Linear Regression: A comprehensive explanation

1. Definition & Purpose

• What it is: Linear regression is a supervised learning algorithm used for predicting a continuous target variable based on one or more independent features.

• Goal: Model the linear relationship between input variables (features) and a single output variable (target) by fitting a linear equation to observed data.

2. Key Concepts

a) Equation:

Simple Linear Regression (one feature):

Y=β0+β1X+ϵ

• Y: Target variable.

• X: Feature.

• β0: Intercept (value of Y when X=0).

• β1: Slope (change in Y per unit change in X).

• ϵ: Error term (residuals).

b) Multiple Linear Regression (multiple features):

Y=β0+β1X1+β2X2+⋯+βnXn+ϵ

c) Best-Fit Line: The line that minimizes the sum of squared residuals (differences between predicted and actual values).

d) Residuals:

• Residual=Yactual−Ypredicted

• Squaring residuals ensures all errors are positive and penalizes larger errors.

3. How It Works: Step-by-Step

a) Data Preparation:

• Features (X): Independent variables (e.g., head size, age).

• Target (Y): Dependent variable (e.g., brain weight).

• Split data into training (80%) and testing (20%) sets.

b) Model Training:

  i. Objective: Find coefficients (β0,β1,…) that minimize the cost function (Mean Squared Error).

ii. Cost Function:

 iii. Closed-Form Solution (Normal Equation):

  iv. X: Feature matrix (with a column of 1s for the intercept).

   v.  Y: Target vector.

c) Prediction: Use the learned coefficients to predict new values:

d) Evaluation:

     i. R-squared (R²): Measures the proportion of variance in Y explained by X.

    ii. Adjusted R²: Adjusts R² for the number of predictors to avoid overfitting.

4. Assumptions

• Linearity: Relationship between features and target is linear.

• Independence: Residuals are uncorrelated.

• Homoscedasticity: Residuals have constant variance.

• Normality: Residuals are normally distributed.

5. Practical Example

Problem: Predict brain weight (Y) from head size (X).

Steps:

1. Load Data: Ensure no missing values or outliers.

2. Split Data: Train on 80%, test on 20%.

3. Reshape Data: Convert features to 2D arrays (required by libraries like sklearn).

4. Train Model: Fit LinearRegression to training data.

5. Predict: Use the model to predict test data.

6. Evaluate: Calculate R² and visualize predictions vs. actual values.

6. Code Implementation (Python)

7. Strengths & Limitations

• Strengths: Simple and interpretable; Fast to train; Works well with linearly separable data.

• Limitations: Assumes linearity; fails on non-linear relationships; Sensitive to outliers; Cannot handle categorical features directly (requires encoding).

All-in-all, Linear regression is a foundational algorithm for predicting continuous outcomes. It models relationships using a linear equation, optimizing coefficients to minimize prediction errors. While simple, understanding its assumptions and limitations is crucial for effective application. Tools like R² and adjusted R² help evaluate performance, and libraries like scikit-learn streamline implementation.

For more on this check out this link: Linear Regression - sample code, whiteboard, summary