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Object Orie’d Data Analysis, Last Time

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  1. Object Orie’d Data Analysis, Last Time • Cornea Data • Images (on the disk) as data objects • Zernike basis representations • Outliers in PCA (have major influence) • Robust PCA (downweight outliers) • Eigen-analysis of robust covariance matrix • Projection Pursuit • Spherical PCA

  2. Cornea Data Cornea: Outer surface of the eye Driver of Vision: Curvature of Cornea Sequence of Images Objects: Images on the unit disk Curvature as “Heat Map” Special Thanks to K. L. Cohen, N. Tripoli, UNC Ophthalmology

  3. PCA of Cornea Data PC2 Affected by Outlier: How bad is this problem? View 1: Statistician: Arrggghh!!!! • Outliers are very dangerous • Can give arbitrary and meaningless dir’ns • What does 4% of MR SS mean???

  4. Robust PCA What is multivariate median? • There are several! (“median” generalizes in different ways) • Coordinate-wise median • Often worst • Not rotation invariant (2-d data uniform on “L”) • Can lie on convex hull of data (same example) • Thus poor notion of “center”

  5. Robust PCA M-estimate (cont.): “Slide sphere around until mean (of projected data) is at center”

  6. Robust PCA M-estimate (cont.): Additional literature: Called “geometric median” (long before Huber) by: Haldane (1948) Shown unique for by: Milasevic and Ducharme (1987) Useful iterative algorithm: Gower (1974) (see also Sec. 3.2 of Huber). Cornea Data experience: works well for

  7. Robust PCA Robust PCA 3: Spherical PCA

  8. Robust PCA Recall M-estimate for Cornea Data: Sample Mean M-estimate • Definite improvement • But outliers still have some influence • Projection onto sphere distorts the data

  9. Robust PCA Useful View: Parallel Coordinates Plot X-axis: Zernike Coefficient Number Y-axis: Coefficient

  10. Robust PCA Spherical PCA Problem: Magnification of High Freq. Coeff’s Solution: Elliptical Analysis • Main idea: project data onto suitable ellipse, not sphere • Which ellipse? (in general, this is problem that PCA solves!) • Simplification: Consider ellipses parallel to coordinate axes

  11. Robust PCA Rescale Coords Spherical PCA Unscale Coords

  12. Robust PCA Elliptical Analysis (cont.): • Simple Implementation, via coordinate axis rescaling • Divide each axis by MAD • Project Data to sphere (in transformed space) • Return to original space (mul’ply by orig’l MAD) for analysis • Where MAD = Median Absolute Deviation (from median) (simple, high breakdown, outlier resistant measure of “scale”)

  13. Robust PCA Elliptical Estimate of “center”: Do M-estimation in transformed space (then transform back) Results for cornea data: Sample Mean Spherical Center Elliptical Center • Elliptical clearly best • Nearly no edge effect

  14. Robust PCA Elliptical PCA for cornea data: Original PC1, Elliptical PC1 • Still finds overall curvature & correlated astigmatism • Minor edge effects almost completely gone Original PC2, Elliptical PC2 • Huge edge effects dramatically reduced • Still find steeper superior vs. inferior

  15. Robust PCA Elliptical PCA for Cornea Data (cont.): Original PC3, Elliptical PC3 • -Edge effects greatly diminished • But some of against the rule astigmatism also lost • Price paid for robustness Original PC4, Elliptical PC4 • Now looks more like variation on astigmatism???

  16. Robust PCA Current state of the art: • Spherical & Elliptical PCA are a kludge • Put together by Robustness Amateurs • To solve this HDLSS problem • Good News: Robustness Pros are now in the game: Hubert, et al (2005)

  17. Robust PCA Disclaimer on robust analy’s of Cornea Data: • Critical parameter is “radius of analysis”, • : Shown above, Elliptical PCA very effective • : Stronger edge effects, Elliptical PCA less useful • : Edge effects weaker, don’t need robust PCA

  18. Big Picture View of PCA Above View: PCA finds optimal directions in point cloud • Maximize projected variation • Minimize residual variation (same by Pythagorean Theorem) Notes: • Get useful insights about data • Shows can compute for any point cloud • But there are other views.

  19. Big Picture View of PCA Alternate Viewpoint: Gaussian Likelihood • When data are multivariate Gaussian • PCA finds major axes of ellipt’al contours of Probability Density • Maximum Likelihood Estimate Mistaken idea: PCA only useful for Gaussian data

  20. Big Picture View of PCA Simple check for Gaussian distribution: • Standardized parallel coordinate plot • Subtract coordinate wise median (robust version of mean) (not good as “point cloud center”, but now only looking at coordinates) • Divide by MAD / MAD(N(0,1)) (put on same scale as “standard deviation”) • See if data stays in range –3 to +3

  21. Big Picture View of PCA E.g. Cornea Data: Standardized Parallel Coordinate Plot

  22. Big Picture View of PCA Check for Gaussian dist’n: Stand’zed Parallel Coord. Plot • E.g. Cornea data (recall image view of data) • Several data points > 20 “s.d.s” from the center • Distribution clearly not Gaussian • Strong kurtosis • But PCA still gave strong insights

  23. Correlation PCA A related (& better known) variation of PCA: Replace cov. matrix with correlation matrix I.e. do eigen analysis of Where

  24. Correlation PCA Why use correlation matrix? Reason 1: makes features “unit free” • e.g. M-reps: mix “lengths” with “angles” (degrees? radians?) • Are “directions in point cloud” meaningful or useful? • Will unimportant directions dominate?

  25. Correlation PCA Alternate view of correlation PCA: Ordinary PCA on standardized (whitened) data I.e. SVD of data matrix Distorts “point cloud” along coord. dir’ns

  26. Correlation PCA Reason 2 for correlation PCA: • “Whitening” can be a useful operation (e.g. M-rep Corp. Call. data) • Caution: sometimes this is not helpful (can lose important structure this way) E.g. 1: Cornea data Elliptical vs. Spherical PCA

  27. Correlation PCA E.g. 2: Fourier Boundary Corp. Call. Data • Recall Standard PC1, PC2, PC3: • Gave useful insights • Correlation PC1, PC2, PC3 • Not useful directions • No insights about population • Driven by high frequency noise artifacts • Reason: whitening has damped the important structure • By magnifying high frequency noise

  28. Correlation PCA Parallel coordinates show what happened: Most Variation in low frequencies Whitening gives major distortion Skews PCA towards noise directions

  29. Correlation PCA Summary on correlation PCA: • Can be very useful (especially with noncommensurate units) • Not always, can hide important structure • To make choice: Decide whether whitening is useful • My personal use of correlat’n PCA is rare • Other people use it most of the time

  30. PCA to find clusters Toy Example (2 clusters)

  31. PCA to find clusters Toy Example (2 clusters) • Dominant direction finds very distinct clusters • Skewer through meatballs (in point cloud space) • Shows up clearly in scores plot • An important use of scores plot is finding such structure

  32. PCA to find clusters Recall Toy Example with more clusters:

  33. PCA to find clusters Best revealed by 2d scatterplots (4 clusters):

  34. PCA to find clusters Caution: there are limitations… • Revealed by NCI60 data • Recall Microarray data • With 8 known different cancer types • Could separate out clusters • Using specialized DWD directions • But can these be found using PCA? • Recall only finds dirn’s of max variation • Does not use class label information

  35. PCA to find clusters Specific DWD directions, for NCI60 Data: Good Separation Of Cancer Types

  36. PCA to find clusters PCA directions, for NCI60 Data: See clusters? PC2? PC1-3?

  37. PCA to find clusters PCA directions, for NCI60 Data: PC2: Melanoma PC1-3 Leukemia

  38. PCA to find clusters PC5-8 directions, for NCI60 Data: See clusters? PC5? Others?

  39. PCA to find clusters PC5-8 directions, for NCI60 Data: PC5: Leukemia & Renal PC6: CNS???

  40. PCA to find clusters PC9-12 directions, for NCI60 Data: See clusters? Maybe Not???

  41. PCA to find clusters PC9-12 directions, for NCI60 Data: Some New Combos ???

  42. PCA to find clusters Off Diagonal PC1-4 & 5-8, NCI60 Data: See clusters? PC1 & 5 PC2

  43. PCA to find clusters Off Diagonal PC1-4 & 5-8, NCI60 Data: PC1 & 5 Renal Leukemia (separated) PC2: Melanoma

  44. PCA to find clusters Off Diagonal PC1-4 & 9-12, NCI60 Data: See clusters? PC2 Others???

  45. PCA to find clusters Off Diagonal PC1-4 & 9-12, NCI60 Data: PC2 Melanoma PC1 Colon???

  46. PCA to find clusters Off Diagonal PC5-8 & 9-12, NCI60 Data: See clusters? PC5 Others???

  47. PCA to find clusters Off Diagonal PC5-8 & 9-12, NCI60 Data: PC5 Leukemia & Renal Others???

  48. PCA to find clusters Main Lesson: PCA is limited a finding clusters • Revealed by NCI60 data • Recall Microarray data • With 8 known different cancer types • PCA does not find all 8 clusters • Recall only finds dirn’s of max variation • Does not use class label information

  49. PCA to find clusters A deeper example: Mass Flux Data • Data from Enrica Bellone, • National Center for Atmospheric Research • Mass Flux for quantifying cloud types • How does mass change when moving into a cloud

  50. PCA to find clusters PCA of Mass Flux Data:

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