Ordinary Least Squares in Python

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Introduction

In this project, you will learn how to implement the Ordinary Least Squares (OLS) method in Python. OLS is a fundamental mathematical optimization technique used in machine learning, particularly in linear regression problems.

ðŸŽŊ Tasks

In this project, you will learn:

  • How to implement the OLS function to calculate the slope (w1) and intercept (w0) of a linear equation based on sample data.
  • How to test the OLS function with sample data and verify the correctness of the results.

🏆 Achievements

After completing this project, you will be able to:

  • Derive the OLS formulas for calculating the slope and intercept of a linear equation.
  • Implement the OLS function in Python without using any external libraries.
  • Test and validate the OLS function with sample data.
  • Understand the importance of the OLS method in machine learning and linear regression problems.

Skills Graph

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Implement the Least Squares Function

In this step, you will implement the Ordinary Least Squares (OLS) function in Python. Follow the steps below to complete this step:

  1. Open the least_squares.py file in your text editor.

  2. Implement the least_squares_function function according to the provided specifications:

    • The function should take two input parameters: x (a list of sample x values) and y (a list of sample y values).
    • The function should calculate the slope (w1) and intercept (w0) of the linear equation using the OLS formulas provided in the challenge description.
    • The function should round the calculated w0 and w1 values to two decimal places before returning them.
    • The function should return the w0 and w1 values in that order.

Here's the completed least_squares_function implementation:

def least_squares_function(x, y):
    """
    Parameters:
    x -- list of values of sample x
    y -- list of values of sample y

    Returns:
    w0 -- linear equation parameter, rounded to two decimal places
    w1 -- linear equation parameter, rounded to two decimal places
    """

    n = len(x)

    ## Calculate the sums needed for the slope (w1) and intercept (w0)
    sum_x = sum(x)
    sum_y = sum(y)
    sum_xy = sum(x_i * y_i for x_i, y_i in zip(x, y))
    sum_x_squared = sum(x_i**2 for x_i in x)

    ## Calculate the slope (w1) and intercept (w0) using the OLS formulas
    w1 = (n * sum_xy - sum_x * sum_y) / (n * sum_x_squared - sum_x**2)
    w0 = (sum_y - w1 * sum_x) / n

    ## Round to two decimal places
    w0 = round(w0, 2)
    w1 = round(w1, 2)

    return w0, w1
  1. Save the least_squares.py file.
âœĻ Check Solution and Practice

Test the Least Squares Function

In this step, you will test the least_squares_function you implemented in the previous step.

  1. You can see the following code at the end of the file to test the least_squares_function:
## Example usage
if __name__ == "__main__":
    x_example = [1, 2, 3, 4]
    y_example = [4, 5, 6, 7]

    result = least_squares_function(x_example, y_example)
    print("w0:", result[0])
    print("w1:", result[1])
  1. Run the least_squares.py script in your terminal:
python3 least_squares.py

You should see the following output:

w0: 3.0
w1: 1.0

This output confirms that the least_squares_function is working as expected.

Congratulations! You have successfully implemented the Ordinary Least Squares function in Python. In the next steps, you can explore how to use this function in a real-world machine learning problem.

âœĻ Check Solution and Practice

Summary

Congratulations! You have completed this project. You can practice more labs in LabEx to improve your skills.

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