15 short videos in the Essence of linear algebra series by 3Blue1Brown on YouTube. First video is Vectors, what even are they?

Here is a simplified PCA code that we used on 6 March 2020: pca_code_part1.R (here’s the original code pca_code.R)

Here is code from Bryan Shader that illustrates how R does PCA: SVAPCA2.R

… and an announcement … for the time-being we’re going to suspend our meetings due to the anticipated spread of COVID19. So there will be no meeting on 13 March 2020, nor during spring break (20 March). We will see from there and maybe we’ll get together via Zoom or something else thereafter.

Let’s follow up on this thread on Twitter, posted by a population genomicist at Ancestry.com

Learning objectives

• Cholesky decomposition in context of non-centered parameterizations of multivariate hierarchical Bayesian models

  • Specifically the multivariate reparameterizations section of this page in the Stan documentation: https://mc-stan.org/docs/2_22/stan-users-guide/reparameterization-section.html

• Random matrix theory

Linear algebra resources and links of interest

Online courses

Linear algebra

• Khan Academy: Linear Algebra

  • Not really a stand alone linear algebra class (more as a supplement), but contains lots of good information.

• Open Courseware: Linear Algebra

  • MIT’s introductory linear algebra course.

• Open Courseware: Matrix Methods in Data Analysis, Signal Processing, and Machine Learning

  • The follow-on course to the introduction linear algebra class. Matrix methods expands on the fundamental concepts discussed and introduces some additional ideas that are also very useful, such as positive-definite matrices, singular value decomposition, etc.

• edx.org: Linear Algebra Courses

  • Free (or paid, if you want a certificate), self-paced linear algebra courses from a number of institutions.

Singular Value Decomposition (SVD)

SVD is a fundamental factorization technique that is used a in wide variety of situations and fields that vary from statistics (e.g., principal component analysis) to fluid flow (e.g., dynamical systems).

• Steve Brunton

  • Singular Value Decomposition YouTube playlist

    • Book associated with lectures: Data Driven Science & Engineering: Machine Learning, Dynamical Systems, and Control

    • Book website

  • Notes

    • I (Jason) have been really enjoying this playlist. Great explanations of SVD with coding examples (Matlab and Python) and applications (e.g., image compression, fast approximations of fluid flow, PCA).

    • What may be of particular interest to some in this group is discussion of Robust Principal Component Analysis (RPCA), which is a means of dealing with matrices that are missing lots of data.

Principal Component Analysis (PCA)

A specific application of SVD focused on finding the rotation of a matrix that maximizes the most variation along a set of axes (columns) that have been standardized.

• MIT’s Open Courseware

  • Statistics for Applications (18.650/18.6501)

    • YouTube playlist

    • Principal Component Analysis

    • Principal Component Analysis (Lecture 19)

    • Principal Component Analysis (cont.)(Lecture 20)

Videos highlighting applications of linear algebra

In addition to the 3Blue1Brown series, there are other videos and series out there that may be useful in illustrating the applications of linear algebra. Below are a couple of examples.

• Zach Star

  • Dear linear algebra students, This is what matrices (and matrix manipulation) really look like

  • The Applications of Matrices | What I wish my teachers told me way earlier

PCA

• Luis Serrano

  • Principal Component Analysis (PCA)

    • One of the better explanations of PCA I’ve seen.

• StatQuest with Josh Starmer (Genetics Dept., University of North Carolina at Chapel Hill)

  • Machine Learning playlist

    • PCA main ideas in only 5 minutes!!!

    • Principal Component Analysis (PCA) clearly explained (2015)

    • Principal Component Analysis (PCA), Step-by-Step

    • PCA - Practical Tips

      • Nice explanations of (among other things):

        • Scaling: so as not to artificially weight data measured at different scales.

        • Centering: SVD expects data at the origin, so PCA expects data to be centered at the origin.

    • PCA in R

      • Code used in the video

• STHDA

  • Principal Component Methods in R: Practical Guide

• Data4Bio

  • Data for Biologists

    • Dimensionality Reduction: High Dimensional Data, Part 1

• Julia Silge

  • Understanding PCA using Stack Overflow data

• Code Emporium

  • Principal Component Analysis (PCA) - The Math You Should Know!

    • A deep dive into the mathematical justification of PCA.