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An ExPosition of Bootstrap and Permutation tests for Principal Components Analyses

An ExPosition of Bootstrap and Permutation tests for Principal Components Analyses. Derek Beaton Joseph Dunlop Hervé Abdi. An ExPosition of Bootstrap and Permutation tests for Principal Components Analyses. Derek Beaton Joseph Dunlop Hervé Abdi. Kinds of Data.

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An ExPosition of Bootstrap and Permutation tests for Principal Components Analyses

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  1. An ExPosition of Bootstrap and Permutation tests for Principal Components Analyses Derek Beaton Joseph Dunlop HervéAbdi

  2. An ExPositionof Bootstrap and Permutation tests for Principal Components Analyses Derek Beaton Joseph Dunlop HervéAbdi

  3. Kinds of Data

  4. An ExPosition of Bootstrap and Permutation tests for Principal Components Analyses Derek Beaton Joseph Dunlop HervéAbdi

  5. An ExPosition of Bootstrap and Permutation tests for Principal Components Analyses Derek Beaton Joseph Dunlop HervéAbdi

  6. An ExPosition of Bootstrap and Permutation tests for Principal Components Analyses Derek Beaton Joseph Dunlop HervéAbdi

  7. An ExPosition of Bootstrap and Permutation tests for Principal Components Analyses Derek Beaton Joseph Dunlop HervéAbdi Daniel Faso

  8. Outline • We have a lot to talk about! • Principal Components Analysis (PCA) • Multiple Correspondence Analysis (MCA) • Bootstrap • Permutation

  9. The SVD • We have a lot to talk about! • Principal Components Analysis (PCA) • Multiple Correspondence Analysis (MCA) • Bootstrap • Permutation

  10. Resampling • We have a lot to talk about! • Principal Components Analysis (PCA) • Multiple Correspondence Analysis (MCA) • Bootstrap • Permutation

  11. An ExPosition of • The SVD • Resampling

  12. An ExPosition of • The SVD • Resampling

  13. The SVD • Root of all evil most multivariate techniques • Is just an eigendecomposition* • Analyses or pre-analyses

  14. Orthogonawesome • The SVD is for rectangular tables • Does two things • Finds the major source of variance • Finds orthogonal slices of your data

  15. PCA = SVD • Center & Scale your data • Then SVD • = PCA! • Quick illustration

  16. Data

  17. Centered & Normed

  18. Find variance

  19. How?

  20. How?

  21. How?

  22. That’s a component!

  23. PCA!

  24. And variables

  25. PCA!

  26. And variables

  27. PCA!

  28. PCA!

  29. Usual visual

  30. An ExPosition of • The SVD • Resampling

  31. Resampling • Why?

  32. Resampling • Why? • Provides a null • Provides a distribution • Provides intervals

  33. First: Folklore • Require > 200 (Guilford, 1954) or > 250 (Cattell, 1978) observations • Require 5:1 observations:measures ratio (Gorsuch, 1983)

  34. More Folklore • Keep components with eigen values > 1 • Scree/elbow “tests”

  35. Fixing Folklore • High dimensional low sample size can be OK (Jung & Marron, 2009; Chi 2012) • Power derived like MANOVA (in some cases; D’Amico et al., 2001)

  36. Fixing Folklore • Sometimes all eigens < 1

  37. We need a null • Resampling can do that! • Bootstrap (Efron & Tibshirani, 1983, Hesterberg 2011, Chernick 2008) • Permutation (Berry et al., 2011) • But really, Fisher & Student did this first.

  38. Permutation • Scrambles data • An exact test of the H0 • Tests an omnibus effect • Tests each component

  39. Permutation r = -0.5

  40. Permutation

  41. Permutation

  42. Permutation

  43. Permutation

  44. Permutation

  45. Permutation r = 0.2

  46. Permutation in R • R> sample(1:4,4,FALSE) 2 3 1 4 • R> sample(1:4,4,FALSE) 3 2 1 4 • R> sample(1:4,4,FALSE) 4 3 2 1 • R> sample(1:4,4,FALSE) 3 4 1 2

  47. Bootstrap • Confidence intervals • Which measures are different from each other • t-like tests • Which measures are important to components?

  48. Bootstrap r = -0.5

  49. Bootstrap

  50. Bootstrap

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