Object Orie’d Data Analysis, Last Time

1. Object Orie’d Data Analysis, Last Time • Gene Cell Cycle Data • Microarrays and HDLSS visualization • DWD bias adjustment • NCI 60 Data Today: More NCI 60 Data & Detailed (math’cal) look at PCA

2. Last Time: Checked Data Combo, using DWD Dir’ns

3. DWD Views of NCI 60 Data • Interesting Question: • Which clusters are really there? • Issues: • DWD great at finding dir’ns of separation • And will do so even if no real structure • Is this happening here? • Or: which clusters are important? • What does “important” mean?

4. Real Clusters in NCI 60 Data • Simple Visual Approach: • Randomly relabel data (Cancer Types) • Recompute DWD dir’ns & visualization • Get heuristic impression from this • Deeper Approach • Formal Hypothesis Testing • (Done later)

5. Random Relabelling #1

6. Random Relabelling #2

7. Random Relabelling #3

8. Random Relabelling #4

9. Revisit Real Data

10. Revisit Real Data (Cont.) Heuristic Results: Strong Clust’s Weak Clust’s Not Clust’s MelanomaC N S NSCLC LeukemiaOvarianBreast RenalColon Later: will find way to quantify these ideas i.e. develop statistical significance

11. NCI 60 Controversy Can NCI 60 Data be normalized? Negative Indication: Kou, et al (2002) Bioinformatics, 18, 405-412. Based on Gene by Gene Correlations Resolution: Gene by Gene Data View vs. Multivariate Data View

12. Resolution of Paradox: Toy Data, Gene View

13. Resolution: Correlations suggest “no chance”

14. Resolution: Toy Data, PCA View

15. Resolution: PCA & DWD direct’ns

16. Resolution: DWD Adjusted

17. Resolution: DWD Adjusted, PCA view

18. Resolution: DWD Adjusted, Gene view

19. Resolution: Correlations & PC1 Projection Correl’n

20. Needed final verification of Cross-platform Normal’n Is statistical power actually improved? Will study later

21. DWD: Why does it work? Rob Tibshirani Query: Really need that complicated stuff? (DWD is complex) Can’t we just use means? Empirical Fact (Joel Parker): (DWD better than simple methods)

22. DWD: Why does it work? Xuxin Liu Observation: Key is unbalanced sub-sample sizes (e.g biological subtypes) Mean methods strongly affected DWD much more robust Toy Example

23. DWD: Why does it work?

24. Xuxin Liu Example Goals: Bring colors together Keep symbols distinct (interesting biology) Study varying sub-sample proportions: Ratio = 1: Both methods great Ratio = 0.61: Mean degrades, DWD good Ratio = 0.35: Mean poor, DWD still OK Ratio = 0.11: DWD degraded, still better Later: will find underlying theory

25. PCA: Rediscovery – Renaming Statistics: Principal Component Analysis (PCA) Social Sciences: Factor Analysis (PCA is a subset) Probability / Electrical Eng: Karhunen – Loeve expansion Applied Mathematics: Proper Orthogonal Decomposition (POD) Geo-Sciences: Empirical Orthogonal Functions (EOF)

26. An Interesting Historical Note The 1st (?) application of PCA to Functional Data Analysis: Rao, C. R. (1958) Some statistical methods for comparison of growth curves, Biometrics, 14, 1-17. 1st Paper with “Curves as Data” viewpoint

27. Detailed Look at PCA • Three important (and interesting) viewpoints: • Mathematics • Numerics • Statistics • 1st: Review linear alg. and multivar. prob.

28. Review of Linear Algebra • Vector Space: • set of “vectors”, , • and “scalars” (coefficients), • “closed” under “linear combination” • ( in space) • e.g. • , • “ dim Euclid’n space”

29. Review of Linear Algebra (Cont.) • Subspace: • subset that is again a vector space • i.e. closed under linear combination • e.g. lines through the origin • e.g. planes through the origin • e.g. subsp. “generated by” a set of vector (all linear combos of them = • = containing hyperplane • through origin)

30. Review of Linear Algebra (Cont.) • Basis of subspace: set of vectors that: • span, i.e. everything is a lin. com. of them • are linearly indep’t, i.e. lin. Com. is unique • e.g. “unit vector basis” • since

31. Review of Linear Algebra (Cont.) Basis Matrix, of subspace of Given a basis, , create matrix of columns:

32. Review of Linear Algebra (Cont.) Then “linear combo” is a matrix multiplicat’n: where Check sizes:

33. Review of Linear Algebra (Cont.) Aside on matrix multiplication: (linear transformat’n) For matrices , Define the “matrix product” (“inner products” of columns with rows) (composition of linear transformations) Often useful to check sizes:

34. Review of Linear Algebra (Cont.) • Matrix trace: • For a square matrix • Define • Trace commutes with matrix multiplication:

35. Review of Linear Algebra (Cont.) • Dimension of subspace (a notion of “size”): • number of elements in a basis (unique) • (use basis above) • e.g. dim of a line is 1 • e.g. dim of a plane is 2 • dimension is “degrees of freedom”

36. Review of Linear Algebra (Cont.) • Norm of a vector: • in , • Idea: “length” of the vector • Note: strange properties for high , • e.g. “length of diagonal of unit cube” =

37. Review of Linear Algebra (Cont.) • Norm of a vector (cont.): • “length normalized vector”: • (has length one, thus on surf. of unit sphere • & is a direction vector) • get “distance” as:

38. Review of Linear Algebra (Cont.) • Inner (dot, scalar) product: • for vectors and , • related to norm, via

39. Review of Linear Algebra (Cont.) • Inner (dot, scalar) product (cont.): • measures “angle between and ” as: • key to “orthogonality”, i.e. “perpendicul’ty”: • if and only if

40. Review of Linear Algebra (Cont.) • Orthonormal basis : • All ortho to each other, • i.e. , for • All have length 1, • i.e. , for

41. Review of Linear Algebra (Cont.) • Orthonormal basis (cont.): • “Spectral Representation”: • where • check: • Matrix notation: where i.e. • is called “transform (e.g. Fourier, wavelet) of ”

42. Review of Linear Algebra (Cont.) • Parseval identity, for • in subsp. gen’d by o. n. basis : • Pythagorean theorem • “Decomposition of Energy” • ANOVA - sums of squares • Transform, , has same length as , • i.e. “rotation in ”

43. Review of Linear Algebra (Cont.) Gram-Schmidt Ortho-normalization Idea: Given a basis , find an orthonormal version, by subtracting non-ortho part

44. Review of Linear Algebra (Cont.) • Projection of a vector onto a subspace : • Idea: member of that is closest to • (i.e. “approx’n”) • Find that solves: • (“least squares”) • For inner product (Hilbert) space: • exists and is unique

45. Review of Linear Algebra (Cont.) • Projection of a vector onto a subspace (cont.): • General solution in : for basis matrix , • So “proj’n operator” is “matrix mult’n”: • (thus projection is another linear operation) • (note same operation underlies least squares)

46. Review of Linear Algebra (Cont.) • Projection using orthonormal basis : • Basis matrix is “orthonormal”: • So = • = Recon(Coeffs of “in dir’n”)

47. Review of Linear Algebra (Cont.) • Projection using orthonormal basis (cont.): • For “orthogonal complement”, , • and • Parseval inequality:

48. Review of Linear Algebra (Cont.) • (Real) Unitary Matrices: with • Orthonormal basis matrix • (so all of above applies) • Follows that • (since have full rank, so exists …) • Lin. trans. (mult. by ) is like “rotation” of • But also includes “mirror images”

49. Review of Linear Algebra (Cont.) Singular Value Decomposition (SVD): For a matrix Find a diagonal matrix , with entries called singular values And unitary (rotation) matrices , (recall ) so that

50. Review of Linear Algebra (Cont.) • Intuition behind Singular Value Decomposition: • For a “linear transf’n” (via matrix multi’n) • First rotate • Second rescale coordinate axes (by ) • Third rotate again • i.e. have diagonalized the transformation