Linear Regression from Scratch (Python + NumPy + Pandas)

Introduction

Linear Regression is a fundamental machine learning algorithm used to predict continuous values.

It models the relationship between a target variable and one or more features by fitting a straight line (or hyperplane):

ŷ = w₁ x₁ + w₂ x₂ + … + wₙ xₙ

where:

• ŷ = predicted output
• xᵢ = input features
• wᵢ = learned weights

This simple version does not include a bias/intercept to focus on the core gradient descent logic.

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Gradient Descent (Training Logic)

We train the model by minimizing Mean Squared Error (MSE):

MSE = (1 / m) Σ₍ᵢ₌₁₎ᵐ (yᵢ - ŷᵢ)²

Step 1 — Find the Gradient

∂MSE / ∂w = (2 / n) · Xᵀ · (X w - y)

This tells how to change weights w to reduce the error.

Step 2 — Update the Weights

w = w - η * gradient

where η is the learning rate.

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Project Architecture (Pipeline)

Data Flow of the Program: house_price.csv

    ↓

Load & Clean

(drop non-numeric, handle NaN, convert dtypes)

    ↓

Train-Test Split

(train_test_split_data)

    ↓

Normalize Features

(normalize_features)

    ↓

Gradient Descent Training

(linear_regression_train → gradient_descent)

    ↓

Predict on Test Data

(linear_regression_predict)

    ↓

Evaluate with MSE

(mean_squared_error)

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How the Program Works

1. Data Preparation

• Load CSV using pd.read_csv()
• Drop non-numeric columns and the Id column
• Drop missing values
• Convert specific columns to integer type

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2. Splitting the Dataset

• train_test_split_data(df, test_size, random_seed)
  • Splits data into train (80%) and test (20%)
  • Returns: X_train, X_test, y_train, y_test

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3. Normalization

• normalize_features(features)
  • Applies Z-score normalization:

x' = (x - μ) / σ

where μ is the mean and σ is the standard deviation.

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4. Model Training

• linear_regression_train(X_train, y_train, learning_rate, epochs)
  • Normalizes data
  • Calls gradient_descent(X, y, learning_rate, epochs)

Core gradient descent update (pseudo-code):

weights = np.zeros(num_features)

for epoch in range(epochs):
    predictions = np.dot(X, weights)
    errors = predictions - y
    gradient = 2 * np.dot(X.T, errors) / len(y)
    weights -= learning_rate * gradient

5. Prediction

• linear_regression_predict(weights, X_test)

  • Normalizes X_test

  • Predicts using:

  • ŷ = X · w

6. Evaluation

• mean_squared_error(predictions, targets)

  • Calculates average squared difference between predictions and actual targets

How to Run

python linear_regression.py

It will:

• Load and clean dataset
• Train a linear regression model
• Predict on test data
• Print final Mean Squared Error (MSE)

Notes

• No bias term included
• Features and targets are normalized separately
• For learning/demo only — not production
• Learning rate and epochs may need tuning

Concepts Covered

• Data Cleaning & Preprocessing
• Train-Test Splitting
• Z-score Normalization (x' = (x - μ) / σ)
• Gradient Descent Optimization (∂MSE/∂w = (2/n) Xᵀ (Xw - y))
• Linear Regression
• Model Evaluation (MSE)