ft_linear_regression

Overview

The project trains a linear regression model to predict car prices based on mileage using a dataset of mileage-price pairs. The dataset is stored in data.csv.

The project includes:

• Training Program: Trains the model and saves parameters for future use.
• Prediction Program: Predicts car prices for given mileage values.
• Precision Evaluation Program: Calculates the precision of the model by comparing predicted prices with actual prices.

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Gradient Descent: Simultaneous Updates

theta1

• The slope (theta1) represents the change in price per km.
• It is normal for theta1 to be a small value if the scale of the input feature (mileage) is much larger than the scale of the target variable.

For example, if theta1 = −0.021, it means: For every 1 km increase, the price decreases by 0.021 currency units.
Over a larger range (e.g., 10,000 km), the effect becomes noticeable: Price decrease = 10,000⋅(−0.021) = −210 currency units.

tmpθ1 = learningRate * (1 / m) * Σ (estimatePrice(mileage[i])-price[i])* mileage[i]

theta0

• The intercept (theta0) represents the base price when mileage is 0.

For example, if theta0 = 8499.53, it is the price when the car is new.

tmpθ0 = learningRate * (1 / m) * Σ(estimatePrice(mileage[i])- price[i])

The parameters are then adjusted:

theta0 -= temp0
theta1 -= temp1

The loop continues until the parameter adjustments are smaller than a defined threshold (1e-3)

if abs(temp0) < epsilon and abs(temp1) < epsilon:
    break

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Why normalization is needed:

• Without normalization, features with large values (mileage) dominate features with smaller values (price).
• Gradient descent may take longer to converge or might not converge at all because the cost function’s shape becomes highly skewed.

Z-Score Normalization Formula:

z = (x - mean) / std

Where:

• z: The normalized value.
• x: The original mileage value.
• mean: The average of all mileage values.
• std: The standard deviation of the mileage values

Reverting Normalization

• After training, the parameters (theta0, theta1) derived from normalized data must be reverted to the original scale for predictions to make sense. This ensures:

1. The slope is scaled correctly for the original mileage range.
2. The intercept with the actual price values.

theta0 = estimated_intercept−(estimated_slope⋅std_mileagemean_mileage)
theta0 = std_mileage/estimated_slope

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Precision Algorithm

• The precision of the algorithm is evaluated using the Mean Absolute Error (MAE) in percentage.
• This is calculated by comparing the predicted prices with the actual prices from the dataset:

total_error = sum(abs((p - a) / a) for p, a in zip(predicted_p, actual_p))
mae_percentage = total_error / len(actual_p) * 100

• Smaller MAE Percentage: Indicates the model is better at predicting prices.
• Larger MAE Percentage: Suggests the model is less precise, potentially due to underfitting or overfitting.

Setup

Install requirements

This project requires Python and the following libraries:

• numpy
• pandas
• matplotlib

Run the program

python3 training.py: Trains the regression model and plots the results in a graph.

python3 training.py -d: Displays a graph of the distribution of the data (no training).

python3 predict.py: Predicts car prices based on user-input mileage.

python3 precision.py: Calculates and displays the precision of the regression model.