This project is a comprehensive, hands-on implementation of Linear Regression using Python and NumPy, built entirely from scratch without relying on high-level machine learning libraries such as scikit-learn, TensorFlow, or PyTorch.
The goal of this project is not just to use linear regression, but to deeply understand how it works internally by manually coding every major component - from data generation to forward propagation, loss calculation, gradient computation, and parameter updates using gradient descent.
It serves as a strong foundation for understanding more advanced machine learning and deep learning models.
Linear regression is one of the most fundamental algorithms in machine learning and statistics. It forms the backbone of more advanced models and is essential for understanding how learning systems optimize themselves using gradient-based methods.
Instead of using prebuilt libraries, this project focuses on:
- Understanding the mathematics behind linear regression
- Learning how gradient descent optimizes model parameters
- Seeing how training loss decreases over time
- Visualizing how predictions improve after training
The primary objectives of this project are:
- Implementation of a linear regression model from first principles
- Manual construction of the gradient descent algorithm
- Understanding how weights and bias influence predictions
- Building intuition about loss functions and error minimization
- Learning how backpropagation works in simple linear models
- Visualization of model performance before and after training
The project is structured as a progressive, task-based notebook that builds the model step by step:
-
Task 1: Introduction
- Introduction to the regression problem
-
Task 2: Dataset
- Review of the linear regression equation and its components
- Creation of synthetic data using NumPy
- Addition of controlled randomness (noise) to simulate real-world data
- Visualization of raw dataset
-
Task 3: Initialize Parameters
- Introduction to model parameters:
- Weight (W)
- Bias (b)
- Building the base structure of a custom
LinearModelclass - Proper initialization of parameters using NumPy arrays
- Introduction to model parameters:
-
Task 4: Forward Pass
- Implementation of the core linear equation:
$y = Wx + b$ - Converting raw input features into predictions
- Verification of model outputs
- Implementation of the core linear equation:
-
Task 5: Compute Loss
- Implementation of the Mean Squared Error (MSE) loss function
- Measuring how far predictions are from true values
- Tracking loss across iterations for performance monitoring
-
Task 6: Backward Pass
- Manual calculation of gradients:
- Partial derivative of loss with respect to W
- Partial derivative of loss with respect to b
- Understanding gradient flow and parameter sensitivity
- Manual calculation of gradients:
-
Task 7: Update Parameters
- Implementation of the gradient descent update rule
- Use of learning rate to control step size
- Continuous improvement of model parameters
-
Task 8: Training Loop
- Building a full training pipeline
- Iterative execution of:
- Forward pass
- Loss computation
- Backward pass
- Parameter update
- Logging and storing training loss
- Visualization of loss decreasing smoothly over 1000 iterations. This confirms that gradient descent is functioning correctly.
-
Task 9: Predictions
- Generating predictions from:
- An untrained model
- A trained model
- Visual comparison of model behavior
- Strong evidence of learning shown through:
- Random scatter from the untrained model
- Tight alignment of trained predictions with ground truth
- Generating predictions from:
- The loss curve shows a consistent downward trend.
- Demonstrates stable and effective parameter updates.
- Confirms that the learning algorithm successfully minimizes error.
The prediction plot clearly shows:
| Model State | Behavior |
|---|---|
| Untrained Model (red points) | Random, widely scattered, inaccurate predictions |
| Trained Model (blue points) | Tightly aligned with true values, forming a strong linear relationship |
This visual comparison validates the correctness of the gradient descent and backpropagation implementation.
This project includes practical implementations of key machine learning mathematics:
- Linear equation modeling
- Mean Squared Error (MSE)
- Partial derivatives and gradients
- Batch gradient descent optimization
- Python
- NumPy for numerical computations
- Matplotlib for data visualization
- Jupyter Notebook as the development environment
- Designed an end-to-end regression pipeline from raw data generation to model evaluation
- Implemented gradient descent optimization manually without external ML libraries
- Built visual diagnostics to evaluate loss trends and prediction quality
- Demonstrated strong understanding of model interpretability and performance analysis
This project demonstrates how predictive models can be built from scratch to understand relationships between numeric variables. The techniques implemented here form the foundation for:
- Sales forecasting
- Trend analysis
- Cost estimation
- Performance prediction
- Business decision support
- Loss decreases steadily as the model learns underlying patterns in the data
- Proper gradient updates are essential for model convergence
- Model predictions significantly improve after training
- Visualization is critical for interpreting model performance