Update pages/Linear Regression.py

pages/Linear Regression.py

@@ -32,64 +32,74 @@ else:
 tab1, tab2, tab3 = st.tabs(["📖 About Linear Regression", "⚙️ Train Model", "📈 Visualize"])
 
 with tab1:
-    st.markdown("""
-    ## 📈 What is Linear Regression?
-
-    **Linear Regression** is a fundamental algorithm in machine learning used to predict continuous numerical values.
-
-    ---
-    ### 🔢 The Linear Equation:
-
-    The general form:
-    $$
-    y = \beta_0 + \beta_1x_1 + \beta_2x_2 + ... + \beta_nx_n + \varepsilon
-    $$
-
-    - **y**: Output (target)
-    - **x₁, x₂, ..., xₙ**: Input features
-    - **β₀**: Intercept
-    - **β₁, ..., βₙ**: Coefficients
-    - **ε**: Error term
+   
 
-    ---
-    ### 🧠 How it Works:
-
-    1. Fit a straight line that minimizes the squared error between predicted and actual values.
-    2. Uses Ordinary Least Squares (OLS) for best-fit line.
+    st.title("📈 Linear Regression - Intuition & Explanation")
     
-    ---
-    ### 🧮 Loss Function: Mean Squared Error (MSE)
-
-    $$
-    MSE = \frac{1}{n} \sum_{i=1}^{n}(y_i - \hat{y}_i)^2
-    $$
-
-    ---
-    ### 📦 Use Cases:
-
-    - Predicting housing prices
-    - Estimating salaries
-    - Forecasting trends
-
-    ---
-    ### ✅ Pros:
-    - Simple and fast
-    - Interpretable
-    - Good baseline for regression tasks
-
-    ### ⚠️ Cons:
-    - Assumes linear relationship
-    - Sensitive to outliers
-    - Doesn't handle multicollinearity well
-
-    ---
-    ### 📌 Assumptions:
-    - Linearity
-    - Homoscedasticity
-    - Independence
-    - Normality of residuals
-
+    st.markdown("""
+    Linear Regression is a **supervised machine learning algorithm** used to predict a continuous target variable based on one or more input features.
+    
+    It tries to **fit a straight line** (or hyperplane) through the data that minimizes the error between actual and predicted values.
+    """)
+    
+    st.subheader("🔹 Simple Linear Regression Formula")
+    
+    st.latex(r'''
+    y = \beta_0 + \beta_1 x + \epsilon
+    ''')
+    
+    st.markdown("""
+    Where:
+    - \( y \): Predicted value  
+    - \( x \): Input feature  
+    - \( \beta_0 \): Intercept  
+    - \( \beta_1 \): Slope of the line  
+    - \( \epsilon \): Error term  
     """)
+    
+    st.subheader("🔹 Multiple Linear Regression")
+    
+    st.latex(r'''
+    y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \dots + \beta_n x_n + \epsilon
+    ''')
+    
+    st.markdown("""
+    This is used when we have more than one independent variable.
+    """)
+    
+    st.subheader("🎯 Objective of Linear Regression")
+    st.markdown("To find the best-fit line by minimizing the **sum of squared errors (SSE)**.")
+    
+    st.latex(r'''
+    SSE = \sum_{i=1}^{n} (y_i - \hat{y}_i)^2
+    ''')
+    
+    st.subheader("📘 Cost Function (Mean Squared Error)")
+    
+    st.latex(r'''
+    J(\beta) = \frac{1}{n} \sum_{i=1}^{n} (y_i - \hat{y}_i)^2
+    ''')
+    
+    st.markdown("""
+    - The algorithm tries to find values of \( \beta \) (coefficients) that **minimize this cost function**.
+    """)
+    
+    st.subheader("📌 Assumptions of Linear Regression")
+    st.markdown("""
+    - **Linearity**: Relationship between input and output is linear  
+    - **Independence**: Observations are independent  
+    - **Homoscedasticity**: Constant variance of errors  
+    - **Normality of errors**  
+    - **No multicollinearity** (for multiple regression)
+    """)
+    
+    st.subheader("💡 When to Use Linear Regression?")
+    st.markdown("""
+    - To predict continuous numeric values (e.g., price, salary, marks)  
+    - To analyze how inputs are related to output  
+    - Easy to implement and interpret
+    """)
+
 
 with tab2:
     st.subheader("⚙️ Train Linear Regression Model")