ACCT 331 LECTURE 14

Transcript Notes: Linear Regression and Polynomial Regression in Machine Learning

Introduction to the Session

• The session discusses various aspects of linear regression and polynomial regression as part of machine learning techniques.

• Announcement: 03/26/2026, Parkinson's school will host a full day of workshops in AI, looking for student volunteers for assistance.

Current Events and AI Trends

• McKinsey's AI shakeup: Replaced 5,000 consultants with 12,000 positions geared towards AI roles.

• Walmart plans to adjust workforce by integrating AI roles, citing a major global transformation.

• JPMorgan aims to provide AI agents for every employee and automate backend processes for enhanced client experiences.

Overview of Linear Regression

• Linear regression defined as the most crucial algorithm in machine learning.

• Not the most sophisticated algorithm, but central to understanding relationships in datasets.

Definition of Linear Regression

• Purpose: To quantify the relationship between a dependent variable ($y$, the target) and independent variables ($x1, x2, … x_n$, the features).

• The relationship can be formulated as: y = f(x) + e Where,

  • $y$ is the dependent variable.

  • $e$ represents the error term.

Application and Example of Linear Regression

• Example: Predicting the price of a home based on features such as the number of bedrooms and square footage.

• Each feature has a significant impact on the predicted outcome (home price).

Key Concepts

• Labeled Data: Essential for machine learning; involves having both input variables ($x$) and target variable ($y$) annotated for learning.

• Prediction Function: An equation that helps in predicting values based on input features.

• Coefficients: Parameters of the model that are estimated from the training data, crucial for making predictions.

Linear Regression Equation

• The general equation is of the form: y{hat} = eta0 + eta1x1 + eta2x2 + … + etanxn Where,

  • $y_{hat}$ = predicted value.

  • $eta0$ = intercept, $etan$ = coefficients for each feature.

Assessment of Fit

• Best fit line is determined using the Least Squares Approach:

  • The method minimizes the sum of the squared errors (residuals).

  • Residual = actual value - predicted value.

  • Residual Sum of Squares (RSS) is calculated as:
    RSS = ext{Sum of } (yi - y{hat})^2

Simple vs. Multiple Linear Regression

• Simple Linear Regression involves one feature ($x$) to predict $y$.

• Multiple Linear Regression involves multiple features ($x1, x2, … x_n$) affecting $y$.

• Example of multiple linear regression can be predicting compensation based on education and job experience.

Visual Representation of Linear Regression

• Data points plotted on a graph with a line fitted through them to represent best fit, aiding in predicting new values based on previous data.

Polynomial Regression

• Polynomial regression is an extension of linear regression, accommodating non-linear relationships by introducing polynomial terms.

Transition to Polynomial Regression

• While linear regression predicts a straight line, polynomial regression can fit curves to data by raising features to a power.

• New features introduced in polynomial regression allow for flexibility in modeling:

  • For example, if $x$ is raised to the power n, it introduces additional predictors.

Formula in Polynomial Regression

• General formula format for polynomial regression will involve transformations of $x$:
  y = eta0 + eta1x + eta2x^2 + … + etanx^n

• Each term corresponding to the given power of $x$ helps create a curve that better fits the data.

Importance and Performance Evaluation

• Performance of polynomial regression is evaluated through error calculations similar to linear regression, using terms like Mean Squared Error to assess model accuracy.

• Evaluating outcomes helps in determining if the model is underfitting or overfitting the data.

Conclusion

• Summary of linear regression, polynomial regression, their differences, and the process of creating effective models for predictive analytics in machine learning.

Introduction to Linear and Polynomial Regression

• This session outlines fundamental concepts of linear and polynomial regression, crucial machine learning algorithms for understanding and modeling relationships in data.

Overview of Linear Regression

• Linear regression is a foundational algorithm in machine learning, though not the most complex, it is essential for quantifying relationships between variables.

Definition and Purpose

• Purpose: To quantify the linear relationship between a dependent variable (y), which is the target we want to predict, and one or more independent variables (x1, x2, \dots, x_n), which are the features.

• The relationship is modeled as: y = f(x) + e

  • y: The dependent variable.

  • e: The error term, representing irreducible error or noise not explained by the model.

Key Concepts for Mastery

• Labeled Data: Essential for training, meaning datasets must include both input features (x) and their corresponding known target values (y).

• Prediction Function: An equation, also known as the hypothesis function (h(x)), used to predict y values based on given input features.

• Coefficients (Parameters): These are the weights (\eta values) estimated from the training data that define the slope and intercept of the regression line, dictating the impact of each feature on the prediction.

The Linear Regression Equation

• The general form for the predicted value (y_{hat}) is: y_{hat} = \eta0 + \eta1x1 + \eta2x2 + \dots + \etanx*n

  • y_{hat}: The predicted value of the dependent variable.

  • \eta*0: The intercept, representing the predicted value of y when all independent variables are zero.

  • \etan: The coefficients for each feature (xn), indicating the change in y_{hat} for a one-unit change in x*n, assuming all other features are constant.

Assessment of Model Fit: The Least Squares Approach

• The Least Squares Approach is the standard method for determining the "best-fit" line in linear regression.

  • Objective: To minimize the Residual Sum of Squares (RSS).

  • Residual: The difference between the actual observed value (yi) and the predicted value (y_{hat}) for a given data point (yi - y_{hat}).

  • RSS Calculation: The sum of the squared residuals across all data points:
    RSS = \text{Sum of } (y_i - y_{hat})^2

  • Minimizing RSS helps ensure that positive and negative errors don't cancel out, giving larger penalties to larger errors, thus finding the line that best approximates the overall trend.

Types of Linear Regression

• Simple Linear Regression: Involves a single feature (x) to predict the target variable (y).

• Multiple Linear Regression: Utilizes multiple features (x1, x2, \dots, x_n) to predict y, allowing for more complex relationships (e.g., predicting compensation based on education and job experience).

Polynomial Regression

• Polynomial regression extends linear regression to model non-linear relationships between variables.

Transition and Concept

• Unlike linear regression which fits a straight line, polynomial regression fits a curve to the data.

• It achieves this by introducing polynomial terms of the original features (e.g., x^2, x^3) as new independent variables in the model.

• This transformation allows the model to capture more complex, curved patterns in the data without changing the linear nature of the model in terms of its coefficients.

Formula in Polynomial Regression

• The general format involves powers of x: y = \eta0 + \eta1x + \eta2x^2 + \dots + \etanx^n

  • Each term (\eta*kx^k) contributes to creating a curve that can better fit the non-linear data patterns than a simple straight line.

Importance and Performance Evaluation

• The performance of polynomial regression, like linear regression, is critically evaluated using error metrics such as Mean Squared Error (MSE).

• Proper evaluation helps identify if the model is underfitting (too simple to capture the underlying patterns) or overfitting (too complex, capturing noise in the training data rather than the true relationship, leading to poor generalization).

Conclusion

• Both linear and polynomial regression are vital tools in machine learning, offering distinct approaches to modeling data relationships. Understanding their equations, assessment methods, and applications is fundamental for effective predictive analytics and building robust models.