Lecture Notes on the Mathematics of Regression

Introduction to Regression

• Discussing the mathematical intuition behind regression problems.
• Previous examples covered simple and multiple linear regression.
• Focus of this lecture: detailed explanation of regression mathematics.

Linear Regression Basics

• Equation of Linear Regression:
  [ y = mx + c ]
  • m: Slope of the line
  • c: Intercept of the line

Example of Regression Application

• Problem statement: Predicting house prices based on size.
• Data points represent size vs. price.
• Goal: Create a best fit line to predict future prices for given sizes.

Understanding the Intercept and Slope

• Intercept (c):
  When size (X) is 0, price (Y) equals c.

  • This point indicates the starting price when the size is zero.

• Slope (m):
  Indicates how much Y (price) changes for one unit change in X (size).

  • Ex: If size increases by 100 sq. ft, how does price change?

Finding the Best Fit Line

• Multiple lines can be drawn; need to minimize errors between actual points and predicted points.
• Cost Function:
  [ J(m, c) = \frac{1}{2m} \sum_{i=1}^{m} (\hat{y} - y)^2 ]
  • ( \hat{y} ): Predicted value
  • ( y ): Actual value
• Objective: Minimize the cost function to find optimal m and c.

Gradient Descent Overview

• Gradient Descent: A method to find the minimum cost function.
• Plotting cost function against slope values (m) shows how cost changes with different m values.
• The goal is to find the m that results in the lowest cost.

Convergence Theorem

• The theorem aids in reaching the global minimum:
  [ m = m - \alpha \frac{dJ}{dm} ]
  • ( \alpha ): Learning rate
  • Process iteratively adjusts m to reduce cost function.

Implementation Steps

• Choose initial m value and calculate cost function.
• Adjust m using gradient descent based on the slope of the cost function.
• Continue until the slope is 0 (indicating optimal m).

Multiple Features in Regression

• If multiple independent features are present, gradient descent will extend to higher dimensions (3D or 4D).

Conclusion

• Understanding the cost function and gradient descent is crucial for implementing linear regression.
• Encouraged to review the video for deeper understanding and practical implementation links will be provided.
• Thank you for attending the lecture!