1. SVD, PCA, AND THE NFL
By: Andrew Zachary

2. Introduction What is PCA? What is SVD? What is the NFL?
Principal Component Analysis
What is SVD?
Singular Value Decomposition
What is the NFL?
National Football League

3. SVD
𝐴=𝑼𝚺 𝐕 ∗

4. Computing the SVD Matrices
Consider, 𝐴 𝐴 𝑇 = 𝑈Σ 𝑉 ∗ 𝑈Σ 𝑉 ∗ 𝑇 = 𝑈Σ 𝑉 ∗ 𝑉Σ 𝑈 ∗ =𝑈 Σ 2 𝑈 ∗ 𝐴 𝐴 𝑇 𝑈=𝑈 Σ 2
And,
𝐴 𝑇 𝐴= 𝑈Σ 𝑉 ∗ 𝑇 𝑈Σ 𝑉 ∗
=𝑉Σ 𝑈 ∗ 𝑈Σ 𝑉 ∗
=𝑉 Σ 2 𝑉 ∗
𝐴 𝑇 𝐴𝑉=𝑉 Σ 2

5. Eigenvalue Problem
𝑈=𝜆 Σ= 𝜎 𝜎 2 Where 𝜆 is the eigenvectors and 𝜎 1 and 𝜎 2 are the eigenvalues of 𝐴 𝐴 𝑇 𝑈=𝑈 Σ 2 Similarly, 𝑉=𝜆 Where 𝜆 is the eigenvectors and 𝜎 1 and 𝜎 2 are the eigenvalues of 𝐴 𝑇 𝐴𝑉 =𝑉 Σ 2

6. Covariance Matrix
𝐶 𝑋 = 1 𝑛−1 𝑋 𝑋 𝑇 Where X is a matrix of data and n is the number of data points. Elements from the covariance matrix Diagonals → Variance (dynamics) Off-diagonals → Covariance (redundancy)

7. PCA
Ideal basis
No Redundancy
Diagonalization

8. NFL NFL Ranking Fantasy Ranking PCA Ranking Single stat
Several Stats (Position Dependent)
PCA Ranking
All available NFL stats

9. NFL Correlation Matrix

10. Correlation vs. Covariance
𝜌 𝐴,𝐵 = 1 𝑁−1 𝑖=1 𝑁 𝐴 𝑖 − 𝜇 𝐴 𝜎 𝐴 𝐵 𝑖 − 𝜇 𝐵 𝜎 𝐵 , 𝑃𝑒𝑎𝑟𝑠𝑜𝑛 𝐶𝑜𝑒𝑓. 𝜌 𝐴,𝐵 = 𝑐𝑜𝑣 𝐴,𝐵 𝜎 𝐴 𝜎 𝐵 𝑅= 𝜌(𝐴,𝐴) 𝜌(𝐴,𝐵) 𝜌(𝐵,𝐴) 𝜌(𝐵,𝐵)

11. Singular Values
PCA Projection:
𝑋 𝑝 =𝑈 Σ 𝑟𝑒𝑑𝑢𝑐𝑒𝑑

12. Results

13. Conclusions PCA is a powerful data analysis tool
PCA is the ideal basis for a dataset
SVD and Covariance Matrix equal PCA
PCA has strong applications in sports

14. References
“Data-Driven Modeling and Scientific Computation, Methods for Complex Systems & Big Data”, J. Nathan Kutz, 2013
“An Introductory application of Principal Components to Cricket Data”, Ananda B. W. Manage et. al., 2013