac{\\partial \\mathcal{L}}{\\partial w}
+    $$
+
+    - \\( \\alpha \\): Learning rate  
+    ⚖️ Choosing Learning Rate:
+    - Too high → overshoots  
+    - Too low → slow convergence  
+    - Recommended: 0.01 or 0.1 for starters
+
+    📦 Types:
+    - **Batch**: Uses full dataset each update  
+    - **Stochastic (SGD)**: Updates per data point  
+    - **Mini-Batch**: Updates per small batch → widely used
+    """)
+    
+with st.expander("🔢 Multiclass Logistic Regression"):
+    st.subheader("1️⃣ Softmax Regression")
+    st.write("""
+    Generalizes binary logistic regression to multi-class using **Softmax Function**:
+    $$
+    \\text{Softmax}(z_i) = \\frac{e^{z_i}}{\\sum_j e^{z_j}}
+    $$
+    Each class gets a probability, and we pick the class with the highest.  
+    Example: If predicted probabilities are:
+
+    ```text
+    [0.02, 0.05, 0.07, 0.10, 0.08, 0.12, 0.10, 0.30, 0.05, 0.11]
+    ```
+
+    → Predict class **7** (highest probability)
+    """)
+
+    st.subheader("2️⃣ One-vs-Rest (OvR) Classification")
+    st.write("""
+    OvR builds one binary classifier **per class**:
+    - Apple vs Not-Apple  
+    - Banana vs Not-Banana  
+    - Orange vs Not-Orange  
+
+    Then pick the class with the highest confidence score.
+    """)
+
+with st.expander("🧲 Regularization in Logistic Regression"):
+    st.write("""
+    Regularization helps reduce **overfitting** by penalizing large weights.
+    - **L1 (Lasso)**: \\( \\lambda \\sum |w| \\) → Feature selection  
+    - **L2 (Ridge)**: \\( \\lambda \\sum w^2 \\) → Smooths weights  
+    - **ElasticNet**: Combines L1 + L2
+
+    ⚖️ Helps balance **bias vs variance**
+    """)
+
+with st.expander("🔍 Detecting Multicollinearity"):
+    st.write("""
+    Multicollinearity = when predictors are highly correlated.  
+    🎯 Check using **VIF (Variance Inflation Factor)**:
+    - \\( \\text{VIF} > 10 \\) → High multicollinearity
+
+    ✅ Reduce it by:
+    - Removing correlated features  
+    - Applying regularization
+    """)
+
+with st.expander("⚙️ Hyperparameters in Logistic Regression"):
+    st.table([
+        ["penalty", "Regularization type ('l1', 'l2', 'elasticnet', None)"],
+        ["dual", "Use dual formulation (for liblinear only)"],
+        ["tol", "Tolerance for stopping criteria"],
+        ["C", "Inverse of regularization strength"],
+        ["fit_intercept", "Whether to include the bias term"],
+        ["solver", "Optimization algorithm ('lbfgs', 'saga', etc.)"],
+        ["multi_class", "'ovr' or 'multinomial' for multi-class"],
+        ["max_iter", "Max iterations for convergence"],
+        ["class_weight", "'balanced' or manual dictionary"],
+        ["n_jobs", "Number of parallel jobs"],
+        ["l1_ratio", "Used with ElasticNet"],
+    ])
+
+st.markdown("---")
+st.markdown("### 📓 Want to see it in action? Open the Colab notebook below:")
+st.markdown("""
+<a href='https://colab.research.google.com/drive/1IiZmfkCcMltcE5-r5PbQqRpj_maR_M-f?usp=sharing' target='_blank'>
+🔗 Open Logistic Regression Notebook
+</a>
+""", unsafe_allow_html=True)
+
+st.success("Logistic Regression is simple yet powerful. Master it before diving into complex models!")