Locally Weighted & Logistic Regression | Stanford CS229: Machine Learning - Lecture 3 (Autumn 2018)

Introduction

This tutorial provides a comprehensive overview of Locally Weighted and Logistic Regression, as presented in the Stanford CS229 Machine Learning course by Andrew Ng. The concepts discussed are fundamental in supervised learning, making this guide relevant for anyone looking to deepen their understanding of regression techniques in machine learning.

Step 1: Recap of Linear Regression

• Understand the basic principles of linear regression:
  • It models the relationship between a dependent variable and one or more independent variables.
  • The objective is to minimize the difference between predicted and actual values using a cost function (often the mean squared error).
• Familiarize yourself with the equation of linear regression:
  • y = β0 + β1x1 + β2x2 + ... + βnxn
  • Where y is the predicted value, β0 is the intercept, and β1, β2,..., βn are the coefficients corresponding to each feature x1, x2,..., xn.

Step 2: Understanding Locally Weighted Regression

• Locally weighted regression (LWR) is a non-parametric technique:
  • It predicts values using a weighted linear regression where nearby points have a greater influence on the prediction.
• Key features of LWR:
  • The weights are determined based on a distance metric (usually Euclidean distance).
  • It allows for better fitting of data that may not be globally linear.
• Implementation steps for LWR:
  1. Choose a query point x.
  2. Compute the weights for each training example based on their distance from x.
  3. Fit a linear regression model using these weights.
  4. Make predictions based on the fitted model.

Step 3: Exploring Probabilistic Interpretation

• Linear regression can be interpreted probabilistically:
  • Assume that the outputs (y) are generated from a linear function plus some normally distributed noise.
• This leads to understanding the likelihood function and how it can be maximized to find the best parameters for the model.

Step 4: Introduction to Logistic Regression

• Logistic regression is used for binary classification problems:
  • It predicts the probability of a binary outcome based on one or more predictor variables.
• Understand the logistic function:
  • It maps any real-valued number into a value between 0 and 1:
  • P(y=1|X) = 1 / (1 + e^(-z)), where z = β0 + β1x1 + ... + βnxn.
• Key points:
  • The output of the logistic function can be interpreted as a probability, which is useful for classification tasks.

Step 5: Implementing Newton's Method

• Newton's method is an optimization technique used to find the maximum likelihood estimates in logistic regression:
  • It iteratively updates the parameters by moving towards the maximum of the likelihood function.
• Steps to apply Newton’s method:
  1. Initialize the parameters (β).
  2. Calculate the gradient and Hessian matrix of the likelihood function.
  3. Update the parameters using the formula:
     • β_new = β_old - (Hessian^-1 * Gradient)
  4. Repeat until convergence (i.e., when updates become negligible).

Conclusion

In this tutorial, we covered key concepts such as linear regression, locally weighted regression, and logistic regression, along with the application of Newton's method for optimization. These techniques are crucial for understanding and implementing supervised learning algorithms effectively. For next steps, consider experimenting with these regression techniques on your own datasets to solidify your understanding.