With all the excitement and energy around AI, it’s easy to lose sight of the foundational mathematics and technology that make these models possible.

As professionals in the data and AI fields, one of the most rewarding ways to deepen your understanding is to code models from the ground up — without using machine learning libraries like scikit-learn, TensorFlow, or PyTorch.

This inspired the creation of the DIY AI Series, where we build machine learning models using pure Python and math. In this installment, we tackle Multiple Linear Regression — a fundamental algorithm used in countless real-world scenarios.

Multiple Linear Regression models the relationship between two or more independent variables and a continuous dependent variable.

Predicting house prices using:

• Number of bedrooms
• Number of bathrooms
• Square footage
• Neighborhood score

1. Linearity
   The relationship between features and target is linear. A one-unit change in a feature leads to a constant change in the outcome.

2. No Multicollinearity
   Features should not be highly correlated with each other (e.g., bedrooms vs. bathrooms). Redundant variables distort coefficient estimation.

3. Homoscedasticity
   Constant variance of residuals across all levels of independent variables. If prediction errors increase with the target value, this assumption is violated.

You may remember the simple linear equation:

[ y = mx + b ]

Multiple Linear Regression generalizes this to:

[ y = B₀ + B₁x₁ + B₂x₂ + ... + Bₙxₙ + \epsilon ]

Where:

• y: target variable
• x₁...xₙ: independent variables (features)
• B₀: intercept (bias term)
• B₁...Bₙ: coefficients (slopes)
• ϵ: error term

We can rewrite it as:

[ \mathbf{y} = \mathbf{X} \boldsymbol{\beta} + \boldsymbol{\varepsilon} ]

To solve for coefficients ( \boldsymbol{\beta} ), we use the Normal Equation:

[ \boldsymbol{\beta} = (\mathbf{X}^T \mathbf{X})^{-1} \mathbf{X}^T \mathbf{y} ]

This gives the optimal linear solution using linear algebra.

1. Prepare the Design Matrix
   Add a column of 1s to the input features to account for the intercept term.

2. Apply the Normal Equation
   Use matrix multiplication to compute coefficients.

3. Predict Target Values
   [ \hat{y} = \mathbf{X} \boldsymbol{\beta} ]

4. Optional: Evaluate
   Calculate R² score to assess model performance.

• Python 3
• NumPy

# Clone the repo
git clone https://github.com/Simacoder/52_Weeks_Challenges.git
cd 52_Weeks_Challenges/week_15

# Install numpy
pip install numpy

# Run the model
python multiple_linear_regression.py

• Simanga Mchunu