Accuracy
Regularization
Efficiency
5/30/2015
Topics
Accuracy
Assume a is "small"
Least squares
Ths SVD lets us solve least squares problems using the direct approach
Doornik's VECM Example
Naive Johansen method can exhibit complete loss of accuracy when \(\delta \approx 10^{-6}\).
Regularization
Subspace projection
Reduce influence of noise by projecting into a "less-noisy" subspace.
SVD is one way to define such subspaces.
Example: exact system
Example: perturbed system
Wild exact solution
Chill projected solution
Thresolded precision matrix
(If you can't see a plot, try reloading the page and it should show up.)
Regularized
Ill-posed problems
P.C. Hansen's PPTSVD
Efficiency
Projection methods turn big data into data
IRLBA directed network centrality
Projected LMM
Turns large \(n\times n\) LMM into a series of easy \(n\times 1\) LMMs.
References
- Fast truncated SVD http://cran.r-project.org/web/packages/irlba/
- In-development version (SVD + symmetric eigenproblems + network problems + more) https://github.com/bwlewis/IRL
- Mike Kane's and my notes on stable solution of GLM http://bwlewis.github.io/GLM/
- Covariance shrinkage example https://bwlewis.github.io/covariance-shrinkage
- Network D3.js plot widget for R https://github.com/bwlewis/networkD3
- The PP-TSVD algorithm http://www.scicomp.dk/static/docs/hansen00-pptsvd.pdf
- Fast projected linear mixed models http://research.microsoft.com/en-us/um/redmond/projects/MSCompBio/Fastlmm