Quadratic Forms | Degrees of Freedom Ch. 3

Quadratic forms are polynomials with terms of degree two. We'll explore what they are in general, how to represent them with matrix multiplication, and how you can express vector lengths as quadratic forms, and how certain statistics like the sample variance can be represented using quadratic forms. This is working towards the definition of "degrees of freedom" we gave in chapter 1 of this series, as "the rank of the projection matrix in the quadratic form of a statistic."

This is Chapter 3 in a series on Degrees of Freedom, or, The Geometry of Statistics, which is trying to rigorously but intuitively explain what is easily the most confusing concept in statistics, Degrees of Freedom. Check out the other videos here:    • Degrees of Freedom  

Chapters:
0:00 Introduction
0:30 Review
1:12 Preview
2:02 Quadratic Form definition
2:34 Graphs of Quadratic Forms
6:30 Review of matrix multiplication
10:33 Representing Quadratic Forms with matrices
12:53 Cross-term matrix entries are arbitrary
13:50 Symmetric matrices in Quadratic Forms
14:26 Quadratic Forms with more than 2 variables
15:04 Review relevant vectors
15:58 Quadratic Forms of these vectors
17:13 Sample Variance Quadratic Form, the hard way
18:24 Graph of Sample Variance Quadratic Form for n=2
19:20 Finding the more general Sample Variance Quadratic Form
19:53 Squared length of the Data vector
20:40 Squared length of the Sample Mean vector
24:20 Graph of the Sample Mean vector squared length
24:48 Graphs of all 3 vector squared lengths combined
25:05 Review and Conclusion


Further Reading/Viewing:
The Essence of Linear Algebra, by 3Blue1Brown:    • Essence of linear algebra  

Quadratic Forms, on Wikipedia: https://en.wikipedia.org/wiki/Quadrat...

Saville, David J., and Graham R. Wood. Statistical Methods: A Geometric Primer. New York, NY: Springer New York, 1996. https://doi.org/10.1007/978-1-4612-07....

Wickens, Thomas D. The Geometry of Multivariate Statistics. Hillsdale, N.J: L. Erlbaum Associates, 1995.

Saville, David J., and Graham R. Wood. Statistical Methods: The Geometric Approach. Corr. 3rd print. Springer Texts in Statistics. New York: Springer, 1997.


Attributions:
'Sunday Smooth' by Scott Buckley - released under CC-BY 4.0. www.scottbuckley.com.au

Other music is from the YouTube Audio Library, by artists Alex Hamlin, E's Jammy Jams, Chris Haugen, and Silent Partner.

Made with Manim: https://www.manim.community/. The source code will be posted at the conclusion of the series.



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