2. Least squares: geometric, algebraic, and numerical aspects#
In this chapter, we introduce the least squares problem and develop the mathematical concepts at its foundation.
Main objectives:
To introduce the various mathematical facets of the least squares problem: the algebra, the geometry, the numerics.
To discuss a key application: regression analysis.
Here is an overview of the chapter courtesy of ChatGPT.
Section 2.1 - Motivating example: predicting sales This section uses a motivating example of predicting product sales based on advertising budgets in different media (TV, radio, newspaper) to introduce regression analysis, demonstrating how statistical models can help businesses allocate advertising spending more effectively to increase sales.
Section 2.2 - Background: review of vector spaces and matrix inverses The section provides a comprehensive review of fundamental linear algebra concepts such as vector spaces, subspaces, linear independence, orthogonality, and matrix inverses, which are essential for understanding and solving problems in data science.
Section 2.3 - The geometry of least squares The section explores the geometric interpretation of least squares, focusing on orthogonal projection and its applications to solving overdetermined systems, thereby providing a deeper understanding of the least squares solution in linear regression.
Section 2.4 - QR decomposition and Householder transformations The section explains the QR decomposition and Householder transformations in the context of linear algebra, detailing their application in obtaining an orthonormal basis and their significance in solving linear least squares problems with improved numerical stability compared to other methods like the Gram-Schmidt process.
Section 2.5 - Application to regression analysis The section outlines the application of regression analysis techniques, specifically focusing on linear and polynomial regression, addressing issues such as overfitting, and introducing bootstrapping methods for assessing the reliability and variability of regression coefficients in predictive models.
MINUTE PAPER: Coming back to these section summaries after going through this chapter, write your own more detailed summaries. What were the most significant, interesting, insightful, surprising, or challenging concepts? \(\ddagger\)
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